ಽሶ⑼ቇࠕ࡯ࠞࠗࡉࠬ
AC0005ቇㇱቇ↢ߩߚ߼ߩ㊂ሶൻቇ⻠⟵ࡁ࡯࠻೨✬቟Ⓧ ᔀ ⪺౏㐿ᣣ2007 ᐕ
2008 ᐕ

11 ᦬
1᦬

8ᣣ
11 ᣣ

╙㧝
╙2

ಽሶ⑼ቇળ✬㓸ᆔຬળߪ‫ޔ‬ఝࠇߚ࠹ࠠࠬ࠻ࠍಽሶ⑼ቇࠕ࡯ࠞࠗࡉࠬߣߒߡ౏
㐿ߒ߹ߔ߇‫ߩߘޔ‬ౝኈߩ৻ಾߩ⽿છߪ⪺⠪ߦ޽ࠅ߹ߔ‫⺒ޕ‬⠪߆ࠄߩ⾆㊀ߥߏᗧ⷗ߪ‫ޔ‬㧔edit-office@j-molsci.jp㧕ߢ㓐ᤨฃߌઃߌߡ߅ࠅ߹ߔ‫ߏޕ‬ᗧ⷗ߪ✬㓸ᆔຬળ߆ࠄ⪺⠪ߦ߅વ߃ߒ‫ߩ࠻ࠬࠠ࠹ޔ‬ౝኈߦ෻ᤋߒߡ޿߈߹ߔ‫ޕ‬⪺⠪⚫੺቟Ⓧ ᔀ㧔޽ߠߺߣ߅ࠆ㧕ᚲዻ㧦᧲ർᄢቇฬ⹷ᢎ᝼ ࿖㓙ᢎ㙃ᄢቇ․છᢎ᝼
ኾ㐷ಽ㊁㧦ಽሶಽశቇ࡮శൻቇ࡮ࠬࡇࡦൻቇ

ᡷ⸓ጁᱧ
╙2

⋡ᰴߢ Coffee Break ߩ㗄⋡߇ 1 ࠞᚲᛮߌߡ޿ߚߩߢㅊടߒߚ‫ޕ‬

PREFACEߎߩ⻠⟵ࡁ࡯࠻ߪ㧘⑳߇᧲ർᄢቇࠍ஗ᐕㅌ⡯ߒߚᐔᚑ 10 ᐕ߹ߢ㧘᧲ർᄢቇℂቇㇱൻቇ⑼ߩቇ↢ߦኻߒߡⴕߞߚ㊂ሶൻቇߩ⻠⟵ߢቇ↢ߦ㈩Ꮣߒߚ߽ߩߢ޽ࠆ㧚⑳㆐ߩൻቇᢎቶߢߪ㧘㊂ሶൻቇߩ⻠⟵ࠍ㧘ቇㇱ 2 ᐕ↢ᓟඨ߆ࠄ 3 ᐕ↢ࠍኻ⽎ߦⴕߞߡ߈ߚ㧚ᢎ⑼ᦠߣߒߡ㧘Atkins ߩ Molecular Quantum Mechanics ࠍ↪޿ߚ߇㧘ቇㇱ 2 ᐕ↢߆ࠄᆎ߹ࠆ⻠⟵ߢߪ㧘Atkins ߩᢎ⑼ᦠࠍߘߩ߹߹↪޿ࠆߩߪ㔍ߒߔ߉ߚ㧚߽ߞߣၮ␆ࠍৼካߦ⺑᣿ߔࠆᔅⷐࠍᗵߓ㧘ߎߩࠃ߁ߥ⻠⟵ࡁ࡯࠻ࠍ૞
ࠆߦ⥋ߞߚ㧚
ߘ߽ߘ߽㧘ߎߩ⻠⟵ࡁ࡯࠻ߪ㧘Atkins ߩᢎ⑼ᦠߩ⵬⿷ࠍ⹜ߺߚ߽ߩߢ޽ࠆߩ
ߢ㧘ో૕ߩ┨ߩ㗅⇟ߪߎߩᧄߦૃߡ޿ࠆ㧚ၮ␆⊛ߥᑼߩዉ಴ߦߘࠇ߶ߤ originalityࠍ⊒ើߔࠆߎߣ߽಴᧪ߕ㧘ᣢሽߩᢎ⑼ᦠߣૃߚㇱಽ߇ᄙߊ޽ࠆߎߣߪ੐ታߢ޽
ࠆ㧚ߒ߆ߒ㧘‛ℂ⊛㧘ൻቇ⊛ᗧ๧ߩ⺑᣿ߥߤߦ㧘⑳ߩ originality ߪ㓐ᚲߦ⷗ࠄࠇ
ࠆߪߕߢ޽ࠆ㧚߹ߚ㧘ቇ↢ߦ⥝๧ࠍᜬߞߡ߽ࠄ߃ࠆߎߣࠍᦼᓙߒߡ㧘޿ࠈ޿ࠈߥᱧผ⊛ߥࠛࡇ࠰࡯࠼ࠍ Coffee Break ߣߒߡઃߌട߃ߚ㧚ၮ␆⊛ߥߎߣߦ㊀ὐࠍ߅޿ߚߚ߼ߦ㧘ൻቇ⠪ߦߣߞߡ߽ߞߣ߽⥝๧߇޽ࠆಽ
ሶߦߟ޿ߡߪ㧘᳓⚛ಽሶߒ߆ᛒࠊߥ޿߁ߜߦᤨ㑆ಾࠇߣߥߞߡߒ߹ߞߡ޿ߚ㧚੹࿁ߎߘߪ߽ߞߣᄢ߈ߥಽሶ߹ߢขࠅᛒ޿ߚ޿ߣᕁ޿ߟߟ㧘⚿ዪ㧘ታⴕߢ߈ߥ
޿ߣ޿߁➅ࠅ㄰ߒߢ޽ߞߚ㧚ߥ߅㧘ߎߩ⻠⟵ࡁ࡯࠻ߦ߅޿ߡߪ㧘⧷⺆ߣᣣᧄ⺆ࠍ㓐ᤨᷙߗߡ૶ߞߡ޿ࠆߎ
ߣࠍ߅⸵ߒ޿ߚߛ߈ߚ޿㧚଀߃߫㧘޽ࠆᤨߦߪ‫ޟ‬ᵄേ㑐ᢙ‫޿⸒ߣޠ‬㧘޽ࠆᤨߦߪ‫ޟ‬wavefunction‫ߦ߁ࠃ߁޿ߣ߁⸒ߣޠ‬㧘㚂የ৻⽾ߒߡ޿ߥ޿㧚ߒ߆ߒ㧘ߘ߽ߘ߽㧘ᚒ‫ߪޘ‬ᣣᏱ↢ᵴߢ߽ᣣᧄ⺆ߣ⧷⺆ࠍᷙߗߡ૶ߞߡ޿ࠆࠊߌߢ㧘ߎߩ⻠⟵ࡁ࡯࠻ࠍ૞ࠆߣ߈߽㧘⹤ߒ⸒⪲ߩߟ߽ࠅߢ㧘޿޿ടᷫߥ⸒⪲㆜޿ࠍߒߡ޿ࠆ߇㧘ᱜᑼߥ⪺ᦠߢߪߥߊߡ㧘නߥࠆ⻠⟵ࡁ࡯࠻ߣ޿߁ߎߣߢ⋡ࠍߟ߱ߞߡ޿ߚߛߌ
ࠇ߫ᐘ޿ߢ޽ࠆ㧚⻠⟵ࡁ࡯࠻ࠍ㈩Ꮣߒߡ޿ߚᤨ߆ࠄ㧘ൊ㆑޿㧘ᑼߩ㑆㆑޿㧘⧷⺆ߩࠬࡍ࡞ߩ㑆㆑޿㧘ṽሼߩ㑆㆑޿ߥߤࠍήߊߔࠆߎߣߪ㕖Ᏹߦ߻ߕ߆ߒ޿ߎߣࠍ∩ᗵߒߡ᧪ߚ㧚੹࿁㧘ߎߩ⻠⟵ࡁ࡯࠻߇ Molecular Science ߩ archives ߣߒߡ౏㐿ߐࠇࠆߣ޿
߁㧘޽ࠅ߇ߚ޿⹤ߦߥߞߚ߇㧘⺒⠪ߩ⊝᭽߆ࠄߩߏป⽿ࠍฃߌߥ߇ࠄ㧘㑆㆑޿㧘ൊ㆑޿㧘ࡒࠬࡊ࡝ࡦ࠻ߥߤࠍ⸓ᱜߔࠆߎߣ߇಴᧪ࠇ߫ᐘ޿ߢ޽ࠆ㧚㧔ᐔᚑ 19 ᐕ 7 ᦬㧘࿡㓙ᢎ㙃ᄢቇߦߡ㧕䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Preface Page 1

INDEX
Preface
Chapter 01 ㊂ሶജቇߩ⺀↢ 㧙 ߘߩᱧผ⊛⢛᥊
1.1㤥૕ユ኿
Coffee Break㧦ultraviolet catastrophe ߦ㑐ߔࠆ joke
Let's Play with Mathematica㧦 Newton ᣇᴺߦࠃࠆᣇ⒟ᑼߩ⸃ᴺ
1.2
1.3

1.4
1.5
1.6

Coffee Break㧦Rayleigh-Jeans law ߦ߅ߌࠆ Rayleigh ߣ Jeans ߩᓎഀశ㔚ലᨐ࿕૕ߩᾲኈ㊂Coffee Break㧦ᾲኈ㊂ߦ㑐ߔࠆ Einstein Model
Coffee Break㧦Dulong-Petit ߩᴺೣCompton ᢔੂߣశߩ☸ሶᕈ
Coffee Break㧦Raman ಽశቇ‛⾰ߩᵄേ⊛ᕈ⾰ 㧙 de Broglie ᵄ᳓⚛ේሶߩࠬࡍࠢ࠻࡞
Coffee Break㧦 cm 1 ߣ޿߁න૏ߩ⊒㖸

Chapter 02 ዊߐߥ☸ሶߩᝄࠆ⥰޿
2.1஍శߒߡ޿ࠆశሶ
Coffee Break㧦৻ᤄ೨ߩ"If it were possible"߇੹ߢߪ2.2⏛᳇ࡕ࡯ࡔࡦ࠻ࠍᜬߞߚේሶ 㧙 Stern-Gerlach Experiment
Chapter 03 Schrödinger ᣇ⒟ᑼ
3.1৻ᰴరߩᵄߩᣇ⒟ᑼ
3.2

ᤨ㑆ߦଐሽߒߥ޿ Schrödinger ᣇ⒟ᑼ
Coffee Break㧦Hückel ߇⹗ࠎߛ Schrödinger ߦߟ޿ߡߩ⹞

Chapter 04 ㊂ሶജቇߦ߅ߌࠆၮᧄ⊛ߥේℂߣ઒ቯ
4.1
ᵄേ㑐ᢙ
4.2

㊂ሶജቇ⊛Ṷ▚ሶ
Coffee Break㧦Hermitian operator ߦ㑐ߔࠆ joke

4.3
4.4

Schrödinger ᣇ⒟ᑼ
Ṷ▚ሶߩ੤឵㑐ଥ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Index Page 1

4.5

Coffee Break㧦Commutator ߦ㑐ߔࠆ joke‛ℂ㊂ߩหᤨ᷹ቯ㧦ਇ⏕ቯᕈේℂ
Coffee Break㧦Heisenberg ߽᥉ㅢߩੱ㧫

Chapter 05 ⷺㆇേ㊂
5.1
5.2

ⷺㆇേ㊂Ṷ▚ሶߩ੤឵㑐ଥ
ⷺㆇേ㊂Ṷ▚ሶߩᭂᐳᮡߦࠃࠆ⴫⃻

5.3
5.4

ⷺㆇേ㊂Ṷ▚ሶߩ࿕᦭㑐ᢙߣ࿕᦭୯
ⷺㆇേ㊂Ṷ▚ሶߩ૞↪

5.5

⃿㑐ᢙ
Coffee Break㧦ᵄേ㑐ᢙߩ phase factor ߦ㑐ߔࠆࡐࡇࡘ࡜࡯ߥ㑆㆑޿
Let's Play with Mathematica㧦 ⃿㑐ᢙߩ Mathematica ߦࠃࠆ⴫⃻5.6

ⷺㆇേ㊂ߩวᚑ
Coffee Break㧦bra ߣ ket

Chapter 06 ਗㅴㆇേ৻ᰴరߩ⥄↱☸ሶ
6.1
6.2
੗ᚭဳࡐ࠹ࡦࠪࡖ࡞ߩਛߩ⥄↱☸ሶ㧔৻ᰴరߩ▫ߩਛߩ☸ሶ㧕
6.3
᦭㒢ߥᷓߐߩ੗ᚭဳࡐ࠹ࡦࠪࡖ࡞
6.4
ࡐ࠹ࡦࠪࡖ࡞ߩოߣ⽾ㅢ
6.5ᲑᏅߩ޽ࠆ੗ᚭဳࡐ࠹ࡦࠪࡖ࡞
6.6
ਃᰴరߩ▫ߩਛߩ☸ሶ
Chapter 07 ⺞๺ᝄേሶ
7.1
7.2
7.3

Hooke ߩᴺೣߦߒߚ߇߁♽ߦߟ޿ߡߩ Schrödinger ᣇ⒟ᑼ
Hermite ߩᓸಽᣇ⒟ᑼ
౹⚖ᢙᴺߦࠃࠆ Hermite ߩᓸಽᣇ⒟ᑼߩ⸃
Let's Play with Mathematica㧦Hermite polynomial

7.4

Hermite ߩᄙ㗄ᑼߦ㑐ߔࠆੑਃߩᕈ⾰

7.5
7.6
7.7

↢ᚑṶ▚ሶߣᶖṌṶ▚ሶࠍ↪޿ߚ⺞๺ᝄേሶߩ⸃
⺞๺ᝄേሶߦ㑐ߔࠆੑਃߩ‛ℂ㊂ߩᐔဋ୯⺞๺ᝄേሶࡕ࠺࡞ߣੑේሶಽሶߦ߅ߌࠆᝄേ
Let's Play with Mathematica㧦 ⺞๺ᝄേሶߩ Mathematica ߦࠃࠆ⴫⃻䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Index Page 2

Chapter 08 ࿁ォㆇേ
8.1
౞๟਄ࠍ⥄↱ㆇേߒߡ޿ࠆ☸ሶ
8.2
8.3
8.4

⃿ߩ⴫㕙ࠍ⥄↱ㆇേߒߡ޿ࠆ☸ሶ೰૕࿁ォሶੑේሶಽሶߩ࿁ォࠬࡍࠢ࠻࡞

Chapter 09 ᳓⚛ේሶ
9.1
9.2
9.3

9.4
9.5
9.6
9.7

᳓⚛ේሶߦ㑐ߔࠆ Schrödinger ᣇ⒟ᑼ
േᓘᵄേ㑐ᢙߩቯᕈ⊛ี๧േᓘᵄേ㑐ᢙߩ౹⚖ᢙᴺߦࠃࠆ⸃
Coffee Break㧦Bohr ߩᖺᤊࡕ࠺࡞㧘቟ࠄ߆ߦ߅⌁ࠅߊߛߐ޿
Let's Play with Mathematica㧦Laguerre polynomial᳓⚛ߩㅪ⛯⁁ᘒṶ▚ሶᴺߦࠃࠆേᓘᵄേᣇ⒟ᑼߩ⸃
᳓⚛㘃ૃේሶ

9.8

ේሶ゠㆏㑐ᢙ
᳓⚛ේሶߩේሶ゠㆏㑐ᢙߩ࿑␜㧔1㧕േᓘᵄേ㑐ᢙߩ⴫␜9.9᳓⚛ේሶߩේሶ゠㆏㑐ᢙߩ࿑␜㧔2㧕 YA ,m T , I ߦ㑐ߔࠆᭂᐳᮡࡊࡠ࠶࠻

9.10

9.11
9.12

9.13

Let's Play with Mathematica㧦ⷺᐲㇱಽߩᵄേ㑐ᢙߩᡆਃᰴరࡊࡠ࠶࠻
᳓⚛ේሶߩේሶ゠㆏㑐ᢙߩ࿑␜㧔3㧕ᐔ㕙਄ߩᵄേ㑐ᢙߩ୯ߩᡆ 3 ᰴర⊛⴫␜
Let's Play with Mathematica㧦ᡆਃᰴరࡊࡠ࠶࠻
᳓⚛ේሶߩේሶ゠㆏㑐ᢙߩ⴫␜㧔4㧕ᐔ㕙਄ߩᵄേ㑐ᢙߩ୯ߩ╬㜞✢࿑᳓⚛ේሶߩේሶ゠㆏㑐ᢙߩ⴫␜㧔5㧕ᵄേ㑐ᢙߘߩ߽ߩߩ୯ߩ
╬୯ᦛ㕙⴫␜
Let's Play with Mathematica㧦ᵄേ㑐ᢙߘߩ߽ߩߩ╬୯ᦛ㕙᳓⚛ේሶߩේሶ゠㆏㑐ᢙߩ⴫␜㧔6㧕ᵄേ㑐ᢙߘߩ߽ߩߩ୯ߩ
ੑਸ਼ߩ╬୯ᦛ㕙⴫␜

Chapter 10 ៨േℂ⺰10.1
㕖❗ㅌ♽ߩ៨േ⺰10.2❗ㅌߒߡ޿ࠆ♽ߦߟ޿ߡߩ៨േ⺰䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Index Page 3

Chapter 11 ᄌಽේℂ
11.1
ᄌಽේℂ
11.2

Coffee Break㧦wavefunction ߩ⇟ภઃߌ✢ᒻᄌಽේℂ㧔Rayleigh-Ritz ߩᣇᴺ㧕
Coffee Break㧦basis ߣ base

Chapter 12 ᤨ㑆ߦଐሽߒߚ៨േ⺰
12.1
12.2
12.3

ᤨ㑆ߦଐሽߒߚ៨േ⺰ੑḰ૏♽ߦߟ޿ߡߩࡕ࠺࡞⸘▚
ಽሶߩ౒㡆

12.4

ᄙḰ૏♽ߦ߅ߌࠆੑḰ૏ㄭૃ

Chapter 13 㔚⏛ᵄߩๆ෼ߣ᡼಴
13.1
㔚⩄ࠍ߽ߞߚ☸ሶߣ㔚⏛ᵄߣߩ⋧੕૞↪ߩࡂࡒ࡞࠻࠾ࠕࡦ
13.2
㔚⏛ᵄߩ៨േ 㧙 ⺃ዉๆ෼ߣ⺃ዉ᡼಴
13.3
ㆫ⒖ㅦᐲ 㧙 Fermi ߩ㤛㊄ೣ13.4
ㆫ⒖෺ᭂሶࡕ࡯ࡔࡦ࠻
Coffee Break㧦ㆇേ㊂Ṷ▚ሶߦ㑐ߔࠆⓍಽߣᐳᮡߦ㑐ߔࠆⓍಽ
13.5
㔚᳇෺ᭂሶㆫ⒖ߩㆬᛯೣ
13.6
⺃ዉๆ෼߅ࠃ߮⺃ዉ᡼಴ߦ㑐ߔࠆ Einstein ߩ B ଥᢙ
13.7
⥄ὼ᡼಴ߦ㑐ߔࠆ Einstein ߩ A ଥᢙ
13.8
13.9

࡟࡯ࠩ࡯
ࠦࡅ࡯࡟ࡦ࠻ಽశ

Chapter 14 㔚ሶࠬࡇࡦ
14.1
Stern-Gerlach ߩታ㛎
14.2
Uhlenbeck ߣ Goudsmit ߩ spin ઒⺑
14.3
14.4
14.5

ࠬࡇࡦ઒⺑ߦࠃࠆ Stern-Garlach ߩታ㛎ߩ⸃㉼ࠬࡇࡦߩᢙቇ⊛⴫⃻৻⥸ߩࠬࡇࡦ㑐ᢙ

14.6
14.7

ࠬࡍࠢ࠻࡞ߩᓸ⚦᭴ㅧ 㧙 ੑ㊀✢ߩ⸃㉼Coffee Break㧦ࠬࡇࡦߩ⊒⷗ߦ઻߁࠼࡜ࡑ
㔚ሶࠬࡇࡦߣ⏛႐ߣߩ⋧੕૞↪ 㧙 㔚ሶࠬࡇࡦߦߟ޿ߡߩ Zeeman ലᨐ

14.8

Coffee Break㧦⏛᧤ኒᐲߩන૏㧦࠹ࠬ࡜㔚ሶࠬࡇࡦ౒㡆䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Index Page 4

14.9‫౒ޟ‬㡆‫ޟ߮ࠃ߅ޠ‬㕖౒㡆‫ߩޠ‬᷹ቯᣇᴺߦߟ޿ߡ

Chapter 15 ᩭࠬࡇࡦCoffee Break㧦㔚⩄ࠍ߽ߚߥ޿☸ሶ߽ࠬࡇࡦࠍᜬߟ
H2 ߩ㆙⿒ᄖࠬࡍࠢ࠻࡞ߣᩭࠬࡇࡦߣߩ㑐ଥ
15.1
15.2

Coffee Break㧦⿒޿ᵄേ㑐ᢙ㧫ᩭࠬࡇࡦߩᓇ㗀㧙ࠝ࡞࠰᳓⚛ߣࡄ࡜᳓⚛
Coffee Break㧦Dennison ߩ⺰ᢥߩ̍Added on Proof"
Coffee Break㧦Dennison ߦ㑐ߔࠆㅺ⹤㧦SiO2 ߪ᳇૕㧫
Coffee Break㧦ᩭࠬࡇࡦߩ⊒⷗ߦ㊀ᄢߥᓎഀࠍᨐߚߒߚ Hori

15.3
15.4
15.5
15.6
15.7
15.8

Let's Play with Mathematica㧦 ᓸಽ⸘▚߇ߎࠎߥߦ߽◲නߦ㊀᳓⚛ᩭߩᩭࠬࡇࡦ
ઁߩᩭ⒳ߩᩭࠬࡇࡦ
ᩭࠬࡇࡦ㑐ᢙߣᩭࠬࡇࡦṶ▚ሶᩭࠬࡇࡦߣ⏛႐ߣߩ⋧੕૞↪㧙ᩭࠬࡇࡦߦߟ޿ߡߩ Zeeman ലᨐᩭ⏛᳇౒㡆㧔NMR㧕ࠬࡇࡦࠬࡇࡦ⚿ว㧙2 ࡊࡠ࠻ࡦ♽ࠍ଀ߣߒߡ
Coffee Break㧦NMR ߩ⊒⷗ߦߟ޿ߡ

Chapter 16 ේሶ᭴ㅧ
16.1
ේሶන૏
16.2
゠㆏ߩ᭎ᔨ
16.3
ࠬࡇࡦ゠㆏ (spin-orbital)
16.4

ࡄ࠙࡝ේℂߣࠬ࡟࡯࠲࡯ⴕ೉ᑼ
Coffee Break㧦Pauliߪታ㛎߇⧰ᚻ㧫

16.5

He ߩၮᐩ⁁ᘒߩࠛࡀ࡞ࠡ࡯㧔I㧕៨േ⺰ߦࠃࠆขࠅᛒ޿

16.6
16.7
16.8

He ߩၮᐩ⁁ᘒߩࠛࡀ࡞ࠡ࡯㧔II㧕ᄌಽᴺߦࠃࠆขࠅᛒ޿
He ේሶߩബ⿠⁁ᘒ
Li ේሶ

16.9
16.10

ᄙ㔚ሶේሶߦ߅ߌࠆ゠㆏ࠛࡀ࡞ࠡ࡯ߣㆤ⭁ലᨐ
Coffee Break㧦atomic orbital energy ߩ㗅ᐨ
Aufbau ේℂ

16.11

὇⚛ේሶߩၮᐩ⁁ᘒ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Index Page 5

Chapter 17 ⟲⺰
Coffee Break㧦Gruppenpest
17.1
17.2

ኻ⒓ᠲ૞㧔symmetry operation㧕ಽሶߩᒻߩኻ⒓ᕈߦࠃࠆಽ㘃
Coffee Break㧦ᚢ࿖ᱞ዁ߩ⚉┨

17.3
17.4

⟲
㘃㧔class㧕

17.5⴫⃻㧔representation㧕߅ࠃ߮ᜰᮡ㧔character㧕
Coffee Break㧦❑ࡌࠢ࠻࡞߆ᮮࡌࠢ࠻࡞߆㧫ᣢ⚂⴫⃻ 㧔irreducible representation㧕

17.6
17.7
17.8
17.9
17.10

Let's Play with Mathematica㧦 ㅒⴕ೉⴫⃻ⴕ೉ߩ⋥੤ᕈᜰᮡ⴫ 㧔character table㧕
Coffee Break ᜰᮡ⴫ߩ㑆㆑޿⟲⺰ࠍಽሶߦㆡ↪ߔࠆ႐วߩゲߩㆬ߮ᣇ߅ࠃ߮ੑਃߩขࠅ᳿߼

17.11
17.12
17.13
17.14
17.15
17.16
17.17

ᣢ⚂⴫⃻߳ߩ◲⚂ᣢ⚂⴫⃻ߩၮᐩߣߥࠆࠃ߁ߥ㑐ᢙߩ૞ࠅᣇ⋥Ⓧ (direct product)⟲ߩ⋥Ⓧ
ኻ⒓Ⓧߣ෻ኻ⒓ⓍⓍಽ୯߇࠯ࡠߦߥࠆ߆ߤ߁߆ߩ್ቯ
ේሶ゠㆏㑐ᢙߩᣢ⚂⴫⃻
ᷙᚑ゠㆏

17.18

ⷙḰᝄേ

Chapter 18 ᢿᾲㄭૃ
18.1
18.2

᳓⚛ಽሶߦ߅ߌࠆᢿᾲㄭૃ㧘ೋᱠ⊛ߥ⠨ኤ㔚ሶߩㆇേߣᩭߩㆇേߩಽ㔌

Chapter 19 ᳓⚛ಽሶ
19.1

ಽሶ゠㆏ᴺ
Coffee Break㧦orbital ߣ޿߁⸒⪲19.2

᳓⚛ಽሶࠗࠝࡦ H2+ߩಽሶ゠㆏⺰
Coffee Break㧦molecular orbital ࠍ⴫ߔ⸥ภ㧘V㧘S㧘Gߩ↱᧪䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Index Page 6

19.3
19.4

᳓⚛ಽሶߩಽሶ゠㆏ᴺߦࠃࠆขࠅᛒ޿
᳓⚛ಽሶߩේሶଔ⚿วᴺߦࠃࠆขࠅᛒ޿㧚Heiter-London ߩᣇᴺ

19.5

Coffee Break㧦৻㊀㗄⁁ᘒߣਃ㊀㗄⁁ᘒߣߤߜࠄ߇ࠛࡀ࡞ࠡ࡯߇㜞޿㧫
ಽሶ゠㆏ᴺߣේሶଔ⚿วᴺߣߩᲧセ
Coffee Break㧦ಽሶ゠㆏ᴺߣේሶଔ⚿วᴺߘࠇߙࠇߩ ․ᓽࠍ␜ߔᲧ༙

19.6

᳓⚛ಽሶߦ㑐ߔࠆⓥᭂߩ⸘▚Appendix 01 SI න૏
A1.1㔚⏛᳇ቇߩන૏
A1.2
A1.3

SI න૏
ၮᧄ⊛ߥ‛ℂ㊂Coffee Break㧦៨᳁᷷ᐲࠍ⧷⺆ߢߥࠎߣ⸒߁㧫A1.4

޽ࠆ⒳ߩ SI ⺃ዉන૏ߦኻߔࠆ․೎ߩฬ⒓ߣ⸥ภ

A1.5
A1.6

Coffee Break㧦Meter ߣ޿߁න૏ߩ↱᧪ߪߏᇚੱߩฬ೨߆ࠄ㧫
Coffee Break㧦ࠛ࡯࡞ࠬ࠹࠶࠼ߩ஧ὼߩ⊒⷗⸘▚ߩ଀㔚⏛᳇ቇߦ߅ߌࠆઍ⴫⊛ߥ޿ߊߟ߆ߩᑼ

Appendix 02 ⴕ೉ߩኻⷺൻ
A2.1
ⴕ೉ߩ࿕᦭୯ߣ࿕᦭ࡌࠢ࠻࡞
Coffee Break㧦secular equation ߩ"secular"ߣߪ㧫
A2.2
A2.3
A2.4

Let's Play with Mathematica㧦Mathematica ࠍ↪޿ߚኻⷺൻ
❗ㅌߩ޽ࠆ႐วߩ࿕᦭ࡌࠢ࠻࡞
ࡂࡒ࡞࠻࠾ࠕࡦߩ࿕᦭㑐ᢙࠍቢో♽ࠍߥߔᵄേ㑐ᢙߩ✢ᒻ⚿วߢ⴫ߔ✢ᒻᄌಽේℂ
Coffee Break㧦ᭂ୯߇ᦨዊ୯ߦߥߞߡ޿ࠆ߆㧫䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Index Page 7

CHAPTER 01㊂ሶജቇߩ⺀↢ 㧙 ߘߩᱧผ⊛⢛᥊
19 ਎♿ߩ⚳ࠅ߆ࠄ 20 ਎♿ߩೋ㗡ߦ߆ߌߡ㧘ߘࠇ߹ߢਁ⢻ߣᕁࠊࠇߡ޿ߚ࠾ࡘ࡯࠻
ࡦജቇߢߪߤ߁ߒߡ߽⺑᣿಴᧪ߥ޿ࠃ߁ߥታ㛎੐ታ߇޽ࠆߎߣ߇᣿ࠄ߆ߣߥߞߚ㧚ߘߩ߁ߜߦ㧘੹ᣣᚒ‫߇ޘ‬ฬ೨ࠍ⍮ߞߡ޿ࠆࠃ߁ߥᄤᚽ㆐߇ᰴ‫ߡࠇ⃻ߣޘ‬㧘ߎࠇࠄߩታ㛎੐ታߩ⸃㉼ߦᚑഞߒߚ㧚ᣂߒ޿⸃㉼ߢߪ㧘ᓥ᧪ㅪ⛯⊛ߥ୯ࠍขࠅᓧࠆߣᕁࠊࠇߡ޿ߚ
ࠛࡀ࡞ࠡ࡯߇ታߪߣ߮ߣ߮ߩ୯ߒ߆ขࠅᓧߥ޿ߣ⠨߃ࠆࠃ߁ߦߥߞߚ㧚ߘ߁޿߁ߎߣ
߆ࠄ㧘࠾ࡘ࡯࠻ࡦജቇߦઍࠆᣂߒ޿ቇ໧ߪ㊂ሶജቇ㧔quantum mechanics㧕ߣ⸒ࠊࠇࠆ
ࠃ߁ߦߥߞߚ㧚ߘࠇߣหᤨߦ㧘㊂ሶജቇߣᲧセߒߡ࠾ࡘ࡯࠻ࡦജቇࠍ⺰ߕࠆᤨߦᓟ⠪ࠍฎౖജቇߣ⸒߁ࠃ߁ߦߥߞߚ㧚ߎߎߢߪ㧘㊂ሶജቇߩ⊒ዷߩ♻ญߣߥߞߚ޿ߊߟ߆
ߩታ㛎ߣߘߩ⸃㉼ࠍᝄࠅ㄰ߞߡߺࠆߎߣߦߒࠃ߁㧚

1.1 㤥૕ユ኿ኅᐸߢ↪޿ࠄࠇࠆ࠲ࡦࠣࠬ࠹ࡦ㔚Ἦߪ㧘㔚ᵹߦࠃߞߡᾲߖࠄࠇߚ࠲ࡦࠣࠬ࠹ࡦ✢߆ࠄశ߇಴ߡ᧪ࠆߎߣࠍ೑↪ߒߚ߽ߩߢ޽ࠆ㧚ߎߩࠃ߁ߦ㧘‛⾰ࠍᾲߔࠆߣశ߇಴ߡ᧪
ࠆ߇ߎߩ⊒శߩ⦡ߪߘߩ‛⾰ߩ᷷ᐲߦࠃߞߡᄌࠊࠆ㧚࠾ࠢࡠࡓ✢ࠍᏎ޿ߚࡅ࡯࠲࡯ߦ
㔚᳇ࠍㅢߔߣ㧘ᆎ߼ߪߊߔࠎߛ⿒⦡ߢ޽ߞߚ߽ߩ߇ߛࠎߛࠎ޽ߑ߿߆ߥ⿒⦡ߦߥࠅ㧘ߟ޿ߦߪ㕍⊕ߊߥࠆߎߣߪ๟⍮ߩߎߣߢ޽ࠈ߁㧚ߎߩࠃ߁ߦ㧘‛⾰߇ᾲߖࠄࠇߚᤨߦ
಴ࠆశߩ⦡ߪ㧘ߘߩ‛ߩ᷷ᐲ਄᣹ߦ઻ߞߡ⿒߆ࠄ㕍⦡ߩᣇะߦᄌൻߔࠆ㧚᷹ቯߐࠇߚ
ࠬࡍࠢ࠻࡞ಽᏓߣ᷷ᐲߣߩ㑐ଥࠍ࿑ 1.1.1 ߦ␜ߔ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 1࿑ 1.1.1㤥૕ユ኿ߩࠬࡍࠢ࠻࡞ಽᏓߩታ㛎୯㧚᷷ᐲ਄᣹ߦ઻ߞߡࡇ࡯ࠢ߇㜞ᝄേᢙ஥㧔ߔߥࠊߜ⍴ᵄ㐳஥㧕ߦࠪࡈ࠻ߒߡ޿ߊ
ߣหᤨߦ㧘ో⊒శᒝᐲߪᕆỗߦჇടߔࠆ㧚ታ㛎⊛ߦ᳞߼ࠄࠇߚ㤥૕ユ኿ߩࠬࡍࠢ࠻࡞ࠍℂ⺰⊛ߦ⺑᣿ߒࠃ߁ߣ޿߁⹜ߺߪ㧘ߘߩᤄ㧘Rayleigh ߣ Jeans ߦࠃߞߡߥߐࠇߚ㧚Rayleigh-Jeans law ߣ⸒ࠊࠇߡ޿ࠆߎߩℂ⺰ߢߪ㧘᷷ᐲ T ߢᝄേᢙQ ߣQ  dQ ߣߩ㑆ߩ⊒శߩユ኿ኒᐲߪᑼ(1.1.1a)ߢ⴫ߐࠇࠆ㧚U Q , T dQ

8S kT 2Q dQ
c3

(1.1.1a)ߎߎߢ㧘 k ߪ Boltzmann constant㧘 c ߪశߩㅦᐲߢ޽ࠆ㧚ߎߩᑼߪ

Q

c

O

ࠍ⠨ᘦߒߡ

U O,T d O

8S kT

O4

dO

ߣߥࠆ㧚
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 2(1.1.1b)

ߎߩℂ⺰ᑼߩ⚿ᨐࠍ࿑ 1.1.2 ߦ␜ߔ㧚࿑ 1.1.2

Rayleigh-Jeans law ߩ⸘▚⚿ᨐ㧚࿑ 1.1.1 ߦ␜ߒߚታ᷹୯ߣߪቯᕈ⊛ߦ߽ᄢ߈ߊ⇣ߥߞߡ޿ࠆ㧚

࿑ 1.1.2 ߩ⚿ᨐߪ࿑ 1.1.1 ߩታ㛎੐ታߣቯᕈ⊛ߦߐ߃߽วࠊߥ޿㧚ታ᷹ߢߪ޽ࠆᵄ㐳ߢᦨᄢ୯ࠍ␜ߔߩߦኻߒߡ㧘Rayleigh-Jeans law ߢߪᵄ㐳߇⍴ߊߥࠆߦߟࠇߡන⺞ߦჇടߔࠆߩߺߢ޽ࠆ㧚ታ㛎⚿ᨐߣߩਇ৻⥌ߪ⚡ᄖ✢ߩ㗔ၞ㧔ߔߥࠊߜᵄ㐳߇⍴޿㗔ၞ㧕ߢ․ߦ㗼⪺ߢ޽ࠆ㧚ᝄേᢙଐሽᕈߦ㑐ߒߡታ㛎୯ࠍᲧセ⊛ࠃߊౣ⃻ߔࠆߩߪนⷞㇱࠍㅢࠅ⿧ߒߡ⿒ᄖ✢ߩ㗔ၞߦ౉ߞߡ߆ࠄߢ޽ࠆ㧚߽ߞߣ߅߆ߒ޿ߎߣߦ㧘ᑼ(1.1.1)ߢߪ

Q 2 ߩଐሽᕈࠍᜬߞߡ޿ࠆߚ߼ߦ㧘߆ߥࠅߩૐ᷷ߢ߽Q ߩᲧセ⊛ᄢ߈ߥนⷞㇱߩశࠍᒝߊ಴ߔߎߣߦߥࠆ㧚ߎ߁޿߁ߎߣ߇ᧄᒰߥࠄ਎ߩਛߦ㑧ߣ޿߁߽ߩ߇ሽ࿷ߒߥ޿ߎߣ
ߦߥࠆ㧚෶㧘ᑼ(1.1.1)ߢ᷷ᐲࠍ㜞ߊߔࠆߣユ኿ᒝᐲߪჇᄢߔࠆ߇㧘ታ㛎ߢ⷗಴ߐࠇߡ
޿ࠆࠃ߁ߦ᷷ᐲᄌൻߦ઻ߞߡ⦡߇ᄌൻߔࠆߎߣߪ⺑᣿ߐࠇߥ޿㧚ᦝߦᑼ(1.1.1)ࠍQ ߦ
ߟ޿ߡ࠯ࡠ߆ࠄή㒢ᄢ߹ߢⓍಽߔࠆߣ⊒ᢔߒߡߒ߹߁㧚ߔߥࠊߜ᦭㒢ߩ᷷ᐲߢή㒢ߩࠛࡀ࡞ࠡ࡯߇಴ࠆߣ޿߁߅߆ߒߥߎߣߦߥߞߡߒ߹߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 3ᔨߩߚ߼㧘᷷ᐲ 8000 K ߦ߅޿ߡߩታ᷹୯ߣ Rayleigh-Jeans law ߩᲧセࠍ࿑ 1.1.3 ߦ␜ߔ㧚࿑ 1.1.3
Rayleigh-Jeans ߩᴺೣߪ㧘⿒ᄖ✢ߩ㗔ၞߢߪᲧセ⊛ࠃߊታ㛎ࠍౣ⃻ߔࠆ߇㧘นⷞㇱߩశߩ㗔ၞߢߪታ㛎⚿ᨐߣቯᕈ⊛ߦߐ߃߽วࠊߥ޿㧚

Rayleigh-Jeans ߩᑼ߇ታ㛎ߣวࠊߥ޿ߎߣ㧔࿑ 1.1.3㧕ߪ㧘Q ߩᄢ߈޿㗔ၞ㧘ߔߥࠊߜ⚡ᄖ✢ߩ㗔ၞߢߎߩᴺೣߩ⎕✋߇⿠߈ࠆߣ޿߁ᗧ๧ߢ"ultraviolet catastrophe"ߣ⸒ࠊࠇࠆ㧚ߒ߆ߒߎߩ"ultraviolet catastrophe"ߣ޿߁⸒⪲ߪ޽߹ࠅᱜ⏕ߢߪߥ޿㧚catastropheߪ⚡ᄖ✢ߩᵄ㐳㗔ၞߢࠃࠅ㗼⪺ߦߪߥࠆ߽ߩߩ㧘଀߃߫࿑ 1.1.2 ߆ࠄ߽ಽ߆ࠆࠃ߁ߦ㧘catastrophe㧔ߔߥࠊߜ㧘ᒝᐲ U ߇⇣Ᏹߦᄢ߈ߊߥߞߡߒ߹߁ߎߣ㧕ߪߔߢߦ⿒ᄖߦㄭ޿นⷞㇱߩ㗔ၞߢߔߢߦ⿠ߎߞߡ޿ࠆ߆ࠄߢ޽ࠆ㧚
Rayleigh-Jeans law ߇ታ㛎੐ታࠍቯᕈ⊛ߦ߽ౣ⃻ߒߥ޿ߣ޿߁ߎߣ߆ࠄ㧘⋥ߜߦ㧘࠾ࡘ࡯࠻ࡦജቇ߇⎕✋ࠍ߈ߚߒߚߣ᳿߼ߟߌࠆࠊߌߦߪ޿߆ߥ޿߆߽ߒࠇߥ޿㧚࠾ࡘ࡯࠻ࡦജቇ߇߅߆ߒ޿ߩߢߪߥߊ㧘Rayleigh-Jeans ߩℂ⺰ߩዷ㐿ߦߤߎ߆㑆㆑޿߇޽ߞ
ߚߩߢߪߥ߆ࠈ߁߆㧚ߒ߆ߒߎߩὐߪᄙߊߩ‛ℂቇ⠪㆐ߦࠃߞߡี๧ߐࠇ㧘ዋߥߊߣ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 4߽ᒰᤨߩ‛ℂቇ⠪ߪℂ⺰ߩ㑆㆑޿ࠍ⷗ߟߌࠆߎߣߪ಴᧪ߥ߆ߞߚࠃ߁ߢ޽ࠆ㧚ߘ߁޿߁ࠊߌߢ㧘㤥૕ユ኿ߩಽᏓߩ໧㗴ߪ㧘࠾ࡘ࡯࠻ࡦജቇߢߪߤ߁ߒߡ߽⺑᣿಴᧪ߥ޿ᦨ
ೋߩ੐଀ߣߒߡੱ‫ߩޘ‬ᵈ⋡ࠍ޽߮ߚ㧚
Rayleigh-Jeans law ࠃࠅ߶ࠎߩዋߒ೨ߦឭ಴ߐࠇߚ Max Planck ߩℂ⺰ߢߪߎߩࠃ߁
ߥ࿎㔍ߪߥ޿㧚Rayleigh-Jeans ߩℂ⺰ߣห᭽ߦ㧘Planck ߽㧘శߩユ኿ߪ‛૕ਛߩ㔚ሶߩᝄേߦ↱᧪ߔࠆߣ⠨߃ߚ㧚ߚߛ㧘Planck ߪᝄേߔࠆ㔚ሶߩࠛࡀ࡞ࠡ࡯ߪߤࠎߥ୯ߢ
߽ขࠅᓧࠆߣ޿߁Ᏹ⼂⊛ߥ⠨߃ࠍᝥߡࠆᔅⷐ߇޽ࠆߣ⠨߃ߚ㧚Physikalische
Gesellschaft ߩ 1900 ᐕ 12 ᦬ 14 ᣣภߢ "Zur Theorie des Gesetzes der Energieverteilung im

Normalspectrum"ߣ㗴ߔࠆ⺰ᢥߢ Planck ߪ㧘ᝄേᢙQ ߢᝄേߔࠆ㔚ሶߩࠛࡀ࡞ࠡ࡯ߪ hQࠍ޽ࠆቯᢙߣߒߡ㧘 hQ ߩᢛᢙ୚ߩ୯ߒ߆ขࠅᓧߥ޿ߣ޿߁㧘ታߦᄢ⢙ߥ઒⺑ࠍዉ౉ߒߚ㧚ߔߥࠊߜ㧘ࠛࡀ࡞ࠡ࡯ E ߪ
E nhQ(1.1.2)ߣ⴫ߐࠇࠆߣ⠨߃ߚࠊߌߢ޽ࠆ㧚⛔⸘ജቇ⊛ᚻᴺߦߎߩ઒⺑ࠍ౉ࠇࠆߎߣߦࠃࠅ㧘
Planck ߪᑼ(1.1.1)ߦઍࠆߴ߈߽ߩߣߒߡᰴᑼࠍឭ໒ߒߚ㧚U Q , T dQ

8S hQ 3c3

1dQ
§ hQ ·
exp ¨¸ 1© kT ¹(1.1.3)h એᄖߩቯᢙߪߔߴߡᑼ(1.1.1)ߣหߓᗧ๧ߢ޽ࠆ㧚Planck ߪ㧘߽ߒ h ߣߒߡ

6.55 u 1034 Js ߣ޿߁୯ࠍ↪޿ࠇ߫㧘਄ᑼ߇ታ㛎⚿ᨐࠍḩ⿷ߦ⺑᣿ߔࠆߣㅀߴߡ޿ࠆ߇㧘ߎߩ୯ߪᦨᣂߩ♖ኒ᷹ቯߦࠃࠆ୯
h 6.626176 u 1034 Js(1.1.4)ߣ㕖Ᏹߦㄭ޿୯ߢ޽ࠆ㧚Planck ߇ዉ౉ߒߚ h ߣ޿߁ቯᢙߪ㧘ߘߩᓟ Planck's constantߣ๭߫ࠇࠆࠃ߁ߦߥࠅ‛ℂ㧘ൻቇߩ޽ࠄࠁࠆಽ㊁ߢ㊀ⷐߥᓎഀࠍṶߕࠆߎߣߦߥࠆ㧚(physics ߣ޿߁න⺆ߦ߽ chemistry ߣ޿߁න⺆ߦ߽ߎߩ㊀ⷐߥ h ߣ޿߁ᢥሼ߇฽߹ࠇߡ
޿ࠆ㧍ൻቇ߿‛ℂࠍኾ᡹ߔࠆቇ↢ߢ޽ࠇ߫ h ߩ୯ߊࠄ޿ߪ⸥ᙘߒߡ߅޿ߡ᰼ߒ޿㧚)[Coffee Break] ultraviolet catastrophe ߦ㑐ߔࠆ joke
"A Random Walk in Science"㧔edited by E. Mendoza, Institute of Physics, 1973㧕 ߣ޿߁㕙⊕޿ᧄ߇޽ࠆ㧚ߎߩਛߦ Physics terms made easy ߣ޿߁ section ߇޽ࠅ㧘‛ℂߩಽ㊁ߢ૶ࠊࠇࠆ⸒⪲ߩ⺑᣿߇౬⺣ߣߒߡ⸥タߐࠇߡ޿ࠆ㧚ߎߩਛߢ ultraviolet catastropheߩ⺑᣿ߣߒߡ bad sunburn ߣߥߞߡ޿ࠆ㧚࠙࠶ࡈ࠶ࡈ㧚ߎ߁޿߁౬⺣ࠍ⻠⟵ߩਛߢ⺑᣿ߒߡ޿ࠆߦ߽߆߆ࠊࠄߕ㧘⹜㛎ߢߎߩ౬⺣ߣૃߚࠃ߁ߥ⃟╵ࠍᦠߊቇ↢߇޿ࠆ߽ߩߛ㧚ᦨㄭߩவ૞ߪ‫ޟ‬ultraviolet catastrophe ߣ޿߁ߩߪ⚡䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 5ᄖ✢ߦࠃߞߡࠝ࠱ࡦጀ߇⎕უߐࠇߡ⊹⤏≸߇Ⴧ߃ࠆߎߣ‫⃟߁޿ߣޠ‬࿁╵ߢ޽ߞߚ㧚‫⵬ޣ‬ㆮ‫ޤ‬Rayleigh-Jeans law ߩዉ಴

Rayleigh-Jeans law ߪᒢᕈ૕ߩℂ⺰ߦࠛࡀ࡞ࠡ࡯╬ಽ㈩ߩᴺೣࠍ↪޿ߡዉ߈಴ߔߎ
ߣ߇಴᧪ࠆ㧚ࠛࡀ࡞ࠡ࡯╬ಽ㈩ߩᴺೣߣ޿߁ߩߪ㧘޽ࠆ᷷ᐲ T ߦ߅޿ߡ㧘1 ߟߩᝄേߩ⥄↱ᐲߦߟ޿ߡ kT ߛߌߩࠛࡀ࡞ࠡ࡯ࠍᜬߟߣ޿߁⠨߃ᣇߢ޽ࠅ㧘ߘࠇ߇ᚑࠅ┙ߚߥ޿ߎߣߪ⃻࿷ߢߪ᣿ࠄ߆ߣߥߞߡ޿ࠆ߇㧘෶หᤨߦ㧘᷷ᐲ߇㜞޿ᭂ㒢ߢߪᱜߒ޿ߎ
ߣ߽ಽ߆ߞߡ޿ࠆ㧚ߎߩ⠨߃ᣇࠍ↪޿ߡ Rayleigh-Jeans law ࠍዉ߈಴ߔߎߣߪ㧘⛔⸘ജቇߩೋᱠߢࠃߊቇ⠌ߔࠆߎߣߢ߽޽ࠆߩߢ㧘એਅߦ◲නߦ╭㆏ࠍㅀߴߡ߅ߎ߁㧚
ᝄേᢙ߇Q ߣQ  dQ ߩ㑆ߦ޽ࠆ࿕᦭ᝄേߩᢙ g (Q )dQ ߇᳞߹ࠇ߫㧘ࠛࡀ࡞ࠡ࡯╬ಽ㈩ߩᴺೣࠃࠅ㧘ߘࠇߦ kT ࠍដߌࠇ߫ߘߩᝄേᢙࠍᜬߟశߩࠛࡀ࡞ࠡ࡯߇᳞߹ࠆߎߣߦ
ߥࠆ㧚⹤ࠍಽ߆ࠅ߿ߔߊߔࠆߚ߼ߦ߹ߕ৻ᰴరߩᒏߩ࿕᦭ᝄേࠍ⠨߃ࠃ߁㧚ਔ┵ࠍ࿕ቯߒߚ㐳ߐ L ߩᒏߦߟ޿ߡ㧘⸵ߐࠇࠆᝄേߩᵄ㐳ߪ 2 L,

2L 2L,
, ... ߣߥࠆ㧚ߎߩᒏߩ਄ࠍ
2
3વࠊࠆᒢᕈᵄߩㅦᐲࠍ c ߣߔࠆߣ⸵ߐࠇࠆᝄേߩᝄേᢙߪ

c 2c 3c,,, ... ߩࠃ߁ߦ㧘
2L 2L 2Lcc
ߩ n ୚ߢਈ߃ࠄࠇࠆ㧚ߎߎߢ n ߪᱜߩᢛᢙ㧚ߔߥࠊߜ㧘ᒏߩ࿕᦭ᝄേߩᝄേᢙߪ
2L
2L
ߩ㑆㓒ࠍ߽ߞߡ╬㑆㓒ߦਗࠎߢ޿ࠆ㧚ᓥߞߡ㧘Q ߣQ  dQ ߩᝄേᢙߩ▸࿐ౝߦߪg Q dQ

dQ
§ c ·
¨ ¸
© 2L ¹(1.1.5)ߛߌߩ࿕᦭ᝄേ߇޽ࠆߎߣ߇ಽ߆ࠆ㧚એ਄ߩߎߣࠍ 3 ᰴరߦ᜛ᒛߒࠃ߁㧚ߔߥࠊߜ৻ㄝ߇ L ߩ┙ᣇ૕ߩਛߩ࿕᦭ᝄേࠍ⠨߃ࠆ㧚ߎߩ႐ว㧘3 ᰴరᝄേߪ n ߩઍࠅߦ 3 ߟߩᱜߩᢛᢙ [ , K , ] ߩ⚵ߢ⇟ภࠍઃߌࠆ
ߎߣ߇಴᧪ࠆ߇㧘ߘߩᝄേᢙߪ

Q [K][ 2 K 2  ] 2

c2L(1.1.6)

ߢਈ߃ࠄࠇࠆ㧚3 ᰴరߩ࿕᦭ᝄേߪ 3 ߟߩᢛᢙߢ⇟ภઃߌߐࠇࠆ߆ࠄ㧘ߎࠇࠍਗߴࠆ
ߦߪ 3 ᰴరߩⓨ㑆ࠍ↪޿ࠆߩ߇ଢ೑ߢ޽ࠆ㧚ߎߩ 3 ᰴరⓨ㑆ߩ 3 ߟߩ⋥੤ᐳᮡ♽ߦ߅
޿ߡ

c
c
c
[, y
K, z
]
2L
2L
2L
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 6x(1.1.7)3§ c ·ߢ޽ࠆࠃ߁ߥὐࠍ⠨߃ࠆ㧚 [ , K , ] ߪᢛᢙߢ޽ࠆ߆ࠄߎࠇࠄߩὐߪ૕Ⓧ ¨¸ ߩⓨ㑆ߩ
© 2L ¹⎴⋚ߩ⋡ߩ਄ߦ޽ࠆߎߣߦߥࠆ㧚ߘߩᤨ㧘ේὐ߆ࠄߩ〒㔌ߪ

x2  y 2  z 2[ 2 K 2  ] 2

c2L(1.1.8)ߣߥࠆ߇㧘ᑼ(1.1.6)ߣᲧߴࠆߣߎߩ〒㔌߇ߘߩᝄേߩᝄേᢙࠍਈ߃ࠆߎߣ߇ಽ߆ࠆ㧚ߎߩࠃ߁ߦ㧘ᝄേᢙࠍ 3 ᰴరⓨ㑆ᐳᮡߢ߽ߞߡ⴫ߔߎߣ߇಴᧪ࠆߣ㧘 g (Q )dQ ߪ⋥ߜߦ᳞߹ࠆ㧚ߔߥࠊߜ㧘ᝄേᢙ߇Q ߣQ  dQ ߣߩ㑆ߦ޽ࠆ࿕᦭ᝄേߩ୘ᢙߪ਄ߩ xyz ⓨ㑆ߦඨᓘQ ߣඨᓘQ  dQ ߣߩ 2 ߟߩ⃿ࠍឬ߈㧘ߘߩ⃿ߢ᜽߹ࠇߚⓨ㑆ߩਛߦ޽ࠆ⎴⋚3§ c ·⋡ߩ୘ᢙࠍᢙ߃ࠆߎߣߦࠃߞߡਈ߃ࠄࠇࠆ㧚1 ߟߩ⎴⋚⋡ߩ૕Ⓧߪ ¨¸ ߢ޽ࠆ߆ࠄ㧘© 2L ¹3§ c ·ߎߩ୘ᢙߪඨᓘQ ߣQ  dQ ߣߩ⃿ߩ㑆ߦ᜽߹ࠇߚⓨ㑆ߩ૕Ⓧ 4SQ dQ ࠍ ¨¸ ߢഀࠇ© 2L ¹2߫ࠃ޿㧚ߚߛߒߎߎߢ [ , K , ] ߪᱜߩᢛᢙߢ޽ࠆ߆ࠄ㧘╙৻⽎㒢ߛߌࠍ⠨߃ࠇ߫ࠃߊ
1
૕Ⓧߪ 4SQ 2 dQ ߩ ߣߥࠆ㧚㧔ߎߩ 8 ߢഀࠆߎߣߦߟ޿ߡߪߎߩ▵ߩ[coffee break]ࠍෳ
8

ᾖ㧕ᓥߞߡ

4SQ 2 dQ 8L3(1.1.9)8c3ߣߥࠆ㧚ߚߛߒ߽߁ 1 ߟ⠨߃ߥߌࠇ߫޿ߌߥ޿ߎߣ߇޽ࠆ㧚శߪᮮᵄߢ޽ࠆ߆ࠄ㧘஍శߩᣇะ߇⇣ߥࠆ 2 ⒳㘃ߩᝄേ߇ሽ࿷ߔࠆ㧚ߘࠇߢᑼ(1.1.9)ߩฝㄝࠍ 2 ୚ߒߡ㧘ᦨ⚳⊛ߦ
g Q dQ

8S L3 2Q dQ
(1.1.10)c3߇ᓧࠄࠇࠆ㧚ࠛࡀ࡞ࠡ࡯╬ಽ㈩ߩᴺೣ߆ࠄࠛࡀ࡞ࠡ࡯ߪߎࠇߦ kT ࠍ߆ߌߚ߽ߩߣߥࠆ㧚g Q dQ㧔ࠛࡀ࡞ࠡ࡯╬ಽ㈩ߩᴺೣߣߪ㧘1 ߟߩ⥄↱ᐲߦߟ޿ߡߩࠛࡀ࡞ࠡ࡯ߪ H

1
kT ߢ޽
2

ࠆߣ޿߁ᴺೣ㧚⹦ߒ޿ߎߣߪ⛔⸘ജቇߩ᝼ᬺߢቇ⠌ߔࠆߎߣߦߥࠆ߇㧘ߣࠅ޽߃ߕߎ
ߩᴺೣߪ᷷ᐲ߇㜞޿ᚲߢߪᚑࠅ┙ߟߣᕁߞߡߘ߁㑆㆑޿ߢߪߥ޿㧕㧚ߔߥࠊߜ㧘න૏
૕Ⓧᒰࠅߩࠛࡀ࡞ࠡ࡯ߪ

1
H g Q dQL3
8SQ 2 kT
dQ
c3䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 7

U Q , T dQ

(1.1.11)ߣߥࠆ㧚ߎࠇ߇ Rayleigh-Jeans law ߢ޽ࠆ㧚‫⵬ޣ‬ㆮ‫ޤ‬Planck ߩᑼ(1.1.3)ߩዉ಴

Planck ߇⠨߃ߚࠃ߁ߦࠛࡀ࡞ࠡ࡯߇ nhQ ߣ޿߁ࠃ߁ߦᢛᢙ n ߩ୚ᢙߢ␜ߐࠇࠆߣߔ
ࠆߣ㧘਄ㅀߒߚࠛࡀ࡞ࠡ࡯╬ಽ㈩ߩᴺೣߪ㜞᷷ᭂ㒢ߢߒ߆ᚑࠅ┙ߚߥߊߥࠆ㧚એਅ㧘

Planck ߩᑼࠍዉ಴ߒߡߺࠃ߁㧚ࠛࡀ࡞ࠡ࡯߇ᢛᢙ n ߩ㑐ᢙߢ޽ࠆߩߢ n ࠍ subscript ߣߒߡ
H n nhQ(1.1.12)ߣ⴫ߔ㧚ߎߩࠛࡀ࡞ࠡ࡯⁁ᘒߩሽ࿷⏕┙ Pn ߪ♽߇ᾲᐔⴧ⁁ᘒߦ޽ࠆߣ߈ߦߪ

Boltzmann ಽᏓೣߢᰴߩࠃ߁ߦ⴫ߐࠇࠆ㧚

Pn

§ H ·
exp ¨  n ¸© kT ¹§ H ·¦n exp ¨©  kTn ¸¹ᓥߞߡ♽ߩᐔဋࠛࡀ࡞ࠡ࡯ H ߪએਅߩࠃ߁ߦ⴫ߔߎߣ߇಴᧪ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 8(1.1.13)

H

¦PH

n n

n

§ Hn ·¦ H exp ¨©  kT ¸¹nn§ Hn ·¦ exp ¨©  kT ¸¹n§ nhQ ·

¦ nhQ exp ¨©  kT ¸¹n§ nhQ ·

¦ exp ¨©  kT ¸¹nw§ nhQ ·ln ¦ exp ¨ ¸
wT© kT ¹n
1wln'╬Ყ⚖ᢙߩ౏ᑼࠍ↪޿ߡ)kT 2
wT§ hQ ·
1  exp ¨ ¸© kT ¹w ª§ hQ · ºln «1  exp ¨ 
kT 2
¸»
wT ¬
© kT ¹ ¼
kT 2§ hQ ·
hQ exp ¨ ¸© kT ¹§ hQ ·
1  exp ¨ ¸© kT ¹hQ§ hQ ·
exp ¨¸ 1© kT ¹(1.1.14)㜞᷷ߩᭂ㒢ߢߪᑼ(1.1.14)ߪ kT ߦ෼᧤ߒ㧘ࠛࡀ࡞ࠡ࡯╬ಽ㈩ߩᴺೣ߇㜞᷷ߩᭂ㒢ߢߪᚑࠅ┙ߟߎߣ߇ಽ߆ࠆ㧚ߘࠇએᄖߩ᧦ઙਅߢߪ(1.1.14)ߩ H ߇ kT ߩઍࠊࠅߦ↪޿ࠄࠇߥߌࠇ߫ߥࠄߥ޿㧚ߘߎߢ㧘ߎߩ H ߩ୯ࠍ(1.1.11)ߩ╙ 1 ߩᑼߦઍ౉ߒߡᰴߩᑼࠍᓧࠆ㧚U Q , T dQ

1
H g Q dQ
L3
8S hQ 31dQ
3c
§ hQ ·
exp ¨¸ 1© kT ¹(1.1.15)߆ߊߒߡ Planck ߩᑼ߇ዉ಴ߐࠇߚ㧚‫ڏ‬Problem 1.1.1
ᑼ(1.1.1)߅ࠃ߮(1.1.3)ߪ޿ߕࠇ߽න૏૕Ⓧᒰࠅߩࠛࡀ࡞ࠡ࡯㧘ߔߥࠊߜ J/m3 ߣ޿߁䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 9න૏ࠍᜬߟߎߣࠍ␜ߖ㧚‫ڏ‬Problem 1.1.2
ᑼ(1.1.3)ߪユ኿ࠛࡀ࡞ࠡ࡯ߩᝄേᢙଐሽᕈࠍ␜ߒߚ߽ߩߢ޽ࠆ㧚ᵄ㐳 O ߦኻߔࠆଐ
ሽᕈߪᰴᑼߢ⴫ߐࠇࠆߎߣࠍ␜ߖ㧚

U O,T dO

8S hc

O

5

1
dO
§ hc ·exp ¨¸ 1© O kT ¹(1.1.16)‫ڏ‬Problem 1.1.3
Planck ߩ⺑߇⊒⴫ߐࠇࠆ೨߆ࠄ Wien displacement law㧔Wien ߩᄌ૏ᴺೣ㧕ߣ๭߫ࠇࠆ⚻㛎ೣ߇⍮ࠄࠇߡ޿ߚ㧚ߘࠇߪ㧘 U O ,T ߇ᭂᄢߣߥࠆᵄ㐳 Omax ߪ

OmaxT

2.90 u103 mK(1.1.17)ߣ⴫ߐࠇࠆߣ޿߁߽ߩߢ޽ࠆ㧚㧔Wien ߪߎߩഞ❣ߦࠃߞߡ 1911 ᐕߩࡁ࡯ࡌ࡞‛ℂቇ
⾨ࠍฃ⾨ߒߚ㧚Wien's displacement law ߦೝ᚝ߐࠇߡᄢ߈ߥഞ❣ࠍߥߒߣߍߚ Planckߪ Wien ߦㆃࠇࠆߎߣ 7 ᐕ㧘1918 ᐕߦࡁ࡯ࡌ࡞‛ℂቇ⾨ࠍฃ⾨ߒߡ޿ࠆ㧚㧕ᑼ(1.1.16)
ࠍ O ߢᓸಽߔࠆߎߣߦࠃߞߡ OmaxT ࠍዉ಴ߒ㧘ᑼ(1.1.17)ߣᲧセߖࠃ㧚

Hint㧦P. W. Atkins ߩ Molecular Quantum Mechanics ߩ page 8 ߢߪ㧘ᑼ(1.1.16)ߦ߅޿
ߡᜰᢙ㑐ᢙߩㇱಽߪߤ߁ߖዊߐ޿ߛࠈ߁ߣ޿߁ߎߣߢήⷞߒߡ㧘ߘࠇࠍ O ߢᓸಽߒߚ
߽ߩࠍ࠯ࡠߣ߅߈㧘 OmaxThc / 5k ࠍዉ߈ߛߒߡ޿ࠆ㧚ߒ߆ߒ㧘ߘߩ⚿ᨐߪᑼ(1.1.17)

ߣ߆ߥࠅ⇣ߥࠆ㧚ߎߎߢ⻉ำߪᜰᢙ㑐ᢙߩㇱಽࠍήⷞߔࠆߥߤߣ޿߁㧘ߌߜߊߐ޿ߎߣߪ⠨߃ߕߦᑼ(1.1.16)ࠍ⋥ធᓸಽߒ⸃޿ߡߺࠃ㧚ߚߛߒᓸಽߒߚ߽ߩࠍ࠯ࡠߣ߅޿ߚᑼߪ⸃ᨆ⊛ߦߪ⸃ߌߥ޿ߩߢ㧘଀߃߫ Newton ᴺߥߤࠍ↪޿ߡ㧔㔚ථߢ◲නߦ෼᧤ߔࠆ㧕⸃ߊߣࠃ߆
ࠈ߁㧚⸃߇᳞߹ߞߚࠄ㧘 h 㧘 c 㧘 k ߩౕ૕⊛ߥ୯ࠍ౉ࠇߡ Wien's displacement law ߣᲧセߒߡߺࠃ㧚ࠊࠅߣ◲නߦ Atkins ߩᢎ⑼ᦠࠃࠅ߽⦟޿⚿ᨐ߇ᓧࠄࠇࠆߎߣࠍታᗵߒߡ
߶ߒ޿㧚
‫ޤ⸼ޣ‬
㧦
Newton ᴺࠍ⍮ࠄߥ޿ੱߪ㧘ᢙቇߩᧄ߿ᖱႎಣℂߩᧄߢീᒝߒߡ߽ࠄ޿ߚ޿߇㧘
essence ߪᰴߩㅢࠅߢ޽ࠆ㧚f x 0 ߩ⸃ࠍᢙ୯⊛ߦ᳞߼ࠆߦ޽ߚࠅ㧘߹ߕ x ߩೋᦼ୯㧘x0 ࠍ઒ቯߔࠆ㧚 f ( x0 ) ߇චಽ࠯ࡠߦㄭߌࠇ߫ x0 ߇ᩮߢ޽ࠆ㧚 f x0 ߇චಽ࠯ࡠߦㄭߊ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 10ߥ޿႐วߪ x1

x0 

f x0
߇ࠃࠅᩮߦㄭ޿㧚ߎߩ x1 ࠍ਄ㅀߩ x0 ߣหߓࠃ߁ߦ⠨߃ߡ㧘
f ' x0ᰴ‫ࠅࠃߦޘ‬ㄭ޿ᩮ x2 , x3 , ... ࠍ᳞߼ߡ޿ߊߣ㧘ߟ޿ߦ f xn ߣ࠯ࡠߣߩᏅ߇චಽߦዊߐ
ߊ㧘޽ࠄ߆ߓ߼᳿߼ߡ߅޿ߚ޽ࠆዊߐߥᢙ H ࠃࠅ߽ዊߐߊߥࠆߣ޿߁ xn ߇᳞߹ࠆ㧚ߎߩ xn ߇ᩮߢ޽ࠆ㧚ߎߩᣇᴺߪ㕖Ᏹߦ෼᧤߇ᣧߊ㧘ࠃ߶ߤ pathological ߥ㑐ᢙߢߥ޿㒢ࠅ 2㧘3 ࿁ߢ෼᧤ߔࠆ㧚[Let’s play with Mathematica] Newton ᣇᴺߦࠃࠆᣇ⒟ᑼߩ⸃ᴺ
Newton ᴺߪᢙ୯Ṷ▚ߩࡊࡠࠣ࡜ࡓߩ⸃⺑ᦠߢࠃߊ଀ߦ಴ߐࠇࠆ߽ߩߢ޽ࠆ㧚ߘࠇߪߎߩṶ▚ߩਛߦ㧘➅ࠅ㄰ߒ㧔Fortran ߢߪ DO loop㧕ߣ߆್ᢿ㧔IF ᢥ㧕ߣ߆ߩၮᧄ⊛ߥᣇᴺࠍ฽ࠎߢ޿ߡ㧘ᩰᅢߥṶ⠌໧㗴ߣߥࠆ߆ࠄߢ޽ࠆ㧚ߒ߆ߒ㧘ࡊࡠࠣ࡜ࡓࠍ⚵߻Ბߦߥࠆߣ㧘߹ߕ f x ߩᓸಽࠍ⸃ᨆ⊛ߦ᳞߼ߥߊߡߪ޿ߌߥ޿ߒ㧘߹ߚ㧘޽ࠆዊߐߥ
ᢙ H ࠍౕ૕⊛ߦ޿ߊߟߦߒߚࠄ޿޿߆߽᳿߼ߥߊߡߪ޿ߌߥ޿ߒ㧘◲නߣߪ޿ߞߡ߽
ᘠࠇߥ޿ੱߦߪߘࠇ߶ߤ◲නߢ߽ߥ޿㧚ᦨㄭߪᄙߊߩ㔚ථߢ⸃ߩㄭૃ⸃ࠍ಴ߒߡߊࠇࠆߩߢߘ߁޿߁߽ߩࠍⓍᭂ⊛ߦ૶ߞߡ㧘㗡ߪ߽ߞߣᄢಾߥ߽ߩߦ૶ߞߚࠄ޿޿ߛࠈ߁㧚㔚ථߢߪ㧘ࠕ࡞ࠧ࡝࠭ࡓ߇ߪߞ߈ࠅߒߥ޿ߩߢߜࠂߞߣਇḩߛߣᕁ߁ੱߦߪ㧘ࡄ࠰ࠦࡦ਄ߢേߊ Mathematica ߣ޿߁࠰ࡈ࠻߇ଢ೑ߢ޽ࠆ㧚੹߿ࡄ࠰ࠦࡦ߇ 10 ਁ౞૏ߢ⾈߃ࠆᤨઍߢ޽ࠆ߆ࠄߎ߁޿߁࠰ࡈ࠻߽Ⓧᭂ⊛ߦ૶߁ߣൻቇ߳ߩℂ⸃ᐲ߽ᷓ߹ࠆ߽ߩ
ߢ޽ࠆ㧚଀߃߫㧘⸃ᨆ⊛ߦߪ⸃ߌߥ޿ f x

exp  x 

Mathematica ߢߪ㕖Ᏹߦ◲නߢ޽ࠆ㧚߹ߕ㑐ᢙࠍቯ⟵ߒࠃ߁㧚
f[x_]:=Exp[-x]+x/5-1
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 11x
 1 0 ߩ⸃ࠍ᳞߼ߚ޿ߣߒࠃ߁㧚
5ᢙ୯⸃ߪᰴߩࠃ߁ߦߒߡ᳞߼ࠆ㧚ߎߎߢ㧘 x ߩೋᦼ୯ߣߒߡ 5 ࠍਈ߃ߡ޿ࠆ㧚
FindRoot[f[x]==0,{x,5}]ߎ߁޿߁߰߁ߦ key in ߒߡ Shift key ࠍ᛼ߒߥ߇ࠄ return ߔࠆߣ⋥ߜߦ{x -> 4.96511}
ߩࠃ߁ߥ⸃߇಴ߡ᧪ࠆ㧚߽ߞߣ♖ᐲࠍ਄ߍߚߌࠇ߫㧘଀߃߫ 10 ᩴߩ♖ᐲ߇᰼ߒߌࠇ
߫
N[FindRoot[f[x]==0,{x,5}],10]ߣߔࠆߣ
{x -> 4.965114232}
ߩࠃ߁ߦ 10 ᩴߩ⸃߇ਈ߃ࠄࠇࠆ㧚ߎߎߢᓧࠄࠇߚ⸃ߩ୯ࠍరߩ㑐ᢙߦઍ౉ߒߡߺࠃ
߁f[4.965114232]⚿ᨐߪ-11
4.93605 10ߣߥࠅ㧘㒢ࠅߥߊ࠯ࡠߦㄭ޿୯߇ᓧࠄࠇߡ㧘⚿ᨐ߇ᱜߒ޿ߎߣ߇ಽ߆ࠆ㧚㧔਄ߩ଀ߩ
಴ജߪ 4.93605 u1011 ߩᗧ๧ߢ޽ࠆ㧚Mathematica ߢߪߎߩ଀ߩࠃ߁ߦ㧘10 ߩ૗ਸ਼ߣ޿
߁ᢙሼࠍ᥉ㅢᚒ‫ ߦ߁ࠃߊᦠ߇ޘ‬1 Ბ਄ߦߕࠄߒߡਈ߃ߡߊࠇࠆ㧚㧕‫ڏ‬Problem 1.1.4
Planck ߩ઒⺑ߢߪ㧘㔚᳇⊛ᝄേߩࠛࡀ࡞ࠡ࡯ߪ E

nhQ ߩߺࠍᜬߟߣ޿߁ߎߣߢ޽

ߞߚ㧚ߔߥࠊߜ 'E hQ 㧚ᓥߞߡQ o 0 ߢ 'E hQ ߢ E ߪታ⾰⊛ߦㅪ⛯ߢ޽ࠆ㧚ߎ߁޿߁ߎߣ߆ࠄ㧘Planck ߩᑼ(1.1.3)ߪQ o 0 ߩૐᝄേᢙ㗔ၞߢ Rayleigh-Jeans ߩᑼ(1.1.1)ߦ
෼᧤ߔࠆߎߣࠍ␜ߖ㧚Hint: exponential function ࠍዷ㐿ߖࠃ㧚‫ڏ‬Problem 1.1.5
߽ߞߣ߽᷷ᐲ߇㜞޿ᤊߩ 1 ߟߣߒߡ⍮ࠄࠇࠆࠪ࡝࠙ࠬ㧔Sirius㧕ߪ㧘߶߷

Omax

260 nm ߩ㤥૕ユ኿ߩࠬࡍࠢ࠻࡞ࠍ␜ߔ㧚ߎߩᤊߩ⴫㕙᷷ᐲࠍផቯߖࠃ㧚‫ڏ‬Problem 1.1.6
ᧄᢥߢㅀߴߚࠃ߁ߦ㧘Planck ߪ㤥૕ユ኿ߩℂ⺰ᑼߣታ㛎ߣߩᲧセ߆ࠄ h ࠍ᳿߼ߚࠊ
ߌߢ޽ࠆ㧚ߎߎߢߪ㧘⚻㛎⊛ߦਈ߃ࠄࠇߚ Wien's displacement law ߣ Planck ߩᑼ(1.1.16)
ߣߩᲧセ߆ࠄ㧘Problem 1.1.3 ߩ⚿ᨐࠍ↪޿ߡ h ࠍ▚಴ߖࠃ㧚᦭ലᢙሼߦ⇐ᗧߒߡ⚿ᨐ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 12ࠍ᳞߼ࠆߎߣ㧚[Coffee Break] Rayleigh-Jeans law ߦ߅ߌࠆ Rayleigh ߣ Jeans ߩᓎഀ
Rayleigh-Jeans law ߣ޿߁ߩߪ Rayleigh ߣ Jeans ߩ 2 ੱߦࠃߞߡ⷗಴ߐࠇߚߩߢ޽ࠈ
߁ߣߪᕁߞߡ߽㧘ߎߩ 2 ੱ߇౒หߢ⷗಴ߒߚߩ߆㧘ోߊ⁛┙ߦ⷗಴ߒߚߩ߆㧘ߘࠇߣ
߽ 1 ੱ߇⷗಴ߒߡ㧘߽߁ 1 ੱ߇ߘࠇࠍ⊒ዷߐߖߚߩߛࠈ߁߆⇼໧ߦᕁ߁ੱ߽޿ࠆߢ޽
ࠈ߁㧚
Lord Rayleigh ߪ㤥૕ユ኿ߦ㑐ߒߡ޿ࠈ޿ࠈߣ⠨߃ߡ޿ߚࠃ߁ߢ㧘ᓐߩᦨೋߩ⺰ᢥߪ
"Remarks Upon the Law of Complete Radiation"ߣ޿߁㗴ߢ Philosophical Magazine ߩ
Series 5, Vol. 49, 539-540 (1900)ߦ಴ߡ޿ࠆ㧚ߒ߆ߒ㧘ߎߎߢ಴ߐࠇߚᑼߪ㧘ᑼ(1.1.1)
ߣߪᄢಽ⇣ߥߞߡ޿ࠆ㧚ߘߒߡ 5 ᐕߚߞߚ 1905 ᐕ㧘Nature 72, 54 (1905)ߦ"The
Dynamical Theory of Gases and of Radiation"ߣ޿߁㗴ߩ⺰ᢥࠍᦠ޿ߡ޿ࠆ߇㧘ߎߩਛߢ㧘U Q ,T

64S kT 2
Q dQc3

ߣ޿߁ᑼ㧘ߔߥࠊߜᑼ(1.1.1)ߩ 8 ୚ߩᄢ߈ߐߩᑼࠍ಴ߒߡ޿ࠆ㧚ߎߩᤨߦߪ߽߁ Planckߩᑼ߇಴ߐࠇߡ޿ߚ߇㧘ߎߩ⺰ᢥߩਛߢ㧘Q ߇ዊߐ޿ᚲߢߪ㧘"This result is eight times
as large as that found by Planck"ߣㅀߴߡ޿ࠆ㧚ታߪߎߩᑼߩዉ಴ߢ Rayleigh ߪߜࠂߞߣ
ߒߚࡒࠬࠍ߅߆ߒߚߩߦ᳇߇ߟ߆ߥ߆ߞߚߩߢ޽ࠆ㧚ߎߩߎߣࠍᜰ៰ߒߚߩߪ Jeansߢ޽ߞߚ㧚Philosophical Magazine ߩ Series 6, Vol. 10, 91-98 (1905) ߦ "On the Partition ofEnergy Between Matter and Aether" ߣ޿߁⺰ᢥࠍᦠ޿ߡ޿ࠆ߇㧘ߎߩ⺰ᢥߩ postscriptߣߒߡᓟߢㅊടߒߚᢥߩਛߢ㧘Jeans ߪᰴߩࠃ߁ߦㅀߴߡ޿ࠆ㧚"This is one-eighth of theamount found by Lord Rayleigh but agrees exactly with that given by Planck for large valueof ǳ. It seems to me that Lord Rayleigh has introduced an unnecessary factor 8 by counting

negative as well as positive value of his integers [, K, ], ..."ߎࠇࠍฃߌߡ㧘Rayleigh ߪ
Nature 72, 243 (1905)ߩ"The Constant of Radiation as Calculated From Molecular Data"ߣ޿߁⺰ᢥߩਛߢ㧘"In connection with similar work of his own, Mr. Jeans has just pointed outthat I have introduced a redundant factor 8 by counting negative as well as positive values of
my integers [, K, ]㧚I hasten to admit the justice of this correction ..." ߣ޿߁߰߁ߦㅀߴ
ߡ޿ࠆ㧚ⷐߔࠆߦ Jeans ߩഞ❣ߪ㧘Rayleigh ߩᑼߩ 8 ߣ޿߁ factor ߩ㑆㆑޿ࠍᜰ៰ߒߚߎߣ
ߢ޽ߞߚࠃ߁ߢ޽ࠆ㧚ߚߞߚߎࠇߛߌߩߎߣߢ᳗ਭߦฬ೨߇ᱷࠆߎߣߦߥߞߚߩߛ߆
ࠄ Jeans ߪߕ޿߱ࠎᓧࠍߒߚߣ⠨߃ࠆߩߪ߭߇ߺߢ޽ࠈ߁߆㧚߽߁ 1 ߟࠦࡔࡦ࠻ࠍⴕ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 13߃߫㧘⃻࿷㧘ᄙߊߩᢎ⑼ᦠߢᛒࠊࠇߡ޿ࠆ Rayleigh-Jeans law ߪ㧘಴ߐࠇߚᒰᤨߪ޽
߹ࠅᵈ⋡ߐࠇߥ߆ߞߚࠃ߁ߢ޽ࠆ㧚ߣ޿߁ߩߪߔߢߦ Planck ߩᑼ߇޽ߞߡߘࠇ߇ታ㛎ߣቢోߦ৻⥌ߒߡ޿ࠆߎߣ߇ಽ߆ߞߡ޿ߚ߆ࠄߢ޽ࠈ߁㧚㧔Rayleigh-Jeans law ߇ታ㛎ߣวࠊߥ޿ߎߣࠍᡷ⦟ߒࠃ߁ߣߒߡ Planck ߩᑼ߇಴ߐࠇߚߣ޿߁⺑᣿߇ߒߡ޽ࠆᢎ
⑼ᦠ߇޽ࠆ߇㧘ᱧผ⊛ߦߪߤ߁߽ߘ߁޿߁ߎߣߢߪߥ޿ࠄߒ޿㧚㧕ߢߪ Rayleigh ߩ⁓޿ߪߤߎߦ޽ߞߚߩߛࠈ߁߆㧫ߤ߁߽㤥૕ユ኿ߩಽᏓߪੑᰴ⊛ߥ⹤ߢ㧘ᧄᒰߦᓐ߇໧
㗴ߦߒߚ߆ߞߚߩߪ㧘⛔⸘ജቇߦ߅ߌࠆ╬ಽ㈩ߩᴺೣ㧔ߎࠇࠍ↪޿ߡᑼ(1.1.1)ࠍዉ಴
ߒߡ޿ࠆ㧕ߣ߆㧘ࠛ࡯࠹࡞ߩሽ࿷ߩ᦭ήߣ߆ߢ޽ߞߚࠃ߁ߢ޽ࠆ㧚ߎߩ޽ߚࠅߩߎߣ
ߪ㧘J. Mehra, H. Rechenberg "The Historical Development of Quantum Theory" (Springer
Verlag) Vol. 1, p. 83㨪 ߦ޿ࠈ޿ࠈߣ⸃⺑߇ߩߞߡ޿ߡ⥝๧ᷓ޿㧚

1.2 శ㔚ലᨐ㊄ዻ⴫㕙ߦ⚡ᄖ✢ࠍᾖ኿ߔࠆߣ㔚ሶ߇㘧߮಴ߒߡ᧪ࠆߎߣ߇ 19 ਎♿ߩ⚳ࠅߏࠈ⷗ߟ߆ࠅ㧘శ㔚ലᨐ㧔photoelectric effect㧕ߣߒߡ⍮ࠄࠇࠆࠃ߁ߦߥߞߚ㧚శ㔚ലᨐߦ㑐ߔࠆታ㛎ߢ㧘એਅߦㅀߴࠆ 2 ߟߩ੐ᨩ߇ᣥ᧪ߩశߩᵄേ⺑ߢ⺑᣿ߔࠆߎߣ߇಴᧪ߥ߆
ߞߚ㧚1 ߟߪ಴ߡ᧪ߚ㔚ሶߩㆇേࠛࡀ࡞ࠡ࡯߇౉኿శߩᒝߐߦή㑐ଥߢ޽ࠆߣ޿߁ߎ
ߣߢ޽ࠆ㧚శ㧘ߔߥࠊߜ㔚⏛ᵄߣ޿߁߽ߩߪߘߩㅴⴕᣇะߦု⋥ߦᝄേߔࠆ㔚႐ߢ޽ࠅ㧘ߘߩᒝᐲߪ㔚႐ߩੑਸ਼ߦᲧ଀ߔࠆ㧚శߩᒝߐ߇Ⴧߔߦߟࠇߡ㔚ሶߪࠃࠅỗߒߊㆇേߔࠆࠃ߁ߦߥࠅ㧘ߟ޿ߦ㧘㊄ዻߩ⴫㕙߆ࠄ㘧߮಴ߒߡ᧪ࠆߣ⠨߃ࠄࠇࠆ㧚ߎߩࠃ߁ߥฎౖ⊛ߥ⠨߃ᣇߢߪ㧘಴ߡ᧪ߚ㔚ሶߩㆇേࠛࡀ࡞ࠡ࡯߇శߩᒝߐߦή㑐ଥߢ޽ࠆߎ
ߣࠍ⺑᣿಴᧪ߥ޿㧚ᦝߦ㧘ߎߩࠃ߁ߥ⠨߃ᣇߢߪ㧘චಽߥᒝߐߩశߢ޽ࠅߐ߃ߔࠇ߫㧘౉኿శߩᵄ㐳ߦ㑐ଥߥߊ㔚ሶ߇㘧߮಴ߔߎߣ߇ᦼᓙߐࠇࠆ㧚ߒ߆ߒታ㓙ߦߪ㧘޽ࠆ㑣
୯Q 0 ࠃࠅ߽ૐ޿ᝄേᢙߩశߢߪ㔚ሶߪ಴ߡ᧪ߥ޿㧚ߘߒߡ㔚ሶߩㆇേࠛࡀ࡞ࠡ࡯ߪਈ
߃ߚశߩᝄേᢙQ ߣߎߩ㑣୯Q 0 ߣߩᏅߦᲧ଀ߔࠆ㧘ߣ޿߁߹ߎߣߦᄸᅱߥ⚿ᨐ߇᷹ⷰߐࠇߚ㧚㧔࿑ 1.2.1㧕

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 14

kinetic energy/eV

4

3

2

1

0
0

2

4

6

8

14

Q /10

10

12

14

Hz

࿑ 1.2.1. శ㔚ലᨐߢ㘧߮಴ߒߚ㔚ሶߩㆇേࠛࡀ࡞ࠡ࡯ߣ౉኿శߩᝄേᢙߣߩ㑐ଥߎߩ໧㗴ࠍ⸃᳿ߒߚߩ߇ Einstein ߢ޽ࠆ㧚ᓐߪ Planck ߩ㊂ሶ઒⺑ࠍขࠅ౉ࠇߚ߇㧘ᦝߦߘࠇࠍ㕖Ᏹߦ㊀ⷐߥὐߢ⊒ዷߐߖߚ㧚਄ㅀߒߚࠃ߁ߦ㧘Planck ߪࠛࡀ࡞ࠡ࡯ߩ㊂ሶൻ㧘ߔߥࠊߜ㧘 E hQ ߣ޿߁᭎ᔨࠍේሶਛߩ㔚ሶߩᝄേߦㆡ↪ߒߚ߇㧘಴ߡ᧪ߚశ
ߘߩ߽ߩߪฎౖ⊛ߥᵄߣߒߡᝄࠆ⥰߁ߣ⠨߃ߚ㧚ߒ߆ߒ Einstein ߪ㧘ߘࠇߦ෻ߒ㧘ユ኿ߘߩ߽ߩ߇ E

hQ ߣ޿߁ࠛࡀ࡞ࠡ࡯ࠍᜬߞߚశߩ☸ሶ㧔⃻࿷ߪ‫ޟ‬శሶ‫ߣޠ‬๭߫ࠇࠆ㧕ߩ㓸߹ࠅߢ޽ࠆߣ⠨߃ߚ㧚ࠛࡀ࡞ࠡ࡯଻ሽߩᴺೣࠃࠅㆇേࠛࡀ࡞ࠡ࡯ߪ

1 2
mv
2

h Q Q 0(1.2.1)ߢ␜ߐࠇࠆߎߣࠍ Einstein ߪ␜ߒߚ㧚ߎߎߢ hQ 0 ߪ‛⾰ߦ࿕᦭ߥ઀੐㑐ᢙߢ޽ࠅ㧘ߘߩ‛⾰ߩࠗࠝࡦൻࡐ࠹ࡦࠪࡖ࡞ߦ⋧ᒰߔࠆ㧚Einstein ߪታ㛎⚿ᨐࠍ⸃ᨆߒߡ h ߩ୯ࠍ᳿ቯߒߚ߇㧘Planck ߩ᳞߼ߚ୯ߣ߶ߣࠎߤหߓ୯߇ᓧࠄࠇߚ㧚‫ޤ⸼ޣ‬㧦࿑ 1.2.1 ߩ❑ゲߪ eV ߣ޿߁ࠛࡀ࡞ࠡ࡯ߩන૏ߢ⴫ߒߡ޽ࠆ㧚eV ߣ޿߁ߩߪ㧘㔚᳇⚛㊂ e ࠍᜬߟ㔚⩄ࠍ㧘㔚࿶ V ߦߐ߆ࠄߞߡ㧘ή㒢㆙ߦㆇ߱ߦⷐߔࠆ઀੐ߢ޽ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 15

eV ߣ J ߣߩ㑐ᢙߪ㧔㜞ᩞߩ‛ℂߢ߽ቇ⠌ߒߚࠃ߁ߦ㧕ᰴߩࠃ߁ߦߒߡ᳞߹ࠆ㧚
1 eV= 1.602 u 1019 C u 1 V

1.602 u 1019 J‫ڏ‬Problem 1.2.1
࿑ 1.2.1 ࠃࠅ㧘Planck constant ࠍ᳞߼ࠃ㧚ߎߩ࿑߆ࠄᓧࠄࠇࠆ㒢ࠅߩᱜ⏕ߐߢ㧘ࠣ࡜ࡈߩ⺒ߺขࠅߩ⺋Ꮕ߹ߢ⠨ᘦߒߡ᳞߼ࠃ㧚ᱜߒ޿୯[ᑼ(1.1.4)]ߣᲧセߒߡߺࠃ㧚

1.3 ࿕૕ߩᾲኈ㊂නේሶ࿕૕ߩࡕ࡞ᾲኈ㊂㧔molar heat capacity㧕߇‛⾰ߩ⒳㘃ߦࠃࠄߕ 3R 㧔 R ߪ gas

constant㧘 R 8.314 JK 1 mol1 㧕ߦ╬ߒ޿ߣ޿߁ߎߣߪฎߊ߆ࠄ Dulong-Petit ߩᴺೣߣߒߡ⍮ࠄࠇߡ޿ߚ㧚ߎߩᴺೣߪࠛࡀ࡞ࠡ࡯ߩ╬ಽ㈩ೣ߆ࠄ◲නߦℂ⸃಴᧪ࠆ㧚ߔߥࠊߜ㧘޽ࠆ㊂ߩ࿕૕߇ N ୘ߩේሶࠍ฽߻ߣߔࠆ㧚⚿᥏ࠍ 1 ߟߩ႙ߣ⠨߃ࠆߣਗㅴㆇേߩ
⥄↱ᐲ߇ 3㧘࿁ォㆇേߩ⥄↱ᐲ߇ 3 ߢ޽ࠆ߆ࠄ⚿᥏ో૕ߩㆇേߩ⥄↱ᐲߪ 3 N  6 ߢ޽ࠆ㧚 N ߪචಽߦᄢ߈޿߆ࠄߎࠇࠍ 3 N ߣ߅޿ߡᏅߒᡰ߃ߥ޿㧚ฎౖ⊛ߥේሶߩㆇേߢ
ߪ㧘ේሶ 1 ୘ᒰߚࠅߩᐔဋߩㆇേࠛࡀ࡞ࠡ࡯ߪ kT 2 㧘ᐔဋߩࡐ࠹ࡦࠪࡖ࡞ࠛࡀ࡞ࠡ
࡯ߪ kT 2 ߢ޽ࠆ߆ࠄ㧘ో૕ߣߒߡ 3NkT ߩᝄേࠛࡀ࡞ࠡ࡯ࠍᜬߟ㧚ᓥߞߡ 1 ࡕ࡞ᒰࠅߩᾲኈ㊂㧘ߔߥࠊߜࡕ࡞ᾲኈ㊂ CV ߪ N ߣߒߡࠕࡏࠟ࠼ࡠቯᢙ N A ࠍ↪޿ 3 N A k

3R ߣߥࠆߎߣ߇⺑᣿ߐࠇࠆ㧚ߒ߆ߒߥ߇ࠄ㧘᷷ᐲ߇ૐߊߥࠆߣ Dulong-Petit ߩᴺೣ߇ᚑࠅ┙ߚߥߊߥࠆ㧚᷷ᐲߩૐਅߦ઻ߞߡ CV ߪዊߐߊߥࠅ㧘ߟ޿ߦ T

0 ߢ CV䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 16

0 ߦ෼᧤ߔࠆ㧚㧔࿑ 1.3.1㧕

25

Cv/J K-1 mol-1

20

15

10

5

0
0

50

100

150

200

250

300

T/K
࿑ 1.3.1㊄ዻ㌁ߩࡕ࡞ᾲኈ㊂ߎߩࠃ߁ߥᾲኈ㊂ߩ⇣Ᏹᕈߦᦨೋߦ⸃᳿ࠍਈ߃ߚߩߪ㧘߹ߚ߽߿޽ߩ Einstein ߢ޽ߞߚ㧚Einstein ߪ㧘⚿᥏ਛߩේሶߩ㧘ᐔⴧ૏⟎ߩ๟ࠅߢߩᝄേ߇㊂ሶൻߒߡ޿ࠆߣ઒ቯ
ߒߚ㧚ߔߥࠊߜ㧘ߔߴߡߩේሶߪᝄേᢙQ ߢᝄേߒ㧘ߘߩࠛࡀ࡞ࠡ࡯ߪ E nhQ ߢਈ߃ࠄࠇࠆߣ⠨߃ߚࠊߌߢ޽ࠆ㧚ߎ߁ߒߡ Einstein ߪᾲኈ㊂߇ૐ᷷ߦߥࠆߣዊߐߊߥࠆ
ߎߣࠍ⺑᣿ߔࠆߎߣ߇಴᧪ߚ㧚

Einstein ߩᾲኈ㊂ߩᑼߪએਅߩࠃ߁ߦߒߡዉ಴ߐࠇࠆ㧚⚿᥏ਛߩ 1 ߟߩේሶ߇ hQ ߩn ୚ߩࠛࡀ࡞ࠡ࡯ߢᝄേߒߡ޿ࠆߩߢ޽ࠆ߆ࠄࡏ࡞࠷ࡑࡦಽᏓࠍߒߡ޿ࠆ⁁ᘒߢߩᐔဋࠛࡀ࡞ࠡ࡯ H ߪ㧘ᑼ(1.1.4)ߢ᳞߼ߚࠃ߁ߦ

H

hQ§ hQ ·
exp ¨¸ 1© kT ¹(1.3.1)ߢ⴫ߐࠇࠆ㧚1 ߟߩේሶߦߟ޿ߡ 3 ߟߩᝄേߩ⥄↱ᐲ߇޽ࠅ㧔 x, y, z ߘࠇߙࠇߩᣇะ
ߩᝄേ㧕㧘߹ߚ㧘⚿᥏߇ N ୘ߩේሶ߆ࠄ಴᧪ߡ޿ࠆ႐วࠍ⠨߃ࠆߣ⚿᥏ో૕ߣߒߡ 3Nߩᝄേߩ⥄↱ᐲ߇޽ࠆߎߣߦߥࠆ㧚㧔ᱜ⏕ߦߪ⚿᥏ో૕ߣߒߡߩਗㅴㆇേߩ⥄↱ᐲ߇ 3䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 17ߢ⚿᥏ో૕ߣߒߡߩ࿁ォㆇേߩ⥄↱ᐲ߇ 3 ߢ޽ࠆ߆ࠄߘࠇࠄࠍᒁ޿ߡᝄേߩ⥄↱ᐲߪ
3 N  6 ߢ޽ࠆ㧚ߒ߆ߒ㧘ࡕ࡞⒟ᐲߩ‛⾰㊂ࠍ໧㗴ߣߒߡ޿ࠆ႐วߦߪචಽࠃ޿ㄭૃߢ
3 N  6 3 N ߣ߅޿ߡᏅߒᡰ߃ߥ޿㧚㧕ᓥߞߡ㧘⚿᥏ో૕ߩᐔဋࠛࡀ࡞ࠡ࡯ E ߪᰴߩࠃ

߁ߦߥࠆ㧚

E

3NhQ
§ hQ ·
exp ¨¸ 1© kT ¹(1.3.2)ᾲኈ㊂ߪ E ࠍ᷷ᐲߦߟ޿ߡᓸಽߒߚ߽ߩߢ޽ࠆߩߢ

§ hQ ·exp ¨¸§ wE ·§ hQ ·© kT ¹3
CV ¨
Nk(1.3.3)¸¨¸
2© wT ¹V© kT ¹ ª§ hQ · º«exp ¨ kT ¸  1»©
¹ ¼¬
1 ࡕ࡞ᒰߚࠅߩᾲኈ㊂㧘ߔߥࠊߜࡕ࡞ᾲኈ㊂㧔molar heat capacity㧕ߪ N N A 㧔 N A ߪࠕࡏࠟ࠼ࡠቯᢙ㧕ߣ߅޿ߡ㧘ߐࠄߦ N A k R ߦ⌕⋡ߔࠆߎߣߦࠃࠅ㧘2§ hQ ·exp ¨¸§ hQ ·© kT ¹
3R ¨¸
2© kT ¹ ª§ hQ · º«exp ¨ kT ¸  1»
©
¹ ¼
¬
2

CV ,m

(1.3.4)

ߎߎߢ Einstein temperature T E ࠍቯ⟵ߔࠆ㧚

TE

hQ
k

(1.3.5)

Einstein temperature ࠍ↪޿ࠆߣᑼ(1.1.4)ߪᰴߩࠃ߁ߦߥࠆ㧚
2

CV ,m

§T ·
exp ¨ E ¸
©T ¹

§T ·
3R ¨ E ¸
2©T ¹ ª§ TE · º«exp ¨ T ¸  1»
© ¹ ¼¬

(1.3.6)ߎߩᦛ✢ߪ T o f ߢ 3R ߦ෼᧤ߒ Dulong-Pettit law ߣหߓߦߥࠆ㧚߹ߚ㧘 T o 0 ߢ 0 ߦ෼᧤ߒታ㛎⚿ᨐࠍ߶߷⺑᣿ߔࠆ㧚ߎߩࠃ߁ߦ E nhQ ߣ޿߁⠨߃ᣇ߇ౣ߮૶ࠊࠇߚࠊߌߢ޽ࠆ㧚੹߹ߢ಴ߡ᧪ߚ㊂ሶൻߣ޿߁᭎ᔨߪ㧘ේሶߩਛߩ㔚ሶߩᝄേ߿㧘߹ߚߪユ኿ߘߩ߽ߩ߇㊂ሶൻߒߡ޿ࠆߣ
޿߁⠨߃ᣇߢ޽ࠅ㧘ℂ⸃ߒߦߊ޿ߥ߇ࠄ߽ߘࠎߥ߽ߩ߆ߥߣᕁ߃ࠆ߆߽⍮ࠇߥ޿㧚ߒ߆ߒ੹࿁ߩࠃ߁ߦ㧘⚿᥏ਛߩේሶߩᝄേߢ㧘ታߦฎౖ⊛ߥ߫ߨߢ⚿߫ࠇߚ☸ሶߩᝄേߣߺߥߖࠆࠃ߁ߥᯏ᪾⊛ߥᝄേ߇หߓࠃ߁ߦ㊂ሶൻߒߡ޿ࠆߣ޿߁⠨߃ᣇߪߥ߆ߥ
߆ℂ⸃ߒ߇ߚ޿ߎߣߢߪ޽ࠆߣᕁ߁㧚ߒ߆ߒߣ߽߆ߊ㧘㊂ሶൻߣ޿߁᭎ᔨ߇޿ࠈ޿ࠈ
ߥ႐วߦ಴ߡ߈ߡ㧘ੱ‫ߪޘ‬ᕁ⠨ߩォ឵ࠍ૛௾ߥߊߐࠇࠆࠊߌߢ޽ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 18‫ڏ‬Problem 1.3.1

Einstein ߩᾲኈ㊂ߩᑼ(1.3.6)ߪ T o f ߢ 3R ߦ෼᧤ߒ㧘߹ߚ㧘 T o 0 ߢ࠯ࡠߦㄭߠߊ
ߎߣߪߔߋಽ߆ࠆ㧚ߢߪߘࠇߙࠇߩ႐ว㧘ߤ߁޿߁㑐ᢙᒻߢᭂ㒢୯ߦㄭߠߊ߆㧚[Coffee Break] ᾲኈ㊂ߦ㑐ߔࠆ Einstein Model
Einstein ߩࡕ࠺࡞ߪᾲኈ㊂߇᷷ᐲߩૐਅߣ౒ߦᷫዋߔࠆߣ޿߁੐ታࠍቯᕈ⊛ߦߪ⺑᣿ߒᓧߚ߇㧘ታߪቯ㊂⊛ߦߪਇචಽߢ޽ߞߚ㧚㕖Ᏹߦૐ᷷ߢߪ㧘ታ㛎ߢߪ㧘߶߷ T 3 ߩ㑐ଥߢ࠯ࡠߦㄭߠߊߩߦ㧘Einstein ߩℂ⺰ߢߪ߽ߞߣᕆỗߦ࠯ࡠߦㄭߠߊ㧚ߘߩὐߦ
㑐ߔࠆℂ⺰ߩਇቢోߐߪᓟߢ Debye ߩℂ⺰ߢ⸃᳿ߐࠇࠆߎߣߦߥࠆ߇㧘ߎߩᤨὐߢ

Einstein ߇ዊߐߥ㘩޿㆑޿ࠍ޽߹ࠅ᳇ߦߔࠆߎߣߥߊ㧘ቯᕈ⊛ߦ⃻⽎ࠍᛠីߒߚߎߣ
߇ࠃ߆ߞߚߩ߆߽ߒࠇߥ޿㧚ᭂૐ᷷ㄭㄝߩࠊߕ߆ߥ㘩޿㆑޿ߦᖠࠎߢ Einstein ߇ᓐߩ⠨߃ࠍ⊒⴫ߔࠆߩࠍᱛ߼ߡߒ߹ߞߡ޿ߚࠄ㧘ᓐߩฬ೨ߪᾲኈ㊂ߩ໧㗴ߦ㑐ߒߡߪᱷࠄ
ߥ߆ߞߚ߆߽ߒࠇߥ޿㧚⎇ⓥߪ෩ኒߢߥߌࠇ߫޿ߌߥ޿߇ᤨߣ႐วߦࠃߞߡߪ෩ኒߐ
ߦߎߛࠊࠅߔ߉ࠆߩ߽⦟ߊߥ޿ߣ޿߁ߎߣ߆߽ߒࠇߥ޿㧚[Coffee Break] Dulong-Petit ߩᴺೣ‫࠭ࠗࠢڏ‬㧦 ‫ޟ‬Dulong-Petit ߩᴺೣ‫߁ࠂߒߢߚࠇߐ⷗⊒ߡߞࠃߦ⺕ߪޠ‬㧚 1, 2, 3 ߩ߁ߜ߆ࠄㆬߴ㧚㧔ᱜ⸃ߪ㧘଀߃߫㧘ጤᵄߩℂൻቇㄉౖࠍ⷗ࠃ㧚㧕

1. Dulong ߇ᦨೋߦ⊒⷗ߒ㧘ᓟߢ Petit ߇ᦝߦ⊒ዷߐߖߚ㧚
2. Dulong ߣ Petit ߩ 2 ੱߩ౒ห⎇ⓥߢ⊒⷗ߐࠇߚ㧚
3. Dulong-Petit ߣ޿߁ฬ೨ߩ 1 ੱߩੱߦࠃߞߡ⊒⷗ߐࠇߚ㧚

1.4 Compton ᢔੂߣశߩ☸ሶᕈశ㔚ലᨐߩታ㛎߆ࠄ␜ໂߐࠇߚࠃ߁ߦశ߇☸ሶߣߒߡߩᕈᩰࠍᜬߟߥࠄ߫㧘శߪㆇ
േ㊂ p ࠍᜬߟߪߕߢ޽ࠆ㧚ߎߩߎߣߪ 1923 ᐕߩ Compton ߩታ㛎ߢ⸽᣿ߐࠇࠆߎߣߦ
ߥࠆ㧚࿑ 1.4.1 ߩࠃ߁ߦ㧘޽ࠆ࿕૕ߦࠃߞߡᢔੂߐࠇߚ㨄✢ࠍಽశེߦ߆ߌ㧘ᢔੂᵄߩᒝᐲࠍᵄ㐳ߩ㑐ᢙߣߒߡ᷹ቯߔࠆ㧚ಽశེߣ޿ߞߡ߽⚿᥏ߢ޽ࠅ㧘Bragg ߩ᧦ઙࠍ೑↪ߒߡ㨄✢ߩᵄ㐳ࠍ᷹ቯߔࠆࠊߌߢ޽ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 19

T

O

O
C
࿑ 1.4.1

ᢔੂⷺ T

O
D

Thomson scattering ߣ Compton scattering㧔a㧕 T 0  㧔b㧕 T z 0

0 ߩᤨߦߪ౉኿㨄✢ߩᵄ㐳 O0 ߩᚲߦ 1 ߟߩ peak ߇᷹ⷰߐࠇࠆ㧚ߎࠇߪ౉኿㨄✢߇ߘߩ߹߹ᢔੂߐࠇߚ߽ߩߢ޽ࠆ㧚ߣߎࠈ߇ T z 0 ߩ႐วߦߪ㧘 O0 ࠃࠅ߽㐳ᵄ㐳ߩ O1 ߦ߽ᢔੂᒝᐲߦ peak ߇⃻ࠇࠆ㧚 O0 ߣ O1 ߩᵄ㐳ߩᏅ 'O

1  cos T

2sin 2T
2

O1  O2 ߪߣ޿߁ T ଐሽᕈࠍ␜ߔ㧚ᵄ㐳ᄌൻࠍ઻ࠊߥ޿ᢔੂࠍ Thomson ᢔੂߣ๭߮㧘ᵄ㐳ᄌൻࠍ઻߁ᢔੂࠍ Comptonᢔੂߣ޿߁㧚ߤ߁ߒߡ౉኿శࠃࠅ߽㐳ᵄ㐳஥ߦᢔੂ߇⃻ࠇࠆ߆ߪ㧘శࠍ☸ሶߣߺߥߔ
ߎߣߦࠃߞߡᆎ߼ߡ⺑᣿ߔࠆߎߣ߇಴᧪ࠆ㧚శሶ߇࿕૕ਛߩ㔚ሶߣⴣ⓭ߒߡ㧘ߘࠇࠍߪߓ߈㘧߫ߔߎߣߦࠃࠅ㧘శሶ⥄りߪࠛࡀ࡞ࠡ࡯ࠍᄬ޿㧘ߘߩಽߛߌᝄേᢙ߇ዊߐߊ
ߥࠅᵄ㐳߇㐳ߊߥࠆߩߢ޽ࠆ㧚એਅߎߩ໧㗴ࠍ߽߁ዋߒቯ㊂⊛ߦᛒߞߡߺࠆ㧚

Planck ߩశ㊂ሶ⺑ߦࠃࠆߣ㧘ᝄേᢙQ ࠍᜬߞߚశߩ energy ߪ E

hQ ߢਈ߃ࠄࠇࠆ㧚৻ᣇ㧘ฎౖ㔚⏛᳇ቇߦࠃࠇ߫㧘energy E ߩ㔚⏛ᵄߪߘߩㅴⴕᣇะߦ

Eߩᄢ߈ߐߩㆇേ
c䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 20

㊂㧔momentum㧕ࠍㆇ߱ߎߣ߇⍮ࠄࠇߡ޿ࠆ㧚ߎߩ 2 ߟߩ㑐ଥᑼࠍวࠊߖࠆߣ㧘శߩᜬߞߡ޿ࠆ momentum ߪ

p

hQ
c

h

(1.4.1)Oߣ޿߁ߎߣߦߥࠆ㧚౉኿㨄✢ߩᜬߟ momentum vector ࠍ p0 ߣߒࠃ߁㧚శሶ߇㔚ሶ㧔ᆎ߼ߪ㕒ᱛߒߡ޿ࠆ
ߣ⠨߃ߡࠃ޿㧕ߣⴣ⓭ߒߡ㧘⥄ಽ⥄りߪ momentum vector p1 ߢ㘧߮෰ࠅ㧘ߘߩ෻〡ߢ㔚ሶߪ momentum vector p ࠍᜬߟߣߒࠃ߁㧚㧔࿑ 1.4.2(a)㧕

࿑ 1.4.2

Compton scattering

momentum ଻ሽߩᑼߪ
p0 p1  p

(1.4.2)ห᭽ߦ energy ଻ሽೣࠃࠅ

p0 c  me c 2

p1c  me c 4  c 2 p 2
2(1.4.3)ߎߎߢ㧘㔚ሶߩ㕒ᱛࠛࡀ࡞ࠡ࡯ me c 2 ࠍ฽߻⋧ኻ⺰⊛ߥ energy ߩ⴫ᑼࠍ↪޿ߡ޿ࠆ㧚

momentum ଻ሽߩᑼߢ㧘3 ߟߩ vector p0 , p1 , p ߪ࿑ 1.4.2(b)ߩࠃ߁ߥਃⷺᒻࠍ૞ࠆߎ
ߣ߆ࠄpp0 2  p12  2 p0 p1 cos T䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 21(1.4.4)ߣߥࠆ㧚ߎߩᑼࠍᑼ(1.4.3)ߩฝㄝߦઍ౉ߒᢛℂߔࠆߣ
me c p0  p1

p0 p1 1  cos T(1.4.5)ᑼ(1.4.1)ࠍઍ౉ߔࠆߎߣߦࠃߞߡᵄ㐳ߩߕࠇ 'O ߣߒߡᰴᑼࠍᓧࠆ㧚hhT
1  cos T2
sin 2'O O1  O
2
me c
me c
ߎߩᑼߪታ㛎⊛ߦ⏕߆߼ࠄࠇߚ㧚ߎ߁޿߁ߎߣߢశ߇ momentumhO(1.4.6)
ࠍᜬߞߚ☸ሶߢ

޽ࠆߎߣ߇ታ㛎⊛ߦߪߞ߈ࠅߒߚࠊߌߢ޽ࠆ㧚‫ڏ‬Problem 1.4.1
⾫᣿ߥ⻉ำߪߎߩ▵ߩ⼏⺰ߢ㧘⚿᥏ౝߩ㔚ሶ߇⥄↱㔚ሶߣߒߡᛒࠊࠇߡ޿ࠆߎߣߦ
ਇḩࠍᜬߞߚߢ޽ࠈ߁㧚㊄ዻౝߩ㔚ሶߪේሶᩭߦ Coulomb ജߢᒁ߈ઃߌࠄࠇߡ޿ࠆࠊ
ߌߢ޽ࠆ㧚ߘߩ energy ߣᾖ኿ߔࠆ X ✢ߩశሶߩ energy ߣᲧߴߡߺࠃ㧚଀߃߫ X ✢ߣߒߡߪ㧘Mo ߩ KD✢ߣߒߡ⍮ࠄࠇࠆ 71 pm ࠍ⠨߃ࠃ㧔 pm 1012 m 㧕㧚㔚ሶ߇࿕૕ߦ᧤❈ߐࠇߡ޿ࠆജߣߒߡߪ㧘ࠝ࡯࠳࡯⊛ߦ㧘  e ߅ࠃ߮ e ߩ㔚⩄߇ 100pm 㔌ࠇߡሽ࿷ߒߡ޿ࠆᤨߩ Coulomb ࡐ࠹ࡦࠪࡖ࡞ߣߒߡߺࠃ㧚‫ڏ‬Problem 1.4.2
ᵄ㐳 71 pm ߩ X ✢߇ graphite ߦ౉኿ߔࠆᤨ㧘ᢔੂⷺ 90qߩ Compton ᢔੂߦࠃࠆᵄ㐳ᄌൻߪ޿ߊࠄ߆㧚‫ڏ‬Problem 1.4.3
ߎߩ▵ߩ⼏⺰ߪ㧘శሶߣⴣ⓭ߔࠆ߽ߩ߇㔚ሶߢߥߊߡ߽หߓࠃ߁ߦㆡ↪಴᧪ࠆ㧚นⷞశߩశࠍಽሶߦᾖ኿ߒߚᤨ㧘శሶߣߘߩಽሶߣߩⴣ⓭ࠍ⠨߃ࠇ߫㧘਄ߢ᳞߼ߚᑼߩme ࠍಽሶߩ⾰㊂ߢ߅߈߆߃ࠇ߫޿޿ߪߕߢ޽ࠆ㧚ᵄ㐳 500 nm ߩశ߇㧘benzene ߦᾖ኿ߒߚᤨߦߤߩ૏ߩᵄ㐳ߩߕࠇ߇޽ࠆ߆ࠍ⸘▚ߖࠃ㧚

[Coffee Break] Raman ಽశቇ
Raman ಽశቇߣ޿߁㕖Ᏹߦ㊀ⷐߥಽశቇ߇޽ࠆ㧚‛⾰ߦశࠍᒰߡࠆߣ㧘ߘࠇࠃࠅ߽
㐳ᵄ㐳߅ࠃ߮⍴ᵄ㐳஥ߦᵄ㐳߇ዋߒߕࠇߚᚲߦᢔੂ߇⃻ࠇࠆ㧚ߎߩᝄേᢙߩߕࠇߪ㧘ಽሶߩᝄേߩࠛࡀ࡞ࠡ࡯ߦ⋧ᒰߔࠆߣ޿߁ߎߣߢ‛⾰ߩหቯߦ߽㧘෶㧘ಽሶᝄേߩ⎇䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 22ⓥߦ߽ᰳ߆ߖߥ޿㊀ⷐߥಽశቇߣߒߡ⊒ዷߒߚ㧚Raman ലᨐߣ⸒ࠊࠇࠆߎߩലᨐߪ

1928 ᐕࠗࡦ࠼ߩ Raman ߦࠃࠅ⊒⷗ߐࠇ㧘Raman ߪߎߩഞ❣ߦࠃࠅ 1930 ᐕߦࡁ࡯ࡌ࡞
‛ℂቇ⾨ࠍฃ⾨ߒߡ޿ࠆ㧚Raman ߩ⊒⷗ߪ Compton ߩ⎇ⓥ߇߈ߞ߆ߌߣߥߞߚࠃ߁
ߢ޽ࠆ㧚Nature, Vol. 121, 501 (1928)ߩ"A New Type of Secondary Radiation"ߣ޿߁㗴ߩ
Raman ߩ⺰ᢥߪᰴߩࠃ߁ߥᢥߢᆎ߹ߞߡ޿ࠆ㧚"If we assume that the X-ray scattering of the 'unmodified' type observed by Prof.
Compton corresponds to the normal or average state of the atoms and molecules, while
the 'modified' scattering of altered wavelength corresponds to their fluctuations from thestate, it would follow that we should expect also the case of ordinary light two types ofscattering, one determined by the normal optical properties of the atoms or molecules,
and another representing the effect of their fluctuations from their normal state. It
accordingly becomes necessary to test whether this is actually the case. ..."
ߎߩᢥࠍ⺒߻ߣ㧘Raman ߪ㧘Compton ߇ X ✢ߢ⷗಴ߒߚ߽ߩ߇นⷞㇱߩశߢ߽޽
ࠆߛࠈ߁ߣ⠨߃ߡታ㛎ߒߚߎߣ߇ಽ߆ࠆ㧚ߘߒߡᧄᒰߦ੍ᗐㅢࠅߩ⚿ᨐ߇ᓧࠄࠇߚࠊ
ߌߢ޽ࠆ㧚ߎߩࠃ߁ߥ߈ߞ߆ߌ߆ࠄⴕߞߚታ㛎߇ࡁ࡯ࡌ࡞⾨ߦߥߞߚߣ޿߁ߎߣߪᗵ
ᘣᷓ޿㧚ߣ޿߁ߩߪ㧘ߎ߁޿߁߈ߞ߆ߌߪ㧘ߒ߫ߒ߫‫ޟ‬㌃㋕⎇ⓥ‫ߢ⪲⸒߁޿ߣޠ‬ᛕ್ߐࠇࠆ⎇ⓥ⸘↹ߣ⚕৻㊀ߩᏅߢ޽ࠆ߆ࠄߢ޽ࠆ㧚㌃㋕⎇ⓥߣ޿߁⸒⪲ߪ‫ޟ‬㌃ߢ⎇ⓥࠍⴕߞߚࠄ߁߹ߊ޿ߞߚ㧚ߘࠇߢߪ㋕ߢ߿ߞߡߺࠃ߁‫ߚߞ޿ߣޠ‬቟ᤃߥ⎇ⓥ⸘↹ߦၮߠ޿ߚ⎇ⓥߢ㕙⊕ߊߥ޿⎇ⓥߩౖဳ⊛ߥ଀ߣߒߡࠃߊᒁ↪ߐࠇࠆ⸒⪲ߢ޽ࠆ㧚‫ޟ‬X ✢߽㔚⏛ᵄ㧘นⷞㇱߩశ߽㔚⏛ᵄ㧘⇣ߥࠆߩߪᵄ㐳㧔෶ߪᝄേᢙ㧕ߩߺߢ޽ࠆ㧚ᓥߞߡ X✢ߦߟ޿ߡ⷗಴ߐࠇߚ⃻⽎߇ߚߣ߃นⷞㇱߦ޽ߞߚߣߒߡ߽ਇᕁ⼏ߢ߽ߥ޿ߒ㕙⊕ߊ߽ߥ޿‫ࠅ⫋ߡߞ޿ߣߛ⺰᥸ߦߜ⋥ߪ⺰⼏ߚߞ޿ߣޠ‬෰ࠆߎߣ߇಴᧪ߥ޿߶ߤߩ㗀߈
ߪ޽ࠆ㧚Raman ߇߶ࠎߩߜࠂߞߣߒߚᕁ޿ઃ߈߆ࠄߎߩࠃ߁ߥታ㛎ࠍߒߚߩ߆㧘෶
ߪ"fluctuation"ߣ޿߁⸒⪲ߢᓐ߇⴫ߒߡ޿ࠆㆫ⒖ߩౝኈߦ㨄✢㗔ၞߣนⷞㇱ㗔ၞߣߢᧄ⾰⊛ߥ㆑޿߇޽ࠆߎߣࠍᗧ⼂ߒߡߎߩታ㛎ࠍ߿ߞߚߩ߆ߪ㧘߽߁ዋߒᱧผ⊛⠨⸽ࠍߒߡߺߥ޿ߣಽ߆ࠄߥ޿߇㧘ߊߛࠄߥ޿ߣ⸒ࠊࠇࠆ⎇ⓥ߽ߔ߫ࠄߒ޿⎇ⓥ߽߈ߞ߆ߌ
ߪ⚕৻㊀ߩᏅߣ޿߁ߎߣߪታߦ㕙⊕޿㧚ߥ߅㧘਄ߦᒁ↪ߒߚ Raman ߩᢥߢ㧘unmodifiedߣ޿߁ߩߪᵄ㐳߇౉኿శߩߘࠇߣᄌࠄߥ޿ Thomson scattering ࠍ␜ߒ㧘modified ߣ޿߁ߩ߇ᵄ㐳߇ᄌൻߒߚ Compton scattering ࠍ␜ߒߡ޿ࠆ㧚modified ߅ࠃ߮ unmodified ߣ޿߁⸒⪲ߪ㧘Compton ߇ Physical Review, Vol 22, 409 (1923) ߩ⺰ᢥߢ↪޿ߡ߅ࠅ㧘ߘߩ⸒⪲ࠍᒁ↪ߒߚ߽ߩߣᕁࠊࠇࠆ㧚߹ߚ Raman ߩ⺰ᢥߩ"fluctuation"ߣ޿߁⸒⪲ߪߥ߆ߥ߆ಽ߆ࠅߦߊ޿㧚‫ߊࠃߣࠆߔߣޠ߉ࠄࠁޟ‬ಽ߆ࠄߥ޿㧚߅ߘࠄߊ⃻࿷ߩ⸒⪲ߢ޿߃߫‫ޟ‬ㆫ⒖‫(ޠ‬transition) ߣ޿߁ߩ߇߽ߞߣ߽ㆡᒰߥ⸒⪲ߢ޽ࠈ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 23

1.5 ‛⾰ߩᵄേ⊛ᕈ⾰ 㧙 de Broglie ᵄశ߇☸ሶ⊛ᕈ⾰ࠍᜬߞߡ޿ࠆߥࠄ߫㧘෻ኻߦ㧘‛⾰߇ᵄേ⊛ᕈ⾰ࠍᜬߞߡ޿ߡ߽ࠃ
޿ߩߢߪߥ߆ࠈ߁߆㧚ߎࠇ߇ de Broglie㧔࠼࡮ࡉࡠࠗ㧕ߩ⠨߃ߚߎߣߢ޽ࠅ㧘㊂ሶജ
ቇߩ⊒ዷߩᄢ߈ߥ♻ญߣߥߞߚ㧚਄ㅀߒߚࠃ߁ߦ㧘Einstein ߪ㧘ᵄ㐳 O ߩశߪ

p

h

O

ߩㆇേ㊂ࠍᜬߟߎߣࠍ␜ߒߚࠊߌߢ޽ࠆ㧚de Broglie ߪߘࠇࠍ෻ኻߦ⠨߃ߡ㧘ㆇേ㊂ p
ࠍᜬߟ‛⾰ߪᵄ㐳hO
p(1.5.1)ߢ␜ߐࠇࠆᵄߣߒߡߩᕈ⾰ࠍᜬߟߣ⠨߃ߚ㧚ߎߩᑼߢ␜ߐࠇࠆ O ߪ de Broglie ᵄ㐳ߣ⸒ࠊࠇߡ޿ࠆ㧚‫଀ڏ‬㗴
శㅦߩ 1㧑ߩㅦᐲࠍᜬߞߡ޿ࠆ㔚ሶߦߟ޿ߡ de Broglie ᵄ㐳ࠍ⸘▚ߖࠃ㧚‫ޤ⸃ޣ‬㧦㔚ሶߩ⾰㊂ me ߅ࠃ߮ㅦᐲ v ߪme

9.11u1031 kg mv

0.01u 3.00 u108 m/s

3.00 u 106 m/s(1.5.2)ߣߥࠆ㧚ᓥߞߡ momentum ߪpme v
9.11u 1031 kg u 3.00 u106 m/s(1.5.3)2.73 u1024 kg m/s

ᓥߞߡ de Broglie wavelength O ߪ
hO
2.43 u 1010 m 243 pm
p(1.5.4)de Broglie ߩ⠨߃ߚߎߣ߇ᱜߒߌࠇ߫㧘㔚ሶ߽శߣหߓࠃ߁ߦ࿁᛬ߔࠆߪߕߢ޽ࠆ㧚਄ߩ଀㗴ߢ␜ߐࠇߚࠃ߁ߦ㧘ㆡᒰߦടㅦߐࠇߚ㔚ሶߩ de Broglie ᵄ㐳ߪ߶߷⚿᥏ਛߩේሶ㑆〒㔌ߦߥࠆߩߢ㧘⚿᥏ࠍ↪޿ߡ㔚ሶߩ࿁᛬߇⷗಴ߐࠇࠆߪߕߢ޽ࠆ㧚⃻ߦ 1925䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 24ᐕߦ Davisson ߣ Germer ߦࠃߞߡ㔚ሶߩ࿁᛬⃻⽎߇⷗಴ߐࠇߚ㧚‛⾰ߩᵄേᕈߪߘߩᓟ᳓⚛ේሶ߿ࡋ࡝࠙ࡓේሶߦߟ޿ߡ߽ታ⸽ߐࠇ㧘⃻࿷ߢߪ㔚ሶ✢ߩߺߥࠄߕਛᕈሶߦࠃࠆ࿁᛬߽㧘‛⾰᭴ㅧߩ⎇ⓥߦࠃߊ೑↪ߐࠇߡ޿ࠆ㧚‫ޣ‬ᵈᗧ‫ޤ‬㧦ᑼ(1.5.2)߅ࠃ߮(1.5.3)ߦ߅޿ߡ㧘ㅦᐲߩ⸥ภ‫ ޟ‬v ‫ޠ‬㧔v ߩࠗ࠲࡝࠶ࠢ૕㧕߇ᝄേᢙߩ
⸥ภ‫ޟ‬Q ‫ޠ‬㧔ࠡ࡝ࠪࡖᢥሼߩ࠾ࡘ࡯㧕ߣૃߡ޿ࠆߩߢ߹߉ࠄࠊߒ޿ߣᕁ߁ੱ߽޿ࠆ߆
߽ߒࠇߥ޿㧚ߒ߆ߒ㧘߽ߞߣ߽ᐢߊ૶ࠊࠇߡ޿ࠆࡈࠜࡦ࠻㧔⧷ሼߪ Times New Roman㧘ࠡ࡝ࠪࡖᢥሼߪ Symbol㧕ߢߘ߁ߥߞߡ޿ࠆߩߢᵈᗧߒߡ⷗ߡ޿ߚߛߊߒ߆ߥ޿㧚ࠃߊ⷗ࠇ߫㆑޿ߪಽ߆ࠆߢ޽ࠈ߁㧚ᔨߩߚ߼㧘࿑ 1.5.1 ߦਔࡈࠜࡦ࠻ࠍ㧘ዋߒᄢ߈ߥሼ૕ߢᲧセߒߡ߅ߊ㧚v

Q⧷ሼ‫ ޟ‬v ‫ࠢ࠶࡝࠲ࠗߩޠ‬૕ࠡ࡝ࠪࡖᢥሼ࠾ࡘ࡯‫ࠢ࠶࡝࠲ࠗߩޠ  ޟ‬૕

࿑ 1.5.1 ⧷ሼ v 㧔ࡧࠖ㧕ߣࠡ࡝ࠪࡖᢥሼQ 㧔࠾ࡘ࡯㧕‫ڏ‬Problem 1.5.1
ᰴߩ߽ߩߦߟ޿ߡ de Broglie ᵄ㐳ࠍ᳞߼ࠃ㧚(a) 200 eV ߩㆇേࠛࡀ࡞ࠡ࡯ࠍᜬߟ㔚ሶ㧘(b) 105 eV ߩㆇേࠛࡀ࡞ࠡ࡯ࠍᜬߟࡊࡠ࠻ࡦ㧘(c) 100 km/h ߩㅦᐲࠍᜬߟ⾰㊂0.14 kg ߩ㊁⃿ߩࡏ࡯࡞

1.6 ᳓⚛ේሶߩࠬࡍࠢ࠻࡞
ߎߩ▵ߢߪ᳓⚛ේሶߩ⊒శࠬࡍࠢ࠻࡞ߦߟ޿ߡቇ⠌ߔࠆ੍ቯߢ޽ࠆ߇㧘ߘߩ೨ߦ㧘৻⥸⊛ߥ introduction ߣߒߡశߩࠬࡍࠢ࠻࡞ߩ᷹ቯߦߟ޿ߡዋ‫ޘ‬ㅀߴߡ߅߈ߚ޿㧚
Section 1.1 ߢ㤥૕ユ኿ߩߎߣࠍቇ⠌ߒߚᤨߦ㧘ᾲߖࠄࠇߚ‛૕ߩࠬࡍࠢ࠻࡞߇಴ߡ
᧪ߚ㧚ࠬࡍࠢ࠻࡞㧘ߔߥࠊߜ㧘శߩᒝᐲߩᵄ㐳ଐሽᕈߪ㧘৻⥸ߦߪ࿑ 1.6.1 ߩࠃ߁ߥಽశེߢ߽ߞߡ᷹ቯߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 25

slit

lens

prism
lens
photo
plate
࿑ 1.6.1 ࡊ࡝࠭ࡓಽశེ

ߔߥࠊߜ㧘ࠬ࡝࠶࠻ࠍㅢߒߡ౉኿ߒߚశߪࡊ࡝࠭ࡓߢಽᢔߐࠇ㧘࡟ࡦ࠭ߢ㓸శߐࠇߚᓟ㧘౮⌀ߩੇ᧼਄ߦᗵశߐࠇࠆ㧚㤥⊕ߩ౮⌀ੇ᧼㧔߹ߚߪࡈࠖ࡞ࡓ㧕ߩ႐วߦߪశ
߇ᒰߞߚᚲ߇㤥ߊߥࠆ㧚శ߇ᒝߌࠇ߫ᒝ޿߶ߤ૛⸘ߦ㤥ߊߥࠆ㧚೨߽ߞߡ㧘ੇ᧼਄ߩ૏⟎ߣᵄ㐳ߣߩ㑐ଥࠍ⺞ߴߡ߅ߌ߫㧘శḮ߆ࠄߩశߩᒝᐲߩᵄ㐳ଐሽᕈ߇ಽ߆ࠆ㧚ߎߩࠃ߁ߥಽశེߪ 20 ਎♿ߩᆎ߼޽ߚࠅ߆ࠄ㕖Ᏹߦᄙߊ↪޿ࠄࠇߚ߇㧘ᦨㄭߢߪ㧘౮⌀ߩੇ᧼ߩઍࠅߦ㧘㔚᳇⊛ߦశߩᒝᐲࠍ᷹ቯߔࠆ⚛ሶ߇૶ࠊࠇࠆ႐ว߇ᄙ޿㧚ߘߩ⚛
ሶߣߒߡߪ㧘శ߇นⷞㇱߩ㗔ၞߦ޽ࠆ႐วߦߪశ㔚ሶჇ୚▤㧔శ㔚ലᨐߦࠃߞߡ಴ߡ
᧪ߚ㔚ሶࠍჇ᏷ߒߡ㔚ᵹߣߒߡขࠅ಴ߔⵝ⟎㧕߇ࠃߊ↪޿ࠄࠇࠆ߇㧘ߘߩઁ㧘ߘࠇߙࠇߩ႐วߦㆡߒߚ޿ࠈ޿ࠈߥ⚛ሶ߇޽ࠆ㧚ࠬࡍࠢ࠻࡞ߪᄙߊߩ႐ว㧘❑ゲߦᒝᐲࠍ㧘ᮮゲߦశߩᵄ㐳߹ߚߪᝄേᢙ㧔߹ߚߪᓟߢㅀߴࠆࠃ߁ߦߎࠇࠄߦᲧ଀ߔࠆ‛ℂ㊂㧕ࠍߣߞߡ⴫ߔ㧚ߚߛ㧘❑ゲߩ‫ޟ‬ᒝᐲ‫߇ޠ‬෩ኒߦ૗ࠍ␜ߔ߆ߪ໧㗴ߢ㧘ߒ߫ߒ߫ߘࠇ߇ߪߞ߈ࠅߣ␜ߐࠇߥ޿႐ว߇ᄙ޿㧚࿑ 1.1.1
ߦ␜ߒߚ㤥૕ユ኿ߩࠬࡍࠢ࠻࡞ߪ㧘න૏ᝄേᢙߩ᏷ dQ ߩਛߢන૏૕Ⓧᒰߚࠅߩࠛࡀ
࡞ࠡ࡯߇ߤࠇߛߌ߆㧘ߣ޿߁ߎߣࠍ␜ߒߡ޿ࠆ㧚৻ᣇ㧘ᑼ(1.1.5)ߢ␜ߒߚಽᏓߢߪ㧘หߓ‛ℂ㊂ࠍන૏ᵄ㐳ߩ᏷ߢ␜ߔߎߣߦߥࠆ㧚ߎߎߢ‫ߪ⪲⸒߁޿ߣޠ࡯ࠡ࡞ࡀࠛޟ‬㧘
1 ୘ߩశሶߩᜬߞߡ޿ࠆࠛࡀ࡞ࠡ࡯ hQ㧔photon energy㧕ߦ㧘ࠛࡀ࡞ࠡ࡯߇ hQ ߣ h(Q  d Q)ߩ㑆ߦ޽ࠆశሶߩᢙࠍ߆ߌߚ߽ߩߢ޽ࠆ㧚❑ゲߪ㧘ߎߩࠃ߁ߦࠛࡀ࡞ࠡ࡯ߢ␜ߔ႐ว
ߩઁ㧘శሶߩᢙߢ␜ߔ႐ว߽޽ࠆ㧚෶㧘ߒ߫ߒ߫නߦᬌ಴ེߩ಴ജߘߩ߽ߩࠍ␜ߔ႐
ว߽޽ࠆ㧚ߎߩࠃ߁ߦ㧘❑ゲߩ⴫␜ᣇᴺߪ޿ࠈ޿ࠈߥᵹ௾߇޽ࠅ㧘ߘ߁޿߁ߎߣ߇⹦
ߒߊี๧ߐࠇߡ޿ߥ޿㧔෶ߪ⴫␜ߐࠇߡ޿ߥ޿㧕ታ㛎⚿ᨐࠍ߽ߣߦశߩᒝᐲࠍቯ㊂⊛ߦขࠅᛒ߁ߦߪᄢ߈ߥ໧㗴ߣߥࠆ߇㧘ߎߎߢߪ޽߹ࠅᷓߊ┙ߜ౉ࠄߥ޿ߎߣߦߒߚ޿㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 26ߐߡ㧘࿑ 1.6.1 ߦ␜ߒߚࠃ߁ߥⵝ⟎ࠍ↪޿ߡ᥉ㅢߩ࠲ࡦࠣࠬ࠹ࡦ㔚⃿ߩశࠍ᷹ቯߒ
ߚ߽ߩࠍ࿑ 1.6.2 ߦ␜ߔ㧚ᮮゲߦశߩᵄ㐳㧘❑ゲߪశ㔚ሶჇ୚▤ߩ಴ജߘߩ߽ߩ㧘ߣ޿߁㧘޿ߚߞߡ޽޿߹޿ߥዤᐲߢ޽ࠆ߇㧘ߣߦ߆ߊᒝᐲࠍ␜ߒߡ޿ࠆ㧚ᗵᐲࠍ߈ߜࠎ
ߣ⵬ᱜߒߡ㧘ᾲߖࠄࠇߚ࠲ࡦࠣࠬ࠹ࡦߩࡈࠖ࡜ࡔࡦ࠻ߩ᷷ᐲ߇ಽ߆ࠇ߫㧘㤥૕ユ኿ߩಽᏓᑼߦㄭ޿߽ߩ߇ᓧࠄࠇࠆߪߕߢ޽ࠆ㧚࿑ 1.6.2
࠲ࡦࠣࠬ࠹ࡦ⃿ߩశߩࠬࡍࠢ࠻࡞ಽᏓ
㧔᧲ർᄢቇℂቇㇱ቟Ⓧ⎇ⓥቶߢߩ᷹ቯ⚿ᨐ㧕੹ᐲߪ㧘৻⥸ߦ૶ࠊࠇߡ޿ࠆⰯశἮߩశߩಽᏓࠍ᷹ⷰߒߡߺࠃ߁㧚⚿ᨐߪ࿑ 1.6.3
ߦ␜ߔ㧚࠲ࡦࠣࠬ࠹ࡦ㔚⃿ߩశߣⰯశἮߩశߩಽᏓߪ㧘৻⷗ߒߡᄢ߈ߊ⇣ߥߞߡ޿ࠆ
ߎߣߦ᳇߇ઃߊߢ޽ࠈ߁㧚ߔߥࠊߜ㧘࠲ࡦࠣࠬ࠹ࡦ㔚⃿ߩశߪ㧘ᐢ޿ᵄ㐳▸࿐ߦࠊߚߞߡ᏷ᐢ޿ࠬࡍࠢ࠻࡞ࠍਈ߃ࠆߩߦኻߒ㧘ⰯశἮߩశߢߪ㕖Ᏹߦࠪࡖ࡯ࡊߥࡇ࡯ࠢ߇
޿ߊߟ߆ሽ࿷ߒߡ޿ߡ㧘᏷ᐢ޿ࠬࡍࠢ࠻࡞ߩነਈߪ߶ߣࠎߤߥ޿㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 27

Relative Intensity

fluorescent lamp

300

400

500

600

700

800

wavelength/nm
࿑ 1.6.3ⰯశἮߩࠬࡍࠢ࠻࡞ಽᏓ
㧔᧲ർᄢቇℂቇㇱ቟Ⓧ⎇ⓥቶߢߩ᷹ቯ⚿ᨐ㧕ߤ߁ߒߡߎ߁޿߁㆑޿߇޽ࠆߩߣ޿߁ߣ㧘శ߇಴ࠆᯏ᭴߇߹ߞߚߊ⇣ߥࠆ߆ࠄߢ޽
ࠆ㧚࠲ࡦࠣࠬ࠹ࡦ㔚⃿ߩ႐วߦߪ㧘࠲ࡦࠣࠬ࠹ࡦߩࡈࠖ࡜ࡔࡦ࠻ߦ㔚ᵹࠍᵹߒߡߘߩ
ᾲߦࠃࠆユ኿ߢ޽ࠅ㧘߶߷㤥૕ユ኿ߩಽᏓߦㄭ޿߽ߩ߇ᓧࠄࠇࠆ㧚࠲ࡦࠣࠬ࠹ࡦߢߥ
ߊߡ㧘଀߃߫࠾ࠢࡠࡓ✢ࠍ↪޿ߡ߽ᄢ૕ૃߚࠃ߁ߥಽᏓ߇ᓧࠄࠇࠆ㧚ࠛࠫ࠰ࡦ߇⊒᣿ߒߚ㔚⃿ߢߪ㧘ᣣᧄ⵾ߩ┻߇૶ࠊࠇߚߣ޿߁ߎߣߢ޽ࠆ߇㧘ᵄ㐳ಽᏓߪ࠲ࡦࠣࠬ࠹ࡦ
㔚⃿ߣૃㅢߞߚ߽ߩߢ޽ߞߚߣᗐ௝ߔࠆ㧚৻ᣇⰯశἮߩశߪ㧘⌀ⓨਛߦࠊߕ߆ߩ᳓㌁ߩ᳇૕߇ሽ࿷ߔࠆ⁁ᘒߢ㧘2 ߟߩ㔚ᭂ㑆ߦ㜞㔚࿶ࠍ߆ߌࠆߎߣߦࠃߞߡ಴ߡ᧪ࠆశߢ޽ࠆ㧚ㄘ᧛ߢ⷗ࠄࠇࠆ⺃ⱌἮߣߒߡ↪޿ࠄࠇࠆ᳓㌁Ἦߣᧄ⾰⊛ߦหߓߢ޽ࠆ㧚߹ߚ㧘૑ቛⴝߩⴝἮߢ⊕ߞ߸޿శࠍ಴ߔ߽ߩ߇
޽ࠆ߇㧘ߘࠇ߽᳓㌁Ἦߢ޽ࠆ㧚ኅᐸ߿ࠝࡈࠖࠬߢ૶߁ⰯశἮߪ㧘ߥࠎߣ߆ᄥ㓁శߦㄭ޿శ߇಴ࠆࠃ߁ߦ޿ࠈ޿ࠈߣᎿᄦߒߡ޿ࠆߩߢ㧘᳓㌁Ἦߘߩ߽ߩߩశߣߪᄢಽ⇣ߥࠆ߇㧘ߘࠇߢ߽࿑ 1.6.3 ߩࠬࡍࠢ࠻࡞ߪ߶ߣࠎߤ᳓㌁Ἦߩࠬࡍࠢ࠻࡞ߣหߓߢ޽ࠆ㧚߹ߚ㧘㜞ㅦ㆏〝ߢ⷗ࠄࠇࠆ㤛⦡ߞ߸޿శḮߪ㧘᳓㌁ߩઍࠅߦ࠽࠻࡝࠙ࡓࠍ౉ࠇߚ߽ߩߢ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 28޽ࠅ㧘෶㧘↸ߩࡀࠝࡦࠨࠗࡦߩ⿒⦡ߩశߪࡀࠝࡦࠟࠬࠍ౉ࠇߚ߽ߩߦ㔚࿶ࠍ߆ߌߚ߽ߩߢ޽ࠆ㧚㧔‫ߪ⪲⸒߁޿ߣޠࡦࠗࠨࡦࠝࡀޟ‬㧘߽ߣ߽ߣߪ㧘ࡀࠝࡦߩ⿒⦡ߩ⊒శࠍ␜ߒߚ߽ߩߢ޽ߞߚ㧚ߒ߆ߒ㧘ᦨㄭߪ㧘ᄛߩ❥⪇ⴝࠍᓀࠆߐ߹ߑ߹ߥశߩߎߣࠍ߹ߣ߼ߡ
‫ߦ߁ࠃ߁޿ߣޠࡦࠗࠨࡦࠝࡀޟ‬㧘㆑ߞߚᗧ๧ߢ૶ࠊࠇࠆࠃ߁ߦߥߞߚ㧚ߘࠇߦ઻޿㧘଀߃߫㧘‫ޟ‬㕍⦡ߩࡀࠝࡦࠨࠗࡦ‫߁޿ߣߤߥޠ‬㧘⑼ቇ⊛ߦߪ޽ࠅᓧߥ޿㧘߅߆ߒߥ⸒⪲߽૶ࠊࠇࠆࠃ߁ߦߥߞߡ߈ߡ޿ࠆ㧚㧕ߎࠇࠄߩ଀ߢ⷗ࠄࠇࠆࠃ߁ߦ㧘⌀ⓨਛߦߏߊࠊߕ߆ߩේሶߩ᳇૕ࠍኽ౉ߒߡ㧘ߘߎߦ㜞㔚࿶ࠍ߆ߌࠆߣ㧘శ߇⊒ߔࠆ㧚ߎࠇߪේሶߩ⊒శࠬࡍࠢ࠻࡞ߣߒߡ 19 ਎♿ᧃ޽ߚࠅߩᤄ߆ࠄᄙߊߩ⑼ቇ⠪ߦࠃߞߡ⎇ⓥߐࠇߚ㧚ߘࠎߥᤄߪ㧘శߩᬌ಴ߪ߽ߞ߬ࠄ౮
⌀ߩੇ᧼ߢⴕߞߡ޿ߚߒ㧘෶㧘㕖Ᏹߦࠪࡖ࡯ࡊߥࠬࡍࠢ࠻࡞ߪ㧘⃻࿷ߢ߽౮⌀ᴺߢࠃ
ࠅᱜ⏕ߦᵄ㐳ߩ᷹ቯ߇಴᧪ࠆߎߣ߽޽ࠅ㧘ේሶࠬࡍࠢ࠻࡞ߪߒ߫ߒ߫ੇ᧼ߩࠗࡔ࡯ࠫ
ߢ࿑␜ߐࠇࠆ㧚࿑ 1.6.4 ߇ߘߩ৻଀ߢ㧘ᮮゲߪᵄ㐳ࠍ␜ߔ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 29

H

Na

Hg

400

500

600

700

Wavelength/nm
࿑ 1.6.4 ੑਃߩේሶߩ㧘นⷞ㗔ၞߦ߅ߌࠆ⊒శࠬࡍࠢ࠻࡞
౮⌀ੇ᧼਄ߩࠬࡍࠢ࠻࡞㧚ᦨㄭߪశᒝᐲࠍ౮⌀ੇ᧼ߢߪߥߊߡ㧘࿑ 1.6.3 ߩ଀ߩࠃ߁ߦ㧘㔚ሶ⊛ߦ᷹ቯߔࠆߩ߇᥉ㅢߢ޽ࠆ߇㧘৻ᤄ೨ߪߎߩࠃ߁ߥ౮⌀ੇ᧼߇໑৻ߩ᷹ቯ⚿ᨐߢ޽ߞߚ㧚౮⌀ੇ᧼਄ߢߪ㧘ᮮゲߪᵄ㐳ߦᲧ଀ߒߡ޿ߥ޿߇㧘ߎߎߢߪᵄ㐳࡝࠾ࠕ࡯ߩࠬࠤ࡯࡞ߦ⋥ߒߡឬ޿ߡ޿ࠆ㧚ࠬࡍࠢ࠻࡞߇⃻ࠇࠆᵄ㐳ߩ૏⟎ߦု⋥ߥ✢ࠍᒁ޿ߡ޿ࠆ㧚᠟ᓇߒߚ౮⌀ੇ᧼ࠍ⃻௝ߔࠇ߫ৼᐲߘߩࠃ߁ߥ߽ߩ߇ᓧࠄࠇࠆࠊߌߢ޽ࠆ㧚߽ߣࠃࠅ❑ゲߪᗧ๧ߪߥߊ㧘ᒝᐲߪಽ߆ࠄߥ޿߇㧘ᄙߊߩ႐ว㧘ᒝᐲࠃࠅ߽⃻ࠇߚࠬࡍࠢ࠻࡞ߩ૏⟎ߦ⥝๧߇޽ࠆߚ߼ߦ
ߎߩࠃ߁ߥ⴫␜߇ߥߐࠇࠆߩߢ޽ࠆ㧚ߎߩ࿑ߢߺࠄࠇࠆࠃ߁ߦ㧘ේሶߩࠬࡍࠢ࠻࡞ߪ
ੇ᧼਄ߢ✢ߦ⷗߃ࠆߎߣ߆ࠄ✢ࠬࡍࠢ࠻࡞㧔line spectrum㧕ߣ๭߫ࠇࠆ㧚ߎߎߢࠬࡍࠢ࠻࡞✢ߩ⃻ࠇࠆ႐ᚲߩ⴫ߒᣇߦߟ޿ߡㅀߴߡ߅ߊ㧚࿑ 1.6.3 ߿࿑ 1.6.4
ߢߪᮮゲߪᵄ㐳 O ߢ޽ࠆ㧚৻ᣇ㧘࿑ 1.1.1 ߢߪᝄേᢙQ ߢ⴫ߒߡ޿ࠆ㧚ߎߩࠃ߁ߦ㧘ࠬࡍࠢ࠻࡞ߪᵄ㐳 O ߆ᝄേᢙQ ߢ߽ߞߡ⴫ߐࠇࠆ߇㧘ߎߩᝄേᢙߪ 10 ߩ૗ਸ਼ߣ޿߁࿃ሶ
߇ߟߊߩߢ㧘นⷞㇱㄭㄝߩశߩᤨߦߪࠊߕࠄࠊߒ޿㧚ߘߎߢߒ߫ߒ߫ᵄᢙ
㧔wavenumber㧕ߣ޿߁න૏߇↪޿ࠄࠇࠆ㧚⸥ภߪ㧘Q ߩ਄ߦᵄ⸥ภ㨪ࠍߟߌߚ㧘Q ߣ޿߁ᢥሼࠍ↪޿ࠆ㧚ᵄᢙߣߪන૏㐳ߐᒰࠅߦᵄ߇૗๟ᦼಽ౉ࠆ߆ߣ޿߁ᢙߢ޽ࠆ㧚ᰴ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 30రߣߒߡߪᒰὼ㐳ߐߩㅒᢙߣ޿߁ߎߣߦߥࠆ㧚SI න૏♽ߢߪ㐳ߐߩන૏ߪ m ߢ޽ࠆ߆ࠄ㧘1 m ਛߩᵄߩᢙߢ߽ࠃ޿ߩߢ޽ࠆ߇㧘ᤄ߆ࠄߩ⠌ᘠߢ 1 cm ਛߩᵄߩᢙߢ⴫ߔߩ
߇৻⥸⊛ߢ޽ࠆ㧚ᦨㄭߪ 1 μm ᒰߚࠅߩᵄߩᢙߢ⴫ߔߎߣ߽ߒ߫ߒ߫޽ࠆ㧚ߘߩ႐วߪ

μm 1 ߣ޿߁න૏ߦߥࠆ߇㧘cm 1 ߣ޿߁න૏߇⃻ታߦߪ߽ߞߣ߽ࠃߊ↪޿ࠄࠇߡ޿ࠆ㧚

‫଀ڏ‬㗴
㧔⸃㧕Q

ᵄ㐳 500 nm ߩశߩᵄᢙࠍ᳞߼ࠃ㧚1

O

1
500 u 109 m

2 u 106 m 1

20000 cm 1

2 μm 1ߐߡ㧘ࠬࡍࠢ࠻࡞ߩ᷹ቯᣇᴺߣߘߩ⴫␜ᣇᴺߦߟ޿ߡㅀߴߚߩߢ㧘ᰴߦ᳓⚛ේሶߩࠬࡍࠢ࠻࡞ߦߟ޿ߡㅀߴࠃ߁㧚ߔߢߦㅀߴߚ᳓㌁Ἦߣหߓࠃ߁ߦ㧘⌀ⓨߦߒߚ᡼㔚▤
ߩਛߦዋ㊂ߩ᳓⚛ࠟࠬࠍ౉ࠇߡ㜞࿶ࠍ߆ߌߡ᡼㔚ߐߖࠆߣ㧘⊒శ߇⷗ࠄࠇࠆ㧚⚡ᄖㇱ㧘นⷞㇱ㧘⿒ᄖㇱߩ㗔ၞߦߟ޿ߡᮨᑼ⊛ߦ␜ߔߣ࿑ 1.6.5 ߩࠃ߁ߦߥࠆ㧚࿑ 1.6.5 ᳓⚛ේሶߩ⊒శࠬࡍࠢ࠻࡞ߩ࿑

นⷞㇱ㧘⚡ᄖㇱ㧘⿒ᄖㇱߦߘࠇߙࠇ૗ᧄ߆ߩ✢ࠬࡍࠢ࠻࡞߇⷗ࠄࠇ㧘ߘࠇߙࠇ⷗ߟ
ߌߚੱߩฬ೨ࠍߣߞߡ Balmer series, Lyman series, Ritz-Paschen series ߣ๭߫ࠇߡ޿ࠆ㧚ߘࠇߙࠇߩ series ߦߟ޿ߡ㧘᷹ⷰߐࠇࠆᵄᢙࠍ⴫ 1.6.1 ߦ␜ߔ㧚⴫ߦ␜ߒߡ޽ࠆ n1 ߣ߆
n2 ߣ޿߁⸥ภߦߟ޿ߡߪߔߋᓟߢ⺑᣿ߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 31

⴫ 1.6.1 ᳓⚛ේሶߩ⊒శࠬࡍࠢ࠻࡞✢ߩᵄᢙ㧔 cm 1 න૏㧕

n2  n1

Ritz-Paschen
series
n1

1
2
3
4
...
...

3

5332
7799
9140
9948
...
...
12175

f

BalmerseriesLyman
series

n1

2

n1 1

15233
20565
23032
24373
...
...
27408

82258
97491
102823105291...
...
109667

⴫ 1.6.1 ߩ⚿ᨐࠍᵈᗧᷓߊ⌑߼ߡߺࠆߣ㧘޿ߊߟ߆ߩⷙೣᕈ߇޽ࠆߎߣ߇ಽ߆ࠆ㧚଀߃߫ Lyman series ߩ 4 ⇟⋡ߩ✢ߣ 3 ⇟⋡ߩ✢ߣߩᵄᢙᏅߪ

105291 cm 1  102823 cm 1 2468 cm 1
Balmer series ߩ 3 ⇟⋡ߩ✢ߣ 2 ⇟⋡ߩ✢ߣߩᵄᢙᏅߪ
23032 cm 1  20565 cm 1 2467 cm 1
Paschen series ߩ 2 ⇟⋡ߩ✢ߣ 1 ⇟⋡ߩ✢ߣߩᵄᢙᏅߪ
7799 cm 1  5332 cm 12467 cm 1ߣ޿߁ࠃ߁ߦ߶߷৻ቯߩ୯ߣߥࠆ㧚ߎߩࠃ߁ߥⷙೣᕈߪⷞὐࠍᄌ߃ߡߺࠆߣ޿ࠈ޿ࠈ
⷗ߟ߆ࠆ㧚଀߃߫ Lyman series ߩ 3 ⇟⋡ߩ✢ߣ Balmer series ߩ 2 ⇟⋡ߩ✢ߣߩᏅߪ
102823 cm 1  20565 cm 1 82258 cm 1
Lyman series ߩ 2 ⇟⋡ߩ✢ߣ Balmer series ߩ 1 ⇟⋡ߩ✢ߣߩᏅߪ
97491 cm 1  15233 cm 1 82258 cm 1ߣ޿ߞߚࠃ߁ߦหߓᵄᢙᏅ߇㓐ᚲߦ⷗ࠄࠇࠆ㧚ߎߩࠃ߁ߥⷙೣᕈߪ㧘ේሶࠬࡍࠢ࠻࡞
৻⥸ߦᤄ߆ࠄ⍮ࠄࠇߡ޿ߚ㧚ߔߥࠊߜ㧘ࠬࡍࠢ࠻࡞ࠍᵄᢙන૏ߢ⴫ߔߣ㧘ߘߩᵄᢙߪᲧセ⊛ዊߐߥᢙߩ࠮࠶࠻หᔒߩᏅߣߒߡ⴫ߐࠇࠆ㧘ߣ޿߁߽ߩߢ Ritz ߩ combination
principle ߣ⸒ࠊࠇߡ޿ߚ㧚ߎߩࠃ߁ߥⓍߺ㊀ߨࠍ⚻ߡ㧘ߟ޿ߦ Balmer ߿ Rydberg ߦࠃࠅ㧘ᰴߩࠃ߁ߥ⚻㛎⊛ߥᑼ߇ਈ߃ࠄࠇߚ㧚

Q

§ 1
1 ·
RH ¨ 2  2 ¸© n1 n2 ¹䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 32(1.6.1)

n1 1 ߇ Lyman series㧘 n1

2 ߇ Balmer series㧘 n1

3 ߇ Paschen series ߦ⋧ᒰߔࠆ㧚Q ߪࠬࡍࠢ࠻࡞ߩ⃻ࠇࠆᵄᢙ㧔Q / c 㧕㧘 RH ߪ Rydberg constant ߣ੔ࠊࠇࠆቯᢙߢ㧘⃻࿷ᱜߒ޿ߣ⹺߼ࠄࠇߡ޿ࠆ୯ߪ

109677.6 cm 1

RHߢ޽ࠆ㧚n1 ߅ࠃ߮ n2 ߪᢛᢙߢ޽ࠆ㧚ߎߩᑼߪℂ⺰⊛ߥᩮ᜚ߥߒߦ⚻㛎⊛ߦዉ߆ࠇߚᑼ
ߢ޽ࠆ߇㧘n1 ߅ࠃ߮ n2 ߣ޿߁ᢛᢙ߇↪޿ࠄࠇߡ޿ࠆߎߣߦᵈ⋡ߔߴ߈ߢ޽ࠈ߁㧚ᢛᢙߣ޿߁ߩߪᚒ‫ੱޘ‬㑆߇‛ࠍᢙ߃ࠆߩߦ↪޿ࠆᢙߢ޽ࠆ㧚⑼ቇߩ⃻⽎ߦᢛᢙ߇⃻ࠇࠆߣ޿߁ߩߪߤ߁޿߁ߎߣߢ޽ࠈ߁߆㧫
ᦨೋߩℂ⺰⊛⸃᳿ߪ Bohr ߦࠃߞߡਈ߃ࠄࠇߚ㧚Rutherford ߇ߔߢߦ⠨߃ߚࠃ߁ߦ㧘

Bohr ߪ㧘᳓⚛ේሶߪᱜߦ⩄㔚ߒߚᩭߩ๟ࠅࠍ㔚ሶ߇ඨᓘ r ߢ࿁ߞߡ޿ࠆ߽ߩߣ⠨߃ߚ㧚㆙ᔃജߣᱜ⽶㔚⩄㑆ߣߩ Coulomb ജߣ߇ߟࠅวࠊߥߌࠇ߫ߥࠄߥ޿ߩߢᰴᑼࠍᓧࠆ㧚

P rZ

1 e2
4SH 0 r 2

2

(1.6.2)

ߎߎߢ Z ߪⷺㅦᐲߢ޽ࠅ P ߪ㔚ሶߩ⾰㊂ me ߣᩭߩ⾰㊂ M ߣߢቯ⟵ߐࠇࠆ឵▚⾰㊂ߢ
޽ࠆ㧚

1

P

1
1

me M

(1.6.3)ㆇേࠛࡀ࡞ࠡ࡯ T ߅ࠃ߮૏⟎ߩࠛࡀ࡞ࠡ࡯ V ߪߘࠇߙࠇᰴߩࠃ߁ߦߥࠆ㧚

T

1 2 2
Pr Z
2

V



e2
4SH 0 2r
1

(1.6.4)1 e2
4SH 0 r(1.6.5)ᓥߞߡోࠛࡀ࡞ࠡ࡯ E ߪ

E T V

1 1 e2
2 4SH 0 r

Tߎߎߢ Bohr ߪ㧘ⷺㆇേ㊂ P r 2Z ߪ

(1.6.6)hߩᢛᢙ୚ߒ߆ขࠅᓧߥ޿ߣ޿߁઒ቯࠍዉ౉ߒߚ㧚
2S

ߔߥࠊߜ

P r 2Znh
2Sn=( n ߪᢛᢙ)(1.6.7)

ߎߎߢᣂߒߊ
=h2S䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 33(1.6.8)ߣቯ⟵ߒߚ㧚㧔 = ߪࠛ࠶࠴ࠬ࡜࠶ࠪࡘ߹ߚߪࠛ࠶࠴ࡃ࡯ߣ⊒㖸ߐࠇࠆ㧚㧕ᑼ(1.6.7)ࠍ↪
޿ࠆߣ

E

T



1 1 2 2n=
2 Pr 2e2
4SH 0 2r
1

(1.6.9)ߎࠇࠃࠅ゠㆏ߩඨᓘߣߒߡ
r

4SH 0 = 2 2nPe2(1.6.10)߇ᓧࠄࠇࠆ߇㧘ߎߩ r ࠍᑼ(1.6.9)ߦઍ౉ߒ㧘

En

P e4

1



4SH 0

2

2n 2 = 2(1.6.11)ࠍᓧࠆ㧚ߎߎߢ E ߪ n ߩ㑐ᢙߢ޽ࠆߩߢ E ߩ subscript ߣߒߡ n ࠍઃߌߡ޽ࠆ㧚ᑼ(1.6.11)
߇⽶ߢ޽ࠆߣ޿߁ߎߣߪ㧘ࡊࡠ࠻ࡦߣ㔚ሶ߇ή㒢㆙ߦ㔌ࠇߡ޿ࠆ႐วߦᲧߴߡࠛࡀ࡞
ࠡ࡯߇ૐ޿㧘ߔߥࠊߜ㧘ߘࠇߛߌ቟ቯߣ޿߁ߎߣߢ޽ࠆ㧚n 1 ߩ႐ว߇ᦨ߽቟ቯߢߘ
ࠇࠍၮᐩ⁁ᘒ㧔ground state㧕ߣ⒓ߔࠆ㧚
Bohr ߪ㧘᷹ⷰߐࠇߚࠬࡍࠢ࠻࡞ߪᑼ(1.6.11)ߢ⴫ߐࠇࠆ 2 ߟߩ⇣ߥߞߚ⁁ᘒ㑆ߩࠛࡀ࡞ࠡ࡯Ꮕࠍ㔚⏛ᵄߣߒߡ᡼಴ߒߚ߽ߩߢ޽ࠆߣ⠨߃ߚ㧚ߔߥࠊߜ'E

P e4 § 11 ·¨¸
2= 2 © n12 n2 2 ¹(1.6.12)P e4 § 1
1 ·
 2¸
2
3 ¨
2
8H 0 ch © n1 n2 ¹

(1.6.13)

1
4SH 0

2

ᵄᢙන૏ߢ⴫ߔߣ

Q

'E
ch

ߎߩࠃ߁ߦ㧘ฎౖജቇߦᑼ(1.6.7)ߩ㊂ሶ᧦ઙࠍ઒ቯߔࠆߎߣߦࠃࠅ⚻㛎⊛ߥᑼ(1.6.1)
ߣหߓ߽ߩ߇ℂ⺰⊛ߦዉ಴ߐࠇߚ㧚ᑼ(1.6.1)ߩ RH 㧘ߔߥࠊߜ Rydberg constant ߪ㧘ᑼ(1.6.12)ߣߩᲧセࠃࠅᰴᑼࠍᓧࠆ㧚RH

P e4
8H 0 2 ch3(1.6.14)ߎߩࠃ߁ߦ RH ߩਛߦߪ឵▚⾰㊂߇฽߹ࠇߡ޿ࠆߩߢ㧘ේሶᩭߩ⾰㊂ߦࠃߞߡᄙዋ⇣ߥߞߚ୯ߣߥࠆ㧚ߔߥࠊߜ㧘Rydberg constant ߪห૏૕ߦࠃߞߡᄙዋ୯߇⇣ߥࠆ㧚ේሶ
ᩭߩ⾰㊂߇ή㒢ᄢߢ޽ࠆߣ޿߁઒ᗐ⊛ߥ႐วࠍ⠨߃ࠆߣ㧘 m ߪ㔚ሶߩ⾰㊂ me ߘߩ߽ߩߣߥࠆ㧚ߎߩࠃ߁ߥ઒ᗐ⊛ߥ᳓⚛ේሶߩ Rydberg constant ࠍ㧘subscript ߣߒߡfࠍߟߌߡ㧘 Rf ߣ৻⥸ߦ⴫ߔ㧚ߔߥࠊߜ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 34

Rf

me e 4
8H 0 2 ch3(1.6.15)

නߦ Rydberg constant ߣ޿߁ߣ㧘 Rf ߩߎߣࠍᗧ๧ߔࠆߎߣ߇ᄙ޿㧚એ਄ߩࠃ߁ߦߒߡ㧘⚻㛎⊛ߥ Ritz ߩ combination principle ߪ Bohr ߩ઒⺑ࠍዉ౉ߔࠆ
ߎߣߦࠃߞߡ৻ᔕߩ⺑᣿߇ߟ޿ߚ㧚ߘߒߡ㤥૕ユ኿ߩࠃ߁ߦ h න૏ߢ⿠ߎࠆઁߩ⃻⽎ߣ߽㑐ㅪߒߡ޿ࠆߩߢੱ‫ߦޘ‬༑ࠎߢฃߌ౉ࠇࠄࠇߚ㧚ේሶᩭߩ๟ࠅࠍ㔚ሶ߇࿁ߞߡ޿
ࠆߣ޿߁ឬ௝ߪ㧘Bohr ߩᖺᤊࡕ࠺࡞ߣߒߡਇṌߩฬࠍᱷߒߚ㧚ߒ߆ߒታߩߣߎࠈ㧘㊂ሶജቇߩ┙႐ߦ┙ߟߣ Bohr ߩᖺᤊࡕ࠺࡞ߪ߶ߣࠎߤోߡߩὐߢ㑆㆑ߞߡ߅ࠅ㧘෶޿ࠈ޿ࠈߥὐߢ⎕✋ࠍ߈ߚߒߡ޿ࠆ㧚ߤ߁޿߁ὐߢ⎕✋ࠍ߈ߚߒߡ޿ࠆߩ߆㧘ߘߩ⺑᣿
ߪ㊂ሶജቇߩೋᱠࠍቇ⠌ߒߡ߆ࠄߩ߅ᭉߒߺߣ޿߁ߎߣߦߒߡߣࠅ޽߃ߕ Chapter 1ࠍ⚳⚿ߔࠆߎߣߦߒࠃ߁㧚‫ڏ‬Problem 1.6.1
RH ߩ୯ࠍౕ૕⊛ߦ⸘▚ߖࠃ㧚߹ߚ㧘ࡊࡠ࠻ࡦߩ⾰㊂ࠍή㒢ᄢߣ⠨߃ࠆߣ Rf ߩ⸘▚୯ߪߤ߁ߥࠆ߆㧚‫ڏ‬Problem 1.6.2
H ߩࠗࠝࡦൻࡐ࠹ࡦࠪࡖ࡞ࠍ⸘▚ߖࠃ㧚‫ޤ⸼ޣ‬㧦ࠗࠝࡦൻࡐ࠹ࡦࠪࡖ࡞ߪ㔚ሶࠍၮᐩ⁁ᘒ߆ࠄή㒢㆙ߦᒁ߈㔌ߔߩߦᔅⷐߥࠛࡀ࡞
ࠡ࡯ߢ޽ࠆ㧚ᓥߞߡ n1 1, n2 f ߣ߅޿ߡࠛࡀ࡞ࠡ࡯Ꮕࠍ⸘▚ߔࠇ߫ࠃ޿㧚⚿ᨐߪᵄᢙන૏ߩߺߢߥߊ eV න૏ߢ߽᳞߼ࠃ㧚᳓⚛ߩࠗࠝࡦൻࡐ࠹ࡦࠪࡖ࡞ߪߒ߫ߒ߫ࠛࡀ
࡞ࠡ࡯ߩၮḰߣߒߡ↪޿ࠄࠇࠆ㧚ᓥߞߡߎߩ୯ࠍ⸥ᙘߒߡ߅ߊߣߥߦ߆ߣଢ೑ߢ޽ࠆ㧚‫ڏ‬Problem 1.6.3
Balmer ߩේ⺰ᢥߢߪ㧘᳓⚛ߩࠬࡍࠢ࠻࡞✢ߩ㧘⃻࿷ Balmer series ߣ๭߫ࠇߡ޿ࠆ

n2ߢ⴫ߐࠇࠆߣ⠨߃ߡ޿ߚ㧚ߎߎߢ n ߪ 2 એ਄ߩᢛᢙ㧘 C ߪ
n2  4ቯᢙߢ޽ࠆ㧚ᑼ(1.6.12)ߢ n1 2 ߣ߅޿ߚ߽ߩ߇ Balmer series ߦ⋧ᒰߔࠆ㧚C ࠍ᳞߼ࠃ㧚♽೉ߩᵄ㐳Oߪ O C‫ڏ‬Problem 1.6.4
᳓⚛ේሶߩ n 1 ߩ゠㆏ߦ޽ࠆ㔚ሶߦߟ޿ߡ㧔a㧕ㆇേࠛࡀ࡞ࠡ࡯㧔J න૏㧕߅ࠃ߮㧔b㧕ㅦᐲ㧔m/s න૏㧕㧔ធ✢ᣇะߩ✢ㅦᐲ㧕ࠍ᳞߼ࠃ㧚‫ڏ‬Problem 1.6.5
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 35

H. C. Urey, F. G. Brickwedde, G. M. Murphy, Phys. Rev. 40 (1932) 464 ߪ㧘Balmer seriesߩㄭறߦᒙ޿ࠬࡍࠢ࠻࡞✢ࠍ⷗಴ߒ㧘ߘࠇࠍ㧘⹜ᢱਛߦࠊߕ߆ߦሽ࿷ߔࠆ㊀᳓⚛ߦࠃࠆߣ⠨߃ߚ㧚Balmer series ߩᦨೋߩ✢ࠬࡍࠢ࠻࡞㧔 n1 2, n2 3 㧕ߦ߅޿ߡ H ߣ D ߩࠬࡍࠢ࠻࡞✢ߩᏅࠍᵄᢙන૏ߢ᳞߼ࠃ㧚‫ڏ‬Problem 1.6.6
n 1 ߩ Bohr ゠㆏ߩඨᓘ㧘ߔߥࠊߜᑼ(1.6.10)ߢ n 1 ߣ߅߈㧘ᦝߦ឵▚⾰㊂ߩઍࠅߦ㔚ሶߩ⾰㊂ me ࠍ↪޿ߚ߽ߩߪ Bohr ඨᓘߣ޿ࠊࠇ㧘ߒ߫ߒ߫㐳ߐߩන૏ߣߒߡ↪޿ࠄࠇࠆ㧚ᑼ(1.6.10)ߦౕ૕⊛ߥ‛ℂ㊂ࠍઍ౉ߒ㧘Bohr ඨᓘࠍ⸘▚ߖࠃ㧚ߥ߅㧘ߎߩ୯ߪ㧘ߒ߫ߒ߫㐳ߐߩၮḰߣߒߡ↪޿ࠄࠇࠆߩߢ⸥ᙘߒߡ߅ߊߎߣ߇ᦸ߹ߒ޿㧚‫ڏ‬Problem 1.6.7
ᰴߩᕈ⾰ࠍᜬߞߚ㔚⏛ᵄߩᵄᢙࠍ᳞߼ࠃ㧚a) ᵄ㐳 200 nmb) ᵄ㐳 1 cmc) ᝄേᢙ 1015 Hzd) ᝄേᢙ 1 GHz 㧔⸼㧦G ߪ 109 ࠍᗧ๧ߔࠆធ㗡⺆ߢ㧘ࠡࠟߣ⊒㖸㧕[Coffee Break]

cm 1 ߣ޿߁න૏ߩ⊒㖸cm 1 ߣ޿߁න૏ࠍ૗ߣ⊒㖸ߒߚࠄ޿޿ߛࠈ߁߆㧚ᤄߪ cm 1 ߩߎߣࠍ Kayser㧔ࠞࠗࠩ࡯㧕ߣ๭ࠎߢ޿ߚᤨઍ߽޽ߞߚ㧚ಽశቇߦᄢ߈ߥ⽸₂ࠍߒߚ Kayser ߩฬ೨ࠍߣߞ
ߚࠊߌߢ޽ࠆ㧚ߒ߆ߒ㧘ߎߩฬ೨ߪ੹ߪ૶ߞߡߪ޿ߌߥ޿ߎߣߦߥߞߡ޿ࠆ㧚ߘߩℂ↱ߪᰴߩࠃ߁ߥߎߣߢ޽ࠆ㧚਎ߩਛߦߪᢙᄙߊߩන૏߇޽ࠅ㧘ಽ߆ࠅߦߊߊߥߞߚߩߢߘࠇࠍᢛℂߔࠆߎߣ߇ⴕ
ࠊࠇߚߩߪ߽߁ᢙචᐕ߽೨ߩߎߣߢ޽ࠆ㧚1 ߟߩන૏ߣߘߩㅒᢙߣ߇೎‫ߩޘ‬ฬ೨ߢ๭߫ࠇࠆߩߪ޿ߚߕࠄߦᷙੂࠍ᜗ߊ߽ߣߦߥࠆߣ޿߁ߩߢߔߴߡᑄᱛߐࠇߚ㧚଀߃߫ᤄߪ㧘㔚᳇ᛶ᛫ߩන૏ࠝ࡯ࡓ㧔ohm㧕:߇޽ࠅߥ߇ࠄ㧘ߘߩㅒᢙࠍ㧘ohm ߩࠬࡍ࡞ࠍㅒ
ߐ߹ߦ⺒ࠎߢ mho㧔ࡕ࡯㧕ߣ޿߁න૏ߢ๭ࠎߢ޿ߚ㧚ߘߒߡ⸥ภߪ‫ޟ‬:‫ࠍޠ‬ㅒߐ߹ߦߒߚ‫ߥ߁ࠃߩޠ ޟ‬ሼࠍ޽ߡߡ޿ߚ㧚ߎࠇߥߤ߹ߐߒߊẂ↹⊛ߢ޽ࠆ߇㧘ߜ߾ࠎߣߒ
ߚቇⴚ㔀⹹ߢ૶ࠊࠇߡ޿ߚᤨઍ߽޽ߞߚ㧚cm 1 ߩන૏ߦߒߡ߽หߓࠃ߁ߥߎߣ߇⸒߃
ࠆ㧚cm ߣ޿߁න૏߇޽ࠅߥ߇ࠄߘߩㅒᢙࠍ Kayser ߣ޿߁ߩߪή㚝ߢಽ߆ࠅߦߊ޿㧘ߣ޿߁ߎߣߢ Kayser ߣ޿߁න૏ߪ㧘mho ߣ޿߁න૏ߣห᭽૶ߞߡ޿ߌߥ޿ߎߣߦߥ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 36ߞߚߩߢ޽ࠆ㧚޽ࠆන૏ߩㅒᢙࠍ೎ߩන૏ߣߒߡ⹺߼ࠆߎߣࠍᑄᱛߒࠃ߁ߣߒߡ㧘ߚߛ 1 ߟ㧘଀ᄖ⊛ߦᑄᱛߐࠇߥ߆ߞߚන૏߇޽ࠆ㧚ߘࠇߪᝄേᢙߩන૏ Hz㧔ࡋ࡞࠷㧕ߢ޽ࠆ㧚s㧔⑽㧕
ߣ޿߁න૏߇޽ࠅߥ߇ࠄߘߩㅒᢙߩ 1/s ࠍ Hz ߣߒߡᣂߒߊቯ⟵ߔࠆߎߣߪ㧘ᗧ๧ߩߥ޿ߎߣߢ޽ࠆߣ޿߁⼏⺰ߪ޽ߞߚࠃ߁ߢ޽ࠆ߇㧘⑽߽ࡋ࡞࠷߽ߤߜࠄ߽ᣣᏱ↢ᵴߦ߹ߢᷓߊ㑐ࠊߞߡ޿ߡ㧘ߤߜࠄ߆ࠍᑄᱛߔࠆߎߣߪߣߡ߽಴᧪ߥ޿ߣ޿߁⚿⺰ߦߥߞߚࠃ߁ߢ޽ࠆ㧚㧔ߥ߅㧘Hz ߣ޿߁න૏߇ߪߞ߈ࠅߣቯ⟵ߐࠇߚᲑ㓏ߢ㧘ᤄ૶ࠊࠇߚࠨ
ࠗࠢ࡞ߣ޿߁න૏ߪ૶ߞߡߪ޿ߌߥ޿ߎߣߦߥߞߚ㧚㧕

cm 1 ߣ⸒߁න૏ࠍ‫ޟ‬ᵄᢙ‫ߣޠ‬㑆㆑ߞߡ޿߁ੱ߽޿ࠆ㧚2000 cm 1 ߩߎߣࠍ‫ޟ‬2000ᵄᢙ㧔⧷⺆ߢ 2000 wavenumber㧕‫ࠆߌ߆ߺߪߦ߹ߚࠍੱ߁޿ߣޠ‬㧚ߒ߆ߒ㧘ߎࠎߥ⸒޿ᣇߪߣࠎߢ߽ߥ޿ߎߣߢ޽ࠆ㧚‫ޟ‬ᵄᢙ‫‛߁޿ߣޠ‬ℂ㊂ߩන૏߇ cm 1 ߢ޽ࠅ㧘‫ޟ‬ᵄᢙ‫ޠ‬ߪ‛ℂ㊂ߢ޽ࠅන૏ߢߪߥ޿㧚ߘࠇߪ㧘‫ޟ‬㐳ߐ‫‛߁޿ߣޠ‬ℂ㊂ߩන૏߇ m ߢ޽ࠅ‫ޟ‬㐳ߐ‫ߪޠ‬න૏ߢߪߥ޿ߒ㧘߹ߚ㧘
‫⾰ޟ‬㊂‫‛߁޿ߣޠ‬ℂ㊂ߩන૏߇ kg ߢ޽ࠅ‫⾰ޟ‬㊂‫ߪޠ‬න૏ߢߪߥ޿ߩߣหߓߎߣߢ޽ࠆ㧚 㐳ߐ߇ 5 m ߣ޿߁ߩࠍ‫ޟ‬5 㐳ߐ‫޽ߢ޿ߥࠊ⸒ߪߣޠ‬ࠈ߁㧚‫ߩ⑳ޟ‬૕㊀ߪ 50 kg ߢ޽ࠆ‫߁⸒ߣޠ‬ઍࠅߦ‫ߩ⑳ޟ‬૕㊀ߪ 50 ⾰㊂‫ߚߞ⸒ߣߤߥޠ‬ࠄߤ߁޿߁ߎߣߦߥࠆߢ޽ࠈ߁߆㧚ߢߪ cm 1 ߪߥࠎߣ⊒㖸ߔࠇ߫޿޿߆㧫ߘߩ߹߹㧘‫ ࠬ࠽ࠗࡑߩ࡞࠻࡯ࡔ࠴ࡦ࠮ޟ‬1 ਸ਼‫ޠ‬ߣ⸒߃߫ࠃ޿㧚㕙ୟߊߐ޿߇ߒࠂ߁߇ߥ޿㧚⧷⺆ߢߪ‫ޟ‬centimeter to the minus first‫ޠ‬,߹ߚߪ‫ޟ‬centimeter to the minus first power‫ࠆ޽ߢޠ‬㧚‫߅ߡࠎߥޠ࡞࠻࡯ࡔ࠴ࡦ࠮࡯ࡄޟ‬߆ߒߥ⸒⪲ࠍ↪޿ࠆੱ߽޿ߥ޿ࠊߌߢߪߥ޿߇㧘ߘ߁⸒ߞߡߪ޿ߌߥ޿㧚⧷⺆ߢߪ

"reciprocal centimeter"ߣ޿߁⸒⪲߽ࠃߊ↪޿ࠄࠇߡ㧘ᱜߒ޿⸒⪲ߢ޽ࠆ㧚2000 cm 1 ߢ
޽ࠇ߫ 2000 reciprocal centimeters ߢࠃ޿㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 1, Page 37

CHAPTER 02ዊߐߥ☸ሶߩᝄࠆ⥰޿ᄤ૕ߦ߅ߌࠆᖺᤊߩㆇേߪ㧘ฎౖജቇߢቢోߦ⸥ㅀ಴᧪ࠆ㧚ࡇ࠶࠴ࡖ࡯ߩᛩߍࠆ㊁⃿ߩࡏ࡯࡞ߩㆇേߪࡃ࠶࠲࡯ߦߣߞߡߪℂ⸃ߒߦߊ޿߽ߩߢ޽ࠆ߆߽ߒࠇߥ޿߇㧘‛ℂቇߩ਎⇇ߢߪฎౖജቇߢ㔍ߥߊ⸃ߌࠆ໧㗴ߢ޽ࠆߪߕߢ޽ࠆ㧚ߒ߆ߒ㧘☸ሶ߇ߤࠎߤࠎዊߐߊߥߞߡ㔚ሶߣ߆ේሶߣ߆ߦߥࠆߣ㧘ߘߩࠃ߁ߥ☸ሶߩᝄࠆ⥰޿ߪฎౖജቇ
ߢߪℂ⸃಴᧪ߥߊߥࠆߎߣࠍᚒ‫ߡ޿ߟߦ♽ߩ߆ߟߊ޿ߪޘ‬ቇ⠌ߒߡ᧪ߚ㧚ᄢ߈ߥ♽ߦߟ޿ߡᚑࠅ┙ߟฎౖജቇ߇ዊߐߥ♽ߢߪᚑࠅ┙ߚߥ޿ߣ޿߁ߣ㧘ߘࠇߢߪ‫ߩߤޟ‬૏ዊߐ޿‫ࠄ߆ࠈߎߣޠ‬ᚑࠅ┙ߚߥߊߥࠆߩߛࠈ߁߆ߣ޿߁⇼໧߇↢ߕࠆ㧚ฎౖജቇߢߪ㧘ᄢ߈ߥ‛૕ߩᕈ⾰ࠍ㧘ߘࠇ߇ዊߐߥ‛૕߇㓸߹ߞߡ಴᧪ߡ޿ࠆߣ⠨߃ߡℂ⸃ߒߡ᧪ߚ㧚ᚒ‫ߪޘ‬ᄢ߈ߥ‛૕ߩᝄࠆ⥰޿ࠍ㧘ߘࠇࠍ᭴ᚑߒߡ޿ࠆዊߐߥ‛૕ߩᝄࠆ⥰޿߆ࠄዉߊߎߣ߇಴᧪ࠆ㧚ߘ߁޿߁ߎߣߢ޽ࠇ߫㧘ዊߐߥ‛૕ߩᝄࠆ⥰޿ߪ㧘ࠃࠅዊߐߥ‛૕ߩᝄࠆ⥰޿߆ࠄዉߊߎߣ߇಴᧪ࠆߪߕߢ޽ࠆ㧚ᄢ߈޿ߣ߆ዊߐ޿ߣ߆
޿߁ᒻኈ⹖߇⋧ኻ⊛ߥዤᐲߢ޽ࠆએ਄㧘਄ߩ⼏⺰ߪή㒢ߦ⛯ߊࠃ߁ߦᕁࠊࠇࠆ㧚ߎߩࠃ߁ߦ⠨߃ߡ޿ߊߣ㧘‫ޟ‬ᄢ߈޿‫ޠ‬㧘‫ޟ‬ዊߐ޿‫ߩޠ‬඙೎ࠍߤߎߢߟߌߡ޿޿߆ಽ߆ࠄߥߊ
ߥߞߡߒ߹߁㧚ߘߎߢ⠨߃ߥߌࠇ߫ߥࠄߥ޿ߎߣߪ㧘ᄢ߈ߐߣ޿߁߽ߩߦኻߒߡ⛘ኻ⊛ߥᗧ๧ࠍਈ
߃ࠆߎߣ߇಴᧪ࠆߢ޽ࠈ߁߆㧘ߣ޿߁ߎߣߢ޽ࠆ㧚޽ࠆ‛૕ߩᕈ⾰ࠍ᷹ⷰߔࠆߦߪߘ
ߩ‛૕ߦ૗╬߆ߩᄖ⇇߆ࠄߩᓇ㗀ࠍਈ߃ߥ޿ࠊߌߦߪ޿߆ߥ޿㧚߽ߒ߽ߎߩᓇ㗀߇ᄢ߈ߊߡߘߩ‛૕⥄りߩᕈ⾰߇ᄌߞߡߒ߹߁ࠃ߁ߢ޽ࠇ߫ߎߩ‛૕ߪ‫ޟ‬ዊߐ޿‫ߥ⷗ߣޠ‬ߔߎߣߦߒࠃ߁㧚෻ኻߦ㧘᷹ቯߦࠃߞߡਈ߃ࠄࠇࠆᄖ⇇߆ࠄߩᓇ㗀߇චಽߦዊߐߊߡ
‛૕ߩᕈ⾰߇ᄌࠆߎߣ߇ߥߌࠇ߫ߘߩ‛ߪ‫ޟ‬ᄢ߈޿‫߁ࠃߒߦߣߎߔߥߺߣޠ‬㧚ฎౖജቇߩ਎⇇ߢߪ㧘‛ߩᕈ⾰ߦᓇ㗀ࠍਈ߃ࠆߎߣߥߊ‛ߩᕈ⾰ࠍ᷹ቯߔࠆᣇᴺ߇
޽ࠆߣ⠨߃ࠄࠇࠆ㧚ߚߣ߃⃻࿷චಽߥᣇᴺ߇ߥ߆ߞߚߣߒߡ߽㧘⑼ቇߩㅴᱠߦࠃߞߡ
ߘߩࠃ߁ߥ᷹ⷰᣇᴺ߇⏕┙ߐࠇࠆߎߣࠍାߓߡࠃߐߘ߁ߢ޽ࠆ㧚ࡇ࠶࠴ࡖ࡯ߩᛩߍࠆ
ࡏ࡯࡞ߩㅦߐߪߘߩࡏ࡯࡞ࠍฃߌࠆᤨߩࠠࡖ࠶࠴ࡖ࡯ࡒ࠶࠻ߩ㖸ߢ߽ߞߡ್ᢿߔࠆ
ߎߣ߇಴᧪ࠆߣࠃߊ⸒ࠊࠇࠆ㧚ߎߩ႐ว㧘ࠠࡖ࠶࠴ࡖ࡯ࡒ࠶࠻ߢฃߌࠆߣ޿߁ߎߣ߇
ࡏ࡯࡞ߩㅦᐲߩ᷹ⷰᚻᲑߢ޽ࠆ߇㧘ߎߩ᷹ⷰߦࠃߞߡࡏ࡯࡞ߩㆇേߪᱛ߹ߞߡߒ߹߁
ࠊߌߢ޽ࠆ㧚ߎ߁޿߁᷹ⷰᣇᴺࠍ↪޿ࠆ㒢ࠅ㧘ࡏ࡯࡞ߩㅦߐߦᓇ㗀ࠍਈ߃ߥ޿ߢ᷹ቯ
ߔࠆߎߣߪ಴᧪ߥ޿㧚ߒ߆ߒᦨㄭߩࠃ߁ߦ Doppler ലᨐࠍ೑↪ߒߡㅦߐࠍ᷹ቯߔࠆ႐
วߦߪ᷹ቯߦࠃߞߡࡏ࡯࡞ߩㅦߐ߇ᄌࠆߣߪ⠨߃ࠄࠇߥ޿㧚ߒ߆ߒ‛૕߇޽ࠆᭂ㒢୯ࠃࠅዊߐߊߥࠆߣ㧘‛ߩᕈ⾰ࠍᄌ߃ࠆߎߣߥߊ᷹ቯߔࠆߎ
ߣߪ಴᧪ߥ޿ߣ޿߁޽ࠆᭂ㒢ߩᄢ߈ߐ߇ሽ࿷ߔࠆ㧚ߎߩᭂ㒢ߪ㧘‛ℂቇߩ਎⇇ߦ࿕᦭䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 1ߩᭂ㒢ߢ޽ߞߡ㧘⑼ቇߩㅴᱠߢᡷ⦟಴᧪ࠆߣ޿߁߽ߩߢߪߥ޿㧚ߎߎߦᄢ߈ߐߦ㑐ߔࠆ⛘ኻ⊛ߥዤᐲ߇޽ࠆ㧚߽ߒ߽᷹ቯߦࠃߞߡ‛ߩᕈ⾰߇ᄌൻߒߡߒ߹߁ࠃ߁ߢ޽ࠇ߫ߘߩ߽ߩߪ‫ޟ‬ዊߐ޿‫ߣޠ‬ቯ⟵ߐࠇ㧘ฎౖജቇߪᚑࠅ┙ߚߥߊߥࠆ㧚ᦝߦߎߩࠃ߁ߥዊߐߥ♽ߦߟ޿ߡߪ㧘᷹ቯߘߩ߽ߩ߇‛ߩᕈ⾰ࠍᄢ߈ߊᄌ߃ࠆߛ
ߌߢߥߊ㧘ߘߩᕈ⾰ࠍోߊ੍⍮಴᧪ߥ޿ᣇะߦᄌ߃ߡߒ߹߁ߩߢ޽ࠆ㧚એਅߦߪ㧘ੑਃߩౕ૕⊛ߥ଀ࠍ޽ߍߥ߇ࠄߎߩࠃ߁ߥߎߣࠍ߽߁ዋߒߊࠊߒߊቇ⠌ߒࠃ߁㧚
2.1 ஍శߒߡ޿ࠆశሶ
޽ࠆశḮ㧔଀߃߫ㅢᏱߩ࠲ࡦࠣࠬ࠹ࡦ㔚⃿㧕߆ࠄ಴ߡ᧪ߚశߩㅴⴕᣇะ㧔੹઒ߦ yゲᣇะߣߒࠃ߁㧕ߦု⋥ߦ 2 ୘ߩ஍శሶࠍ⟎߈㧘2 ୘⋡ߩ஍శሶߩᓟߢశߩᒝᐲࠍ᷹
ቯߔࠆߣ޿߁ታ㛎♽ࠍ⠨߃ࠆ㧔࿑ 2.1.1㧕
㧚ߎߎߢ↪޿ࠆ஍శሶߣ޿߁ߩߪ㧘޽ࠆ․ቯ
ߩᣇะߩ㔚᳇ࡌࠢ࠻࡞ࠍᜬߞߚశߩߺࠍㅢㆊߔࠆࠃ߁ߥ⚛ሶߢ޽ࠆ㧚⹤ࠍಽ߆ࠅ߿ߔ
ߊߔࠆߚ߼ߦ㧘╙ 1 ߩ஍శሶࠍㅢㆊߒߚశߪ㔚᳇ࡌࠢ࠻࡞߇ z ゲᣇะߦᝄേߒߡ޿ࠆ
ߣߒࠃ߁㧚

࿑ 2.1.1
஍శߒߚశሶߩታ㛎䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 2ߎ߁޿߁႐ว㧘╙ 1 ߩ஍శሶߪߘߩ‫ޟ‬஍శᣇะ߇ z ゲߢ޽ࠆ‫޿޿ߣޠ‬㧘ߘࠇࠍㅢㆊ
ߒߚశߪ‫ ޟ‬z ゲߦ஍శߒߡ޿ࠆ‫ޠ‬㧘ߣ޿߁ࠃ߁ߦ⴫⃻ߔࠆ㧚╙ 2 ߩ஍శሶߪశߩㅴⴕ
ᣇะ㧘ߔߥࠊߜ y ゲߦ㑐ߒߡ࿁ォ಴᧪ࠆࠃ߁ߦߥߞߡ޿ࠆ㧚ߘߒߡߎߩ႐ว㧘࿁ォⷺᐲߪ㧘ߘߩ஍శᣇะ߇ z ゲߣߥߔⷺᐲ T ߢ߽ߞߡ⴫ߔߎߣߦߒࠃ߁㧚ߎߩࠃ߁ߥⵝ⟎ࠍ↪޿㧘2 ୘⋡ߩ஍శሶࠍㅢㆊߒߚశߩᒝߐࠍ㧘╙ 2 ߩ஍శሶߩ࿁
ォⷺᐲ T ߩ㑐ᢙߣߒߡ᷹ቯߔࠆ㧚᷹ቯߩ⚿ᨐ㧘శߩᒝᐲߪ cos 2 T ߦᲧ଀ߔࠆߣ޿߁ߎ
ߣ߇ಽ߆ࠆ㧚ߎߩታ㛎੐ታߪฎౖജቇߢ◲නߦ⺑᣿಴᧪ࠆ㧚╙ 1 ߩ஍శሶࠍㅢㆊߒߚ
ᓟߩశߪ㧘਄ㅀߒߚࠃ߁ߦ㧘 z ゲߦ஍శߒߡ޿ࠆ㧚ᓥߞߡ╙ 2 ߩ஍శሶߩ஍శᣇะ߇

z ゲᣇะ㧘ߔߥࠊߜ T

0 ߢ޽ࠇ߫ߘߩశߪߘߩ߹߹ㅢㆊߔࠆ㧚ߒ߆ߒ㧘 T

S
2

ߢ޽ࠇ

߫శߪోߊㅢㆊߒߥ޿㧚ߘߒߡਛ㑆ߩⷺᐲߢ޽ࠇ߫ㅢㆊߔࠆశߩᝄ᏷ߪߘߩ஍శᣇะ
ߩ኿ᓇߔߥࠊߜ cos T ߦᲧ଀ߔࠆ߆ࠄశߩᒝᐲߣߒߡߪ cos 2 T ߦᲧ଀ߔࠆࠊߌߢ޽ࠆ㧚࿑ 2.1.2
஍శߒߚ 1 ୘ߩశሶߦߟ޿ߡߩታ㛎ߐߡ㧘਄ㅀߩታ㛎ߪ㧘Ᏹ⼂⊛ߥᒝߐߩశࠍ↪޿ߡ㧘శߩᒝߐߪ㧘శ㔚ലᨐߦࠃߞߡ
↢ߓߚ㔚ᵹߦࠃߞߡ᷹ቯߒߡ޿ࠆߣ⠨߃ߡࠃ޿㧚ߘߎߢᰴߦ㧘శߩᒝߐࠍߤࠎߤࠎᒙ
ߊߒߡ޿ߞߡ㧘ߟ޿ߦ㧘╙ 2 ߩ஍శሶࠍㅢㆊߒߚశ㧔߿ߪࠅ z ゲߦ஍శߒߡ޿ࠆ㧕߇శሶߚߞߚ 1 ୘ߢ޽ࠆ႐วࠍ⠨߃ߡߺࠆ㧚ߎ߁޿߁႐วߪశᒝᐲߩ᷹ቯߦߪ․೎ߩᎿ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 3ᄦ߇ᔅⷐߢ޽ࠆ߇㧘޿ࠊࠁࠆశሶ⸘ᢙᴺ㧔photon counting㧕ߣ޿ࠊࠇߡ޿ࠆᣇᴺߢ㧘1୘ߩశሶߢ߽᷹ቯߪน⢻ߢ޽ࠆ㧚ߎߩࠃ߁ߦ㧘 z ゲߦ஍శߒߚ 1 ୘ߩశሶ߇╙ 2 ߩ஍శሶߦ޽ߚࠆߣ㧘ߘߩశሶߪ஍
శሶࠍㅢㆊߔࠆߢ޽ࠈ߁߆㧘ߒߥ޿ߢ޽ࠈ߁߆㧫 T 0 ߩ႐วߪ⇼໧ߩ૛࿾ߪߥ޿㧚ߘߩశሶߪ╙ 2 ߩ஍శሶߢๆ෼ߐࠇࠆߎߣߥߊㅢㆊߔࠆ㧚 T

S
2ߩ႐ว߽߿ߪࠅ᣿ࠄ

߆ߢ޽ࠈ߁㧚ߘߩశሶߪ╙ 2 ߩ஍శሶߢๆ෼ߐࠇߡߒ߹ߞߡㅢㆊߔࠆߎߣߪߥ޿㧚ߢߪᰴߦ߽ߞߣ߽㊀ⷐߥ໧㗴ࠍ⠨߃ߡߺࠆ㧚߽ߒ߽╙ 2 ߩ஍శሶ߇శሶߩ஍శᣇะ
㧔 z ゲᣇะ㧕ߣߩߥߔⷺᐲ T ߇ 0 ߢ߽S2ߢ߽ߥ߆ߞߚࠄߤ߁ߥࠆߢ޽ࠈ߁߆㧫߹ߐ߆

1 ୘ߩశሶߩ৻ㇱಽ߇ㅢㆊߒߡᱷࠅ߇ๆ෼ߐࠇࠆߥࠎߡ޿߁ߎߣߪ޽ࠅᓧߥ޿㧚⠨߃ࠄࠇࠆߎߣߪㅢㆊߔࠆ߆ߒߥ޿߆ߩ޿ߕࠇ߆৻ߟߢ޽ࠆ㧚⠨߃ߡ߽ߎࠇએ਄ߪಽ߆ࠄ
ߥ޿߆ࠄታ㛎⚿ᨐࠍ⷗ߡߺࠃ߁㧚ታ㛎ߩ⚿ᨐߪߎ߁ߢ޽ࠆ㧚శሶߪ޽ࠆ႐วߦߪㅢㆊߒ㧘޽ࠆ႐วߦߪㅢㆊߒߥ޿㧚ߢߪ㧘ߤ߁޿߁႐วߦㅢㆊߒ㧘ߤ߁޿߁႐วߦㅢㆊߒߥ޿ߩߢ޽ࠈ߁߆㧫ᱷᔨߥ߇ࠄ
ㅢㆊߔࠆ߆ߒߥ޿߆ߪታ㛎ࠍ߿ߞߡߺࠆ߹ߢಽ߆ࠄߥ޿㧚޽ࠆ․ቯߩశሶ߇஍శሶࠍ
ㅢㆊߔࠆ߆ߒߥ޿߆ߣ޿߁ᧂ᧪ߩㆇേࠍ᳿߼ࠆ‛ℂᴺೣߪ૗߽ߥ޿ߩߢ޽ࠆ㧚
1 ୘ߩశሶߦߟ޿ߡߪ㧘ߎߩࠃ߁ߦ㧘஍శሶࠍㅢㆊߔࠆ߆ߒߥ޿߆ࠍ᳿߼ࠆ‛ℂᴺ
ೣߪ૗߽ߥ޿㧚ߒ߆ߒ㧘หߓ᧦ઙਅߢ૗࿁߽ታ㛎ࠍ߿ߞߡߺࠆ㧚޽ࠆ႐วߦߪㅢㆊߒ㧘޽ࠆ႐วߦߪㅢㆊߒߥ޿߇㧘ㅢㆊߔࠆశሶߩഀวߪ߶߷ cos 2 T ߦㄭ޿㧚ߘߒߡታ㛎ߩ࿁ᢙ߇Ⴧ߃ࠆߦߟࠇߡߘߩഀวߪ cos 2 T ߦㄭߠߊ㧚1 ୘ߩశሶߩᝄࠆ⥰޿ࠍ੍᷹ߔࠆ
ߎߣߪ಴᧪ߥ޿߇㧘ߎߩࠃ߁ߦ㧘1 ୘ߩశሶߦߟ޿ߡᄙᢙ࿁ߩታ㛎ࠍⴕߞߚ႐ว㧘෶ߪ㧘ᄙᢙߩశሶߦߟ޿ߡታ㛎ࠍⴕߞߚ႐ว㧘⛔⸘⊛ߦߪฎౖ⺰ߩ੍᷹ߦᓥ߁ߎߣߦߥ
ࠆ㧚ߎߩታ㛎ߩ႐ว㧘஍శሶߪ౉኿ߒߚశ߇޽ࠆ․ቯᣇะߦ஍శߒߡ޿ࠆ߆ุ߆ࠍ᷹ⷰ
ߔࠆታ㛎ⵝ⟎ߢ޽ࠆ㧚ߎߩࠃ߁ߥታ㛎ࠍ㊂ሶജቇߩ⸒⪲ߢ‫ⷰޟ‬᷹‫ޠ‬㧔measurement㧕ߣ޿߁㧚ߎߩ measurement ߩ⚿ᨐߪ㧘޽ࠆ․ᱶߥ႐วߦߪᚒ‫ ߪޘ‬100%ߩ⏕ታߐࠍ߽ߞ
ߡ੍᷹ߔࠆߎߣ߇಴᧪ࠆ߇㧘ߘࠇߦ෻ߒߡ㧘ߤ߁޿߁⚿ᨐߦߥࠆ߆ 100%ߩ⏕ታߐߢ߽ߞߡߪ⚿ᨐࠍ੍᷹಴᧪ߥ޿႐ว߽޽ࠆ㧚਄ㅀߒߚታ㛎ߩ଀ߢߪ㧘╙ 2 ߩ஍శሶ߇╙ 1 ߩ஍శሶߣᐔⴕߢ޽ࠆ႐ว㧔 T߹ߚߪု⋥ߢ޽ࠆ႐ว㧔 T

S
2

0㧕

㧕㧘ᚒ‫ ߪޘ‬100%ߩ⏕ታߐࠍ߽ߞߡታ㛎⚿ᨐࠍ੍⍮ߔࠆߎߣ߇಴᧪ࠆ㧚ߎߩࠃ߁ߦ 100%ߩ⏕ታߐࠍ߽ߞߡ⚿ᨐࠍ੍⍮ߔࠆߎߣ߇಴᧪ࠆ႐ว㧘㊂ሶജቇߩ⸒⪲ߢ‫ޟ‬శሶߪߎߩ measurement ߦኻߒߡ eigenstate ߦ޽ࠆ‫޿޿ߣޠ‬㧘ߘߩᤨߩታ㛎⚿ᨐࠍߘߩ᷹ቯߦኻߒߡߩ eigenvalue ߣ⸒߁㧚T

0 ߩ႐วߪ㧘శሶߪߎߩ

measurement ߦኻߒߡ‫ޟ‬parallel eigenstate‫ࠅ޽ߦޠ‬㧘ߘߩ eigenvalue ߪ‫ޟ‬parallel‫޽ߢޠ‬䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 4ࠆ㧘ߣ⸒߁ߎߣߦߒࠃ߁㧚ห᭽ߦ㧘 TS
2ߩ႐วߪ㧘శሶߪߎߩ measurement ߦኻߒߡ‫ޟ‬perpendicular eigenstate‫ࠅ޽ߦޠ‬㧘ߘߩ eigenvalue ߪ‫ޟ‬perpendicular‫ࠆ޽ߢޠ‬㧘ߣ⸒߁ߎߣߦߔࠆ㧚৻ᣇ㧘╙ 2 ߩ஍శሶ߇శሶߩ஍శᣇะߣᐔⴕߢ߽ု⋥ߢ߽ߥ޿ᤨ㧘ߔߥࠊߜ T ߇ 0 ߢ߽S2ߢ߽ߥ޿႐วߪ㧘100%ߩ⏕ታߐߢ߽ߞߡ⚿ᨐࠍ੍⍮ߔࠆߎߣߪ಴

᧪ߥ޿㧚ߎߩ႐วߪ㧘శሶߪߎߩ measurement ߦߟ޿ߡߩ eigenstate ߦߪߥ޿ࠊߌߢ
޽ࠆ㧚
㊂ሶജቇߢߪߤࠎߥ measurement ߢ߽ߘࠇߦ⋧ᒰߔࠆṶ▚ሶߢ⴫ߔߩ߇᥉ㅢߢ޽
ࠆ㧚਄ㅀߩታ㛎ߪ㧘శሶ߇ z ゲ஍శߢ޽ࠆ߆ุ߆ࠍ⺞ߴࠆ measurement ߢ޽ࠆ߇㧘ߎߩ measurement ࠍ઒ߦ A ߣ޿߁Ṷ▚ሶߢ⴫ߔߎߣߦߔࠆ㧚శሶ߇㧘޽ࠆ eigenstate ߦ޽ࠆߣ޿߁ߎߣߪ㧘ㅢᏱ㧘ࠡ࡝ࠪࡖᢥሼߩࡊࠨࠗ㧔ᄢᢥሼ㧦 < 㧘ዊᢥሼ㧦\ 㧕߹ߚߪࡈࠔࠗ㧔ᄢᢥሼ㧦) 㧘ዊᢥሼ㧦I 㧕ߣ޿ߞߚᢥሼࠍ↪޿ߡ⴫ߔ㧚଀߃߫ parallel eigenstateߪ < parallel ߣ㧘perpendicular eigenstate ߪ < perpendicular ߣ޿ߞߚࠃ߁ߦ⴫ߔߎߣߦߒࠃ߁㧚ߘߒߡశሶ߇ߎߩ A ߣ޿߁ measurement ߦኻߒߡ eigenstate ߦߥߞߡ޿ࠆߣ޿߁ߎߣࠍ㧘A< parallel

Oparallel < parallelA< perpendicular(2.1.1)

Operpendicular < perpendicularߩࠃ߁ߦ⴫ߔ㧚 Oparallel ߅ࠃ߮ Operpendicular ߪ eigenvalue ߢ޽ࠆ㧚 T ߇ 0 ߢ߽

S
2

ߢ߽ߥ޿ᤨ

ߦߪ㧘శሶߪߎߩ measurement ߦኻߒߡ eigenstate ߦߪߥߞߡ޿ߥ޿ߩߢ㧘਄ߩࠃ߁ߥᑼߢ⴫ߔߎߣߪ಴᧪ߥ޿㧚ߎߩࠃ߁ߥ႐ว㧘㊂ሶജቇߩ਎⇇ߢߪశሶߪ㧘 < parallel ߣ< perpendicular ߣߩ㊀ߨวࠊߖߢ⴫ߐࠇࠆߣ⠨߃ࠆ㧚એ਄ߩࠃ߁ߥ㊂ሶജቇ․᦭ߩ⸒⪲ߪ㧘ߎࠇ߆ࠄ➅ࠅ㄰ߒߡ↪޿ࠆߎߣߦߥࠈ߁߇㧘< ߿\ ࠍ↪޿ߚ⴫⃻ߦᘠࠇⷫߒ߻ߎߣ߇
ᄢಾߢ޽ࠈ߁㧚ᆎ߼ߪಽ߆ࠅߦߊ޿᭎ᔨߢ޽ࠈ߁߇㧘ߣࠅ޽߃ߕ㧘ฎౖജቇߩ਎⇇ߢߪᐳᮡߣㆇേ㊂ߢ߽ߞߡ⁁ᘒࠍⷙቯߒߚߩߦኻߒ㧘㊂ሶജቇߩ਎⇇ߢߪߘߩઍࠅߦ <߿\ ߣ޿߁߽ߩߢ⁁ᘒࠍⷙቯߒߡ޿ࠆ㧘ߣ޿߁ߊࠄ޿ߦℂ⸃ߒߡ߅ߊߎߣߦߒࠃ߁㧚ߐߡ㧘 T

0 ߢ߽ߥߊ TS
2ߢ߽ߥ޿႐วߩታ㛎ߦᚯߞߡ߽߁ዋߒ⹦ߒߊߺߡߺࠃ

߁㧚╙ 1 ߩ஍శሶߩ஍శᣇะࠍ z ゲߩะ߈ߦߒߡ߅޿ߚߚ߼ߦߘߩ஍శሶࠍㅢߞߚశ
ሶߪߔߴߡ z ゲ஍శߢ޽ࠆ㧚╙ 2 ߩ஍శሶߩ஍శᣇะࠍ z ゲߩะ߈ߦߔࠇ߫㧘ߔߴߡߩశሶߪ╙ 2 ߩ஍శሶࠍㅢㆊߔࠆߎߣ߆ࠄ㧘ߘߩߎߣ߇ᱜߒ޿ߎߣ߇ಽ߆ࠆ㧚ߒ߆ߒ

0 T 

Sߩ႐วߪߤ߁ߥࠆߢ޽ࠈ߁߆㧚଀߃߫ T2䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 5S4ߩ႐วߦߪ cos 2

S
4

1
߆ࠄಽ߆
2

ࠆࠃ߁ߦ㧘ඨಽߩᢙߩశሶߪߎߩ௑޿ߚ஍శሶߦᐔⴕߢ޽ࠅ㧘ᱷࠅߩඨಽߪߘࠇߦု
⋥ߢ޽ࠆ㧚ߘࠇߪߤ߁޿߁ߎߣ߆ߣ޿߁ߣ㧘 z ゲ஍శߩశሶߦߟ޿ߡߩ measurement ࠍⴕߞߚᤨߩ⚿ᨐ߇㧘z ゲ߆ࠄ

S4ߛߌ௑޿ߚᣇะߦᐔⴕ෶ߪု⋥ߩ޿ߕࠇ߆ߣߥࠆ㧚ߔߥࠊߜ㧘measurement ࠍⴕ߁ߎߣߦࠃߞߡశሶߩᕈ⾰߇ߔߞ߆ࠅᄌߞߡߒ߹ߞߚߩߢ޽ࠆ㧚ߎࠇ߇ᧄ┨ߩ౨㗡ߢߩߴߚ‫ޟ‬᷹ቯߦࠃߞߡᕈ⾰߇ᄌߞߡߒ߹߁‫޽ߢ଀৻ߩߣߎ߁޿ߣޠ‬ࠆ㧚ߎߩࠃ߁ߥߎߣ߇⿠ߎࠆ႐ว㧘☸ሶ߇චಽߦዊߐߊ㧘ฎౖജቇߢߪ⺑᣿߇ߟ߆ߥ
޿ߣ್ᢿߐࠇࠆࠊߌߢ޽ࠆ㧚᷹ቯߔࠆߎߣߦࠃߞߡᕈ⾰߇ᄌࠆߣ޿߁ߎߣࠍ߽߁ዋߒߪߞ߈ࠅߐߖࠆߚ߼ߦ㧘ዋ‫߇޿ߥࠇߒ߽߆޿ߤߊޘ‬㧘╙ 3 ߩ஍శሶࠍ↪޿ߚ߽߁৻ߟߩታ㛎ࠍ⠨߃ߡߺࠃ߁㧚㧔࿑ 2.1.3㧕

࿑ 2.1.3

3 ୘ߩ஍శሶࠍ↪޿ߚታ㛎╙ 1 ߩ஍శሶߢ z ゲ஍శߩశሶߩߺࠍㆬᛯߒߚ㧚ߎߩశሶߦߟ޿ߡ㧘 z ゲߦ㑐ߒߡS
4ߛߌ௑޿ߚ╙ 2 ߩ஍శሶࠍ↪޿ߡ᷹ቯࠍⴕߞߚ㧚ඨಽߩᢙߩశሶߪߎߩ╙ 2 ߩ஍శሶࠍㅢㆊߒߚࠊߌߢ޽ࠆ㧚ߎߎ߹ߢߪ੹߹ߢߩታ㛎ߣหߓߢ޽ࠆ㧚╙ 2 ߩ஍శሶߦๆ
෼ߐࠇߡߒ߹ߞߚశሶߦߟ޿ߡߪ߽߁ታ㛎ߔࠆߎߣߪ಴᧪ߥ޿߆ࠄ޽߈ࠄ߼ࠆߣߒ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 6ߡ㧘╙ 2 ߩ஍శሶࠍㅢㆊߒߚశሶߦߟ޿ߡ╙ 3 ߩ஍శሶߢ᷹ቯࠍⴕ߁㧚ߎߩ႐ว㧘╙
3 ߩ஍శሶߪ╙ 1 ߩߘࠇߣหߓߊ஍శᣇะࠍ z ゲᣇะߦߒߡ߅ߊ㧚╙ 3 ߩ஍శሶߪ╙2 ߩߘࠇߣߪ

S
4ߛߌ௑޿ߡ޿ࠆ߆ࠄ㧘ౣ߮ඨಽߩᢙߩశሶߪㅢㆊߒඨಽߪๆ෼ߐࠇߡ

ߒ߹߁㧚ߔߥࠊߜ㧘╙ 2 ߩ஍శሶࠍㅢㆊߒߚశሶߪ z ゲᣇะߩ஍శࠍ⺞ߴࠆߣ޿߁╙3 ߩ᷹ቯߩ eigenstate ߦߪߥߞߡ޿ߥ޿㧚ᚒ‫ߦ߆⏕ߪޘ‬㧘╙ 1 ߩ஍శሶߢ㧘 z ゲᣇะߩ஍శࠍ⺞ߴࠆߣ޿߁᷹ቯߩ eigenstate ࠍㆬ߮಴ߒߚߪߕߢ޽ࠆ㧚㧔߽ߒ╙ 2 ߩ஍శ᧼ࠍ
z ゲᣇะߦߒߡ᷹ቯߔࠇ߫ߘ߁⚿⺰ߐࠇࠆߢ޽ࠈ߁㧕ߘࠇߥߩߦ㧘߽߁ 1 ࿁หߓ᷹ቯ
ࠍⴕ߁೨ߦ 1 ߟઁߩ᷹ቯࠍⴕߞߚߚ߼ߦ㧘೨ߦᜬߞߡ޿ߚᕈ⾰߇ᄌߞߡߒ߹ߞߚࠊߌ
ߢ޽ࠆ㧚ฎౖജቇߢߪ೔ᐩ⠨߃ࠄࠇߥ޿ࠃ߁ߥߎ߁޿߁․ᱶߥ਎⇇㧘ߎࠇ߇㊂ሶജቇߢ޽ࠆ㧚[Coffee Break] ৻ᤄ೨ߩ"If it were possible"߇੹ߢߪ
1944 ᐕ⊒ⴕߩ Eyring, Walter, Kimball ߩ"Quantum Chemistry"ߩ 6 㗁ߦᰴߩࠃ߁ߥ⸥
ㅀ߇޽ࠆ㧚
"If it were possible to perform the experiment with a single photon we could not
possibly obtain the complete diffraction pattern; at most one gain of the emulsion on theplate would be sensitized."ߎߩᢥ߽ዊߐߥ☸ሶߢ޽ࠆ photon ߩ․ᱶߥᝄࠆ⥰޿ࠍㅀߴߚ߽ߩߢ޽ࠆ߇㧘ߎߎߢ㧘ߜࠂߞߣ⇣ߥߞߚⷰὐ߆ࠄ㕙⊕޿ߣᗵߓࠆߎߣ߇޽ࠆ㧚 "if it were possible"ߩ"were"ߪ"if I were a bird"ߥߤߣห᭽ߦ subjunctive mood ߢ޽ࠅ㧘⛘ኻߦ޽ࠅߘ߁߽ߥ޿઒ቯ
ࠍᗧ๧ߒߡ޿ࠆ㧚"we could not obtain"ߩ"could"߽නߥࠆㆊ෰ᒻߢߪߥ޿ߎߣߪ᣿ࠄ߆
ߢ㧔ߣߎࠈ߇ߎࠇࠍ⸶ߖߣ޿߁ߣᄙߊߩᄢቇ↢ߪㆊ෰ᒻߣߒߡ⸶ߔࠆ㧘ߣ޿߁ߩ߇ᖤ
ߒ޿⃻ታߢ޽ࠆ㧕subjunctive ߩ↪ᴺߢ޽ࠆ㧚ߎߩᢥ߆ࠄ߽ಽ߆ࠆࠃ߁ߦ㧘1944 ᐕߦߪ
1 ୘ߩ photon ߦߟ޿ߡߩታ㛎ࠍⴕ߁ߎߣߪ㧘"if it were possible"ߣ޿߁↪ᴺࠍ↪޿ࠆ߶ߤߢోߊਇน⢻ߥߎߣߣᕁࠊࠇߡ޿ߚߩߢ޽ࠆ㧚ߒ߆ߒ㧘⃻࿷ߢߪ㧘޿ࠊࠁࠆ photon
counting ᴺߣ⸒ࠊࠇࠆᣇᴺࠍ↪޿ߡ㧘1 ୘ߩ photon ߩ᷹ቯ߽㔍ߥߊⴕ߁ߎߣ߇಴᧪ࠆ㧚ታ㛎ᛛⴚߩㅴᱠߪ޽ࠅ߇ߚ޿߽ߩߛ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 7

2.2 ⏛᳇ࡕ࡯ࡔࡦ࠻ࠍᜬߞߚේሶ 㧙 Stern-Gerlach Experimentߎߎߢߪ 1922 ᐕߦ⊒⴫ߐࠇߚ Stern ߣ Gerlach ߦࠃࠆ㌁ේሶߩ⏛᳇ࡕ࡯ࡔࡦ࠻᷹ቯ
ߩታ㛎ࠍ߽ߣߦߒߡੑਃߩߎߣࠍ⠨߃ߡߺࠆ㧚Stern ߣ Gerlach ߩታ㛎ߦࠃߞߡ㧘Ag
ේሶ߇⏛᳇ࡕ࡯ࡔࡦ࠻ࠍᜬߞߡ޿ࠆߎߣ㧘ߔߥࠊߜ㧘Ag ේሶߪ⏛⍹ߩᕈ⾰ࠍᜬߞߡ޿ࠆߣ޿߁ߎߣ߇␜ߐࠇߚ㧚⃻࿷ߩᚒ‫ߩޘ‬⍮⼂ߦࠃࠆߣ㧘Ag ේሶߩ㔚ሶ㈩⟎ߪ
1s 2 2 s 2 2 p 6 3s 2 3 p 6 3d 10 4s 2 4 p 6 4d 10 5s ߢ޽ࠅ㧘㐽Ზ᭴ㅧߩᄖ஥ߦ 1 ୘ߩ 5s 㔚ሶࠍᜬߞߡ޿
ࠆ㧚ᓥߞߡ㧘Ag ේሶ߇⏛᳇ࡕ࡯ࡔࡦ࠻ࠍᜬߞߡ޿ࠆߣ޿߁ߎߣߪ㧘 5s 㔚ሶ߇⏛᳇ࡕ࡯ࡔࡦ࠻ࠍᜬߞߡ޿ࠆߣ޿߁ߎߣߣหߓߎߣߢ޽ࠆ㧚ߎ߁޿߁ߎߣߦߟ޿ߡߪߕߞߣ
ᓟߢ spin ߦߟ޿ߡቇ⠌ߔࠆᤨߦ➅ࠅ㄰ߒㅀߴࠆ߇㧘ᒰ㕙ߎߩߎߣߪ޽߹ࠅ໧㗴ߣߥࠄߥ޿㧚㌁ߩ㊄ዻࠍࠝ࡯ࡉࡦߢ㧘㌁ߩᴣὐએ਄ߦᾲߒߡ㧘ࠝ࡯ࡉࡦߦ৻ࡩᚲⓣࠍ޽ߌߡ߅ߊ
ߣߘߩⓣ߆ࠄ Ag ߩේሶ߇㘧߮಴ߒߡ᧪ࠆ㧚੹઒ߦ Ag ේሶߩㅴⴕᣇะࠍ y ゲߣߒࠃ߁㧚ߘߒߡߎߩㅴⴕᣇะߦု⋥ߩᣇะ㧔઒ߦ z ゲߣߒࠃ߁㧕ߦ⏛႐ࠍ߆ߌࠆ㧚Ag ේሶ߇ Pmag ߛߌߩᄢ߈ߐߩ⏛᳇ࡕ࡯ࡔࡦ࠻ࠍᜬߞߡ޿ࠆߣߔࠆߣ ( Pmag ) z B ߛߌߩ

potential ࠍฃߌࠆ㧚㧔⏛᳇ࡕ࡯ࡔࡦ࠻ߪࡌࠢ࠻࡞ߢ޽ࠅᄢ߈ߐߣᣇะࠍᜬߞߡ޿ࠆ㧚⏛᳇ࡕ࡯ࡔࡦ࠻ߩᣇะߪ⏛⍹ߩౝㇱߢߪ S ᭂ߆ࠄ N ᭂߦะ߆ߞߡᱜߦߣࠆ㧚ᄢ߈ߐߪ⏛⩄ߩᄢ߈ߐߣ㧘⏛ᭂ㑆ߩ〒㔌ߣߩⓍߢਈ߃ࠄࠇࠆ㧚㧕ߎߎߢ ( Pmag ) z ߪ⏛᳇ࡕ࡯ࡔࡦ࠻
ߩ z ᚑಽ㧘 B ߪ⏛᧤ኒᐲߢ޽ࠆ㧚ߎߎߢ↪޿ࠆ⏛⍹߇߽ߒ߽࿑ 2.2.1 ߩ(a)ߩࠃ߁ߦᐔ
ߢᐔⴕߢ޽ࠇ߫⏛᧤ኒᐲߪ z ゲ਄ߤߎߢ߽หߓߢ޽ࠆ㧔⏛႐ߩ┵ߦ߅ߌࠆࠊߕ߆ߩᏅ
ߪ޽ࠆ߇㧕㧚ߎߩ႐ว Ag ේሶߪ⏛႐ߦࠃࠅജࠍฃߌߥ޿㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 8

(a) homogeneous magnetic field   (b) inhomogeneous magnetic field࿑ 2.2.1ᢧ⏛႐ߣਇᢧ⏛႐ߒ߆ߒ㧘߽ߒ߽࿑ 2.2.1 ߩ(b)ߩࠃ߁ߦ㧘1 ߟߩᭂ߇ውߞߡ޿ࠇ߫㧘⏛᧤ኒᐲߪ z ゲ਄ߩ႐ᚲߦࠃࠅ
B B0  kz(2.2.1)ߩࠃ߁ߦᄌൻߔࠆ㧚㧔⏛႐ߪ⏛⍹ߩᄖ஥ߢߪ N ᭂ߆ࠄ S ᭂ߳ะ߆ߞߡᱜߦߣࠆ㧚ᓥߞ
ߡ࿑ 2.2.1(b)ߩ႐วߦߪ k ! 0 ߢ޽ࠆ㧕ߎߩࠃ߁ߥ⏛႐ࠍਇᢧ⏛႐㧔inhomogeneous

magnetic field㧕ߣ޿߁㧚ߎߩਇᢧ⏛႐ߩਛߦ߅߆ࠇߚ Ag ේሶߪ㧘
 Pmag'Emagz

B0  kz(2.2.2)

ߛߌߩ potential energy ࠍᜬߟ㧚Ag ේሶߦ௛ߊജߪߎߩ potential energy ࠍ z ߦߟ޿ߡᓸ
ಽߒ㧘╓ภࠍᄌ߃ߚ߽ߩߢ޽ࠆ߆ࠄ

w'EmagPmag z k(2.2.3)wzߣ⴫ߐࠇࠆ㧚Ag ේሶ߇ߤߩࠃ߁ߥജࠍฃߌࠆ߆ߪ Ag ߩㅴⴕᣇะߩవߦ࿑ 2.2.2 ߩࠃ߁ߦࠟ࡜ࠬ᧼ࠍ߅޿ߡ߅ߌ߫ Ag ߇߱ߟ߆ߞߚᚲ߇㤥ߊߥࠆߩߢಽ߆ࠆߢ޽ࠈ߁㧚Fmag䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 9࿑ 2.2.2 Stern-Gerlach ታ㛎ߩⵝ⟎߹ߕ㧘ߤ߁޿߁⚿ᨐߦߥࠆߩ߆㧘ฎౖജቇߩ┙႐߆ࠄ੍ᗐߒߡߺࠃ߁㧚㧔a㧕߽ߒ߽

Ag ේሶ߇⏛᳇ࡕ࡯ࡔࡦ࠻ࠍᜬߞߡ޿ߥߌࠇ߫ߚߛ৻ࡩᚲߦࠬࡐ࠶࠻߇಴᧪ࠆߢ޽ࠈ
߁㧚㧔࿑ 2.2.3(a)㧕 㧔b㧕߽ߒ߽ Ag ේሶ߇ Pmag ߩ⏛᳇ࡕ࡯ࡔࡦ࠻ࠍᜬߞߡ޿ߚߣߔࠆ
ߣߘߩ z ᚑಽߪPmag zPmag cos T(2.2.4)ߢ⴫ߐࠇࠆ㧚ߎߎߢ T ߪ㌁ේሶߩ⏛᳇ࡕ࡯ࡔࡦ࠻ߩᣇะߣ z ゲߣߩߥߔⷺᐲߢ޽ࠆ㧚ࠝ࡯ࡉࡦߢᾲߖࠄࠇߡࠬ࡝࠶࠻߆ࠄ಴ߡ᧪ߚ Ag ේሶߪࡃ࡜ࡃ࡜ߩᣇะࠍะ޿ߡ޿ࠆ
ߪߕߢ޽ࠆ㧚ᓥߞߡ Pmag ߽ᒰὼࡃ࡜ࡃ࡜ߩᣇะࠍะ޿ߡ޿ࠆ㧚ᓥߞߡ㧘ⓨ㑆ゲ z ߣߩߥߔⷺᐲ T ߪ޿ࠈ޿ࠈߥ୯ߣߥࠅ㧘 cos T ߪ+1 ߆ࠄ1 ߹ߢߩ㑆ߩ޿ࠈ޿ࠈߥ୯ࠍߣࠆ
ߣ⠨߃ࠄࠇࠆ㧚ߘࠇߢ㧘ࠟ࡜ࠬ᧼ߩ਄ߦߪ cos T
cos T

1 ߦ⋧ᒰߔࠆ Pmagz

1 ߦ⋧ᒰߔࠆ Pmagz

 P mag ߆ࠄ Pmag ߹ߢㅪ⛯⊛ߦ Ag ߩ☸ሶ߇ߊߞߟߊߎߣߦߥࠆߢ

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 10޽ࠈ߁㧚㧔࿑ 2.2.3(b)㧚
㧕

(a)

(b)
࿑ 2.2.3ฎౖ⺰ߦࠃࠆ੍ᗐ

ߒ߆ߒ㧘Stern-Gerlach ߩታ㛎ߪ਄⸥(a)㧘(b)޿ߕࠇߩ੍ᗐߣ߽ᄢ߈ߊ⇣ߥߞߡ޿ߚ㧚ታ㛎ߢߪߥࠎߣࠟ࡜ࠬ᧼਄ਛᔃ߆ࠄ╬〒㔌ߩ 2 ߟߩὐߦߩߺࠬࡐ࠶࠻߇㓸ਛߒߚ㧚ߘߩਛ㑆ߦߪోὼ Ag ේሶߪ᧪ߡ޿ߥ޿ߎߣ߇ಽ߆ߞߚ㧚㧔࿑ 2.2.4㧕䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 11࿑ 2.2.4
Stern-Gerlach ߩታ㛎ߩ⚿ᨐߎߩታ㛎߆ࠄᰴߩࠃ߁ߥߎߣ߇ಽ߆ࠆ㧚
㧔1㧕 Ag ේሶߪ⏛᳇ࡕ࡯ࡔࡦ࠻ࠍᜬߞߡ޿ࠆ㧚㧔2㧕 ߘߩ⏛᳇ࡕ࡯ࡔࡦ࠻ߩ z ᚑಽߪએਅߦㅀߴࠆ 2 ߟߩ୯ߩ޿ߕࠇ߆ 1 ߟߢߒ߆޽ࠅᓧߥ޿㧚
㧔3㧕

Ag ේሶߩ㘧ⴕㅦᐲ㧘ࠟ࡜ࠬ᧼߹ߢߩ〒㔌㧘ਇᢧ⏛႐ߩᄢ߈ߐ k 㧘2 ߟߩࠬࡐ࠶࠻ౝߩ〒㔌ߥߤ߆ࠄ Pmag ߩᄢ߈ߐࠍታ㛎⊛ߦ᳿߼ࠆߎߣ߇಴᧪ࠆ㧚⚿ᨐߪzPmag zr PBߢ޽ߞߚ㧚ߎߎߢ PB ߪ Bohr magneton ߢ㧘

PB

e=
2me

9.284 u1024 J/Tߢ⴫ߐࠇࠆ㧚 e ߪ㔚ሶߩ㔚⩄ߢ㧘߹ߚ me ߪ㔚ሶߩ⾰㊂ߢ޽ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 12(2.2.5)ߎߩታ㛎⚿ᨐߪታߦᄸ⇣ߢ޽ࠆ㧚Ag ේሶߪࡃ࡜ࡃ࡜ߩᣇะࠍะ޿ߡ޿ࠆߪߕߥߩ
ߦ⏛᳇ࡕ࡯ࡔࡦ࠻ߩ z ᚑಽߪ  PB ߹ߚߪ  PB ߩ޿ߕࠇ߆ߩ୯ߒ߆ขࠅᓧߥ޿㧚޽ࠆ႐วߦߪਇᢧ⏛႐ߩᣇะߦᐔⴕ㧔parallel㧕ߢ޽ࠅ㧘޽ࠆ႐วߦߪ෻ᐔⴕ㧔antiparallel㧕ߢ޽ࠆ㧚ߘߩਛ㑆ߣ޿߁ߎߣߪ޽ࠅᓧߥ޿㧚ߐߡ㧘Stern-Gerlach ߩታ㛎ࠍᦝߦ⊒ዷߐߖ㧘2 ⚵ߩਇᢧ⏛႐ࠍ↪޿ࠆታ㛎ࠍⴕߞߡߺࠆ㧚╙ 1 ߩਇᢧ⏛႐ࠍ਄ߣหߓࠃ߁ߦ z ゲᣇะߦ߆ߌࠆ㧚Ag ේሶߪ parallel㧘

antiparallel ߩ 2 ߟߦಽ߆ࠇࠆߎߣߪ਄ㅀߒߚߣ߅ࠅߢ޽ࠆ㧚ߘߩ߁ߜ parallel ߩ߽ߩߩߺࠍขࠅ಴ߒߡ╙ 2 ߩਇᢧ⏛႐ߦㅢߔ㧚㧔࿑ 2.2.5㧕

࿑ 2.2.5
Stern-Gerlach ߩታ㛎ࠍ 2 ࿁⛯ߌߡⴕ߁㧔ߘߩ 1㧕㧔ᱜ⏕ߦ⸒߁ߣ㧘╙ 1 ߩਇᢧ⏛႐ࠍㅢㆊᓟ㧘Ag ේሶߩㅴⴕᣇะߪ y ゲ߆ࠄዋߒߕࠇߡ
޿ࠆ㧚ߎߩ߹߹ߢߪ⠨߃ߦߊ޿ߩߢߎࠇࠍ૗╬߆ߩᣇะߢ y ゲᣇะߦㅴⴕߔࠆࠃ߁ୃ
ᱜߒߚ਄ߢ╙ 2 ߩਇᢧ⏛႐ߦㅢߔߣ⠨߃ࠆ㧚㧕╙ 2 ߩਇᢧ⏛႐߽╙ 1 ߩ߽ߩߣหߓߊ zゲᣇะߦ߆ߌࠆߣߒࠃ߁㧚╙ 2 ߩਇᢧ⏛႐ߩᓟࠈߦߪࠟ࡜ࠬ᧼ߩઍࠅߦ㧘Ag ේሶ 1୘ߢ߽ᔕ╵ߔࠆࠃ߁ߥࠞ࠙ࡦ࠲࡯ࠍ⟎߈㧘╙ 2 ߩਇᢧ⏛႐ࠍㅢࠆ Ag ේሶ 1 ୘ 1 ୘ߦ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 13ߟ޿ߡߘߩ゠〔߇ಽ߆ࠆࠃ߁ߦߒߡ߅ߊ㧚ߎߩࠃ߁ߥⵝ⟎ߢታ㛎ߒߚ႐ว㧘Ag ේሶߪ╙ 2 ߩਇᢧ⏛႐ߦࠃߞߡߤ߁ߥࠆߢ޽ࠈ߁߆㧫ߎߩታ㛎ߩ⚿ᨐߪᰴߩࠃ߁ߦ⠨߃ࠇ੍᷹߫಴᧪ࠃ߁㧚╙ 1 ߩਇᢧ⏛႐ࠍㅢߞߚᓟ㧘⏛᳇ࡕ࡯ࡔࡦ࠻ߩ z ᚑಽ߇ z ゲߦ parallelߩ߽ߩߩߺࠍขࠅ಴ߒߡ╙ 2 ߩታ㛎ࠍߒߚࠊߌߢ޽ࠆ㧚ߎࠇࠄ Ag ේሶߩ⏛᳇ࡕ࡯ࡔࡦ࠻ߩ z ᚑಽߪ㧘ᒰὼ㧘╙ 2 ߩਇᢧ⏛႐ߦߟ޿ߡ߽ parallel ߢ޽ࠆ㧚ߢ޽ࠆ߆ࠄ㧘╙ 2ߩਇᢧ⏛႐ߦߟ޿ߡߪߔߴߡߩ Ag ේሶ߇ parallel ߩᣇะߦⴕߊߪߕߢ޽ࠆ㧚ታ㛎⚿ᨐߪ߹ߐߒߊߘ߁ߥߞߡ޿ࠆ㧚ߎߩ႐ว㧘ᚒ‫ ߪޘ‬Ag ේሶߩㅴ߻ᣇะࠍ 100%ߩ⏕߆ࠄߒߐߢ੍⍮ߔࠆߎߣ߇಴᧪ࠆ㧚ᰴߩታ㛎ߢߪ㧘╙ 2 ߩਇᢧ⏛႐ࠍ T S ߛߌ࿁ォߒߡ㧘ਇᢧ⏛႐ߩᣇะࠍ  z ゲߩᣇะߣߔࠆ㧚ߎߩ႐ว╙ 1 ߩਇᢧ⏛႐ࠍㅢߞߚ Ag ේሶߪ╙ 2 ߩਇᢧ⏛႐ߦࠃߞߡߤ߁
ߥࠆߢ޽ࠈ߁߆㧫╙ 2 ߩਇᢧ⏛႐ߩᣇะߪ㧘ߘߩਇᢧ⏛႐ࠍㅢㆊߒߚ Ag ේሶߩ⏛᳇ࡕ࡯ࡔࡦ࠻ߩ z ᚑಽߦ antiparallel ߣߥࠆ㧚ߔߥࠊߜ㧘ోߡߩ Ag ේሶߪ㧘╙ 2 ߩਇᢧ⏛႐ߦኻߒߡ antiparallel ߩᣇะߦᦛࠆߪߕߢ޽ࠆ㧚ታ㛎ߢߪ⏕߆ߦߘ߁ߥߞߡ޿ࠆ㧚ߎߩ႐วߦ߽ᚒ‫ ߪޘ‬100%ߩ⏕߆ࠄߒߐࠍᜬߞߡ⚿ᨐࠍ੍᷹ߔࠆߎߣ߇಴᧪ࠆ㧚ᦨᓟߦ߽ߞߣ߽㊀ⷐߥታ㛎ߣߒߡ㧘╙ 2 ߩਇᢧ⏛႐ߩᣇะࠍ z ゲ߆ࠄ࠯ࡠߢ߽ S ߢ߽ߥ޿ⷺᐲ T ߛߌ௑ߌߡߺࠆ㧚㧔࿑ 2.2.6㧕਄ㅀߩ 2 ߟߩታ㛎ߩࠃ߁ߦ㧘 z ゲᣇะߦ߆ߌߚ╙ 1 ߩਇᢧ⏛႐ߦࠃߞߡ parallel ߣߥߞߚ߽ߩߛߌࠍขࠅ಴ߒߡ╙ 2 ߩਇᢧ⏛႐ߦㅢߔ㧚ߎߩ႐ว㧘╙ 2 ߩਇᢧ⏛႐ߦ౉ࠆ Ag ේሶߩ⏛᳇ࡕ࡯ࡔࡦ࠻ߩ z ᚑಽߪ㧘╙

2 ߩਇᢧ⏛႐ߦߟ޿ߡ parallel ߢ߽ antiparallel ߢ߽ߥ޿㧚parallel ߢ߽ antiparallel ߢ߽ߥ޿߆ࠄߤߎ߆ਛ㑆ߩᣇะߦⴕߊߢ޽ࠈ߁߆㧫ߒ߆ߒߘ߁޿߁ߎߣߪ㧘parallel ߆
antiparallel ߆㧘2 ߟߩ߁ߜߩ޿ߕࠇ߆ߦߒ߆ߥࠄߥ޿ߣ޿߁ Stern-Gerlach ߩታ㛎⚿ᨐߣ⍦⋫ߔࠆ㧚ߘࠇߢߎߩవߤ߁⠨߃ߡ޿޿߆ಽ߆ࠄߥߊߥߞߡߒ߹߁㧚ታ㛎ࠍ߿ߞߡߺࠆߣ㧘޽ࠆ႐วߦߪ parallel ߦߥࠅ㧘޽ࠆ႐วߦߪ antiparallel ߣߥࠆ㧚ߢߪߤ߁޿
߁႐วߦ parallel ߢ޽ࠅ㧘ߤ߁޿߁႐ว antiparallel ߦߥࠆߩߢ޽ࠈ߁߆㧚parallel ߦߥࠆ߆ antiparallel ߦߥࠆ߆ߪߤߩࠃ߁ߥⷙೣߢ᳿߹ࠆߩߢ޽ࠈ߁߆㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 14

࿑ 2.2.6
Stern-Gerlach ߩታ㛎ࠍ 2 ࿁⛯ߌߡⴕ߁㧔ߘߩ 2㧕ߎߩ႐ว㧘୘‫☸ߩޘ‬ሶߩⴕ߈వࠍ᳿߼ࠆ‛ℂᴺೣߣ⸒߁߽ߩߪ૗߽ߥ޿ߩߢ޽ࠆ㧚╙ 2 ߩਇᢧ⏛႐ߦࠃߞߡ parallel ߦߥࠆ߆ antiparallel ߦߥࠆ߆ߪ㧘ታ㛎ࠍ߿ߞߡߺࠆ
߹ߢಽ߆ࠄߥ޿㧚ߎߩࠃ߁ߦ㧘1 ୘ 1 ୘ߩ☸ሶߦߟ޿ߡߩ⚿ᨐࠍ੍᷹ߔࠆߎߣߪ಴᧪
ߥ޿߇㧘㕖Ᏹߦᄙᢙߩ☸ሶߦߟ޿ߡታ㛎ࠍⴕ߃߫㧘parallel ߦߥࠆ☸ሶߩᢙߩഀวߪ

cos 2

T
2

ߣߥࠆ㧚ᓥߞߡ antiparallel ߦߥࠆ☸ሶߩഀวߪ sin 2

T
2

ߢ޽ࠆ㧚

ߎߩታ㛎߆ࠄ㧘⏛᳇ࡕ࡯ࡔࡦ࠻ߩ⏛႐ᣇะߩᚑಽߩ⛔⸘⊛ߥᐔဋ୯ࠍ᳞߼ࠆߎߣ߇
಴᧪ࠆ㧚parallel ߦߥߞߚ☸ሶߦߟ޿ߡߪ PB ߩ㧘߹ߚ antiparallel ߦߥߞߚ☸ሶߦߟ޿ߡߪ  PB ߩ⏛᳇ࡕ࡯ࡔࡦ࠻㧔⏛႐ᣇะߩᚑಽ㧕ࠍᜬߞߡ޿ࠆࠊߌߢ޽ࠆ߆ࠄ㧘ߘࠇߙ
ࠇߩ୯ߦ㊀ߺࠍ߆ߌߡ㧘ᐔဋߣߒߡߪ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 15

Pmag

 P B cos 2T2

 P B sin 2

T2(2.2.6)

PB cos T߇ᓧࠄࠇࠆ㧚ߎߩ⚿ᨐߪታߦ㕙⊕޿㧚ߔߥࠊߜ㧘ᐔဋ୯ߪฎౖജቇߢߩ੍ᗐ㧔ᑼ(2.2.4)㧕
ߣ৻⥌ߔࠆ㧚୘‫☸ߩޘ‬ሶߩㆇേࠍ੍᷹ߔࠆߎߣߪ಴᧪ߥ޿߇㧔ߘࠇࠍ᳿߼ࠆ‛ℂᴺೣ
ߪߥ޿߆ࠄ㧕ᢙᄙߊߩ☸ሶߦߟ޿ߡߩᐔဋ୯ࠍ᳿߼ࠆߎߣߪ಴᧪ߡ㧘ߘࠇߪฎౖ⺰ߩ੍ᗐߣ৻⥌ߔࠆߩߢ޽ࠆ㧚਄ㅀߩታ㛎ߢ㧘╙ 2 ߩਇᢧ⏛႐߇╙ 1 ߩߘࠇߣᐔⴕߢ޽ࠆ႐ว㧔 T 0 㧕ߣ෻ᐔⴕߢ޽ࠆ႐ว㧔 T S 㧕ߦߪ㧘☸ሶߩㆇേ⚿ᨐ㧘ߔߥࠊߜ⏛႐ᣇะߩ⏛᳇ࡕ࡯ࡔࡦ࠻ߩ
᷹ቯߦኻߔࠆ⚿ᨐࠍ 100%ߩ⏕߆ࠄߒߐߢ੍᷹ߔࠆߎߣ߇಴᧪ࠆ㧚ߎߩ႐ว㧘☸ሶߪ᷹ቯߦኻߒߡ eigenstate ߦ޽ࠆߣ޿߁㧚੹઒ߦ⏛႐ᣇะߩ⏛᳇ࡕ࡯ࡔࡦ࠻ࠍ᷹ቯߔࠆ
ታ㛎ࠍṶ▚ሶ A ߢ⴫ߔߎߣߦߒࠃ߁㧚parallel ߅ࠃ߮ antiparallel ߘࠇߙࠇߩ႐วߩ

eigenstate ࠍ < parallel ߅ࠃ߮ < antiparallel ߢ␜ߒ㧘eigenvalue ࠍߘࠇߙࠇ Oparallel ߅ࠃ߮ Oantiparallel ߣ⴫ߔߎߣߦߒࠃ߁㧚ߘ߁ߔࠆߣᑼ(2.1.1)ߣหߓߊ
A< parallel
A< antiparallel

Oparallel < parallel
Oantiparallel < antiparallel(2.2.7)

ߣ޿߁ᑼߢ☸ሶ߇᷹ቯ A ߩ eigenstate ߢ޽ࠆߎߣࠍ␜ߔ㧚৻ᣇ㧘 T ߇ 0 ߢ߽ S ߢ߽ߥ޿ᤨߦߪ 100%ߩ⏕߆ࠄߒߐࠍ߽ߞߡߪ᷹ቯ⚿ᨐࠍ੍᷹ߔࠆߎߣߪ಴᧪ߥ޿㧚ߔߥࠊߜᑼ(2.2.7)ߩࠃ߁ߦ

operator (wavefunction) = eigenvalue u wavefunctionߣ޿߁ᒻߦߪ⴫ߔߎߣ߇಴᧪ߥ޿㧚ߎ߁޿߁႐วߦߪ㧘᷹ቯ୯ߩ‫ޟ‬ᐔဋ‫ߩޠ‬୯ࠍ⠨߃ࠆߎߣ߇಴᧪ࠆ㧚wavefunction < ߢ߽ߞߡ⴫ߐࠇࠆ☸ሶߦߟ޿ߡ㧘operator A ߢ⴫ߐࠇࠆ‛ℂ㊂ߩᐔဋ୯  A ! ߪ A!

³ < * A<dW

(2.2.8)ߢ᳞߼ࠄࠇࠆ㧚ᐔဋ୯㧔average value ߹ߚߪ mean value㧕ߪ㧘߹ߚ㧘ᦼᓙ୯㧔expectation

value㧕ߣ߽⸒ࠊࠇࠆ㧚਄ㅀߩታ㛎ߩᗧ๧ࠍ߽߁ዋߒᷓߊ⠨߃ࠆߚ߼ߦ 3 ⚵ߩਇᢧ⏛႐ࠍ↪޿ߚታ㛎ࠍ⠨߃ߡߺࠃ߁㧔࿑ 2.2.7㧕㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 16

࿑ 2.2.7 Stern-Gerlach ߩታ㛎ࠍ 3 ࿁⛯ߌߡⴕ߁╙ 1 ߅ࠃ߮╙ 2 ߩਇᢧ⏛႐ߪ࿑ 2.2.5 ߣోߊหߓߣߔࠆ㧚╙ 2 ߩਇᢧ⏛႐ߦࠃߞߡ
parallel ߣߥߞߚ߽ߩࠍ╙ 3 ߩਇᢧ⏛႐ߦㅢߔߩߢ޽ࠆ߇㧘╙ 3 ߩਇᢧ⏛႐ߩᣇะߪ╙1 ߣหߓߊ z ゲߣߔࠆ㧚ታ㛎⚿ᨐߪߤ߁ߥࠆߢ޽ࠈ߁߆㧫╙ 1 ߩਇᢧ⏛႐ߢ parallelߦߥߞߚ߽ߩߩ߁ߜ㧘╙ 2 ߩਇᢧ⏛႐ߦࠃࠅ parallel ߦߥߞߚ߽ߩߩഀวߪ cos 2޽ࠆ㧚หߓ⠨߃ߢ㧘╙ 3 ߩਇᢧ⏛႐ߦࠃߞߡ parallel ߦߥࠆ߽ߩߩഀวߪ cos 2

antiparallel ߦߥࠆ߽ߩߩഀวߪ cos 2T2

u sin 2

T
2

T
2

T
2

u cos 2ߢT
2㧘ߣߥࠆߪߕߢ㧘ታ㛎⚿ᨐ߽⏕߆ߦߘ߁ߥ

ߞߡ޿ࠆ㧚ߣߦ߆ߊ㧘╙ 3 ߩਇᢧ⏛႐㧔 z ゲᣇะ㧕ࠍㅢߒߡಽ߆ߞߚߎߣߪ㧘޽ࠆഀวߩ☸ሶߦߟ޿ߡߪ⏛᳇ࡕ࡯ࡔࡦ࠻ߩ z ᚑಽ߇ z ゲߦᐔⴕ㧘ᱷࠅ߇෻ᐔⴕߣ޿߁ߎߣ
ߢ޽ࠆ㧚ߎࠇߪߤ߁޿߁ߎߣߢ޽ࠈ߁߆㧫ᚒ‫ߪޘ‬㧘╙ 1 ߩਇᢧ⏛႐ߢ⏛᳇ࡕ࡯ࡔࡦ࠻
ߩ z ᚑಽ߇ z ゲߦᐔⴕ㧔ߔߥࠊߜ Pmagz

 PB 㧕ߥ߽ߩߛߌࠍขࠅ಴ߒߡታ㛎ߒߚߪߕߢ޽ࠆ㧚ߘࠇߥߩߦߤ߁ߒߡᦨ⚳⊛ߦ z ゲߦ෻ᐔⴕ㧔ߔߥࠊߜ Pmagz

 PB 㧕ߩ߽ߩ߇޽ࠆߩߢ޽ࠈ߁߆㧫ߎߩታ㛎੐ታߪ㧘‫ޟ‬᷹ቯߦࠃߞߡᕈ⾰߇ᄌࠆ‫߁޿ߣޠ‬㧘ฎౖജቇߩ਎⇇ߢߪାߓࠄ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 17ࠇߥ޿ࠃ߁ߥߎߣ߇޽ࠆߎߣࠍ␜ߒߡ޿ࠆ㧚ฎౖജቇߩ਎⇇ߢߪ㧘᷹ቯߔࠆߎߣߦࠃ
ߞߡᕈ⾰߇ᄌࠆߣ޿߁ߎߣߪ⠨߃ߥߊߡࠃ޿㧚ዋߥߊߣ߽᷹ቯᛛⴚߩㅴᱠߦᓥߞߡߘ
ߩᣇะߦะ߆ߞߡ޿ࠆߎߣࠍᚒ‫޿ߥࠊ⇼ߪޘ‬㧚ߒ߆ߒ㧘㊂ሶജቇߩ਎⇇ߢߪ᷹ቯߔࠆ
ߎߣߦࠃߞߡᕈ⾰߇ᄌߞߡߒ߹߁ߩߢ޽ࠆ㧚෶ߪ㧘᷹ቯߔࠆߎߣߦࠃߞߡᆎ߼ߡᕈ⾰߇ቯ߹ࠆߣ߽⸒޿ᓧࠆ㧚ᧄ⻠⟵ߢᚒ‫߇ޘ‬ീᒝߒࠃ߁ߣߒߡ޿ࠆ㊂ሶജቇ෶ߪ㊂ሶൻቇߣߪ㧘ߎߩࠃ߁ߦਇᕁ
⼏ߥቇ໧ߢ޽ࠆ㧚ฎౖ⺰ߦᘠࠇⷫߒࠎߢ᧪ߚੱ㆐ߦߣߞߡߪᆎ߼ߪ㕙ߊࠄ߁߆߽ߒࠇ
ߥ޿߇㧘ീᒝߔࠇ߫ߣߡߟ߽ߥߊ㕙⊕޿ቇ໧ߢ޽ࠆ㧚ߎߩ㕙⊕޿ቇ໧ߩ਎⇇ࠍ㧘౒ߦᭉߒ߽߁ߢߪߥ޿߆㧚‫ڏ‬Exercise2.2.1
0 d x d a ߩ▸࿐ߢേ޿ߡ޿ࠆ☸ሶߩ wavefunction ߇\ x

Nx a  xߢ⴫ߐࠇࠆߣߔࠆ㧚a

㧔1㧕 ³\ * x \ x dx 10ߣߥࠆࠃ߁ߦ N ࠍ᳿ቯߖࠃ㧚ߎߩࠃ߁ߥᠲ૞ࠍⷙᩰൻ㧔normalization㧕ߣ޿޿㧘 N ࠍⷙᩰൻቯᢙ㧔normalization factor㧕ߣ⸒߁㧚
㧔2㧕਄ߢቯ⟵ߒߚ☸ሶߦߟ޿ߡ㧘ᰴߦ␜ߔ‛ℂ㊂ࠍ᷹ቯߒߚᤨ㧘100 % ߩ⏕߆ࠄߒߐߢ⚿ᨐ߇᳿߹ࠆ߆ุ߆ࠍี๧ߖࠃ㧚᳿߹ࠆ႐วߪ eigenvalue ࠍ㧘᳿߹ࠄߥ޿႐วߪ

average ࠍ᳞߼ࠃ㧚a) x ᐳᮡ㧔operator = x 㧕
w
㧕
wx

b)

ㆇേ㊂㧔operator = i=c)ㆇേࠛࡀ࡞ࠡ࡯㧔operator = =2 w 2㧕
2m wx 2

‫ޤ⸃ޣ‬
㧦
a

(1)

N ³ x 2 a  x dx 1220ߎࠇࠍ⸃޿ߡ

30a5䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 18

N2ߎߩ⸃ߪછᗧߩታᢙ T ࠍ↪޿ߡ30
a5ߣ⴫ߐࠇࠆ㧚ߎߩ phase factor eiT ߦ⴫ࠇࠆ T ߪߤ߁ߣߞߡ߽ㅢᏱߩ႐วߪᏅߒᡰ߃ߥ
޿㧚ࠊߑࠊߑ㕙ୟߥ୯ࠍߣࠆᔅⷐ߽ߥ޿ߩߢ T 0 ߩ႐วࠍߣߞߡ
N

eiT

30
a5
ߣߔࠆߩ߇᥉ㅢߢ޽ࠆ߇㧘 T
N

S ߣߒߡ

30
a5
ߣߒߡ߽ࠃ޿ߒ㧘߹ߚ㧘છᗧߩ T ߢߪ N ߪⶄ⚛ᢙߣߥࠆ߇㧘ߘ߁ߥߞߡ߽޿ߞߎ߁ߦ
N

߆߹ࠊߥ޿㧚
㧔2㧕

a㧕x\ x ߪ\ x ߩቯᢙ୚ߦߪߥࠄߥ޿㧚ᓥߞߡ x ᐳᮡߪ 100%ߩ⏕߆ࠄߒߐߢߪ᳿
߹ࠄߥ޿㧚aaverage value

30 3
2
x a  x dx
5 ³
a 0
a
2

b)

i =

w
\ x
wx

i=

30
2 x  aa5

ฝㄝߪ\ x ߩቯᢙ୚ߦߪߥߞߡ޿ߥ޿㧚ᓥߞߡㆇേ㊂ߪ 100%ߩ⏕߆ࠄߒߐߢߪ᳿߹
ࠄߥ޿㧚Ⓧಽࠍ⸘▚ߔࠆߎߣߦࠃࠅ
average value 0
ࠍᓧࠆ㧚

c)

ห᭽ߦ⸘▚ߔࠆߎߣߦࠃࠅ㧘average value

5h 2
4S 2 ma 2ࠍᓧࠆ㧚‫ڏ‬Problem 2.2.1
0 d x d a ߩ▸࿐ߢേ޿ߡ޿ࠆ☸ሶߩ wavefunction ߇䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 19

\ x

N sin

Sx
a

ߢ⴫ߐࠇࠆߣߔࠆ㧚(1) normalization factor N ࠍ᳞߼ࠃ㧚(2) ᰴߩ‛ℂ㊂ࠍ᷹ቯߒߚᤨ㧘100%ߩ⏕߆ࠄߒߐߢ߽ߞߡ⚿ᨐ߇᳿߹ࠆ߆ุ߆ࠍ
ี๧ߖࠃ㧚᳿߹ࠆ႐วߪ eigenvalue ࠍ㧘᳿߹ࠄߥ޿႐วߪ average ࠍ᳞߼ࠃ㧚a) x ᐳᮡb) ㆇേ㊂c) ㆇേࠛࡀ࡞ࠡ࡯

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 2: Page 20

CHAPTER 03
Schrödinger ᣇ⒟ᑼ

㊂ሶജቇߦ߅ߌࠆ߽ߞߣ߽ၮᧄ⊛ߥᣇ⒟ᑼߪ Schrödinger ᣇ⒟ᑼߢ޽ࠆ㧚Schrödinger
ᣇ⒟ᑼߩ⸃ߪᵄേ㑐ᢙ㧔wavefunction㧕ߣ⸒ࠊࠇࠆ㧚ߎߩ┨ߢߪ㧘Schrödinger ᣇ⒟ᑼ
ߩਛߢ㧘ᤨ㑆ࠍᄌᢙߣߒߡ฽߹ߥ޿߽ߩߦߟ޿ߡ⚫੺ߔࠆ㧚ߎߩ㧘ᤨ㑆ߦଐሽߒߥ޿
Schrödinger ᣇ⒟ᑼ㧔time independent Schrödinger equation㧕ߩ⸃ߪቯᏱ⁁ᘒߩᵄേ㑐ᢙ
㧔stationary state wavefunction㧕ߣ⸒ࠊࠇࠆ㧚

3.1 ৻ᰴరߩᵄߩᣇ⒟ᑼ
㊂ሶജቇߩ໧㗴ߦ౉ࠆ೨ߦ㧘ฎౖജቇߦ߅ߌࠆᵄߩᣇ⒟ᑼࠍᓳ⠌ߒߡ߅ߎ߁㧚࿑
3.1.1 ߦ␜ߔࠃ߁ߦ㧘2 ߟߩቯὐ㑆ߦ᳓ᐔߦᒛࠄࠇߚ♻ߩㆇേࠍ⠨߃ߡߺࠃ߁㧚࿑ 3.1.1ਔ┵ࠍ࿕ቯߒߚ♻ߩㆇേ♻ߩ᳓ᐔ૏⟎߆ࠄߩᄌ૏㧘ߔߥࠊߜᝄ᏷ࠍ U x, t ߣߔࠆ㧚ߎߎߢ x ߪ᳓ᐔᣇะߩᐳᮡ
ߢ t ߪᤨ㑆ߢ޽ࠆ㧚ฎౖജቇߩ␜ߔߣ߅ࠅ U x, t ߪᰴߩࠃ߁ߥᵄേᣇ⒟ᑼߢ⴫ߐࠇࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 3: Page 1

w 2U x, t
wx 2

1 w U x, t
v2
wt 2
2

ߎߎߢ v ߪᵄߩㅦᐲߢ޽ࠆ㧚 U x, t

U 0, t0

U L, t0(3.1.1)ߪᰴߩࠃ߁ߥႺ⇇᧦ઙࠍḩߚߔ㧚

(3.1.2)ᑼ(3.1.1)ߪ㧘ᄌᢙಽ㔌ᴺߦࠃߞߡ⸃ߊߎߣ߇಴᧪ࠆ㧚ߔߥࠊߜ㧘U x, tߪ㧘 x ߛߌߩ

㑐ᢙ X x ߣ t ߛߌߩ㑐ᢙ T t ߣߩⓍߢ⴫ߐࠇࠆߣ઒ቯߔࠆ㧚ߔߥࠊߜU x, tX x T t(3.1.3)ᑼ(3.1.3)ࠍᑼ(3.1.1)ߦઍ౉ߔࠆߎߣߦࠃࠅᰴᑼࠍᓧࠆ㧚
1

d2X x

X x

dx 2

2
1 d T tv 2T t dt 2

(3.1.4)

Ꮐㄝߪ x ߩߺߩ㑐ᢙ㧘ฝㄝߪ t ߩߺߩ㑐ᢙߢ޽ࠆ㧚 x ߣ t ߣߪ߅੕޿ߦ⁛┙ߥᄌᢙߢ޽
ࠆ߆ࠄ㧘ฝㄝߣᏀㄝߣߪ⁛┙ߦᄌࠅᓧࠆ㧚ߘࠇࠄ߇╬ߒ޿ߣ޿߁ߎߣߪฝㄝ߽Ꮐㄝ߽
౒ߦ޽ࠆቯᢙߦ╬ߒ޿ߣ޿߁ߎߣࠍ␜ߒߡ޿ࠆ㧚ߎߩቯᢙࠍ઒ߦ k ߣ߅ߎ߁㧚
1

d2X x

X x

dx 2

2
1 d T tv 2T t dt 2

k(3.1.5)k(3.1.6)

ߎࠇࠄߩᑼ߆ࠄᰴߩࠃ߁ߥᓸಽᣇ⒟ᑼ߇ᓧࠄࠇࠆ㧚d2X x
 kX xdx 2

0(3.1.7)

d 2T t kv 2T t 0(3.1.8)dt 2
ߎࠇࠄߪ޿ߕࠇ߽৻ᄌᢙߩᓸಽᣇ⒟ᑼߢ޽ࠆ߆ࠄኈᤃߦ⸃ߌࠆ㧚߽ߞߣ߽㧘 k ߩ୯ߪ߽ߣࠃࠅ㧘ߘߩ╓ภߐ߃⊝⋡⸃ࠄߥ޿ࠊߌߢ޽ࠆ㧚ߘࠇߢ߹ߕ
k ! 0 ߣ⠨߃ߡี๧ߒߡߺࠃ߁㧚ߎߩ႐วታᢙ㧔real㧕ߩቯᢙ E ࠍ↪޿ k E 2 ߣ⴫ߐࠇࠆ㧚ᑼ(3.1.7)ߩ⸃ߪ
X x

c1e E x  c2 e  E x䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 3: Page 2(3.1.9)ߎߎߢႺ⇇᧦ઙᑼ(3.1.2)ࠍ⠨߃ࠆ㧚 T t ߇࠯ࡠߦߥࠆߎߣߪߥ޿߆ࠄ

X 0

0

X L

0

(3.1.10)ߣ߅ߌࠆ㧚ᑼ(3.1.9)߇ߎߩ᧦ઙࠍḩ⿷ߔࠆߚ߼ߦߪ c1

0 ߢߥߌࠇ߫ߥࠄߥ޿߇㧘c2

ߘࠇߪ X x ߇ x ߩ޿ߚࠆᚲߢ࠯ࡠߣ޿߁ߎߣ㧘ߔߥࠊߜ㧘 U x, t ߇ x ߩ޿ߚࠆᚲߢ
࠯ࡠߣ޿߁ࠊߌߢ㧘ᗧ๧ߩߥ޿⸃ߢ޽ࠆ㧚ߎ߁޿߁⸃ߩߎߣࠍ⧷⺆ߢ trivial solutionߣ޿߁⸒⪲ߢ⴫ߔ㧚ᰴߦ k0 ߩ႐วࠍี๧ߒߡߺࠃ߁㧚ᑼ(3.1.7)ߩ⸃ߪ

X x

c1 x  c2(3.1.11)ߢ޽ࠆ߇㧘Ⴚ⇇᧦ઙᑼ(3.1.10)ࠍ⠨ᘦߔࠆߣ߿ߪࠅ c1

c2

0 ߣ޿߁ trivial solution ߒ߆ᓧࠄࠇߥ޿㧚
ᦨᓟߦ k  0 ߣ⠨߃ߡี๧ߒߡߺࠃ߁㧚ౣ߮ real ߩቯᢙ E ࠍ↪޿ߡ k

 E 2 ߣ߅ߎ߁㧚ᑼ(3.1.5)ߩ⸃ߪ
X x

c1eiE x  c2 e  iE x

(3.1.12)

A cos E x  B sin E xߣߥࠆ㧚 X x

0 ߣ޿߁Ⴚ⇇᧦ઙ߆ࠄ A 0 ࠍᓧࠆ㧚 X L

0 ߣ޿߁᧦ઙ߆ࠄߪ

B sin E L 0ࠍᓧࠆ㧚߽ߒ B(3.1.13)0 ߢ޽ࠇ߫ X xࠇߥ޿㧚ᓥߞߡ sin E L

0 ߣߥߞߡߒ߹޿㧘߿ߪࠅ trivial solution ߒ߆ᓧࠄ0 ߢߥߌࠇ߫ߥࠄߥ޿㧚ߔߥࠊߜE L nSn 1, 2, 3, ...

(3.1.14)

㧔Problem 3.1.1 ߅ࠃ߮ Problem 3.1.2 ࠍෳᾖ㧕㧚߆ߊߒߡ㧘⸃ X x ߪ

X x

B sin

nS x
L

n 1, 2, 3, ...

(3.1.15)ߩࠃ߁ߦ᳞߹ࠆ㧚หߓࠃ߁ߦߒߡ T t ࠍ᳞߼ߡߺࠃ߁㧚ᑼ(3.1.8)ߪ

d 2T t E 2 v 2T t
2dt0䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 3: Page 3(3.1.16)ߎߩ⸃ߪ
D cos Znt  E sin Znt

T tߎߎߢ ZnEv

nS v
ߢ޽ࠆ㧚T tL

(3.1.17)ߦ㑐ߔࠆႺ⇇᧦ઙߪߥ޿߆ࠄ㧘 D ߅ࠃ߮ E ࠍߎࠇએ਄᳿߼ࠆߎߣߪ಴᧪ߥ޿㧚ᓥߞߡ U x, t ߪᰴߩࠃ߁ߦߥࠆ㧚U x, tX x T tnS x ·§¨ B sin¸ D cos Znt  E sin Znt
L ¹©
nS x ·§¨ sin¸ F cos Zn t  G sin Zn t
L ¹©

(n 1, 2, 3, ...)

(3.1.18)

F cos Zn t  G sin Znt ߪ A cos Z t  I ߣ⴫ߐࠇ㧘෶ 1 ߟ 1 ߟߩ n ߦߟ޿ߡ A ߿ I ߩ୯߽ᄌࠅᓧࠆߩߢߘࠇࠄߩᄌᢙߩ subscript ߣߒߡ n ࠍߟߌ㧘U n x, tAn sinnS x
cos Zn t  In
L

(3.1.19)

߇ 1 ߟߩ⸃ߣߥࠆ㧚⚿ዪ㧘৻⥸⸃ߣߒߡߪ

U x, t¦ U x, t(3.1.20)

n

n

ࠍᓧࠆ㧚
‫ڏ‬Problem 3.1.1
ᑼ(3.1.14)߅ࠃ߮ᑼ(3.1.15)ߢߪ n

0 ࠍ㒰޿ߡ޽ࠆ㧚૗᡿߆㧚‫ڏ‬Problem 3.1.2
ᑼ(3.1.14)߅ࠃ߮ᑼ(3.1.15)ߢߪ n ߩ⽶ߩ୯ࠍ㒰޿ߡ޽ࠆ㧚૗᡿߆㧚

3.2 ᤨ㑆ߦଐሽߒߥ޿ Schrödinger ᣇ⒟ᑼ

Schrödinger equation ߪ㊂ሶജቇߦ߅޿ߡ߽ߞߣ߽ၮᧄ⊛ߥᣇ⒟ᑼߢ޽ࠆ߇㧘ߘࠇࠍ޽ࠆၮᧄ⊛ߥේℂ߆ࠄዉ಴ߔࠆߣ޿߁ߎߣߪ಴᧪ߥ޿㧚ߘࠇߪ Newton ߩ╙ 2 ᴺೣ㧔⾰
㊂×ടㅦᐲ=ജ㧕ࠍዉ಴ߔࠆߎߣ߇಴᧪ߥ޿ߩߣૃߡ޿ࠆ㧚ߘ߁޿߁ࠊߌߢ Schrödinger
equation ߪ㧘㊂ሶജቇߦ߅ߌࠆၮᧄ⊛ߥ઒ቯߣߺߥߐࠇࠆߴ߈߽ߩߢ޽ࠆ㧚ߎߩࠃ߁ߦ㧘Schrödinger ᣇ⒟ᑼߪዉ಴ߔࠆߎߣߪ಴᧪ߥ޿߇㧘ߎߩ઒ቯߪ߽ߞߣ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 3: Page 4߽ࠄߒ޿ߥ㧘ߣߥࠎߣߥߊᗵߓࠆߎߣ߇಴᧪ࠇ߫ᄙዋߩℂ⸃ߦᓎ┙ߟ߆߽ߒࠇߥ޿㧚ߘࠇߢ㧘ฎౖജቇߦ߅ߌࠆ 1 ᰴరߩᵄߩᣇ⒟ᑼ(3.1.1)ࠍ↪޿ߡ⠨߃ߡߺࠃ߁㧚ߚߛ㧘ߎߎߢߪ U x, t ߩઍࠅߦࠡ࡝ࠪࡖᢥሼߩ <㧔ࡊࠨࠗߩᄢᢥሼ㧕ࠍ↪޿ࠆߎߣߦߔࠆ㧚w 2 < x, t
wx 2

1 w < x, t
v2
wt 2
2

(3.2.1)਄ㅀߒߚࠃ߁ߦߎߩᑼߪᄌᢙಽ㔌ߦࠃߞߡ⸃ߊߎߣ߇಴᧪ࠆ㧚ߎߩ႐ว < x, t ࠍ x ߛߌߩ㑐ᢙߣ t ߦ㑐ߔࠆනᝄേߩᑼߣߩⓍߢ⴫ߔߎߣ߇಴᧪ࠆ㧚 x ߩ㑐ᢙߪ೨▵ߢߪX x ߣ߅޿ߚ߇㧘ߎߎߢߪࠡ࡝ࠪࡖᢥሼߩ\ 㧔ࡊࠨࠗߩዊᢥሼ㧕ࠍ↪޿ߡ\ x ߣ߅ߎ߁㧚ߔߥࠊߜ
< x, t\ x eiZt(3.2.2)ᑼ(3.2.2)ࠍᑼ(3.2.1)ߦઍ౉ߔࠆߣ

w 2\ x Z 2(3.2.3)
 2\ x 0
wx 2v
ࠍᓧࠆ㧚ߎߎߢ de Broglie ߩ‛⾰ᵄߩ⠨߃ࠍዉ౉ߔࠆ㧚☸ሶߩోࠛࡀ࡞ࠡ࡯ߪ㧘kinetic
energyp2
ߣ potential energy U x ߣߩ๺ߢ⴫ߐࠇࠆ㧚
2m

p2
U x
2m
ᑼ(3.2.4)ࠍ p ߦߟ޿ߡ⸃ߊߣ
E

2m ^ E  U x `p(3.2.4)(3.2.5)de Broglie ߩᑼߦࠃࠆߣ㧘ㆇേ㊂ p ࠍᜬߟ☸ሶߩᵄ㐳 O ߪ

O

h
p

h
2m ^ E  U x `ߢ⴫ߐࠇࠆ㧚ߎߎߢ㧘ⷺᝄേᢙߣᝄേᢙߩ㑐ଥᑼ
Z 2SQ߅ࠃ߮ㅦᐲ v ߣᝄേᢙQ ߣߩ㑐ଥᑼv QOࠍ↪޿ࠆߣᑼ(3.2.3)ߩ

Z2v2ߪᰴߩࠃ߁ߦ⴫ߐࠇࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 3: Page 5(3.2.6)

Z2

4S 2

v2

O2

2m ^ E  U x `(3.2.7)

=2

ߎߎߢ

h
2S

=

ߣ߅޿ߚ㧚
ᑼ(3.2.7)ࠍᑼ(3.2.3)ߦઍ౉ߔࠆߣᰴᑼࠍᓧࠆ㧚=2 d \ x
U x \ x
2m dx 22

E\ x

(3.2.8)‫ޣ‬ᵈᗧ‫ޤ‬㧦ᑼ(3.2.1)߆ࠄᑼ(3.2.7)߹ߢߩ㑆ߢ㧘ㅦᐲ‫ ޟ‬v ‫ߣޠ‬ᝄേᢙ‫ޟ‬Q ‫ߩޠ‬ሼ૕߇ૃߡ޿ࠆߩߢ
㑆㆑ࠊߥ޿ࠃ߁ߦᵈᗧߒߡ߶ߒ޿㧚ߥ߅㧘ߎߩߎߣߦߟ޿ߡߪ㧘Chapter 01 ߩ࿑ 1.5.1
߇⸶ߦ┙ߟ߆߽ߒࠇߥ޿㧚ߎࠇ߇৻ᰴరߩ♽ߦߟ޿ߡߩ time independent Schrödinger equation ߢ޽ࠅ㧘⾰㊂ m ߩ☸ሶ߇ potential U x ߩ߽ߣߢㆇേߔࠆ᦭᭽ࠍ⴫ߔᑼߢ޽ࠆ㧚ߎߩᲑ㓏ߢߪ\ x ߩᗧ๧ߪ޽޿߹޿ߢߪ޽ࠆ߇㧘޽ࠆᗧ๧ߢߪ㧘ᑼ(3.2.2)ߩ␜ߔࠃ߁ߦ㧘‛⾰ᵄߩᝄ᏷ࠍ␜
ߒߡ޿ࠆࠊߌߢ㧘\ x ߪᵄേ㑐ᢙ㧔wavefunction㧕ߣ๭߫ࠇࠆ㧚ߔߢߦㅀߴߚࠃ߁ߦ㧘
ᑼ(3.2.8)ߪ time independent ߥᑼߢ޽ࠅ㧘ߘߩ⸃\ x ߪ stationary state wavefunction ߣ⸒ࠊࠇࠆ㧚ᤨ㑆ߦଐሽߒߚ Schrödinger equation ߦߟ޿ߡߪߒ߫ࠄߊᓟߢቇ⠌ߒࠃ߁㧚
Schrödinger equation (3.2.8)ߪ㧘ߒ߫ߒ߫ᰴߩࠃ߁ߥ࿕᦭୯ᣇ⒟ᑼߣߒߡ⴫⃻ߐࠇࠆ㧚
H\ x

E\ x

(3.2.9)=2 d 2U x
2m dx 2(3.2.10)

ߚߛߒ

H



ߔߥࠊߜṶ▚ሶ H ߩ࿕᦭㑐ᢙ߇\ x ߢ޽ࠅ㧘ߘߩ࿕᦭୯߇ E ߢ޽ࠆ㧚㧔ߎߩ࿕᦭୯ᣇ⒟ᑼߪߔߢߦᑼ(2.1.1)ߢ߅ߥߓߺߩ߽ߩߢ޽ࠆ㧚㧕 H ߪ㧘࿕᦭୯ߣߒߡࠛࡀ࡞ࠡ࡯ࠍ
ਈ߃ࠆṶ▚ሶߢ㧘Hamiltonian operator ߣ๭߫ࠇࠆ㧚ߎߎߢߪ৻ᰴరߩᐳᮡߒ߆⠨߃ߥ߆ߞߚ߇㧘ਃᰴరߩ႐ว߽ోߊห᭽ߢ޽ࠆ㧚ߔߥࠊߜ㧘䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 3: Page 6

H\ x, y, zHE\ x, y, z

(3.2.11)=2 § w 2
w2
w2 ·


¨
¸  U x, y , z
2m © wx 2 wy 2 wz 2 ¹(3.2.12)=2 2
  U x, y , z2mߎߩ┨ߩ౨㗡ߦㅀߴߚࠃ߁ߦ㧘 Schrödinger equation ߪℂ⺰⊛ߦዉ಴ߐࠇᓧࠆ߽ߩ
ߢߪߥ޿ߎߣࠍౣᐲᒝ⺞ߒߡ߅߈ߚ޿㧚਄ߢㅀߴߚߎߣߪ㧘ฎౖ⊛ߥᵄേᣇ⒟ᑼߦ de
Broglie ߩ‛⾰ᵄߩ⠨߃ࠍขࠅ౉ࠇࠆߎߣߦࠃߞߡ㧘 Schrödinger equation ߇ߥࠎߣߥ
ߊ߽ߞߣ߽ࠄߒߊᕁ߃ࠆߎߣࠍᗧ࿑ߒߚߛߌߩ߽ߩߢ޽ࠆߎߣࠍᢿߞߡ߅߈ߚ޿㧚

Schrödinger equation ߪ㧘Schrödinger ߦࠃߞߡ⸒޿಴ߐࠇߚᤨߪනߥࠆ઒⺑ߢߒ߆ߥ߆
ߞߚ߇㧘⃻࿷ߢߪ߽ߞߣ߽ၮᧄ⊛ߥᑼߣߒߡ⹺߼ࠄࠇߡ޿ࠆ㧚[Coffee Break] Hückel ߇⹗ࠎߛ Schrödinger ߦߟ޿ߡߩ⹞
Erwin Schrödinger ߪᓐߩឭ಴ߒߚ઒⺑ߦኻߒߡߤߩ⒟ᐲ⏕ାࠍᜬߞߡ޿ߚߩߢ޽
ࠈ߁߆㧫Physics Today ߩ 1976 ᐕ 12 ᦬ภߦ Bloch equation ߢ᦭ฬߥ Bloch ߇"Heisenberg
and the early days of quantum mechanics"ߣ޿߁⹤ࠍタߖߡ޿ࠆ߇㧘ߘߩਛߦ޽ࠆᝌ⹤߇
㕙⊕޿㧚޽ߩ Hückel ߇㧘ߘߩᒰᤨߪ Debye-Hückel ℂ⺰ߦขࠅ⚵ࠎߢ޿ߡ߹ߛ Hückel
ߩಽሶ゠㆏߹ߢߪ⎇ⓥߒߡ޿ߥ߆ߞߚ㗃㧘Schrödinger ࠍߜ߾߆ߒߡญߕߐࠎߛߣ޿߁
⹞߇ᒁ↪ߐࠇߡ޿ࠆ㧚
Gar Manches rechnet Erwin schon
Mit seiner Wellenfunktion
Nur wissen möchte man gerne wohl
Was man sich dabei vorstell'n soll.
ߎࠇࠍ⷗ࠆߣ㧘Schrödinger ⥄り㧘ᓐߩ wavefunction ߩᗧ๧ߔࠆߎߣ߇⸃ࠄߥ޿ߎߣߢ
ᖠࠎߢ޿ߚࠄߒ޿ߣ޿߁ߎߣ߇ᗐ௝ߟߊ㧚Bloch ߪ਄ߦᒁ↪ߒߚ Hückel ߩ࠼ࠗ࠷⺆ߩ⹞ߩ⧷⸶ࠍ⹜ߺߡ޿ࠆ߇㧘ᣣᧄ⺆⸶ߪ⻉ำߦ⹜ߺߡ߽ࠄ߅߁㧚޿ߕࠇߦߒߡ߽㧘ߎߩࠃ߁ߥ⨥⋡᳇ߢ Schrödinger ࠍߜ߾߆ߒߚ Hückel ߇㧘ᓟߢ

Schrödinger equation ࠍ↪޿ߡ Hückel molecular orbital ߥࠆ߽ߩࠍᒻᚑߒߚߩߪ㕙⊕޿㧚㔚⸃⾰ṁᶧߩℂ⺰ࠍ߿ߞߡ޿ߚᒰᤨߪ⥄ಽ߇ߘ߁޿߁ߎߣࠍⴕ߁ߥߤߣᕁߞߡ߽޿
ߥ߆ߞߚߢ޽ࠈ߁߇㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 3: Page 7

CHAPTER 04㊂ሶജቇߦ߅ߌࠆၮᧄ⊛ߥේℂߣ઒ቯߎߎ߹ߢߩቇ⠌ߢ㧘⻉ำߪ㧘㊂ሶജቇߩቇ⠌ߦߪ੹߹ߢߣߪ⇣ߥߞߚ᭎ᔨ߇ᔅⷐߢ
޽ࠆߎߣࠍ૗ߣߥߊಽ߆ߞߡ᧪ߚߢ޽ࠈ߁㧚ߎߩ┨ߢߪ㧘㊂ሶജቇߦ߅ߌࠆၮᧄ⊛ߥ᭎ᔨࠍ৻⚵ߩ઒ቯߣߒߡዉ౉ߒ㧘ߘߩ઒ቯ߆ࠄዉ߆ࠇࠆ޿ߊߟ߆ߩቯℂߦߟ޿ߡ⠨߃ߡߺࠃ߁㧚ߘߩขࠅᛒ޿ߪ㧘ᐞ૗ቇߦ߅޿ߡ޿ߊߟ߆ߩ౏ℂࠍዉ౉ߒ㧘ߘߩ౏ℂ߆ࠄዉ߆ࠇࠆߎߣࠍ⺰ℂ⊛ߦ⠨߃ࠆᣇᴺߣૃߡ޿ࠆ㧚ᆎ߼ߦዉ౉ߒߚ઒ቯ㧘෶ߪ౏ℂ߇ᱜߒ޿߆ߤ߁߆ߪᦨ⚳⊛ߦߪታ㛎ߣᲧߴߡߺߡ⍦⋫߇޽ࠆ߆ߤ߁߆ߣ޿߁ߎߣࠍี๧
ߒߡᆎ߼ߡಽ߆ࠆߎߣߢ޽ࠆ㧚ߎߎߢߪታ㛎ߩߎߣ߹ߢࠍㅀߴࠆߎߣߪ಴᧪ߥ޿߇㧘ߎߎߢዉ౉ߔࠆ઒ቯߪ㧘㊂ሶജቇ߇⺀↢ߒߡ߆ࠄඨ਎♿ߦࠊߚࠆ⎇ⓥߢ㧘૗ߩ⍦⋫߽⷗಴ߐࠇߡ޿ߥ޿ߎߣߛߌࠍㅀߴߡ߅ߎ߁㧚એਅᧄ┨ߢቇ⠌ߔࠆߎߣߪ㧘ߔߢߦ೨┨߹ߢߦ᭎ⷐࠍㅀߴߚߎߣߢ޽ࠆ㧚ᧄ┨ߢߪ㧘ߘࠇࠄࠍ߽߁৻ᐲ㧘ߚߛߒ㧘೨ࠃࠅߪ෩ኒߦขࠅᛒ߁ߎߣࠍ⹜ߺࠆ㧚

4.1 ᵄേ㑐ᢙฎౖജቇߩ਎⇇ߢߪ☸ሶߩㆇേߪ Newton ߩᣇ⒟ᑼߢ⸥ㅀߐࠇࠆࠊߌߢ㧘޽ࠆᤨೞߦ߅ߌࠆ☸ሶߩᐳᮡߪ㧘ߘࠇࠃࠅ߽೨ߩᤨೞߩᐳᮡߣㅦᐲ߇ಽ߆ࠇ߫ቢోߦ⸥ㅀ಴᧪
ࠆ㧚ߒ߆ߒ㧘㊂ሶജቇߢᛒ߁㔚ሶ㧘ේሶ㧘ಽሶߥߤߩࠃ߁ߥዊߐߥ☸ሶߢߪ㧘ߘ߁޿߁੐߇ᚑࠅ┙ߚߥߊߥࠆߎߣࠍ㧘Chapter 2 ߢੑਃߩታ଀ߢ߽ߞߡ⷗ߡ᧪ߚ㧚ߎߎߢᰴߦㅀߴࠆ઒ቯ߇ዉ౉ߐࠇࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 1

઒ቯ 1㧦
㊂ሶജቇߢᛒ߁☸ሶߩ⁁ᘒߪ㧘ᐳᮡ x, y, z ߅ࠃ߮ᤨ㑆 t ߩ㑐ᢙߢ޽ࠆߣߎࠈߩ‫ޟ‬ᵄേ㑐ᢙ‫ \ޠ‬x, y, z , t ߦࠃߞߡቢోߦ⸥ㅀߐࠇࠆ㧚ߎߩ☸ሶ߇૏⟎ x, y, z 㧘ᤨೞ t ߦ߅޿ߡᓸዊ૕Ⓧ dxdydz ਛߦ⷗಴ߐࠇࠆ⏕₸ߪ\ x, y, z , t *\ x, y, z , t dxdydz ߢਈ߃ࠄࠇ
ࠆ㧚ߎߎߢ\ x, y, z , t * ߪ\ x, y, z , t 㧔৻⥸ߦⶄ⚛ᢙ㧕ߩⶄ⚛౒ᓎࠍ␜ߔ㧚ᵄേ㑐ᢙߩੑਸ਼߇⏕₸ࠍ⴫ߔߣ޿߁ߎߣ߆ࠄ㧘ᵄേ㑐ᢙߦ޿ߊߟ߆ߩ೙㒢߇ߟߌࠄࠇࠆ㧚߹ߕ⠨߃ߡ޿ࠆోⓨ㑆ߩߤߎ߆ߦߘߩ☸ሶࠍ⷗಴ߔ⏕₸ߪ 1 ߢ޽ࠆ߆ࠄ㧘\ x, y, z , t *\ x, y, z , tࠍⓨ㑆ᐳᮡߩᄌᢙߩోၞߢⓍಽߒߚ୯ߪ 1 ߢߥߌࠇ߫ߥࠄߥ

޿㧚ߔߥࠊߜ㧘³\ x, y, z, t *\ x, y, z, t dW 1

(4.1.1)ߎߎߢdW

dxdydzߪᓸዊ૕Ⓧߢ޽ࠅ㧘Ⓧಽߪ\ x, y, z , t ߇ขࠅᓧࠆᄌၞో૕ߦߟ޿ߡⴕ߁㧚ߔߥࠊߜ㧘․ߦᢿࠄߥ޿㒢ࠅ\ x, y, z , t ߪ x, y, z ߩోၞߢቯ⟵ߐࠇߡ޿ࠆߣ⠨߃ߡࠃߊ㧘

³ dW

f

f

f

f

f

f

³ dx ³ dy ³ dz

(4.1.2a)ߩߎߣߢ޽ࠆ㧚หߓⓨ㑆㗔ၞߢ㧘߽ߒᐳᮡ߇ᭂᐳᮡߩ r , T , I ߢ⴫ߐࠇߡ޿ࠆߣߔࠆߣ

dW

r 2 sin T drdT dI

ߢ޽ࠅ
f

S

2S

0

0

³ dW ³ r dr ³ sin T dT ³ dI
20

(4.1.2b)ߣߥࠆ㧚ᑼ(4.1.1)ߩࠃ߁ߦ㧘ੑਸ਼ߒߡⓍಽߔࠆߣ 1 ߦߥࠆࠃ߁ߥ wavefunction ࠍ‫ⷙޟ‬ᩰൻߐࠇߡ޿ࠆ㧔normalized㧕‫ߔ⴫ߢ⪲⸒߁޿ߣޠ‬㧚߽ߒ߽㧘޽ࠆ wavefunction \ ߩੑਸ਼ࠍⓍ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 2ಽߒߚߣߎࠈ 1 ߢߪߥ޿߇޽ࠆ᦭㒢ߩ୯ߣߥߞߚߣߔࠆ㧚ߘߩᤨߪ޽ࠆቯᢙ N ࠍដߌߡ㧘 N\ ߣߒߡⷙᩰൻߔࠆߎߣ߇಴᧪ࠆ㧚ߔߥࠊߜ³ N\ * N\ dW 1

(4.1.3)

1N2(4.1.4)ࠃࠅ

³\ *\ dWߎߩࠃ߁ߦߒߡ᳿߼ߚ N ࠍⷙᩰൻ࿃ሶ㧔normalization factor㧕ߣ⸒߁㧚ߣߦ߆ߊᄢಾߥߎߣߪ㧘wavefunction ߪ㧘ߘߩੑਸ਼ࠍⓍಽߒߚᤨߦ᦭㒢ߩ୯ߦߥࠄ
ߥߌࠇ߫޿ߌߥ޿ߣ޿߁ߎߣߢ޽ࠆ㧚ߎߩࠃ߁ߥⓍಽ߇น⢻ߢ޽ࠆߚ߼ߦߪ㑐ᢙ߇৻ଔߢ߆ߟㅪ⛯ߢߥߌࠇ߫ߥࠄߥ޿㧚ߎߩ 3 ߟߩ᧦ઙ㧘ߔߥࠊߜ‫৻ޟ‬ଔ‫ޠ‬㧘‫ޟ‬ㅪ⛯‫ޠ‬㧘‫ޟ‬Ⓧ
ಽน‫᧦߁޿ߣޠ‬ઙ߇ឥߞߚ㑐ᢙࠍ㧘Eyring, Walter, Kimball ߩ"Quantum Chemistry"㧔p.26㧕
ߢߪ‫ޟ‬class Q ߦዻߔࠆ‫ࠆ޿ߡߒ⴫ߢ⪲⸒߁޿ߣޠ‬㧚ߒ߆ߒߎߩ⸒⪲ߪ޽߹ࠅ popularߢߪߥ޿ࠃ߁ߢ޽ࠆ㧚৻⥸ߦߪ"well behaved"ߣ޿߁ᄙዋ޽޿߹޿ߐࠍᜬߞߚ⸒⪲ߢ⴫⃻ߒߡ޿ࠆ㧚

‫ޤ⿷⵬ޣ‬
㧦\ ߇ Schrödinger ᣇ⒟ᑼࠍḩߚߔߣ޿߁ߎߣࠍ⠨߃ࠆߣ㧘\ ߪ 2 㓏ᓸಽ߇น⢻ߣ޿߁᧦ઙ߽ᔅⷐߣߥߞߡ᧪ࠆ㧚ߒ߫ߒ߫㧘ᢎ⑼ᦠߢ\ ߩ 1 㓏ᓸಽ߇ㅪ⛯㧘ߣ޿ߞߚ᧦ઙ
߇޽ߍࠄࠇߡ޿ࠆ߇ߎߩߎߣࠍ⸒޿ߚ޿ߚ߼ߢ޽ࠈ߁㧚‫ޟ‬1 㓏ᓸಽ߇ㅪ⛯ߢ޽ࠇ߫ 2㓏ᓸಽ߇น⢻ߢ޽ࠆ‫ߊߥߣࠎߥߣޠ‬⠨߃ߡ޿ࠆߩߢ޽ࠈ߁㧚ߒ߆ߒߎߩ⠨߃ᣇߪᱜ⏕ߢߪߥ޿㧚ᢙቇߩᢎ⑼ᦠࠍ⷗ࠆߣ㧘‫ࠆ޽ޟ‬㑐ᢙ߇ᓸಽน⢻ߢ޽ࠇ߫ㅪ⛯ߢ޽ࠆ߇㧘ㅪ⛯ߢ޽ߞߡ߽ᓸಽน⢻ߣߪ㒢ࠄߥ޿‫߇ߣߎ߁޿ߣޠ‬ㅀߴࠄࠇߡ޿ࠆ㧚ᢙቇߩ਎⇇ߢߪߘ߁޿߁ߎߣߦߥࠈ߁߇㧘‛ℂ߿ൻቇߩ਎⇇ߢߪ㧘ߘࠎߥߦ߅߆ߒߥ㑐ᢙߪ಴ߡ᧪ߥ
޿ߣ޿߁ߩߢ޽ࠈ߁߆㧘ߒ߫ߒ߫ᓸಽน⢻ߣ޿߁ߎߣࠍㅪ⛯ߣ޿߁ߎߣߢ⟎߈឵߃ߡߒ߹ߞߡ޿ࠆ㧚ߘߒߡᦝߦ㧘\ ߇ㅪ⛯ߢ޽ࠇ߫ 1 㓏ߩᓸಽ߽ㅪ⛯ߢ޽ࠆߣ޿߁߰߁ߦ
ᄢ㔀ᛠߦ⠨߃ߡ޿ࠆࠃ߁ߢ޽ࠆ㧚ᢙቇ⊛ߦߪ߅߆ߒ޿ᚲ߇ߚߊߐࠎ޽ࠆ߇㧘ታ⾰⊛ߦߪߎߩ૏ߩ޽޿߹޿ߐߢℂ⸃ߒߡ߅޿ߡᏅߒᡰ߃ߥ޿ߣ޿߁ߎߣߩࠃ߁ߛ㧚well
behaved ߣ޿߁޽޿߹޿ߥ⸒⪲߽㧘ߎߩ⒟ᐲߩ޽޿߹޿ߥ᭎ᔨࠍ⴫ߔ⸒⪲ߣߒߡߪㆡ
ᒰߥߩ߆߽ߒࠇߥ޿㧚‫ڏ‬Exercise 4.1.1
ᰴߩฦ㑐ᢙߪࠞ࠶ࠦ㨇(1) e  x㨉ౝߦ␜ߐࠇߚᄌၞߢ well behaved ߢ޽ࠆ߆ุ߆㧚> 0, f @䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 3㨇⸃㨉৻ଔ㧘ㅪ⛯㧚෶ e2 x ߪⓍಽน㧚ᓥߞߡ well behaved ߢ޽ࠆ㧚> f, f @(2) e  x㨇⸃㨉 e2 x ߪ x o f ߢ⊒ᢔߒߡߒ߹ߞߡⓍಽਇน⢻㧚ᓥߞߡ well behaved ߢߪߥ޿㧚(3) e  x

> f, f @2㨇⸃㨉৻ଔ㧘ㅪ⛯㧘߆ߟⓍಽน㧚ᓥߞߡ well behaved ߢ޽ࠆ㧚> 1, 1@(4) sin 1 x㨇⸃㨉ᄙଔ㑐ᢙߢ޽ࠆߩߢ well behaved ߢߪߥ޿㧚‫ڏ‬Problem 4.1.1
ᰴߩฦ㑐ᢙߪࠞ࠶ࠦౝߦ␜ߐࠇߚᄌၞߢ well behaved ߆ุ߆㧚(1) eix> 0, 2S @(2) xe  x> 0, f @(3)

1
x> 0, f @>0, f @

(4) sinh x(5) e 2 x sinh x> 0, f @‫ڏ‬Problem 4.1.2
ᰴߩฦ㑐ᢙߪࠞ࠶ࠦౝߦ␜ߐࠇߚ㗔ၞߢቯ⟵ߐࠇߡ޿ࠆ㧚ⷙᩰൻߖࠃ

(1) sinnS x
a>0, a @ ,§ x2 ·(2) exp ¨  2 ¸© 2a ¹(3) xe  x

n 1, 2, ... 㧚> f, f @> 0, f @䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 4

4.2 ㊂ሶജቇ⊛Ṷ▚ሶ઒ቯ 2㧦޽ࠄࠁࠆ᷹ቯน⢻ߥ‛ℂ㊂ߦኻߒߡ✢ᒻ㧔linear㧕߆ߟࠛ࡞ࡒ࡯࠻㧔Hermitian㧕ߥṶ▚ሶ߇ኻᔕߠߌࠄࠇࠆ㧚‛ℂ㊂ߣṶ▚ሶ㧔operator㧕ߣߩኻᔕࠍ㧘ᐳᮡ x, y, z ߣㆇേ㊂ px , p y , pz ߦߟ޿ߡ␜ߔ
ߣ⴫ 4.2.1 ߅ࠃ߮⴫ 4.2.2 ߩࠃ߁ߦߥࠆ㧚⴫ 4.2.1‛ℂ㊂ߣṶ▚ሶߣߩኻᔕ 㧔1㧕
㧦ᐳᮡ

‛ℂ㊂

Ṷ▚ሶ

x

x 㧔 x ࠍដߌࠆ㧕yy 㧔 y ࠍដߌࠆ㧕z

z 㧔 z ࠍដߌࠆ㧕䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 5⴫ 4.2.2‛ℂ㊂ߣṶ▚ሶߣߩኻᔕ 㧔2㧕
㧦ㆇേ㊂

‛ℂ㊂

Ṷ▚ሶ

px

i =

w
wx
w
i=
wy

py

i=

pz

w
wz

ฎౖജቇߢߪ㧘ߔߴߡߩ‛ℂ㊂ߪ x, y, z , px , p y , pz ߢ߽ߞߡ⴫ߔߎߣ߇಴᧪ࠆ㧚ᓥߞߡ㧘਄ߩኻᔕ⴫ࠍ↪޿ߡߔߴߡߩ‛ℂ㊂ߦߟ޿ߡߩṶ▚ሶࠍ᳞߼ࠆߎߣ߇಴᧪ࠆ㧚଀߃߫ px ߪ㧘
2

w ·§
w ·
§
¨ i= ¸¨ i= ¸
wx ¹©
wx ¹
©

=2

w2
wx 2

ߦኻᔕߔࠆ㧚ห᭽ߦߒߡ㧘଀߃߫ⷺㆇേ㊂ Lx , Ly , Lz ߦߟ޿ߡߪ⴫ 4.2.3 ߩࠃ߁ߦߥࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 6

⴫ 4.2.3‛ℂ㊂ߣṶ▚ሶߣߩኻᔕ 㧔3㧕
㧦ⷺㆇേ㊂

‛ℂ㊂

Lx

ypz  zp y

Ly

zpx  xpz

Lz

xp y  ypx

Ṷ▚ሶ

§ w
w ·
i= ¨ y  z ¸
wy ¹
© wz
w ·
§ w
i = ¨ z  x ¸
wz ¹
© wx
§ w
w ·
i = ¨ x  y ¸
wx ¹
© wy

ߥ߅㧘‛ℂ㊂ࠍ⴫ߔṶ▚ሶߪ linear ߢ߆ߟ Hermitian ߢ޽ࠆ㧚operator A ߇✢ᒻ㧔linear㧕ߢ޽ࠆߣ޿߁ߎߣߪ
A \ MA c\

A\  AMcA\(4.2.1a)(4.2.1b)ߩ 2 ߟߩᕈ⾰߇౒ߦḩߚߐࠇࠆߎߣߢ޽ࠆ㧚ߎߎߢ\ ߅ࠃ߮ M ߪછᗧߩ wavefunction㧘
c ߪછᗧߩቯᢙߢ޽ࠆ㧚‫ޤ⸼ޣ‬㧦linear ߢߥ޿ operator ߪ޿ࠈ޿ࠈ޽ࠆ߇㧘଀߃߫㑐ᢙࠍੑਸ਼ߔࠆߣ޿߁ operator ߇ߘߩ৻଀ߢ޽ࠆ㧚ߘߩࠃ߁ߥ operator ࠍ Sq ߣ޿߁⸥ภߢ⴫ߔߣ

Sq \  M\ 2  M 2  2\M

Sq\  SqM \ 2  M 2ߣ޿߁ࠊߌߢᑼ(4.2.1a)ߪᚑࠅ┙ߚߥ޿㧚ߐࠄߦ㧘ᑼ(4.2.1b)߇ᚑࠅ┙ߚߥ޿ߎߣ߽᣿⊕ߢ޽ࠆ㧚৻ᣇࠛ࡞ࡒ࡯࠻ 㧔Hermitian㧕 ߣ޿߁ߩߪ㧘³\ * AMdW³ M *A\ dW * ³ M A *\ * dW䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 7(4.2.2)ߣߥࠆ operator A ߩߎߣߢ޽ࠆ㧚ߎߎߢⓍಽߪᄌᢙߩోၞߦߟ޿ߡⴕ߁㧚਄ߩ⴫ߦ␜ߒߚߔߴߡߩ operator ߪ Hermitian ߢ޽ࠆ㧚ߢߪߘࠇࠍߤ߁߿ߞߡ⸽᣿
ߔࠆ߆㧫ߚ޿ߡ޿ߩ႐วߪㇱಽⓍಽࠍⴕ߁ߎߣߦࠃߞߡ⸽᣿಴᧪ࠆ㧚଀߃߫

momentum operator i=wࠍ⠨߃ߡߺࠃ߁㧚޽ࠆ wavefunction \ ߅ࠃ߮ M ߦߟ޿ߡ
wx

f

f

w ·
w\ *
f
§
(4.2.3)³f\ * ¨© i= wx ¸¹M dx i= \ *M f  i= f³ M wx dxߣ⴫ߔߎߣ߇಴᧪ࠆ㧚\ *\ ߩⓍಽ߇᦭㒢ߢ޽ࠆߚ߼ߦߪ㧔ߔߥࠊߜ wavefunction ߇
well-behaved ߢ޽ࠆߚ߼ߦߪ㧕\ ߪ rf ߩᭂ㒢ߢ࠯ࡠߣߥࠄߥߌࠇ߫޿ߌߥ޿㧚M ߦߟ޿ߡ߽หߓߎߣ߇⸒߃ࠆ㧚ᓥߞߡฝㄝߩ╙৻㗄ߪ࠯ࡠߣߥߞߡᶖ߃ࠆ㧚⚿ዪ
f

f

f

w ·
w ·
w ·
§
§
§
(4.2.4)¸M dx ³ M ¨ i= ¸\ * dx ³ M ¨ i= ¸ *\ * dx
wx ¹
wx ¹
wx ¹
©
f
f ©
f ©
ߣߥࠅ㧘໧㗴ߩ momentum operator ߇ Hermitian ߢ޽ࠆߎߣ߇⸽᣿ߐࠇߚ㧚㊂ሶജቇߢ↪޿ࠆ operator ߩᄢㇱಽ㧔߽ߜࠈࠎోߡߢߪߥ޿㧕ߪ Hermitian ߢ޽ࠅ㧘ߘߩᕈ⾰ߦߟ޿ߡ⹦ߒߊቇ⠌ߔࠆߎߣ߇ᔅⷐߢ޽ࠆ߇㧘ߘߩ೨ߦ㧘઒ቯ 3 ߣ઒ቯ 4 ߩ
2 ߟߩ㊀ⷐߥ઒ቯߦߟ޿ߡㅀߴࠃ߁㧚³ \ * ¨ i=઒ቯ 3㧦޽ࠆ♽ߦߟ޿ߡ㧘operator A ߦ⋧ᒰߔࠆ‛ℂ㊂ࠍ᷹ቯߔࠆᤨ㧘޿ߟ߽ 100%ߩ⏕߆ࠄߒߐߢหߓ⚿ᨐ a ߇ᓧࠄࠇࠆᤨ
A\ a\(4.2.5)ߣ⴫ߔ㧚ߎߩ႐ว\ ߪ operator A ߩ࿕᦭㑐ᢙ㧔eigenfunction㧕ߣ޿޿㧘a ࠍ operator Aߩ࿕᦭୯㧔eigenvalue㧕ߣ⸒߁㧚৻⥸ߦߪ㧘1 ߟߩ operator A ߇޿ߊߟ߆ߩ eigenfunction ߅ࠃ߮ eigenvalue ࠍᜬߟߩ߇᥉ㅢߢ޽ࠆ㧚ߔߢߦᚒ‫ ߪޘ‬Section 2.1 ߢశሶ߇ parallel eigenstate ߦ޽ࠆ႐วߣ

perpendicular eigenstate ߦ޽ࠆ႐วߣࠍቇ⠌ߒߚ㧚ߎߩ႐วߦߪ 2 ߟߩ eigenfunction ߇޽ߞߚࠊߌߢ޽ࠆ㧚ߎߩࠃ߁ߦ㧘৻⥸ߦ㧘eigenfunction ߿ eigenvalue ߇ⶄᢙ୘޽ࠆߩ
ߢ㧘ᰴߩࠃ߁ߦ subscript ࠍߟߌߡ඙೎ߔࠆ㧚A\ n a\ n(4.2.6)⇣ߥߞߚ࿕᦭㑐ᢙߦኻߔࠆ࿕᦭୯߇หߓ႐ว߽޽ࠆ㧚ߎߩࠃ߁ߥ⁁ᘒࠍ‫❗ޟ‬ㅌ
㧔degeneracy㧕߇޽ࠆ‫❗ޟ߆ߣޠ‬ㅌߒߡ޿ࠆ㧔degenerate㧕‫ߔ⴫ߢ⪲⸒߁޿ߣޠ‬㧚ߎߩ❗ㅌߣ޿߁߽ߩ߇߿ߞ߆޿ߥ߽ߩߢ㧘㊂ሶജቇߩಽ㊁ߢߪ㧘ߒ߫ߒ߫❗ㅌ߇޽ࠆ႐ว
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 8ߩขࠅᛒ޿ߦ㗡ࠍᖠ߹ߔ߽ߩߢ޽ࠆߎߣߪએਅ߅޿߅޿᣿ࠄ߆ߣߥߞߡ޿ߊߢ޽ࠈ
߁㧚

઒ቯ 4㧦
wavefunction \ ߢ⸥ㅀ಴᧪ࠆ♽ߦߟ޿ߡ㧘operator A ߦኻᔕߔࠆ‛ℂ㊂ࠍ᷹ቯߔࠆ
႐ว㧘100%ߩ⏕߆ࠄߒߐߢߪ⚿ᨐ߇᳿߹ࠄߥ޿ߎߣ߇޽ࠆ㧚ߎߩ႐วߦߪ㧘‛ℂ㊂
ߩᐔဋ୯㧔average value ߹ߚߪ mean value㧕 A ߪ

A³\ * A\ dW(4.2.7)

ߢ⴫ߐࠇࠆ㧚ߥ߅㧘ߎߩ㊂ߪ A ߦኻߔࠆᦼᓙ୯㧔expectation value㧕ߣ߽⸒ࠊࠇࠆ㧚ߎߎߢㅀߴߚ average value ߦߟ޿ߡߪ㧘ߔߢߦ Section 2.2 ߢ⹦ߒߊㅀߴߚߎߣߥߩߢ㧘ߎࠇએ਄ߩ⺑᣿ߪਇⷐߢ޽ࠈ߁㧚ߐߡ㧘‛ℂ㊂ߦ⋧ᒰߔࠆ operator ߪ Hermitian ߢ޽ࠆߣ޿߁ߎߣࠍߔߢߦㅀߴߚࠊ
ߌߢ޽ࠆ߇㧘ߎߩ Hermitian operator ߦ㑐ߒߡ㊀ⷐߥᕈ⾰ࠍએਅߦ߹ߣ߼ߡ߅ߎ߁㧚ቯℂ㧦

1 ߟߩ Hermitian operator ߩ eigenvalue ߪታᢙ㧔real㧕ߢ޽ࠆ㧚‫⸽ޣ‬᣿‫ޤ‬㧦A ࠍ Hermitian operator㧘\ ࠍߘߩ eigenfunction ߣߒ㧘߹ߚ a ࠍ eigenvalue ߣߔࠆ㧚A\ a\(4.2.8)ߎߩ complex conjugate ࠍขࠆ㧚A *\ * a *\ *(4.2.9)ߎࠇࠃࠅ

³\ * A\ dW a ³\ *\ dW(4.2.10)³\ A *\ * dW a * ³\\ * dW

(4.2.11)߹ߚ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 9A ߇ Hermitian ߢ޽ࠆ߆ࠄߎߩ 2 ߟߪ╬ߒߊߥߌࠇ߫޿ߌߥ޿㧚ߔߥࠊߜ a* a ߢ޽ࠆ㧚ߎߩߎߣߪ a ߇ real ߢ޽ࠆߎߣࠍ␜ߔ㧚એ਄ߢ⸽᣿⚳ࠅ㧚
߽߁ 1 ߟ㧘㊀ⷐߥቯℂࠍᰴߦ␜ߔ㧚

ቯℂ㧦

Hermitian operator ߩ eigenfunction ߪ߅੕޿ߦ⋥੤ߒߡ޿ࠆ㧚‫⸽ޣ‬᣿‫ޤ‬㧦

Hermitian operator A ߩ eigenfunction \ n ߅ࠃ߮\ m ࠍ⠨߃ࠆ㧚eigenvalue ࠍߘࠇߙࠇan ߅ࠃ߮ am ߣߔࠆ㧚
A\ n

an\ n

A\ m

am\ m

(4.2.12)ᑼ(4.2.12)ߩ 1 ⇟⋡ߩᑼߦᏀ߆ࠄ\ m * ࠍដߌ㧘ోⓨ㑆ߦߟ޿ߡⓍಽߔࠆ㧚෶㧘2 ⇟⋡ߩᑼߩ complex conjugate ࠍߣߞߚᑼߦᏀ߆ࠄ\ n ࠍដߌߡหߓߊⓍಽߔࠆ㧚

³\ *A\ dW a ³\ *\ dW³\ A *\ * dW a * ³\ \ * dW
m

n

n

m

n

m

n

m

n

(4.2.13)mㄝ‫ޘ‬ᒁ߈▚ࠍߒߡ

³\ *A\ dW  ³\ A *\ * dW
m

n

n

m

an  am * ³\ m *\ n dW

(4.2.14)

operator A ߪ Hermitian ߢ޽ࠆ߆ࠄ਄ᑼߩᏀㄝߪ࠯ࡠߢ޽ࠆ㧚ᓥߞߡan  am * ³\ m *\ n dW

0(4.2.15)

ߎߎߢ\ n ߣ\ m ߣ߇╬ߒ޿႐วߣ╬ߒߊߥ޿႐วߣࠍಽߌߡ⠨߃ߡߺࠆ㧚߽ߒ nߢ޽ࠇ߫㧘ⷙᩰൻ᧦ઙࠃࠅᏀㄝߩⓍಽߩ㗄ߪ 1 ߢ޽ࠆ㧚ᓥߞߡan an *

m

(4.2.16)ࠍᓧࠆ㧚ߎࠇߪ‫ޟ‬Hermitian operator ߩ eigenvalue ߇ real ߢ޽ࠆ‫߁޿ߣޠ‬਄ㅀߩቯℂࠍ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 10೎ߩᣇᴺߢ⸽᣿ߒߚߎߣߦߥࠆ㧚ᰴߦ n z m ߩ႐วࠍ⠨߃ߡߺࠆ㧚߽ߒ߽ an z am ߢ޽ࠇ߫㧔ߔߥࠊߜ\ n ߣ\ m ߣ߇❗
ㅌߒߡ޿ߥߌࠇ߫㧕³\ *\ dWm0n(4.2.17)\ n ߣ\ m ߣߩ㑆ߦᑼ(4.2.17)ߩࠃ߁ߥ㑐ଥ߇޽ࠆᤨ㧘\ n ߣ\ m ߣߪ⋥੤ߒߡ޿ࠆ
㧔orthogonal㧕ߣ޿߁㧚ߎߩࠃ߁ߦ㧘ዋߥߊߣ߽❗ㅌߒߡ޿ߥ޿႐วߦߪ㧘Hermitian
operator ߩ eigenfunction ߪ߅੕޿ߦ orthogonal ߢ޽ࠆߎߣ߇⸽᣿ߐࠇߚ㧚ᦨᓟߦ❗ㅌ߇޽ࠆ႐ว㧘ߔߥࠊߜ૗୘߆ߩ eigenvalue ߇╬ߒ޿႐วࠍ⠨߃ࠆ㧚◲නߩߚ߼㧘2 ߟߩ eigenfunction \ 1 ߣ\ 2 ߣ߇ห৻ߩ eigenvalue a1 ࠍᜬߟߣߒࠃ߁㧚ߔߥ
ࠊߜ

A\ 1

a1\ 1

A\ 2

a1\ 2

(4.2.18)

ߐߡ㧘\ 1 ߣ\ 2 ߣߩ✢ᒻ⚿ว I ࠍ⠨߃ߡߺࠃ߁㧚I

c1\ 1  c2\ 2(4.2.19)

ߎߩ I ߽߹ߚ operator A ߩ eigenfunction ߢ޽ࠅ㧘ߘߩ eigenvalue ߪ a1 ߢ޽ࠆ㧚ߘࠇߪ
AI

A c1\ 1  c2\ 2
c1 A\ 1  c2 A\ 2
a1c1\ 1  a1c2\ 2(4.2.20)

a1I߆ࠄ᣿ࠄ߆ߢ޽ࠆ㧚ోߊหߓ⠨߃ߢ㧘\ 1 ߣ\ 2 ߣ߆ࠄ਄ߣߪ⇣ߥߞߚᒻߩ✢ᒻ⚿วࠍ૞ࠇ߫㧘ߘࠇ߽෶

eigenvalue a1 ࠍᜬߟߎߣ߇ಽ߆ࠆ㧚ߘߎߢ\ 1 ߣ\ 2 ߩ✢ᒻ⚿วߣߒߡ 2 ߟߩ⇣ߥߞߚ
wavefunction I1 ߅ࠃ߮ I2 ࠍ㧘 I1 ߣ I2 ߇⋥੤ߔࠆࠃ߁ߦ㧘ߔߥࠊߜ³ I *I dW12

0(4.2.21)߇ᚑࠅ┙ߟࠃ߁ߦㆬ߱ߣଢ೑ߢ޽ࠆ㧚ߢߪ㧘ౕ૕⊛ߦߤ߁ߔࠆ߆ࠍ⠨߃ߡߺࠃ߁㧚◲නߩߚ߼ߦ㧘 I1 ߪ\ 1 ߘߩ߹߹ߣߒ㧘䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 11

I2 ߪ\ 1 ߣ \ 2 ߣߩᰴߦ␜ߔࠃ߁ߥ✢ᒻ⚿วߣߔࠆ㧚
I1 \ 1I2 \ 2  c\ 1(4.2.22)

I1 ߣ I2 ߣߪ޿ߕࠇ߽ⷙᩰൻߐࠇߡ޿ࠆߣ⠨߃ࠆ㧚 c ߇᳞߻ߴ߈ቯᢙߢ޽ࠆ㧚 c ߪ㧘I1
ߣ I2 ߣ߇⋥੤ߔࠆߣ޿߁᧦ઙ߆ࠄ᳞߼ࠆ㧚³ I *I dW ³\ * \  c\ dW ³\ *\ dW  c 0
1

2

1

2

1

1

2

(4.2.23)ߔߥࠊߜ㧘߽ߒ߽

c ³\ 1 *\ 2 dW

(4.2.24)ߩࠃ߁ߦㆬߴ߫ I1 ߣ I2 ࠍ orthogonal ߦߥࠆࠃ߁ߦ᳿߼ࠆߎߣ߇಴᧪ࠆߎߣ߇ಽ߆ߞ
ߚ㧚ߎߩขࠅᛒ޿ߪ㧘Schmidt orthogonalization ߣ⸒ࠊࠇ㧘ߎߩᣇᴺࠍ↪޿ߡ㧘৻⥸ߦn ୘ߩ❗ㅌߒߚ wavefunction ߩ⚵߆ࠄ n ୘ߩ߅੕޿ߦ orthogonal ߥ wavefunction ࠍ૞ࠆߎߣ߇಴᧪ࠆ㧚߆ߊߒߡ㧘ߚߣ߃❗ㅌ߇޽ߞߚߦߖࠃ㧘Hermitian operator ߩ

wavefunction ࠍ߅੕޿ߦ orthogonal ߦߔࠆߎߣ߇಴᧪ࠆ㧚ߘࠇߙࠇߩ wavefunction ߇ⷙᩰൻ㧔normalized㧕ߐࠇߡ޿ߡ㧘߅੕޿ߦ⋥੤㧔orthogonal㧕ߒߡ޿ࠆ wavefunctionߩ⚵ࠍⷙᩰ⋥੤♽㧔orthonormal set㧕ߣ޿߁⸒⪲ߢ⴫ߔ㧚‫ڏ‬Exercise 4.2.1\1 1 㧘 \2

x ߪ 0 d x d 1 ߩ▸࿐ౝߢ⋥੤ߒߡ޿ߥ޿㧚Schmidt ߩ⋥੤ൻߩᣇᴺࠍ↪

޿ߡ orthonormal set ࠍ૞ࠇ㧚‫ޤ⸃ޣ‬
㧦

I1 \ 1 1
I2 \ 2  c\ 1xcߣߔࠆ㧚ᓥߞߡ

0³ I *I dx
12

1³ x  c dx
0

1
c
2
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 12

ᓥߞߡ

I2

x

1
2

ߎࠇߢ I1 ߣ I2 ߪ߅੕޿ߦ⋥੤ߒߚ㧚ᓟߪߘࠇߙࠇߦߟ޿ߡ normalization ࠍⴕ߃߫ࠃ
޿㧚㧔 I1 ߪߔߢߦ normalize ߐࠇߡ޿ࠆ㧕ߘߩ⚿ᨐߣߒߡᰴߩ set ࠍᓧࠆ㧚

I1 1
1·
§
12 ¨ x  ¸
2¹©
㧔એ਄ߢ⸃⚳ࠊࠅ)

I2

1 ߟߩ Hermitian operator ߩ eigenfunction ߩߔߴߡߩ⚵߆ࠄߥࠆ orthonormal set ߩ㊀
ⷐߥ․ᓽߣߒߡᰴߩቯℂ߇޽ࠆ㧚ቯℂ㧦છᗧߩ㑐ᢙߪ㧘޽ࠆ Hermitian operator ߩ eigenfunction ߩ⚵ߢ޽ࠆ orthonormal setߢዷ㐿಴᧪ࠆ㧚ߔߥࠊߜ㧘

¦ c\

<

i

i

c1\ 1  c2\ 2  ...

(4.2.25)1ߚߛߒߎߎߢ㧘છᗧߩ㑐ᢙ < ߣߘࠇࠍዷ㐿ߔࠆ orthonormal set ߪหߓⓨ㑆㗔ၞߢቯ⟵ߐࠇߡ޿ߡหߓႺ⇇᧦ઙࠍ᦭ߒߡ޿ࠆߎߣ߇ᔅⷐߢ޽ࠆ㧚‫⸽ޣ‬᣿‫ޤ‬㧦ߎߩࠃ߁ߦዷ㐿಴᧪ࠆߣ޿߁ߎߣࠍ⸽᣿ߔࠆߦߪ㧘ዷ㐿ଥᢙ c1 , c2 , ... ߩߔߴߡ߇᳞
߹ࠆߎߣࠍ␜ߖ߫ࠃ߆ࠈ߁㧚਄ᑼߦᏀ߆ࠄ\ j * ࠍ߆ߌߡోⓨ㑆ߢⓍಽߔࠆߣᰴᑼߦࠃ
ࠅ c j ߇᳞߹ࠆ㧚ߔߥࠊߜ㧘³\ * <dW ¦ c ³\ *\ dW
j

i

i

j

i

cj

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 13㧔એ਄ߢ⸽᣿⚳ࠊࠅ㧕㧚ᑼ(4.2.25)ߩࠃ߁ߦછᗧߩ㑐ᢙࠍዷ㐿಴᧪ࠆ wavefunction ߩ⚵ࠍቢో♽㧔completeset㧕ߣ⸒߁㧚৻⥸ߦ޽ࠆ wavefunction ߩ⚵߇ complete set ߢ޽ࠆ߆ߤ߁߆ߩ⸽᣿ߪ߻
ߕ߆ߒ޿߇㧘ㅢᏱߪ㧘޽ࠆ Hermitian operator ߩ࿕᦭㑐ᢙߩߔߴߡߩ⚵ߪ complete setࠍߥߒߡ޿ࠆߣ⠨߃ࠄࠇߡ޿ࠆ㧚‫ڏ‬Problem 4.2.1
߽ߒ߽ operator A ߇³\ * AM dW

 ³ M A *\ * dWࠍḩߚߖ߫ A ߪ anti-Hermitian㧔෻ࠛ࡞ࡒ࡯࠻㧕ߢ޽ࠆߣ⸒߁㧚
w
w w
,
,
ߪߘࠇߙࠇ anti-Hermitian operator ߢ޽ࠆߎߣࠍ⸽᣿ߖࠃ㧚wx wy wz‫ڏ‬Problem 4.2.2=2 w 2
ߪ Hermitian ߢ޽ࠆߎߣࠍ⸽᣿ߖࠃ㧚㧔ࡅࡦ࠻㧦ㇱ
2m wx 2
ಽⓍಽࠍ 2 ࿁➅ࠅ㄰ߖ߫ࠃ޿㧕ㆇേࠛࡀ࡞ࠡ࡯Ṷ▚ሶ ‫ڏ‬Problem 4.2.3
ᰴߦ␜ߔฦ㗄ߦߟ޿ߡ㧘 f x

ߪ operator A ߩ eigenfunction ߢ޽ࠆ㧚eigenvalue ࠍ᳞߼ࠃ㧚ߎߎߢ Z ߅ࠃ߮ a ߪቯᢙߢ޽ࠆ㧚

bg

A

f x

(a)

d2
dx 2

cos Z x(b)ddt

exp iZt

(c)
(d)

d2
d
2 3
2
dx
dx
w
wy

‫ڏ‬Problem 4.2.4䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 14

exp ax

x 2 exp ay

cos ax cos by cos cz ߪ Laplacian operator  2

w2
w2
w2


ߩ eigenfunction ߢ
wx 2 wy 2 wz 2޽ࠆߎߣࠍ␜ߖ㧚߹ߚ eigenvalue ࠍ᳞߼ࠃ㧚‫ڏ‬Problem 4.2.5
ᰴߩฦ㗄ߦ␜ߔ operator ࠍ A ߣߔࠆᤨ㧘 A2 ࠍ᳞߼ࠃ㧚

(a)

d2
dx 2

(b)

d
x
dx

(c)

d2
d
 2x 1
2
dx
dx

Hint㧦Ṷ▚ߪ㑐ᢙ f x ߦ૞↪ߔࠆߎߣߦᵈᗧ㧚‫ڏ‬Problem 4.2.6\1x, \ 2x 2 㧘ߪ 0 d x d 1 ߩ▸࿐ߢ⋥੤ߒߡ޿ߥ޿㧚Schmidt ߩ orthogonalization

ߩᣇᴺࠍ↪޿ߡ orthonormal ߥ wavefunction ߩ⚵ࠍ᳞߼ࠃ㧚‫ڏ‬Problem 4.2.7

(a)
(b)

§d
·§ d
·
¨  x ¸¨  x ¸ ࠍ᳞߼ࠃ㧚
© dx
¹© dx
¹
§d
·§ d
·
¨  x ¸¨  x ¸ ࠍ᳞߼ࠃ㧚
© dx
¹© dx
¹

‫ڏ‬Problem 4.2.8(a)A, B ߇ߤࠎߥ operator ߢ߽㧔linear ߢ߽ nonlinear ߢ߽㧕 A  B

2ᚑࠅ┙ߟߎߣࠍ⸽᣿ߖࠃ㧚(b)ߤߩࠃ߁ߥ᧦ઙߩ߽ߣߢ A  B

2A2  2 AB  B 2 ߇ᚑࠅ┙ߟ߆㧚‫ڏ‬Problem 4.2.9
ᰴߩ operator ߪ linear ߆㧘߹ߚߪ nonlinear ߆㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 15

2

B A ߇

(a)

d
3x
dx 2

(b)

³ dx

(c)

exp

2

2

‫ڏ‬Problem 4.2.10
ᑼ(4.2.1a)ߪᚑࠅ┙ߟ߇㧘ᑼ(4.2.1b)ߪᚑࠅ┙ߚߥ޿ operator ߩ଀ࠍ޽ߍࠃ㧚‫ڏ‬Problem 4.2.11

2 ߟߩ linear ߥ operator ߩⓍߪ linear ߆ non-linear ߆㧫‫ڏ‬Problem 4.2.12

Laplace transform operator L ߪf³ exp  px f x dx

Lf x

(4.2.26)0ߣቯ⟵ߐࠇࠆ㧚(a)

L ߪ linear ߆ non-linear ߆㧚

(b)

f x

1 ߩ႐วߩᑼ(4.2.27)ࠍ᳞߼ࠃ㧚

(c)

f x

exp ax ߩ႐ว㧔ߚߛߒ a  p ߣߔࠆ㧕ߩᑼ(4.2.27)ࠍ᳞߼ࠃ㧚[Coffee Break] Hermitian operator ߦ㑐ߔࠆ joke
"A Random Walk in Science"㧔The Institute of Physics, 1973㧕ߣ⸒߁ᧄߩ"Physics TermEasy"ߣ޿߁౬⺣㓸ߦࠃࠆߣ"Hermitian Operator"ߩᗧ๧ߣߒߡ"Recluse surgeon"ߣቯ⟵ߐࠇߡ޿ࠆ㧚ߎࠇࠍ⷗ߡߔߋ╉߃ࠆ᭽ߛߞߚࠄำߪ߆ߥࠅߩ⧷⺆ജ߇޽ࠆߣ⸒߃ࠆ㧚╉߃ߥ޿ੱߪㄉᦠࠍᒁ޿ߡ⷗ࠆߣ⦟߆ࠈ߁㧚recluse ߣ޿߁⸒⪲ߪ hermit ߣหߓᗧ๧ߢ
޽ࠆ㧚operator ߣ޿߁⸒⪲ߪ⑼ቇߩಽ㊁ߢߪṶ▚ሶߢ޽ࠆ߇㧘ߘߩઁߤࠎߥᗧ๧ߦ૶
ࠊࠇߡ޿ࠆ߆ࠍㄉᦠߢ⺞ߴࠇ߫ߎߩ౬⺣ߩᗧ๧߇ಽ߆ࠆߢ޽ࠈ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 164.3 Schrödinger ᣇ⒟ᑼ

઒ቯ 5㧦

wavefunction < x, y, z , t ߩᤨ㑆ଐሽᕈߪᰴߩ time-dependent Schrödinger equation ߦࠃߞߡ␜ߐࠇࠆ㧚

H<

i=

w
<
wt

(4.3.1)

ߎߎߢ H ߪH=2 2
  V x, y , z
2m(4.3.2)bߢ޽ࠅ Hamiltonian operator ߣ⸒߁㧚 V x , y , z , tg ߪ potential ࠍ␜ߔ㧚߹ߚ

w2
w2
w2


wx 2 wy 2 wz 2

2

(4.3.3)bg․ᱶߥ႐วߣߒߡ㧘potential V x , y , z , t ߇ᤨ㑆ࠍ޽ࠄࠊߦ฽߹ߥ޿႐วࠍ⠨߃ߡߺࠃ߁㧚ߘߩࠃ߁ߥ potential ߩ଀ߪᢙᄙߊ޽ࠆ㧚଀߃߫

1. 2 ߟߩ㔚⩄ q1 ߣ q2 ߇〒㔌 r 㔌ࠇߡሽ࿷ߔࠆ႐ว㧦 V

q1q2
4SH 0 r2. ࡃࡀቯᢙ k ߩࡃࡀ߇ᐔⴧ૏⟎ࠃࠅ x ߛߌિ߫ߐࠇߚ႐ว㧦 V
3. ⾰㊂ m ߩ‛૕߇࿾㕙ࠃࠅ h ߛߌ਄ⓨߦሽ࿷ߔࠆ႐ว㧦 V

1 2
kx
2

mgh

ߎߩ଀ߩࠃ߁ߦ V ߇ᤨ㑆ࠍ฽߹ߥ޿႐วߦߪ㧘Section 3.1 ߢⴕߞߚࠃ߁ߥᄌᢙಽ㔌ߩ
ᣇᴺࠍㆡ↪಴᧪ࠆ㧚ߔߥࠊߜ\ x, y , z I t< x, y , z , tbb gⓍߢ⴫ߔ㧚ᑼ(4.3.4)ࠍᑼ(4.3.1)ߦઍ౉ߒ㧘ਔㄝࠍ \ b x , y , z gI bt g ߢഀࠆߣ㧘g(4.3.4)bgߩࠃ߁ߦ < x , y , z , t ࠍⓨ㑆ᐳᮡ x , y , z ߩߺߩ㑐ᢙ \ x , y , z ߣᤨ㑆 t ߩߺߩ㑐ᢙ I t ߣߩ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 17

1
H\ x, y, z
\ x, y , zi=

1 dI tI t dt(4.3.5)Ꮐㄝߪᐳᮡߩߺߩ㑐ᢙ㧘ฝㄝߪᤨ㑆ߩߺߩ㑐ᢙߢ޽ࠆ㧚ߘࠇ߇߅੕޿ߦ╬ߒ޿ߣ޿߁
ߎߣߪߘࠇߙࠇ߇޽ࠆቯᢙߦ╬ߒߊߥߌࠇ߫޿ߌߥ޿ߎߣࠍ␜ߒߡ޿ࠆ㧚ߘߩቯᢙࠍ

E ߣ߅ߎ߁㧚ߔࠆߣ
H\ x, y, z

E\ x, y, z

(4.3.6)

dI t
1
EI t
(4.3.7)dti=
ᑼ(4.3.6)ߪ㧘
time-independent Schrödinger equation ߣ⸒ࠊࠇߡ޿ߡ㧘ᚒ‫ ߦߢߔߪޘ‬Section
3.2 ߢ߅ߥߓߺߩ߽ߩߢ޽ࠆ㧚ߎߩ႐ว㧘separation constant E ߪ㧘Hamiltonian operator
H ߩ eigenvalue ߢ޽ࠅ㧘energy ߩ dimension ࠍᜬߟ㧚৻ᣇ㧘ᑼ(4.3.7)ߩ⸃ߪ

I t§ iEt ·
constant u exp ¨ 
¸
© = ¹

bg

ߢ޽ࠆ߇㧘ߎߩ႐ว I t ߇

bg bg 1I t *I tߣߥࠆࠃ߁ߦ਄ᑼߩ constant ࠍㆬ߱ߣI t

§ iEt ·
exp ¨ ¸
© = ¹

(4.3.8)ߣߥࠆ㧚⚿ዪ

< x, y , z , t§ iEt ·¸
© = ¹

(4.3.9)\ x, y, z *\ x, y, z

(4.3.10)\ x, y, z exp ¨ ᓥߞߡ< x, y , z , t * < x, y , z , t߆ࠄ㧘⏕₸ಽᏓߪᤨ㑆ߦଐሽߒߥ޿ߎߣ߇ಽ߆ࠆ㧚

b

g

b

g

t ࠍᄌᢙߣߒߡ฽߻ < x , y , z , t ߽㧘 t ࠍ฽߹ߥ޿ \ x , y , z ߽㧘ߤߜࠄ߽ wavefunctionbgߣ๭߫ࠇߡ޿ࠆ߇㧘ߪߞ߈ࠅ඙೎ࠍߟߌߚ޿႐วߦߪ \ x , y , z ࠍ"stationary-state

wavefunction"ߣ޿߁⸒⪲ߢ⴫ߔ㧚⠨߃ߡ޿ࠆ♽ߩ potential ߇ਈ߃ࠄࠇࠇ߫㧘time dependent Schrödinger equation (4.3.1)bg

߹ߚߪ time independent Schrödinger equation (4.3.6)ࠍ↪޿ߡ wavefunction < x , y , z , t ߹䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 18bg

ߚߪ \ x , y , z ࠍ᳿߼ࠆߎߣ߇಴᧪ࠆ㧚߽ߒ߽ potential V ߇ᤨ㑆 t ߩ㑐ᢙߢ޽ࠇ߫ timedependent Schrödinger equation ࠍ૶ࠊߑࠆࠍᓧߥ޿㧚ߒ߆ߒ㧘potential ߇ᤨ㑆ߩ㑐ᢙߢߥ޿႐วߦߪ㧘wavefunction ࠍᤨ㑆ߦଐሽߒߚᒻߣߒߡ߽㧘ᤨ㑆ߦଐሽߒߥ޿ᒻߣߒ
ߡ߽㧘޿ߕࠇߢ߽⴫ߔߎߣ߇಴᧪ࠆ㧚ߎߩ޽ߚࠅߩ⠨߃ᣇߪ㧘ᆎ߼ߩ߁ߜߪࠃߊಽ߆
ࠄߥ޿߆߽ߒࠇߥ޿㧚߽߁ዋߒߚߞߡ߆ࠄṶ⠌ࠍㅢߒߡ޿ࠈ޿ࠈߣ⠨߃ߡߺࠆߎߣߦ
ߒߚ޿㧚

4.4 Ṷ▚ሶߩ੤឵㑐ଥ

2 ߟߩ operator A ߣ B ߇㧘޽ࠆ㑐ᢙ\ ߦ AB ߣ޿߁ᒻߢ૞↪ߒߚᤨ㧘ߎࠇߪ߹ߕ B ࠍ\ ߦ૞↪ߒ㧘ߘߩ⚿ᨐߦᦝߦ A ࠍ૞↪ߔࠆߣ޿߁ᗧ๧ߢ޽ࠆ㧚ߎߩࠃ߁ߥ operator ߣㅢᏱߩᄌᢙߣߩ㆑޿ߪ㧘operator ߩ႐วߪൎᚻߦ㗅ᐨࠍ౉ࠇ឵߃ࠆߎߣߪ಴᧪ߥ޿ߎ
ߣߢ޽ࠆ㧚߽ߒ߽
AB\ BA\(4.4.1)ߢ޽ࠇ߫ A ߣ B ߪ commutable㧔㨬੤឵น⢻㨭෶ߪ㨬น឵㨭㧕ߢ޽ࠆߣ⸒߁㧚෻ኻߦ㧘߽ߒAB z BAߢ޽ࠇ߫ A ߣ B ߪ noncommutable㧔㨬㕖น឵㨭㧕ߢ޽ࠆߣ⸒߁㧚଀߃߫ A

d
㧘B
dx

ࠍដߌࠆ㧕ߣߔࠆߣ

AB\

d
x\
dx

BA\

x

\ x

d\
dx

(4.4.2)৻ᣇd\dx(4.4.3)

ᓥߞߡ A ߣ B ߪ noncommutable ߢ޽ࠆ㧚2 ߟߩ operator ߩ੤឵㑐ଥߪ㧘ߒ߫ߒ߫commutator
A, B AB  BA
(4.4.4)ߢ߽ߞߡ⴫ߐࠇࠆ㧚਄ߩ଀ߩ႐วߪ

ªd º«¬ dx , x »¼ 1

(4.4.5a)

߹ߚߪ

ª d º
«¬ x, dx »¼ 1ߣ޿߁ߎߣߦߥࠆ㧚ᑼ(4.4.5)ࠃࠅ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 19(4.4.5b)

x㧔 x

wº
ª
«¬ x, i= wx »¼
ߔߥࠊߜ

i=

> x, p x @ i =ߎߩࠃ߁ߦᐳᮡߣㆇേ㊂ࠍ⴫ߔṶ▚ሶ߇߅੕޿ߦน឵ߢߥ޿ߣ޿߁ߎߣߪ㕖Ᏹߦ㊀ⷐߥᗧ๧ࠍᜬߟ߇㧘ߎߩߎߣߦߟ޿ߡߪ Section 4.5 ߢ⹦ߒߊㅀߴࠃ߁㧚੤឵น⢻ߥṶ▚ሶߦߟ޿ߡߪᰴߩ㊀ⷐߥቯℂ߇޽ࠆ㧚ቯℂ㧦߽ߒ߽ 2 ߟߩ Hermitian operator A ߅ࠃ߮ B ߇น឵ߢ޽ࠇ߫㧘ਔ⠪ߦ౒ㅢߩ

eigenfunction ߇ሽ࿷ߔࠆ㧚‫⸽ޣ‬᣿‫ޤ‬㧦
operator A ߩ eigenfunction ࠍ\ 㧘eigenvalue ࠍ a ߣߔࠆ㧚A\ a\(4.4.6)

߹ߚ
AB

BA

ߢ޽ࠆ߆ࠄ
AB\

BA\

Ba\

aB\

(4.4.7)

ߔߥࠊߜA B\

b g ab B\ gࠃࠅ B\ ߽߹ߚ operator A ߩ eigenfunction ߢ޽ࠅߘߩᤨߩ eigenvalue ߪหߓߊ a ߢ޽ࠆ㧚ߎࠇ߆ࠄవߪ a ߣ޿߁ eigenvalue ࠍᜬߟ wavefunction ߦ❗ㅌ߇޽ࠆ႐วߣ❗ㅌ߇ߥ޿႐วߣߦಽߌߡ⠨߃ࠆᔅⷐ߇޽ࠆ㧚߹ߕ❗ㅌ߇ߥ޿႐วࠍ⠨߃ࠆ㧚ߎߩ႐ว B\ ߪනߦ\ ߩቯᢙ୚ߢߥߌࠇ߫޿ߌߥ
޿㧚ߘߩቯᢙࠍ b ߣ߅ߊߣB\ b\(4.4.8)ߣߥࠆ㧚ߔߥࠊߜ㧘 b ߪ operator B ߩ eigenvalue ߢ޽ࠆ㧚ߘߒߡ\ ߪ operator A ߣ Bߦ౒ㅢߥ eigenfunction ߢ޽ࠆ㧚ᰴߦ❗ㅌ߇޽ࠆ႐วࠍ⠨߃ࠆ㧚◲නߩߚ߼㧘 A ߩ eigenvalue ߩ߁ߜ a1 ߣ a2 ߩ 2 ߟ߇䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 20❗ㅌߒߡߘߩ୯ߪ a ߢ޽ࠆߣߒࠃ߁㧚ߎߩ႐วߩ eigenfunction ࠍ I1 ߅ࠃ߮ I2 ߣߔࠆ㧚
ߔߥࠊߜ
AI1

a1I1

aI1

AI2

a2I2

aI2

(4.4.9)operator A ߪ✢ᒻߣ⠨߃ߡ޿ࠆ߆ࠄ I1 ߅ࠃ߮ I2 ߩછᗧߩ linear combination ߣߒߡ
c1I1  c2I2 ࠍ\ ߣߔࠆߣA\

A c1I1  c2I2
c1 AI1  c2 AI2(4.4.10)c1aI1  c2 aI2a\ߎࠇࠃࠅ\ ߽߹ߚ eigenvalue a ࠍᜬߟ A ߩ eigenfunction ߢ޽ࠆߎߣ߇ಽ߆ࠆ㧚ߎߎߢc1 ߅ࠃ߮ c2 ࠍߤ߁ㆬ߱߆ߣ޿߁ߣ B\b\ ߇ᚑࠅ┙ߟࠃ߁ߦߔࠆ㧚ߎߩࠃ߁ߥ c1 ߅ࠃ߮ c2 ࠍㆬ߱ߎߣ߇น⢻ߢ޽ࠇ߫❗ㅌ߇޽ࠆ႐ว߽ቯℂ߇⸽᣿ߐࠇߚߎߣߦߥࠆ㧚ౕ૕⊛ߦߤ߁߿ߞߡ c1 ߅ࠃ߮ c2 ࠍ᳞߼ࠆ߆ࠍએਅߦ␜ߔ㧚 B\ b\ ࠃࠅ
c1 BI1  c2 BI2 bc1I1  bc2I2
Ꮐ߆ࠄ I1 * ߹ߚߪ I2 * ࠍ߆ߌߡⓍಽߔࠆߣ

B11  b c1  B12 c2

0

B21c1  B22  b c2

0(4.4.11)ߎߎߢBnm³ I * BI dWnmm, n 1, 2(4.4.12)ߣቯ⟵ߒߚ㧚ߎߎߢ࿕᦭୯ b ߪᧂ⍮ߢ޽ࠆ߆ࠄ㧘c1 , c2 ߅ࠃ߮ b ߩ 3 ߟ߇ᧂ⍮ᢙߢ޽ࠆ㧚
ᑼ(4.4.11)ߩㅪ┙ᣇ⒟ᑼ߆ࠄᒻᑼ⊛ߦ c1 㧘 c2 ࠍ᳞߼ࠆߣ
0B12
0 B22  b
(4.4.13)c1
B11  b
B12
B21
B22  b䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 21

B11  b 0
B210
B11  b
B12
B21
B22  bc2

(4.4.14)਄ߩ 2 ߟߩᑼߢ㧘߽ߒಽᲣߩⴕ೉ᑼ㧔determinant㧕߇࠯ࡠߢߥ޿ߣߔࠆߣ㧘ಽሶߪ࠯ࡠߢ޽ࠆ߆ࠄ㧘 c1 c2 0 ߣ޿߁ trivial solution ߒ߆ᓧࠄࠇߥ޿㧚ᓥߞߡᗧ๧ߩ޽ࠆ⸃
߇ᓧࠄࠇࠆߚ߼ߦߪಽᲣߩ determinant ߇࠯ࡠߢߥߌࠇ߫޿ߌߥ޿㧚ߔߥࠊߜ
B11  b

B12

B21

B22  b(4.4.15)0ߎߩࠃ߁ߥᑼࠍ᳗ᐕᣇ⒟ᑼ㧔secular equation㧕ߣ⸒߁㧚ᑼ(4.4.15)ߪ b ߦ㑐ߔࠆ 2 ᰴᣇ⒟ᑼߢ޽ࠆߩߢߘࠇ߆ࠄ 2 ߟߩ b ߩ⸃ࠍ᳞߼㧘ߘࠇߙࠇߩ b ࠍᑼ(4.4.11)ߦઍ౉ߒߡ㧘c1 ߣ c2 ߦ㑐ߔࠆㅪ┙ᣇ⒟ᑼࠍ૞ࠆ㧚ߘ߁ߔࠆߣㅪ┙ᣇ⒟ᑼߩ 2 ߟߩᑼ߇ోߊหߓᑼߦ
ߥߞߡߒ߹߁㧚ᧂ⍮ᢙ߇ 2 ߟߢᑼ߇ 1 ߟߒ߆ߥ޿߆ࠄ૗߆߽߁ 1 ߟ᧦ઙ߇ߥ޿ߣ⸃ߌ
ߥ޿㧚ߘߎߢ\ ߇ⷙᩰൻߐࠇߡ޿ࠆߣ޿߁᧦ઙ
2c1  c22

c1 * c1  c2 * c2

1

(4.4.16)ࠍട߃ߡ c1 ߣ c2 ࠍ᳞߼ࠆ㧚ߎߎߢㅀߴߚᣇᴺߪઍᢙቇߩ⸒⪲ߢߪ㧘ⴕ೉㧔matrix㧕
ª B11
«B
¬ 21

B12 º
B22 »¼ࠍኻⷺൻߔࠆ㧘ߣ޿߁ߎߣߢ޽ࠅ㧘㊂ሶൻቇߩಽ㊁ߢ޿ࠈ޿ࠈߥᚲߦ➅ࠅ㄰ߒ↪޿ࠄࠇࠆᄢಾߥᢙቇ⊛ᚻᴺߢ޽ࠆ㧚ߣߦ߆ߊ㧘ߎ߁޿ߞߚᣇᴺߢ㧘 A\

a\ ߅ࠃ߮ B\b\ ߣߥࠆࠃ߁ߥ\ ࠍ I1 ߣ I2 ߩ✢ᒻ⚿วߣߒߡ᳞߼ࠆߎߣ߇಴᧪ߚࠊߌߢ޽ࠆ㧚‫ڏ‬Exercise 4.4.1
ᄌၞ 0 d I d 2S ߢ operatord2ߪ❗ㅌߒߚ eigenfunction \ 1
dI 2

1

S

sin mI ߣdd2
ߪ 2 ߣ౒ㅢߩ eigenfunction ࠍᜬߜᓧࠆ߆ߤ߁dI
S
dI
߆ࠍี๧ߒ㧘߽ߒ㧘ᜬߜᓧࠆ႐วߦߪ eigenfunction ߣ eigenvalue ࠍ᳞߼ࠃ㧚\2

1

cos mI ࠍᜬߟ㧚operator i‫ޤ⸃ޣ‬㧦߹ߕ㧘੤឵㑐ଥࠍ⺞ߴࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 22

ª d2
d º
« dI 2 , i dI »¬¼

0ߢ޽ࠆ߆ࠄቯℂߦࠃࠅਔ⠪ߦ౒ㅢߩ eigenfunction ߪሽ࿷ߔࠆ㧚ᰴߦ operator i

d
ߦߟ
dI

޿ߡߩ eigenvalue ߣ eigenfunction ࠍ᳞߼ࠆ㧚ᑼ(4.4.12)ߩฦⴕ೉ⷐ⚛ߪએਅߩࠃ߁ߦߥ
ࠆ㧚

B11

1

2S

d

sin mI i
sin mI dI
S ³
dI

0

0

B22

1

2S

d

cos mI i
cos mI dI
S ³
dI

0

0

B12

1

2S

d

sin mI i
cos mI dI
S ³
dI

im

0

B21

1

2S

d

cos mI i
sin mI dI
S ³
dI

im

0

߆ߊߒߡᑼ(4.4.15)ߩ secular determinant ߪª b im º« im b » 0¬¼ߣߥࠆ㧚ߔߥࠊߜ

b2  m2ߎࠇࠍ⸃޿ߡ b0 m ߅ࠃ߮ b m ࠍᓧࠆ㧚ߔߥࠊߜ㧘operator i

d
ߦߟ޿ߡߩ
dI

eigenvalue ߪ m ߅ࠃ߮ m ߣ޿߁ߎߣߦߥࠆ㧚ᰴߦߘࠇߙࠇߩ eigenvalue ߩ႐วߦߟ
޿ߡ eigenfunction ࠍ᳞߼ࠃ߁㧚(i) b  m ߩ႐ว
ᑼ(4.4.11)ߪߤߜࠄ߽
 mc1  imc2 0ߣߥࠆ㧚ᓥߞߡ
c1 ic2
c1* ic2 *ߎߩ 2 ߟߩᑼࠍㄝ‫ޘ‬ដߌߡ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 23

c1 * c1

c2 * c2

normalization condition (4.4.16)
c1 * c1  c2 * c2 1
ߣวࠊߖߡ

1
2

c1 * c1
c1

1
2

c2

1
i
2

ߔߥࠊߜ࿕᦭୯ m ߦኻᔕߔࠆ࿕᦭㑐ᢙߪ1
sin mI  i cos mI2ߣߥࠆ㧚(ii) b  m ߩ႐ว
ᑼ(4.4.11)ߪߤߜࠄ߽
mc1  imc2 0਄ߣหߓࠃ߁ߦߒߡ
1
c1
2

c2



1
i
2

ߔߥࠊߜ㧘࿕᦭୯  m ߦኻᔕߔࠆ࿕᦭㑐ᢙߪ1
sin mI  i cos mI
2ߣߥࠆ㧚⚿ᨐࠍ߹ߣ߼ࠆߣᰴߩࠃ߁ߦߥࠆ㧚

wavefunction

d2ߦߟ޿ߡߩ࿕᦭୯
dI 2

1
sin mI  i cos mI
2

m2

m

1
sin mI  i cos mI
2

m2

m

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 24idߦߟ޿ߡߩ࿕᦭୯
dI

੤឵น⢻ߥ 2 ߟߩ Hermitian operator ߦ㑐ߒߡߪᦝߦᰴߩ㊀ⷐߥቯℂ߇޽ࠆ㧚ቯℂ㧦߽ߒ߽ 2 ߟߩṶ▚ሶ A ߣ B ߩ࿕᦭㑐ᢙߣߒߡ orthonormal ߥ wavefunction ߩ⚵ ^\ i `߇ᓧࠄࠇࠆߥࠄ߫ A ߣ B ߣߪ߅੕޿ߦ੤឵น⢻ߢ޽ࠆ㧚‫⸽ޣ‬᣿‫ޤ‬㧦\ i ߪ A ߅ࠃ߮ B ߩ࿕᦭㑐ᢙߢ޽ࠆ߆ࠄ
A\ i

ai\ i

B\ i

bi\ i

޽ࠆછᗧߩ wavefunction M ࠍ\ i ߢዷ㐿ߒࠃ߁㧚

¦ c\

M

i

i

i

ߎࠇߦṶ▚ሶ A, B ࠍ૞↪ߐߖࠆ㧚> A, B @M

AB  BA ¦ ci\ ii¦c a b b a \
i

i i

i i

i

0

i

ߔߥࠊߜ㧘

> A, B @0㧔⸽᣿⚳ࠊࠅ㧕ᦝߦᰴߩቯℂߪ㕖Ᏹߦᔕ↪ଔ୯ߩᄢ߈޿߽ߩߢ޽ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 25

ቯℂ㧦Ṷ▚ሶ A ߣ B ߇੤឵น⢻ߢ޽ࠅ㧘߆ߟ Hermitian ߢ޽ࠆߣߔࠆ㧚߹ߚ㧘\ i ߅ࠃ߮\ jߪ operator A ߩ eigenfunction ߢ޽ࠅ㧘ߘߩ႐วߩ eigenvalue ࠍߘࠇߙࠇ ai ߅ࠃ߮ a j ߣߔࠆ㧚ߔߥࠊߜ㧘
A\ i

ai\ i

A\ j

a j\ jߎߩ႐ว㧘 ai ߣ a j ߣ߇╬ߒߊߥ޿㒢ࠅⓍಽ ³\ i * B\ j dW ߪ࠯ࡠߢ޽ࠆ㧚‫⸽ޣ‬᣿‫ޤ‬㧦

³\ * BA\ dWi

a j ³\ i * B\ j dWj৻ᣇ㧘³\ * BA\ dW³\ * AB\ dW ( '> A, B @ 0)
³ B\ A *\ * dW (' A ߪ Hermitaian)a * ³\ *B\ dW
a ³\ *B\ dW ( ' Hermitian operatorߩeigenvalueߪ real)
i

j

i

j

j

i

i

i

i

i

j

j

ᓥߞߡ
ai  a j ³\ i * B\ j dW0ߔߥࠊߜ㧘 ai z a j ߥࠄ߫ ³\ i * B\ j dW

0 ߢ޽ࠆ㧚
㧔⸽᣿⚳ࠊࠅ㧕䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 26‫ڏ‬Problem 4.4.1
ᰴߩ੤឵㑐ᢙࠍ⸽᣿ߖࠃ㧚
AB, C
A, C B  A B, Ca)
A, BC
A, B C  B A, Cb)

(4.4.17)(4.4.18)‫ڏ‬Problem 4.4.2
ᰴߦ␜ߔ operator ߩ⚵ A ߣ B ߦߟ޿ߡ commutator

A, B ࠍ᳞߼ࠃ㧚㧔૞↪ߔࠆ߽ߩ

ߪ wavefunction ߢ޽ࠅ㧘ᒰὼ well-behaved ߢ޽ࠆߣ⠨߃ࠆ㧚㧕

A

B

(a)

d
dx

d2
d
2
2
dx
dx

(b)

x

d
dx

(c)

w
wx

w
wy

(d)

d2
dx2

x

(e)

d2
x
dx 2

d
 x2
dx

[Coffee Break] Commutator ߦ㑐ߔࠆ joke೨ߦ߽ᒁ↪ߒߚ"A Random Walk in Science"ߣ޿߁ᧄߩਛߩ"Physics terms madeeasy"ߦࠃࠆߣ"Commutator"ߩᗧ๧ߣߒߡ"A student who drives to school"ߣਈ߃ࠄࠇߡ
޿ࠆ㧚ߎߩ౬⺣ಽ߆ࠆ߆ߥ㧫䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 27

4.5 ‛ℂ㊂ߩหᤨ᷹ቯ㧦ਇ⏕ቯᕈේℂ߅੕޿ߦน឵ߢߥ޿ operator ߦኻᔕߔࠆ 2 ߟߩ‛ℂ㊂ࠍหᤨߦ᷹ቯߔࠆ႐วߩ᷹ቯ
ߩ߫ࠄߟ߈ࠍ⠨߃ߡߺࠆ㧚ቯℂ㧦
2 ߟߩ‛ℂ㊂ a ߅ࠃ߮ b ߦኻᔕߔࠆ Hermitian operator A ߅ࠃ߮ B ߩ commutator ࠍ iC
ߣ߅ߊ㧚ߔߥࠊߜ

> A, B @ iCߎߩ႐ว㧘޽ࠆ normalized wavefunction \ ߢ⴫ߐࠇࠆ☸ሶߩ㓸วߦߟ޿ߡߩ A ߩ᷹ቯ୯ a ߅ࠃ߮ B ߩ᷹ቯ୯ b ߩᮡḰ஍Ꮕߩᄢ߈ߐ G a ߅ࠃ߮ G b ߪᰴᑼߩࠃ߁ߦ೙㒢ߐࠇࠆ㧚

G aG b t

1\ * C\ dW
2³

ߎߩቯℂߩ⸽᣿ߪᰴߩࠃ߁ߦߒߡߥߐࠇࠆ㧚਄ㅀߒߚࠃ߁ߦ㧘2 ߟߩ operator A ߅ࠃ
߮ B ߦߟ޿ߡ

> A, B @ iC

(4.5.1)ߣ߅ߊ㧚⯯ᢙߩ i ࠍߟߌߚߩߪ㧘> x, px @ i= ߣߩ㘃ૃᕈࠍ⠨߃ߚ߹ߢߩߎߣߢ޽ࠆ㧚ߎߎߢ޽ࠆ wavefunction \ ࠍ⠨߃ࠆ߇㧘\ ߪᔅߕߒ߽ A ߿ B ߩ eigenfunction ߣߪ㒢ࠄߥ޿㧚 a ߅ࠃ߮ b ࠍ⁛┙ߦ᷹ቯߔࠆ႐วߩᐔဋ୯ A ߅ࠃ߮ B ߪAB³\ * A\ dW³\ * B\ dW(4.5.2)᷹ቯ୯ߩ߫ࠄߟ߈ࠍ⠨߃ࠆᤨ㧘ᐔဋ୯߆ࠄߩ߫ࠄߟ߈ࠍ⠨߃ࠆߩ߇ଢ೑ߢ޽ࠆ㧚ߘߎߢᰴߩࠃ߁ߥ operator ߣߒߡ 'A ߅ࠃ߮ 'B ࠍቯ⟵ߔࠆ㧚
'A

A A

'B

B B

(4.5.3)'A ߣ 'B ߣߩ੤឵㑐ଥ߇ A ߣ B ߣߩߘࠇߣหߓߢ޽ࠆߎߣߪᰴߩࠃ߁ߦߒߡ◲නߦ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 28㧕ಽ߆ࠆ㧚㧔 'A ߅ࠃ߮ 'B ߪනߥࠆᢙߢ޽ࠆ߆ࠄ㧘ߤࠎߥ operator ߣ߽น឵ߢ޽ࠆ㧚> 'A, 'B @ª¬ A  A , B  B º¼
> A, B @  ª¬ A , B º¼  ª¬ A, B º¼  ª¬ A , B º¼

> A, B @(4.5.4)iC
ߐߡ㧘ᰴߩࠃ߁ߥⓍಽ I ࠍ⠨߃ࠆ㧚
I

2³ D'A  i'B \ dW

ߎߎߢ D ߪછᗧߩታᢙ㧚ᒰὼ
I t0(4.5.5)(4.5.6)ߢ޽ࠆ㧚෶㧘ᑼ(4.5.5)ߩ integrand ߪᰴߩࠃ߁ߥᗧ๧ߢ޽ࠆ㧚

D'A  i'B \2^ D'A  i'B \ ` *^ D'A  i'B \ `^ D'A  i'B \ *`^ D'A  i'B \ `(4.5.7)‫ޤ⸼ޣ‬㧦
Ⓧಽߪ"integral"㧘ⵍⓍಽ㑐ᢙߪ"integrand"ߢ޽ࠆ㧚ߤ߁߽ integrand ࠍ integral ߣหߓߛߣᕁߞߡ޿ࠆቇ↢ߐࠎ߇ᄙ޿ࠃ߁ߥߩߢ㧘ᵈᗧߒߡ߶ߒ޿㧚
operator A ߅ࠃ߮ B ߪ Hermitian ߢ޽ࠆ㧔ߥߗߥࠄߘࠇࠄߪ‛ℂ㊂ߦ⋧ᒰߔࠆ߆ࠄ㧕
ᓥߞߡᑼ(4.5.7)ߩ integral ߪᰴߩࠃ߁ߦߥࠆ㧚I

D ³ 'A *\ * D'A  i'B \ dW  i ³ 'B *\ * D'A  i'B \ dW
D ³ \ * 'A D'A  i'B \ dW  i ³\ * 'B D'A  i'B \ dW

(4.5.8)³\ * D'A  i'B D'A  i'B \ dW‫ޤ⸼ޣ‬㧦

Hermitian operator ߦ㑐ߔࠆᑼ(4.2.2)ࠍᄌᒻߒߚᑼߢ޽ࠆ ³ A\ * M dW↪޿ߡ޿ࠆ㧚ߎߩᑼ߽ Hermitian operator ߩቯ⟵ᑼߣ߽ߺߥߐࠇࠆ㧚㧕ᦝߦ㧘 I ࠍᰴߩࠃ߁ߦዷ㐿ߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 29³\ * AM dW ࠍ

³\ *^DI`'A  'B  iD 'A'B  iD 'B'A \ dW2

2

2

D 2 'A  'B  iD > 'A, 'B @2

D 2 'A2

 D C  'B

2

(4.5.9)2

ߎࠇߪ D ߣ޿߁ real ߩᢙߦߟ޿ߡߩ 2 ᰴᣇ⒟ᑼߢ޽ࠆ㧚 D ߩ୯ߦࠃࠄߕᑼ(4.5.6)߇ᚑ
ࠅ┙ߟߚ߼ߦߪ㧘ߔߥࠊߜ㧘ᑼ(4.5.9)߇ᱜߢ޽ࠆߚ߼ߦߪ 2 ᰴᣇ⒟ᑼ(4.5.9)ߩ್೎ᑼ
߇⽶ߢߥߌࠇ߫ߥࠄߥ޿㧚C2

 4 'A

2

2

2

'B

2

d0

(4.5.10)

ߔߥࠊߜ

'A

'B

t

1
2
C
4

(4.5.11)ߎߎߢᏀㄝߩᐔဋ୯ߪᑼ(4.5.3)ࠍ↪޿ߡᰴߩࠃ߁ߦ⴫ߐࠇࠆ㧚

'A

2

A A

2

A2  2 A A  A
2A2  2 A  A
A2  A2

2

2

ߎࠇߪ A2 ߩ᷹ቯ୯㧘ߔߥࠊߜ‛ℂ㊂ a 2 ߩ᷹ቯߦ㑐ߔࠆᮡḰ஍Ꮕߢ޽ࠆ㧚ߎߩᐔᣇᩮ
ࠍ G a ߣ⴫ߔ㧚G a ߪ᷹ቯ୯ a ߩ߫ࠄߟ߈ࠍ⴫ߔዤᐲߢ root mean square deviation ߣ⸒ࠊࠇࠆ㧚Ga

A2  A

2(4.5.12)

ߣ߅ߊߣ
'A

2

Ga

2

ห᭽ߦ

Gb

B2  B

2

ߣ߅ߊߣ
'B

2

Gb

2

ߣߥࠆ㧚߆ߊߒߡ㧘ᰴᑼࠍᓧࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 30(4.5.13)

G aG b t

1
C
2

(4.5.14)ߎߩࠃ߁ߦߒߡቯℂ߇⸽᣿ߐࠇߚ㧚ߎߩᑼߪ‛ℂ㊂ a ߣ b ࠍหᤨ᷹ቯߔࠆ႐วߩਇ⏕ቯᕈࠍ␜ߔ߽ߩߢ㧘৻⥸ߦ‫ޟ‬ਇ⏕ቯᕈේℂ‫ࠆ޿ߡࠇࠊ⸒ߣޠ‬㧚ਇ⏕ቯᕈේℂߩౕ૕଀ߩ߁ߜߢ߽ߞߣ߽㊀ⷐߥ߽ߩߪ㧘૏⟎ߣㆇേ㊂ߩ᷹ቯߦ㑐ߔࠆ߽ߩߢ޽ࠆ㧚ߎߩ႐ว㧘 A

x 㧔 x ߪ‫ ޟ‬x ࠍដߌࠆ‫ ߁޿ߣޠ‬operator㧕㧘Bpx 㧔 px ߪ

x ᣇะߩㆇേ㊂ operator㧕ߣ߅ߊߣ㧘ߔߢߦቇ⠌ߒߚࠃ߁ߦ

> x, p x @ i =ߢᑼ(4.5.1)ߩᒻߣߥࠆ㧚 C(4.5.15)= 㧘ߔߥࠊߜ㧘 C ߪ = ߣ޿߁ᢙࠍដߌࠆߣ޿߁ operator ߢ޽ࠆ߆ࠄ㧘ᒰὼߘߩᦼᓙ୯ߪ C= ߢ޽ࠆ㧚߆ߊߒߡ1
2

G xG px t =(4.5.16)ߣ޿߁㧘ࠃߊ⍮ࠄࠇߚᒻߩ Heisenberg ߩਇ⏕ቯᕈේℂ㧔uncertainty principle㧕߇ᓧࠄࠇࠆ㧚߽߁ 1 ߟ㊀ⷐߥ uncertainty principle ߣߒߡᰴߩᑼ߇޽ࠆ㧚

1
2

G EG t t =(4.5.17)ߔߥࠊߜ㧘ࠛࡀ࡞ࠡ࡯ E ߣᤨ㑆 t ߩหᤨ᷹ቯߦਇ⏕ቯᕈ߇޽ࠆߣ޿߁ߎߣߢ޽ࠆ㧚ߎߩᑼߩዉ಴ߪᰴߩࠃ߁ߦߒߡߥߐࠇࠆ㧚߹ߕ㧘ࠛࡀ࡞ࠡ࡯ E ࠍਈ߃ࠆṶ▚ሶࠍ᳞߼ߥߌࠇ߫޿ߌߥ޿㧚ᤨ㑆ߦଐሽߒߥ޿
Schrödinger equation (3.2.11)ߣᤨ㑆ߦଐሽߔࠆ Schrödinger equation (4.3.1)ߣߩᲧセ߆ࠄ

E o i=

w
wt(4.5.18)

ߣ޿߁ኻᔕ߇ᓧࠄࠇࠆ㧚ᤨ㑆ࠍਈ߃ࠆṶ▚ሶߣ޿߁߽ߩߪߥ޿㧚ߒ߆ߒ㘃ផ߆ࠄߘࠇ
ࠍ t ߣ߅ߊߎߣߦߒࠃ߁㧚ߘ߁ߔࠆߣ E ߣ t ߣߩ੤឵㑐ଥߪᑼ(4.4.5)ࠍ↪޿ߡ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 31

ª w º«¬i= wt , t »¼ i=ߣߥࠆ㧚߆ߊߒߡᑼ(4.5.17)߇ᓧࠄࠇࠆ㧚৻ᣇ㧘ᑼ(4.5.17)ߪᰴߩࠃ߁ߦ⠨߃ߡ߽ዉ಴ߔࠆߎߣ߇಴᧪ࠆ㧚> E, t @(4.5.19)

2

E

px
2m

GE

px
Gpx
m

ࠃࠅ

ߢ޽ࠆ߇㧘ߎߎߢ

GE

px
Gxߪㅦᐲߢ޽ࠆ߆ࠄ
ߣ߅ߌࠆ㧚ᓥߞߡ
Gt
m

Gx
G px
Gt

ߔߥࠊߜ

G EG t G xG px߆ߊߒߡᑼ(4.5.16)ߣߩᲧセ߆ࠄᑼ(4.5.17)߇ᓧࠄࠇࠆ㧚[Coffee Break] Heisenberg ߽᥉ㅢߩੱ㧫ਇ⏕ቯᕈේℂߣ౒ߦਇṌߩฬࠍᱷߒߚ Heisenberg ߪߘߩઁ㊂ሶജቇߩಽ㊁ߢᢙ‫ޘ‬ߩఝࠇߚᬺ❣ࠍᱷߒߚᄤᚽߢ޽ࠅ㧘ಠੱ߆ࠄ⷗ࠇ߫␹᭽ߩࠃ߁ߦ߽ᕁ߃ࠆ߇㧘ߘߩ
Heisenberg ߽㧘㋦⫾㔚ᳰߩℂ⺰ߩߎߣࠍ૗߽⍮ࠄߥ޿ߣ߆޿߁ℂ↱ߢࡒࡘࡦࡋࡦᄢቇ
ߢඳ჻ߩቇ૏ࠍขᓧߔࠆߎߣߦᄬᢌߒߚߣ߆㧘ጊࠍᱠ޿ߡ޿ߡ㧘ᷧ⼱ߦ⪭ߞߎߜߡ㧘߽߁ዋߒߢ๮ࠍ⪭ߔߣߎࠈߛߞߚߣ޿߁⹤ࠍ⡞ߊߣ㧘߿ߞ߬ࠅߎߩੱ߽ੱ㑆ߛߞߚߩ߆ߣ޿߁ⷫߒߺࠍᗵߓߐߖࠆ㧚߽ߞߣ߽㧘⼱ߦ⪭ߜߚߎߣߢ޽ߩ᦭ฬߥ⺰ᢥ߇ᦠߌߚ
ߣ޿߁ߎߣߪ߿ߪࠅಠੱߣߪ㆑߁ߣ޿߁ߎߣ߆㧚He ේሶߩ㔚ሶ⁁ᘒߩℂ⺰ࠍᛒߞߚߎߣߢ᦭ฬߥ Egil A. Hylleraas ߇㧘"Reminiscencesfrom Early Quantum Mechanics of Two-Electron Atoms"ߣ޿߁ᕁ޿಴⹤ࠍ Reviews ofModern Physics ߩ Vol. 35, p.421 (1963)ߦᦠ޿ߡ޿ࠆ㧚ߎߩ⹤ߩਛߦ Heisenberg ߩߎߣ߇タߞߡ޿ࠆ㧚㕙⊕޿ߩߢߘߩㇱಽࠍోߡએਅߦᛞ㍳ߔࠆ㧚

Heisenberg, perhaps the most wonderful of Sommerfeld's "Wunderkinder" had not
succeeded in acquiring a doctor's degree in Munich. The main reason for this misfortune
is said to be some unnecessary stubbornness, together with a complete lack of䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 32

knowledge of the theory of the lead accumulator. Max Born obviously did not pay somuch attention to the lead accumulator, so in a very short time Heisenberg foundhimself a doctor of Georgia Augusta University of Göttingen. Then he felt a deep desire
for complete freedom and went to Norway, walking in the mountains for several weeks
entirely alone and with no connection with his family. This might have cost him his life.
One day, when trying to pass a stream, he fell into the water and had a very narrowescape. Back in some hotel he wrote his famous paper. How much the cold bath may
have contributed to clearing his mind, I cannot tell.
‫ڏ‬Problem 4.5.1
Heisenberg ߩ uncertainty principle ߪ㧘޽ࠆᕁ⠨ታ㛎ߩࡕ࠺࡞ࠍ↪޿ߡዉ಴ߒߡ޿ࠆ႐ ว ߇ ࠃ ߊ ᢎ ⑼ ᦠ ߢ ߺ ߆ ߌ ࠄ ࠇ ࠆ 㧚 ଀ ߃ ߫ Eyring, Walter, Kimball ߩ "Quantum
Chemistry"ߢߪ㧘㗼ᓸ㏜ߢ૏⟎ߣㆇേ㊂ࠍ᷹ⷰߔࠆ႐วࠍᛒߞߡ޿ࠆ㧚ߎߩᣇᴺࠍቇ
⠌ߒ㧘਄⸥ߩ⚿ᨐߣᲧセߒߡߺࠃ㧚․ߦᑼ(4.5.16)ߩ⚿ᨐߣߤ߁⇣ߥࠆ߆㧫‫ڏ‬Problem 4.5.2
ᰴߩฦ႐ว㧘2 ߟߩ‛ℂ㊂ߪหᤨߦᱜ⏕ߦ᷹ቯน⢻߆ุ߆㧘ℂ↱ࠍㅀߴߡ್ᢿߖࠃ㧚(a) ᐳᮡߣㆇേࠛࡀ࡞ࠡ࡯
(b) x ᐳᮡߣ y ᣇะߩㆇേ㊂(c) x ᐳᮡߣ y ᐳᮡ(d) x ᣇะߩㆇേ㊂ߣ y ᣇะߩㆇേ㊂‫ڏ‬Problem 4.5.3(a) 㔚ሶߩ૏⟎߇ 100 pm એౝߩ♖ᐲߢಽ߆ߞߡ޿ࠆ႐วߩㅦᐲߩਇ⏕ቯᕈࠍ᳞߼ࠃ㧚(b) 㔚ሶߩㅦᐲ߇ 1 mm/s એౝߩ♖ᐲߢಽ߆ߞߡ޿ࠆ႐วߩ૏⟎ߩਇ⏕ቯᕈࠍ᳞߼
ࠃ㧚‫ڏ‬Problem 4.5.4(a) ࡊࡠ࠻ࡦߩ૏⟎߇ 100 pm એౝߩ♖ᐲߢಽ߆ߞߡ޿ࠆ႐วߩㅦᐲߩਇ⏕ቯᕈࠍ᳞߼ࠃ㧚(b) ࡊࡠ࠻ࡦߩㅦᐲ߇ 1 mm/s એౝߩ♖ᐲߢಽ߆ߞߡ޿ࠆ႐วߩ૏⟎ߩਇ⏕ቯᕈࠍ᳞߼ࠃ㧚‫ڏ‬Problem 4.5.5
ࠛࡀ࡞ࠡ࡯ߩਇ⏕ቯߐ߇㧘ᵄᢙන૏ߢ a) 1000 cm 1 㧘b) 1 cm 1 㧘c) 103 cm 1 ߩ႐䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 33วߩߘࠇߙࠇߦߟ޿ߡ㧘ᤨ㑆ߩਇ⏕ቯᕈࠍ᳞߼ࠃ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 4: Page 34

CHAPTER 05
ⷺㆇേ㊂ฎౖജቇߩ਎⇇ߢ߽㧘ᖺᤊߩ゠㆏ㆇേ߿೰૕ߩ࿁ォㆇേࠍ⺰ߕࠆᤨߦⷺㆇേ㊂߇㊀
ⷐߥ‛ℂ㊂ߢ޽ࠆ߇㧘㊂ሶജቇߩ਎⇇ߢߪⷺㆇേ㊂ߪࠃࠅ㊀ⷐߢ޽ࠆ㧚․ߦ㧘㊂ሶജቇߩൻቇ߳ߩᔕ↪ߩ㕙ࠍ⠨߃ࠆߣ㧘߽ߞߣ߽ၮᧄ⊛ߥൻቇ⒳ߢ޽ࠆ᳓⚛ේሶߩᵄേ㑐ᢙࠍ⺰ߕࠆ႐ว㧘ⷺㆇേ㊂ߪ㊀ⷐߢ߆ߟၮᧄ⊛ߥ‛ℂ㊂ߢ޽ࠆ㧚ߎ߁޿߁ߎߣߢ㧘㊂ሶജቇߩീᒝߦ㧘ⷺㆇേ㊂ߩ໧㗴ࠍㆱߌߡㅢࠆࠊߌߦߪ޿߆ߥ
޿㧚ߒ߆ߒ㧘ߒ߫ߒ߫㧘‛ℂ⊛ࠗࡔ࡯ࠫ߇ߪߞ߈ࠅߒߥ޿߁ߜߦ߿߿ߎߒ޿ᢙᑼࠍ޿
ߓߊࠄߥߊߡߪ޿ߌߥ޿ߩߢ㧘ߎߩ޽ߚࠅߢ㊂ሶജቇߩീᒝ߇ህߦߥߞߡߒ߹߁ቇ↢⻉ำ߇ᄙ޿ߎߣ߽੐ታߢ޽ࠆ㧚߹޽㧘㊂ሶജቇߩ㕙⊕ߐࠍℂ⸃ߔࠆߚ߼ߩᦨೋߩ㑐㐷
߇ⷺㆇേ㊂ߩ໧㗴ߣ߽⸒߃ࠃ߁㧚ߎߎࠍㆊ߉ࠆߣ㧘ᓟߪᕆߦᭉߦߥࠆ߆ࠄ㧘ߥࠎߣ߆ߒߡ߇ࠎ߫ߞߡീᒝߒߡ߽ࠄ޿ߚ޿㧚ߥ߅㧘ⷺㆇേ㊂ߦ㑐ߒߡߪ㧘߅޿߅޿᣿ࠄ߆ߦߒߡ޿ߊ߇㧘㊂ሶജቇ߹ߚߪ㊂ሶൻ
ቇߩᢎ⑼ᦠߩᄙߊߢ㧘ታߦ㊀ⷐߥ㑆㆑޿ࠍ‽ߒߡ޿ࠆ㧚ᧄ┨ߢߩᛒ޿ߪ㧘ᄢ╭ߦ߅޿ߡ㧘ઁߩᢎ⑼ᦠߣૃߡ޿ࠆ߇㧘⢄ᔃߥᚲߪ㧘߈ߜࠎߣ߅ߐ߃ߡ㧘෩ኒߦขࠅᛒߞߡ޿
ࠆߟ߽ࠅߢ޽ࠆ㧚ઁߩᢎ⑼ᦠߣᲧߴߥ߇ࠄᵈᗧᷓߊቇ⠌ߒߡ߶ߒ޿ߣ㗿ߞߡ޿ࠆ㧚

5.1 ⷺㆇേ㊂Ṷ▚ሶߩ੤឵㑐ଥ
ⷺㆇേ㊂Ṷ▚ሶߪߔߢߦChapter 4ߢዉ౉ߒߚ㧚ᔨߩߚ߼ߦౣ߮⴫ߒߡ߅ߊ㧚
Lx
Ly
Lz

§ w
w ·
i= ¨ y  z ¸
wy ¹
© wz
w ·
§ w
i= ¨ z  x ¸
wz ¹
© wx
§ w
w ·
i= ¨ x  y ¸
wx ¹
© wy

(5.1.1)

Lx , Ly , Lz ࠍᚑಽߣߔࠆvector Lx , Ly , Lz ࠍ L ߣᦠߊߎߣߦߒࠃ߁㧚ߔߥࠊߜ
L iLx  jLy  kLz(5.1.2)bg

Lx , Ly , Lz ߥߤߩ㑆ߩ੤឵㑐ଥࠍ⺞ߴࠃ߁㧚છᗧߩ㑐ᢙ \ x , y , z ߦ㧘଀߃߫㧘Lx ߣ Ly ߩ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 1

Ⓧࠍ૞↪ߒߡߺࠆ㧚

Lx Ly\

w
w · § w\w\ ·2§
i= ¨ y  z ¸¨ z
x¸
wy ¹ © wx
wz ¹
© wz
w\
w 2\
w 2\
w 2\
w 2\ ·2§
i= ¨ y yz
 xy 2  z 2
 zx
¸
wzwx
wz
wywx
wywz ¹© wx

Ly Lx\

w\ · § w
w ·2 § w\
i= ¨ zx¸¨ y  z ¸
wz ¹ © wz
wy ¹
© wx
2
w 2\
w 2\
w 2\ ·
2 § w\
2 w \
i
x
xz
xy
z
yz
= ¨




¸
wzwy
wz 2
wxwy
wxwz ¹© wy

ᓥߞߡLx Ly  Ly Lx \

w\
w\ ·
2§
 i= ¨ y
x
¸
wy ¹
© wx
§ w
w ·
i= i= ¨ x  y ¸\
wx ¹
© wy

i=Lz\

\ ߪછᗧߢ޽ࠆ߆ࠄ Lx Ly  Ly Lx

i=Lz ߢߥߌࠇ߫ߥࠄߥ޿㧚ઁߩ⚵ߺวߖߦߟ޿ߡ߽

ห᭽ߢ޽ߞߡ㧘commutatorࠍ↪޿ߡᦠߊߣᰴᑼߩࠃ߁ߦߥࠆ㧚ª¬ Lx , Ly º¼ i=Lzª¬ Ly , Lz º¼ i=Lx

(5.1.3)> Lz , Lx @ i=Ly‫ڏ‬Problem 5.1.1
L2

Lx 2  Ly 2  Lz 2

(5.1.4)ߣߒߚᤨ㧘 L2 ߪ Lx , Ly , Lz ߩ޿ߕࠇߣ߽commutable㧘ߔߥࠊߜª¬ L2 , Lx º¼ª¬ L2 , Ly º¼ª¬ L2 , Lz º¼

ߣߥࠆߎߣࠍ⸽᣿ߖࠃ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 2(5.1.5)

Hint㧦operator A, B, C ߦߟ޿ߡ৻⥸⊛ߦᚑࠅ┙ߟᰴᑼࠍ↪޿ࠆߣଢ೑ߢ޽ࠆ㧚A  B, Cb A  BgC  Cb A  Bg

AB, CABC  ACB  ACB  CAB

A, C  B, C
A B, C  A, C B(5.1.6)(5.1.7)

5.2 ⷺㆇേ㊂Ṷ▚ሶߩᭂᐳᮡߦࠃࠆ⴫⃻⋥੤ᐳᮡ x, y, z ߩઍࠅߦᭂᐳᮡ polar coordinate r , T , I ߇ߒ߫ߒ߫↪޿ࠄࠇࠆ㧚࿑ 5.2.1ᭂᐳᮡਔᐳᮡ㑆ߩᄌ឵ߪ㧘࿑ 5.2.1 ߆ࠄ߽᣿ࠄ߆ߥࠃ߁ߦ
x r sin T cos I
y r sin T sin I
z r cos T(5.2.1)ߣߥࠆ㧚ㅒߩᄌ឵ߪ
r2

x2  y2  z 2
z

cos T
tan I

x2  y2  z 2yx䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 3(5.2.2)ߎࠇࠄࠃࠅᰴߩ㑐ଥ߇ᓧࠄࠇࠆ㧚wr
wx

w x2  y 2  z 2wxxr

sin T cos Iwrwy

w x2  y 2  z 2wyyr

sin T sin I

wrwz

w x2  y2  z 2
wz

z
r

cos T

wT
wx
wT
wy
wT
wz

1 w cos T sin T wx
1 w cos T sin T wy
1 w cos T sin T wz

cos T cos Ir
cos T sin I
r
sin T

r

wI
wx
wI
wy
wI
wz

w tan Iwx
w tan Icos 2 Iwy
w tan Icos 2 Iwz

cos 2 Isin I
r sin Tcos I
r sin T(5.2.3)(5.2.4)(5.2.5)

0

ᓥߞߡ
w
wx

wr w wT w wI w
wx wr wx wT wx wIw 1
w 1 sin I wsin T cos I  cos T cos I
wr r
wT r sin T wI

(5.2.6)w
wy

wr w wT w wI w

wy wr wy wT wy wI
w 1
w 1 cos I w

sin T sin I  cos T sin I
wr r
wT r sin T wI

(5.2.7)w
wz

wr w wT w wI w
wz wr wz wT wz wIw 1w
cos T  sin T
wr r
wT(5.2.8)ߎࠇࠄߩᑼࠍᑼ(5.1.1)ߦઍ౉ߔࠆߎߣߦࠃߞߡᰴߩᑼࠍᓧࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 4

Lx
Ly

§
w
w ·
i= ¨  sin I cot T cos I
¸
wT
wI ¹
©
§
w
w ·
i= ¨ cos I cot T sin I
¸
wT
wI ¹
©

(5.2.9)

w
wI

Lz

i =

L2

­ 1 w §w ·1 w2 ½
= 2 ®

sin
T
¾
¨
¸
wT ¹ sin 2 T wI 2 ¿¯ sin I wT ©

߹ߚ

(5.2.10)‫ڏ‬Problem 5.2.1
ᑼ(5.2.9)ᑼࠍ⏕߆߼ࠃ㧚‫ڏ‬Problem 5.2.2
ᑼ(5.2.9)߅ࠃ߮ᑼ(5.2.10)ࠍ↪޿ߡᑼ(5.1.5)ߩ commutation relation ߇ᚑࠅ┙ߟߎߣࠍ⏕߆߼ࠃ㧚

5.3 ⷺㆇേ㊂Ṷ▚ሶߩ࿕᦭㑐ᢙߣ࿕᦭୯
ⷺㆇേ㊂Ṷ▚ሶ Lx , Ly , Lz ߪ߅੕޿ߦน឵ߢߥ޿߆ࠄ㧘ߎࠇࠄߔߴߡߦ౒ㅢߥ࿕᦭㑐ᢙߪ৻⥸ߦߪሽ࿷ߒߥ޿㧚ߒ߆ߒ㧘ᑼ(5.1.5)ߢ⷗ߚࠃ߁ߦ㧘 Lx , Ly , Lz ߪ޿ߕࠇ߽ L2 ߣน឵ߢ޽ࠆ߆ࠄ㧘ߎࠇࠄ 3 ߟߩ߁ߜߩ޿ߕࠇ߆ 1 ߟߣ L2 ߣߩ౒ㅢߥ࿕᦭㑐ᢙߪሽ࿷ߔ
ࠆ㧚 Lx , Ly , Lz ߩਛߩߤࠇࠍㆬࠎߢ߽޿޿ߩߢ㧘ߎߎߢߪ Lz ࠍㆬ߱ߎߣߦߒࠃ߁㧚ߔߥࠊߜ㧘ª¬ Lz , L2 º¼

0(5.3.1)ߦ⌕⋡ߒߡ㧘 Lz ߣ L2 ਔ⠪ߦ౒ㅢߥ࿕᦭㑐ᢙࠍቯ⟵ߒࠃ߁㧚ߎߩ࿕᦭㑐ᢙࠍ YA ,m ߣ⴫ߔߎߣߦߔࠆ㧚ߘߒߡ L2 ߦኻߔࠆ࿕᦭୯ࠍ kA ߣߒ㧘 Lz ߦߟ޿ߡߩ࿕᦭୯ࠍ km ߣߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 5ߔߥࠊߜ
L2YA ,mkAYA , m(5.3.2)

LzYA ,m

kmYA ,m

(5.3.3)ᑼ(5.3.2)ࠍᦠ߈⋥ߒߡLx  Ly  Lz YA ,m2

2

2

kAYA ,m

(5.3.4)߹ߚ㧘ᑼ(5.3.3)ߦᦝߦ Lz ࠍ૞↪ߒߡ2

2

Lz YA ,mkm YA ,m(5.3.5)ㄝ‫ޘ‬ᒁ߈▚ࠍߒߡLx  Ly YA ,m2

2

kA  km YA ,m2(5.3.6)

߆ߊߒߡ㧘 YA ,m ߪ L2 㧘 Lz ߅ࠃ߮ Lx  Ly ߩ࿕᦭㑐ᢙߦߥߞߡ޿ࠆߎߣ߇ಽ߆ࠆ㧚 Lx ߅22

ࠃ߮ Ly ߪ޿ߕࠇ߽Hermitianߢ޽ࠆ߆ࠄߘߩ࿕᦭୯ߪrealߢ޽ࠆ㧚ᓥߞߡ Lx  Ly ߩ࿕
22᦭୯ߪ⽶ߦߪߥࠄߥ޿㧚ߔߥࠊߜkA t km2(5.3.7)ߐߡ㧘ߎߎߢᣂߒߊ㧘operator L ߅ࠃ߮ L ࠍቯ⟵ߔࠆ㧚L

Lx  iLy

L

Lx  iLy(5.3.8)

ߎࠇߣ Lz ߣߩ੤឵㑐ଥࠃࠅᰴᑼࠍᓧࠆ㧚> Lz , L @ > Lz , Lx @  i ª¬ Lz , Ly º¼(5.3.9)> Lz , L @ > Lz , Lx @  i ª¬ Lz , Ly º¼(5.3.10)ᑼ(5.1.3)ߩ੤឵㑐ଥߣᑼ(5.3.9)ࠃࠅ
Lz L

L Lz  =YA ,m ߦᑼ(5.3.11)ߩਔㄝࠍ૞↪ߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 6(5.3.11)

Lz LYA ,mL Lz  = YA ,mkm  = LYA ,m(5.3.12)ߎߩߎߣ߆ࠄ㧘㧔a㧕 LYA ,m ߪ Lz ߩ࿕᦭㑐ᢙߢ޽ࠅ㧘ߘߩ࿕᦭୯ߪ km  = ߢ޽ࠆ߆㧘߹ߚߪ㧘㧔b㧕 LYA ,m ߪ࠯ࡠߢ޽ࠆ߆㧘ߩ޿ߕࠇ߆ߢ޽ࠆߣ޿߃ࠆ㧚หߓࠃ߁ߦ㧘ᑼ(5.3.10)ࠃࠅ
Lz L

L Lz  =(5.3.13)ߎߩਔㄝࠍ YA ,m ߦ૞↪ߔࠆ㧚

Lz LYA , m

L Lz  = YA ,mkm  = LYA ,m(5.3.14)ᓥߞߡ㧘㧔a㧕LYA ,m ߪ Lz ߩ࿕᦭㑐ᢙߢߘߩ࿕᦭୯ߪ km  = ߢ޽ࠆ߆㧘߹ߚߪ㧘㧔b㧕LYA ,mߪ࠯ࡠߢ޽ࠆ߆㧘ߩ޿ߕࠇ߆ߢ޽ࠆ㧚਄ㅀߒߚࠃ߁ߦ Lz ߪ L2 ߣน឵ߢ޽ࠆ㧚ᓥߞߡ LYA ,m ߽ LYA ,m ߽㧘޿ߕࠇ߽ L2 ߩ࿕᦭㑐ᢙߢ޽ࠅ㧘ߘߩ࿕᦭୯ߪ kA ߢ޽ࠆ㧚ߎ߁޿߁ߎߣߢ㧘 L2 ߣ Lz ߩ࿕᦭㑐ᢙߩ♽೉߇ሽ࿷ߔࠆ㧚ߎߎߢ L2 ߦ㑐ߔࠆ࿕᦭୯ߪߔߴߡ kA ߢ޽ࠅ㧘 Lz ߦ㑐ߔࠆ࿕᦭୯ߪ
(5.3.15)..., km  2=, km  =, km , km  =, km  2=, ...
ߢ޽ࠆ㧚ߎߩ♽೉ߪਔ஥ߢㅜ⛘߃ࠆ㧚ߣ޿߁ߩߪ㧘ᑼ(5.3.7)߇␜ߔࠃ߁ߦ Lz ߦߟ޿ߡߩ࿕᦭୯ߩੑਸ਼ߪ kA ࠃࠅ߽ዊߐ޿߆ࠄߢ޽ࠆ㧚ߘߎߢ㧘 Lz ߩ࿕᦭୯ߩਅ㒢ࠍ kmmin 㧘ߘߩᤨߩ࿕᦭㑐ᢙࠍ YA ,mmin ߣߒࠃ߁㧚ߔߥࠊߜ

LzYA ,mmin

kmmin YA ,mmin 㧔 m ߩਅ㒢㧕

(5.3.16)ห᭽ߦ㧘 Lz ߩ࿕᦭୯ߩ਄㒢ࠍ kmmax 㧘ߘߩᤨߩ࿕᦭㑐ᢙࠍ YA ,mmax ߣߒࠃ߁㧚

LzYA ,mmax

kmmax YA ,mmax 㧔 m ߩ਄㒢㧕

(5.3.17)ߐߡ㧘ߎߩ਄㒢ߩ࿕᦭㑐ᢙ YA ,mmax ߦ L ࠍ૞↪ߒߚ߽ߩ㧘 LYA ,mmax ࠍ⠨߃ࠃ߁㧚਄ㅀߒ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 7ߚࠃ߁ߦ㧔a㧕 LYA ,mmax ߪ Lz ߩ࿕᦭㑐ᢙߢ޽ࠅߘߩ࿕᦭୯ߪ kmmax  = ߢ޽ࠆ߆㧘߹ߚߪ㧔b㧕 LYA ,mmax ߪ࠯ࡠߢ޽ࠆ߆㧘ߩ޿ߕࠇ߆ߢ޽ࠆ㧚 kmmax ߇ Lz ߩ࿕᦭୯ߩᦨᄢߣ᳿߼ߚ
ߩߢ޽ࠆ߆ࠄ㧘ߘࠇࠃࠅ߽ᄢ߈ߥ㧘kmmax  = ߇࿕᦭୯ߣߥࠆߪߕߪߥ޿㧚ߔߥࠊߜ㧔a㧕ߪ޽ࠅᓧߥ޿㧚ᓥߞߡ㧔b㧕߇ᚑࠅ┙ߚߥߌࠇ߫ߥࠄߥ޿㧚ߔߥࠊߜLYA , mmax

0(5.3.18)

ోߊหߓࠃ߁ߦߒߡ

LYA , mmin

0(5.3.19)ࠍᓧࠆ㧚ᰴߦᑼ(5.3.18)ߦ L ࠍ૞↪ߔࠆ㧚
L LYA ,mmax

Lx  i Ly

Lx  iLy YA ,mmax^L  L  i L L  L L ` Y
2

x

2

y

x

y

y

A , mmaxx

Lx 2  Ly 2  =Lz YA ,mmax(5.3.20)

L  Lz  =Lz YA ,mmax
2

2kA  kmmax 2  =kmmax YA ,mmax
0ߎߎߢ਄߆ࠄ3ⴕ⋡ߩᑼߢߪᑼ(5.1.3)ࠍ↪޿ߡ޿ࠆ㧚ᓥߞߡ

kA

kmmax  =kmmax
2(5.3.21)ోߊหߓࠃ߁ߦ㧘ᑼ(5.3.19)ߦ L ࠍ૞↪ߔࠆ㧚
L LYA ,mmin

Lx  iLy

Lx  iLy YA ,mmin^L  L  i L L  L L ` Y
2

x

2

y

x

y

y

x

A , mminLx 2  Ly 2  =Lz YA , mmin
L  Lz  =Lz YA ,mmin
2

2kA  kmmin 2  =kmmin YA , mmin0ߎࠇࠃࠅ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 8(5.3.22)

kA

kmmin  =kmmin
2(5.3.23)

ᑼ(5.3.21)ߣᑼ(5.3.23)ࠃࠅ

kmmax  =kmmax

kmmin  =kmmin

22(5.3.24)ߔߥࠊߜ

kmmax  kmmin

kmmax  kmmin  =0(5.3.25)

kmmax ! kmmin ߣ઒ቯߒߚࠊߌߢ޽ࠆ߆ࠄ਄ᑼߩ2⇟⋡ߩ᜝ᒐౝ߇࠯ࡠߦߥࠆߎߣߪߥ޿㧚ᓥߞߡ kmminkmmax(5.3.26)

ᑼ(5.3.15)ߦ⌕⋡ߔࠆߣ㧘 kmmax  kmmin ߪ = ߩᢛᢙ୚ߢ޽ࠆߎߣ߇ಽ߆ࠆ㧚ߎߩᢛᢙࠍ 2A

1 3
ߣ߅ߎ߁㧚ߎߎߢ A䋽0䋬 , 1, , 2, ... ߢ޽ࠆ㧚ᓥߞߡ㧘ᑼ(5.3.26)ࠍ↪޿ߡ
2 2
kmmax

A=

kmmin

A=

(5.3.27)ࠍᓧࠆ㧚 km ߩ୯ߪ㧘਄ᑼߢ␜ߐࠇߚ਄㒢ߣਅ㒢ߩ㑆ߦ޽ࠆࠊߌߢ㧘ᒰὼkm

m= A d m d  A(5.3.28)

A ߇ᢛᢙߥࠄ߫ m ߽ᢛᢙߢ޽ࠅ㧘 A ߇ඨᢛᢙߥࠄ߫ m ߽ඨᢛᢙߣ޿߁ߎߣߦߥࠆ㧚ᦝߦ㧘ᑼ(5.3.21)߹ߚߪᑼ(5.3.23)ࠃࠅ
kA

A A  1 =2ࠍᓧࠆ㧚߆ߊߒߡ㧘ᑼ(5.3.2)߅ࠃ߮ᑼ(5.3.3)ߩ kA ߅ࠃ߮ km ߇᳞߹ߞߚࠊߌߢ޽ࠆ㧚߹ߣ߼ࠆߣ㧘
L2YA ,mA A  1 = 2YA ,m(5.3.29)

LzYA ,m

m=YA ,m

(5.3.30)ߎߩ⚿ᨐߪ㧘੤឵㑐ଥߩߺߦࠃߞߡᓧࠄࠇߚ߽ߩߢ޽ࠆ㧚ᓥߞߡ㧘ⷺㆇേ㊂ߣห᭽ߥ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 9੤឵㑐ଥࠍ␜ߔߤࠎߥṶ▚ሶߦߟ޿ߡ߽ห᭽ߥ⚿ᨐ߇ᓧࠄࠇࠆ㧚㕖Ᏹߦၮᧄ⊛ߢ㧘߆ߟ㧘㊀ⷐߥᑼߢ޽ࠆ߆ࠄ㧘਄ߩ 2 ߟߩᑼߪ⸥ᙘߒߡ߅ߊߎߣ߇ᔅⷐߢ޽ࠆ㧚

5.4 ⷺㆇേ㊂Ṷ▚ሶߩ૞↪
ᑼ(5.3.30)㧘ߔߥࠊߜ

LzYA ,m

m=YA ,m

(5.4.1)ߪ㧘 YA ,m ߦኻߔࠆ Lz ߩ૞↪ࠍ␜ߒߚ߽ߩߢ޽ࠆ㧚ߢߪ㧘 Lx ߿ Ly 㧔߹ߚߪ L ߿ L 㧕ࠍ૞↪ߒߚࠄߤ߁ߥࠆߢ޽ࠈ߁߆㧫ߘࠇߪ߹ߛ᳞߼ߡ޿ߥ޿ߩߢએਅߢ⠨߃ߡߺࠆ㧚
ᑼ(5.3.14)ߢ␜ߒߚࠃ߁ߦ㧘LYA ,m ߪ Lz ߩ࿕᦭㑐ᢙߢ޽ࠅ㧘ߘߩ࿕᦭୯ߪ m ߇⸵ߐࠇࠆᦨૐߩ୯ A ߢ޽ࠆ႐วࠍ㒰޿ߡ m  1 = ߢ޽ࠆ㧚㧔ᑼ(5.3.14)ߩ km ߪ m= ߢ޽ࠆߎߣ
ߪߔߢߦ␜ߒߚ㧚㧕 Lz ߦߟ޿ߡߩ࿕᦭୯߇ m= ߢ޽ࠆࠃ߁ߥ㑐ᢙࠍ YA ,m ߣቯ⟵ߒߡ᧪ߚࠊߌߢ޽ࠆ߆ࠄ㧘࿕᦭୯߇ m  1 = ߢ޽ࠆࠃ߁ߥ㑐ᢙߪ YA , m 1 ߹ߚߪߘߩቯᢙ୚ߢ޽ࠆ㧚ᓥߞߡߎߩቯᢙࠍ N  ߣ߅޿ߡLYA ,mN YA ,m 1

m z A

(5.4.2)ߣߥࠆ㧚 N  ߪ㧘YA ,m 㧔ᒰὼ YA ,m 1 ߽㧕߇ⷙᩰൻߐࠇߡ޿ߥߌࠇ߫ߥࠄߥ޿ߣ޿߁᧦ઙ߆
ࠄ᳞߼ࠆ㧚ߘߩߚ߼ߦ㧘ᑼ(5.4.2)ߩਔㄝߢ㧘ߘߩ complex conjugate ࠍߣࠅోⓨ㑆ߦߟ޿ߡⓍಽߔࠆ㧚

³ LY

 A ,m*LYA ,m d W

N * N  ³ YA , m*YA , m d W

N * N 

Lx ߅ࠃ߮ Ly ߪ Hermitian ߢ޽ࠆߎߣࠍ↪޿ߡ਄ߩᑼࠍᦠ߈⋥ߔ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 10(5.4.3)N * N *

³ L Y L Y dW
³ ^ L  iL Y ` ^ L  iL Y ` dW
 A,m

 A ,m

*

x

A ,m

y

x

A,m

y

³ Y L ^ L  iL Y ` dW  i ³ Y L ^ L  iL Y ` dW
³ Y ^L *  L * i L * L *  L * L * `Y dW
*

*

A ,m

x

x

A ,m

y

2

A ,m

*

A ,m

y

*

x

y

2

x

*

(5.4.4)

*

A ,m

*

y

x

y

y

x

A,m

ߎߩᑼߩ complex conjugate ࠍขࠆ 㧚

N  N *³ Y ^L  L  i L L  L L `Y dW³ Y ^ L  L  =L `Y dW
*

A ,m

*

A ,m

2

2

x

y

2

x

y

y

x

A ,m

2

z

^A A  1  m  m` =
2

A,m

z

(5.4.5)

2^A A  1  m m  1 ` =

2

߆ߊߒߡ
N

A A  1  m m  1 = exp iJ A ,m(5.4.6)߹ߚߪ
LYA ,mA A  1  m m  1 = exp iJ A , m YA ,m 1(5.4.7)

ߎߎߢ J A ,m ߪછᗧߩታᢙߢ޽ࠅ㧘ᑼ(5.4.5)߆ࠄ᳿߼ࠆߎߣߪ಴᧪ߥ޿㧚ᑼ(5.4.2)ߢᑼ

(5.3.19)ࠍ⠨ᘦߒߡ m ߇ᦨૐߩ୯㧘ߔߥࠊߜ㧘 m A ߩᤨࠍ㒰ᄖߒߡ޽ࠆ㧚ߒ߆ߒ㧘
ᑼ(5.4.7)ߩฝㄝߪ m A ߢ࠯ࡠߦߥࠆߩߢ㧘ᑼ(5.3.19)ࠍ൮฽ߒߡ޿ࠆߎߣߦߥࠆ㧚ోߊห᭽ߦߒߡ
LYA , m

A A  1  m m  1 = exp iG A ,m YA ,m 1(5.4.8)߇ᓧࠄࠇࠆ߇㧘ߘࠇߪ⻉ำߩṶ⠌ߦߒߡ߅ߎ߁㧔Problem 5.4.1㧕
㧚ߎߎߢ G A ,m ߪછᗧߩታᢙߢ޽ࠆ㧚
ᑼ(5.4.7)߅ࠃ߮ᑼ(5.4.8)ߦ⴫ࠇࠆታᢙ㧘 J A ,m ߣ G A ,m ߪછᗧߩታᢙߢ޽ࠆ߆ࠄ㧘ߤ߁޿߁߰߁ߦ᳿߼ߡ߽޿޿㧚ߤ߁᳿߼ߡ߽޿޿ߩߢ㧘ߎߎߢߪߔߴߡߩ A 㧘 m ߦߟ޿ߡJ A,m

0 㧘 G A,m
LrYA , m

0 ߣ߅ߊߎߣߦߔࠆ㧚ߘ߁ߔࠆߣ㧘ᑼ(5.4.7)ߣᑼ(5.4.8)ࠍ߹ߣ߼ߡA A  1  m m r 1 =YA ,m r1

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 11(5.4.9)ߣ⴫ߖࠆ㧚
ߎߩࠃ߁ߦ㧘ߔߴߡߩ A ߅ࠃ߮ m ߦߟ޿ߡ J A ,m0 ߅ࠃ߮ G A ,m0 ߣ߅޿ߡଥᢙ N  ߣ

N  ࠍታᢙߦߣߞߚߩߪ㧘น⢻ߥ޿ߊߟ߆ߩㆬᛯߩ߁ߜߩ 1 ߟߦߔ߉ߥ޿㧚ߎߩᲑ㓏ߢߪ㧘છᗧߩታᢙࠍ࠯ࡠߣ߅޿ߚߩߢ޽ࠆ߆ࠄ㧘߽ߜࠈࠎ㧘ߘࠇߢ߆߹ࠊߥ޿ࠊߌߢ
޽ࠆ㧚ߒ߆ߒ㧘ᰴߦ YA ,m ߩ㑐ᢙᒻࠍౕ૕⊛ߦ᳞߼ࠆᲑ㓏ߢߪߎߩߎߣߪ㊀ⷐߢ޽ࠅ㧘߆ߥࠅߜ߾ࠎߣ⠨߃ߡ޿ߊᔅⷐ߇޽ࠆ㧚‫ڏ‬Problem 5.4.1
ᑼ(5.4.8)ࠍዉ಴ߖࠃ㧚

5.5 ⃿㑐ᢙ

YA ,m ࠍቯ⟵ߔࠆⷺㆇേ㊂Ṷ▚ሶ(5.2.9)߅ࠃ߮(5.2.10)ߪ޿ߕࠇ߽㧘polar coordinate ߦ߅޿ߡ㧘 r ৻ቯߩ⃿㕙਄ߢ T 㧘 I ߩ㑐ᢙߦ૞↪ߒߡߎࠇࠍหߓ⃿㕙਄ߩ㑐ᢙߦᄌ឵ߔ
ࠆ㧚ߎ߁޿߁ߎߣߢ YA ,m ߪ⃿㑐ᢙ spherical function ߣ⸒ࠊࠇࠆ㧚ߢߪ YA ,m ߩౕ૕⊛ߥᒻࠍ᳞߼ߡߺࠃ߁㧚

YA ,m ߪ T ߣ I ߩ㑐ᢙߢ޽ࠆߩߢ㧘ᰴߩࠃ߁ߦ T ߛߌߩ㑐ᢙ 4A ,m T ߣ I ߛߌߩ㑐ᢙ) m I ߩⓍߢ⴫ߐࠇࠆߣ઒ቯߔࠆ㧚

YA , m T , I

4A ,m T ) m I(5.5.1)

LzYA ,m T , I

m=YA ,m T , I(5.5.2)ߘߒߡߦ߅޿ߡ㧘Ꮐㄝߩ Lz ߦ polar coordinate ࠍ↪޿ߚᑼ(5.2.9)

Lz

i =

w
wI

(5.5.3)

ࠍ↪޿ࠆ㧚

i =

w
^4A,m T ) m I ` m= ^4A,m T ) m I `wI䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 12(5.5.4)

ߔߥࠊߜ

w
)m I
wI

im) m I

(5.5.5)ߎߩᓸಽᣇ⒟ᑼߩ⸃ߪ
)m I

N exp imI

(5.5.6)

ߢ޽ࠆ㧚 N ߪⷙᩰൻߩ᧦ઙࠃࠅ᳞߼ࠆ㧚 ) m I ߪ৻ଔߢߥߌࠇ߫ߥࠄߥ޿߆ࠄ㧘)m I) m I  2S

(5.5.7)

߹ߚߪ
eimI
ߔߥࠊߜ

e

im I  2S(5.5.8)e i 2S m 1
ߎࠇߪ m ߇ᢛᢙߩᤨߩߺᚑࠅ┙ߟ㧚normalization condition ࠃࠅ
2S³ ) I ) I dI
*

m

m

0

(5.5.9)2SN

2³ dI 1(5.5.10)

0

ߔߥࠊߜ
N

1
2S

߆ߊߒߡ ) m I ߪ
)m I

1 imI
e
2S

(5.5.11)ߣ᳞߹ߞߚ㧚
ᰴߦ 4A ,m T ߩᒻࠍ᳞߼ߡߺࠃ߁㧚ᑼ(5.2.9)ࠃࠅ

§ w
w ·=eiI ¨ i cot T
¸
wI ¹
© wT
ߢ޽ࠆ߇㧘ߎࠇࠍL

YA ,A T , I

4 A ,A T(5.5.12)

1 iAI
e
2S

ߦ૞↪ߔࠆ㧚ߔߥࠊߜ
§ w
w ·
iAI
 i cot T¨¸ 4A ,A T e
w
T
w
I
©
¹

(5.5.13)0䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 13(5.5.14)ߎࠇࠃࠅd
4A ,A T  A cot T 4A ,A T
dT

0(5.5.15)

߹ߚߪ

d 4A ,A T

cos T
dT
sin T

(5.5.16)

A ln sin T  constant(5.5.17)A

4A ,A TਔㄝࠍⓍಽߒߡ
ln 4A ,A T

߹ߚߪ
4A ,A T

N A ,A sin A T(5.5.18)

ߎߎߢ㧘normalization factor N A ,A ࠍ᳞߼ࠆߦ޽ߚࠅ㧘polar coordinate ߦ߅ߌࠆᓸዊ૕ⓍߪdW

r 2 sin T drdT dI(5.5.19)ߢ޽ࠆߎߣߦᵈ⋡ߒߡ1
N A ,A 2S³ sin T sin T dT
2A

(5.5.20)0ߣߥࠆ㧚ߎߎߢㇱಽⓍಽࠍ↪޿㧘
S

2A³ sin T sin T dT
0SS sin 2 A T cos T  2A ³ cos 2T sin 2A 1 T dT
0

0

S

S

0

0

2A ³ sin 2 A 1 T dT  2A ³ sin 2A 1T dTߣߥࠆ߇㧘1 ⴕ⋡ߩᏀㄝߣ 2 ⴕ⋡ߩ╙ 2 㗄ࠍᢛℂߒߡS

2 A 1³ sin T dT
0S

2A
sin 2A 1 T dT
2A  1 ³0(5.5.21)⚿ዪ㧘normalization factor (5.5.20)ߪ

1
N A ,A 2

S

2A
sin 2 A 1 T dT
2A  1 ³0(5.5.22)ߣߥࠆ߇㧘ߎߩᑼߦᑼ(5.5.21)ࠍᦝߦ A  1 ࿁➅ࠅ㄰ߒߡ↪޿ࠆ㧚ߘ߁ߔࠆߎߣߦࠃߞߡ

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 14

1
N A ,A 2

2A 2A  2 2A  4  S
sin T dT2A  1 2A  1 2A  3  ³0

2 2A A !

2(5.5.23)

2A  1 !

N A ,A ߪ਄ᑼߩㅒᢙߩᐔᣇᩮߦ㧘ੑਸ਼ߔࠆߣ 1 ߦߥࠆ૗ࠄ߆ߩ phase factor ࠍដߌߚ߽ߩ
A

ߢ޽ࠆ߇㧘ߎߎߢߪߘߩ phase factor ࠍ㧘ᓟ߆ࠄߩଢቱߩߚ߼ߦ 1 ߣ߅ߊ㧚⚿ዪ㧘ᑼ(5.5.18)ߪ

4 A ,A T1

2A  1 ! 1
sin A T
2
2A A !

A

ߎߩࠃ߁ߦߒߡ㧘4A ,m T

(5.5.24)ߩ߁ߜ㧘m ߩᦨᄢߥ႐วߩ 4A ,A T ߩᒻ߇᳞߹ߞߚ㧚m ߇ Aࠃࠅዊߐ޿႐วߦߪ㧘 YA ,A T , I߆ࠄᆎ߹ߞߡ L ࠍ૗࿁߆૞↪ߒߡ޿ߌ߫ࠃ޿㧚ߔߥࠊߜ㧘ᑼ(5.4.9)ߦ㧘 L ߩ polar coordinate ߦࠃࠆ⴫␜ࠍ↪޿ߡ§ ww ·=e iI ¨ 
 i cot T¸ YA ,m T , I
wI ¹
© wT

LYA ,m T , I(5.5.25)

A A  1  m m  1 =YA ,m 1 T , Iࠍ↪޿ࠇ߫ࠃ޿㧚ߎߩᣇᴺߢ YA ,m T , I ߅ࠃ߮ 4A ,m T ߩ৻⥸ᑼࠍ᳞߼ࠆߎߣߪน⢻ߢ޽ࠅ㧘ᄙߊߩᢎ⑼ᦠ㧔଀߃߫ Condon, Shortley "The Theory of Atomic Spectra", page 52㧕ߦਈ߃ࠄࠇߡ޿ࠆ㧚ߒ߆ߒ㧘৻⥸ᑼࠍ᳞߼ࠆߩߪ޿ߐߐ߆㕙ୟߢ߽޽ࠆ߆ࠄ㧘߻ߒࠈA ߣ m ߩౕ૕⊛ߥ୯ࠍ౉ࠇߡ਄ㅀߩᣇᴺߢ᳞߼ߡߺࠆߩ߇ᭉ߆߽ߒࠇߥ޿㧚એਅߦੑਃߩ଀ࠍ␜ߘ߁㧚‫ڏ‬Exercise 5.5.1
40,0 T ߅ࠃ߮ Y0,0 T , I ࠍ᳞߼ࠃ㧚‫ޤ⸃ޣ‬㧦
ᑼ(5.5.24)ߢ A
40,0 T0 ߣ߅޿ߡ
12

ᓥߞߡ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 15

Y0,0 T , I

40,0 T

12S

14S‫ڏ‬Exercise 5.5.2
41,m T߅ࠃ߮ Y1,m T , I ࠍ m 1, 0,  1 ߦߟ޿ߡ᳞߼ࠃ㧚‫ޤ⸃ޣ‬㧦
ᑼ(5.5.24)ߦ A 1 ࠍઍ౉ߒߡ
41,1 T

3
sin T
4



ᓥߞߡ
3
1 iI
e
sin T
4
2S
ᰴߦ㧘ᑼ(5.5.25)ࠍ↪޿ߡ

Y1,1 T , I· 1 iT§ w
w ·§ 3e  iI ¨ Tesin i cot T¨¸¸¸
wI ¹ ¨© 4
© wT
¹ 2S

2Y1,0 T , I

ߔߥࠊߜY1,0 T , I

3
1
cos T
2
2S

41,0 T

3
cos T
2

߹ߚߪ

Y1,1 T , I ߅ࠃ߮ 41,1 T ࠍ᳞߼ࠆߚ߼ߦ਄ߢ᳞߼ߚ Y1,0 T , I ߦᦝߦ L ࠍ૞↪ߔࠆ㧚

§ w
1
w · 3
e  iI ¨ cos T i cot T¸
2S
wI ¹ 2© wT

2Y1, 1 T , IߔߥࠊߜY1,1 T , I

3
1  iI
e
sin T
4
2S

41,1 T

3
sin T
4

߹ߚߪ

ߎߩࠃ߁ߦߒߡ᳞߼ߚ 4A ,m T ࠍ㧘A 1, 2, 3 ߦߟ޿ߡ⴫ 5.5.1 ߦ߹ߣ߼ߡ߅ߊ㧚ᦝߦ㧘䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 16ᔨߩߚ߼ YA ,m T , I ߦߟ޿ߡ߽⴫ 5.5.2 ߦ߹ߣ߼ߡ߅ߊ㧚ߎࠇࠄߩ㑐ᢙߪߎࠇ߆ࠄ޿ࠈ޿ࠈߥߣߎࠈߢ಴ߡ᧪ࠆߎߣߦߥࠆ߇㧘․ߦൻ
ቇࠍቇ⠌ߔࠆ⠪ߦߣߞߡߪ᳓⚛ේሶߩᵄേ㑐ᢙߢ㗫❥ߦ૶߁ߎߣߦߥࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 17

4A ,m T ߩౕ૕⊛⴫⃻㧔 A

⴫ 5.5.1A01
240,0 T

A 1
3
cos T
2

41,0 T41,r1 T

3
sin T
2

B

A

2

4 2,0 T5
3cos 2 T  1
2 2

4 2, r1 T

B

15
sin T cos T
2
15 2
sin T
4

4 2, r2 T

A 3

7
2 cos3 T  3sin 2 T cos T
2 2

43,0 T43,r1 TB

21
sin T 5cos 2 T  1
4 2

43,r2 T43,r3 T

105 2
sin T cos T
4
B

35 3
sin T
4 2

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 18

0, 1, 2, 3 㧕⴫ 5.5.2

YA , m T, I ߩౕ૕⊛⴫⃻㧔 A

A0, 1, 2, 3 㧕01
4S

Y0,0 T , I

A 1
Y1,0 T , I

Y1,r1 T , I

B

3
cos T
4S

3
sin T exp riI
2 2S
A

Y2,0 T , I
Y2, r1 T , I
Y2, r2 T , I

B

3 z
4S r

5
4 S

B

3 x r iy
8S r

23cos 2 T  1

5 3z 2  1
16S r 215 z x r iy
8S
r2

15
sin T cos T exp riI
2 2S

B

15
sin 2 T exp ri 2I
4 2S

15 x r iy
32S
r2䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 192

A

Y3,r1 T , I

B

2 cos T  3sin T cos T
3

4 S
21
8 S

Y3,r2 T , I

Y3,r3 T , I

7 z 5 z  3r
16S
r3
2

7

Y3,0 T , I

32

sin T 5cos 2 T  1 exp riI

B

B

35
8 S

sin 3 T exp ri3I2
2
21 x r iy 5 z  r
64S
r3

105 z x r iy
32S
r3

105
sin 2 T cos T exp ri 2I
4 2S
B

35 x r iy
r3
64S

3

‫ڏ‬Problem 5.5.1
Y2, m T , I ߅ࠃ߮ 4 2,m T ࠍߔߴߡߩน⢻ߥ m ߦߟ޿ߡ᳞߼ࠃ㧚‫ڏ‬Problem 5.5.2
Y3,m T , I ߅ࠃ߮ 43,m T ࠍߔߴߡߩน⢻ߥ m ߦߟ޿ߡ᳞߼ࠃ㧚‫ڏ‬Problem 5.5.3
ᰴߩṶ▚ࠍⴕ߃㧚a)

LxY1,1 T , I
LxY1,0 T , I
LxY1, 1 T , I䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 20

2

2

b)
LyY1,1 T , I
LyY1,0 T , I
LyY1, 1 T , I‫ڏ‬Problem 5.5.4

px , p y , pz ࠍᰴߩࠃ߁ߦቯ⟵ߔࠆ㧚

px
py
pz



1
^Y1,1 T , I  Y1,1 T , I `2

1
i ^Y1,1 T , I  Y1,1 T , I `2
Y1,0 T , I㧔ߎߎߢ px , p y , pz ߪߘࠇߙࠇ㑐ᢙ x, y, z ߣห╬ߢ޽ࠆߎߣࠍ⏕߆߼ࠃ㧚㧕

ª¬ A xAyª px ºA z º¼ «« p y »»«¬ pz »¼

ߩṶ▚ࠍⴕߥ޿⚿ᨐߪ px , p y , pz ࠍ↪޿ߡ⴫ߖ㧚‫ڏ‬Problem 5.5.5
Y1,1 T , I , Y1,0 T , I , Y1,1 T , I ߪ L2 ߩ࿕᦭㑐ᢙߢ߅੕޿ߦ❗ㅌߒߡ޿ࠆ㧚 L2 ߣ Lx ߣߪน឵ߢ޽ࠅ㧔ᑼ(5.1.5)㧕෶㧘ߤߜࠄ߽ࠛ࡞ࡒ࡯࠻Ṷ▚ሶߢ޽ࠆ߆ࠄ㧘Section 4.4 ߢቇ⠌ߒߚቯℂߦࠃࠅ㧘 L2 ߅ࠃ߮ Lx ߩਔ⠪ߦ౒ㅢߥ eigenfunction ߇ሽ࿷ߔࠆ㧚ߎߩ

eigenfunction ࠍ Y1,1 T , I , Y1,0 T , I , Y1,1 T , I ߩ✢ᒻ⚿วߣߒߡ⴫ࠊߖ㧚㧔ࡅࡦ࠻㧦
Section 4.4 ߢቇ⠌ߒߚߎߣ㧘߅ࠃ߮ Exercise 4.4.1 ࠍෳ⠨ߦߒߡ 3 ᰴߩ matrix ߩኻⷺൻࠍⴕ߃߫ࠃ޿㧚Problem 5.5.3 ߩ⚿ᨐ߽ᔅⷐ㧕䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 21‫ڏ‬Problem 5.5.6
਄ߩ໧㗴ߣห᭽ߦ⠨߃ߡ㧘 L2 ߅ࠃ߮ Ly ߩਔ⠪ߦ౒ㅢߥ eigenfunction ࠍ᳞߼ࠃ㧚㧔ᵈᗧ㧦ߎߩ႐วଥᢙ߇ⶄ⚛ᢙߣߥࠆߩߢ normalization ߩ႐ว᳇ࠍߟߌࠆߎߣ㧕[Coffee Break] ᵄേ㑐ᢙߩ phase factor ߦ㑐ߔࠆࡐࡇࡘ࡜࡯ߥ㑆㆑޿

4A ,m T ߩౕ૕⊛ߥᒻߢᵈᗧߔߴ߈ߎߣ߇ 2 ߟ޽ࠆ㧚1 ߟߪᑼ(5.5.24)߆ࠄಽ߆ࠆࠃ

߁ߦ㧘 4 A ,A T

A

ߪ 1 ߣ޿߁ phase factor ࠍᜬߟࠃ߁ߦቯ⟵ߐࠇߡ޿ࠆߣ޿߁ߎߣߢ

޽ࠆ㧚ߔߥࠊߜ㧘A ߇ᄸᢙߩᤨ 4 A ,A T ߦߪ⽶ߩ╓ภ߇ߟߊߎߣߦߥࠆ㧚߽߁ 1 ߟߪ m߇ᱜߩ႐วߣ⽶ߩ႐วߣߩ㑐ଥߢ޽ࠆ㧚৻⥸ᑼߩᒻߢ᳞߼ࠆߎߣߪ⋭⇛ߒߚ߇㧘

4A ,m T

1 4 A ,  m Tmߣ޿߁㑐ଥߦ޽ࠆ㧚ߔߥࠊߜ m ߇ᄸᢙߩ႐วߦߪ 4A ,m T ߣ 4A , m T ߣߪ╓ภ߇෻ኻߦߥࠆࠊߌߢ޽ࠆ㧚ߤ߁ߒߡߎ߁޿߁ phase factor ߇ߟߊߩߢ޽ࠈ߁߆㧫ߘࠇߪᑼ(5.4.7)߿ᑼ(5.4.8)ߢ phase factor ߩ J A ,m ߿ G A ,m ࠍ࠯ࡠߣ߅޿ߚߎߣߩ⚿ᨐߥߩߢ޽ࠆ㧚A߿ m ߩ୯ߦ߆߆ࠊࠄߕᑼ(5.4.9)ࠍ↪޿ࠆߣߔࠆߣ㧘ߤ߁ߒߡ߽਄ߩࠃ߁ߥ phase factor߇ᔅⷐߣߥߞߡ᧪ࠆ㧚ߎߩࠃ߁ߥ phase ߩߣࠅᣇߪ㧘ฎߊ㧘Condon ߣ Shortley ߇ᱧผߦᱷࠆฬ⪺ߢ޽ࠆ"The Theory of Atomic Structure"ߢߣߞߡ޿ࠆᣇᴺߢ޽ࠅ㧘ߒ߫ߒ߫Condon-Shortley ߩ phase ߣ⸒ࠊࠇࠆ㧚৻ ᣇ 㧘 ో ߊ ⇣ ߥ ߞ ߚ phase factor ࠍ ข ࠆ ߎ ߣ ߪ น ⢻ ߢ ߪ ޽ ࠆ 㧚 ଀ ߃ ߫ 㧘
4A ,m T

4A ,  m T ߦߥࠆࠃ߁ߦᑼ(5.4.7)߅ࠃ߮ᑼ(5.4.8)ߩ J A ,m ߿ G A ,m ࠍ᳿߼ࠆߎߣߪ಴

᧪ࠆ㧚ߒ߆ߒ㧘ߘߩ႐วߦߪ㧘ߘࠇߙࠇߩ A ߅ࠃ߮ m ߦߟ޿ߡ⇣ߥߞߚ phase factorࠍ↪޿ߥߌࠇ߫޿ߌߥ޿ߩߢ㧘ᑼ(5.4.9)߇ⶄ㔀ߥᒻߣߥࠅ㕙ୟߦᕁ߃ࠆ㧚㊂ሶൻቇߩᢎ⑼ᦠߩᄙߊ߇ߎߩὐߢ⥌๮⊛ߥ㑆㆑޿ࠍ‽ߒߡ޿ࠆ㧚଀߃߫ Lr ߩ૞↪ߦߟ޿ߡߪᑼ(5.4.9)ߩᒻࠍណ↪ߒߡ޿ߥ߇ࠄ㧘 YA , m T , I䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 22

YA , m T , Iߣߒߡ޿ࠆ႐

ว߇ߒ߫ߒ߫޽ࠆߩߢ޽ࠆ㧚଀߃߫㧘Eyring, Walter, Kimball ⪺ߩ"Quantum Chemistry"Aߢߪ㧘page 47 ߢߎߩ lecture note ߣหߓߊ 4 A ,A T ߦ 1 ߣ޿߁ phase factor ࠍߟߌߡ޿ࠆߦ߽߆߆ࠊࠄߕ㧘ᄸᅱߥߎߣߦ page 59 ߩ⴫ߢߪߎߩ factor ߇㒰߆ࠇߡ޿ࠆ㧚ߘߒߡ෶㧘page 58 ߦߪ 4A , m T

4A , m T ߣ޿߁⍦⋫ߒߚᑼߐ߃ਈ߃ߡ޿ࠆ㧚ߎ߁޿߁ߎߣߢ㧘page 59 ߩ⴫ߢߪ A ߇ᄸᢙ㧘m ߇ᱜߩᄸᢙߩ㗄ߩ╓ภ߇㑆㆑ߞߡਈ߃ࠄࠇߡ޿
ࠆ㧚ߎߩ㑆㆑޿ߪ㧘޿ࠈ޿ࠈߥᚲߦ㊀ⷐߥ㑆㆑޿ࠍᒁ߈⿠ߎߔߎߣߦߥࠆߩߢᵈᗧࠍⷐߔࠆ㧚ߎߩ໧㗴ߩ㊀ⷐᕈࠍࠃࠅߪߞ߈ࠅߐߖࠆߚ߼ߦౕ૕⊛ߥ଀ࠍ޽ߍߡߺࠃ߁㧚ߎߩ

lecture note ߢขࠅ਄ߍߚ Condon-Shortley ᵹߣ㧘Eyring, Walter, Kimball ߩ"Quantum
Chemistry" ߢ↪޿ࠄࠇߡ޿ࠆ Eyring-Walter-Kimball ᵹ㧔઒ߦߘ߁ฬઃߌߚ㧕ߣࠍ㧘
41,m Tߩ 3 ߟߩ㑐ᢙߢᲧߴߡߺࠃ߁㧚Condon-Shortley ᵹ

Eyring-Walter-Kimball ᵹ

41,1 T



3
sin T
4



3
sin T
4

41,0 T



3
cos T
2



3
cos T
2

41,1 T



3
sin T
4



3
sin T
4

ߎߩࠃ߁ߦ㧘41,1 Tߦ߅޿ߡ 2 ߟߩᵹ௾ߢ╓ภ߇෻ኻߦߥߞߡ޿ࠆ㧚ዋ‫޿ߤߊޘ‬߆߽ߒࠇߥ޿߇㧘ߘࠇߙࠇߩᵹ௾ߦߟ޿ߡ⃿㑐ᢙ Y1,m T , I ࠍᦠ߈ਅߒߡߺࠃ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 23

Condon-Shortley ᵹ

Eyring-Walter-Kimball ᵹ

Y1, 1 T, I3
sin T eiI
8S

3
sin T eiI
8S

Y1,0 T, I



3
cos T
4S



3
cos T
4S

Y1, 1 T, I3
sin T e  iI
8S

3
sin T e  iI
8S

ߎߩ 2 ߟߩᵹ௾ߢ Y1,0 T , I ߪߤߜࠄ߽หߓᑼߢ⴫ߐࠇࠆ߆ࠄ㧘ߎߩᑼߦ L ࠍ૞↪ߔࠆߎߣߦࠃߞߡ㧘ᑼ(5.4.9)ߔߥࠊߜ

LY1,0 T , I

2=Y1,1 T , I߇ᚑࠅ┙ߟ߆ߤ߁߆ࠍี๧ߒߡߺࠃ߁㧚 L ߪߔߢߦᑼ(5.5.12)ߢ

§ w
w ·=eiI ¨ i cot T
¸
wI ¹
© wI
ߣߥࠆߎߣ߇ಽ߆ߞߡ޿ࠆߩߢߘࠇࠍ↪޿ࠆ㧚ߢߪታ㓙ߦ⸘▚ߒߡߺࠃ߁㧚L

LY1,0 T , I

§ w
w · 3
=eiI ¨cos T
 i cot T¸
wI ¹ 4S© wI=eiI

3 d cos T
4S dT

=eiI

3
 sin T2

3
sin T eiI
8S
ߔߥࠊߜ㧘Condon-Shortley ߩ phase ࠍ↪޿ࠆ㒢ࠅ
 2=LY1,0 T , I

2=Y1,1 T , Iߩᑼ߇ᚑࠅ┙ߟ㧚ߒ߆ߒ㧘Eyring-Walter-Kimball ᵹߩ phase ࠍ↪޿ࠆߣ㧘 L ߩ૞↪ߪLY1,0 T , I

 2=Y1,1 T , I䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 24ߣቯ⟵ߒߥߌࠇ߫޿ߌߥ޿ߎߣ߇ಽ߆ࠆ㧚ߎࠇߪᑼ(5.4.9)ߩ৻⥸ᑼߣ⍦⋫ߔࠆ㧚ߔߥࠊߜ㧘ᑼ(5.4.9)߇ A ߿ m ߩ୯ߦ߆߆ࠊࠄߕᚑࠅ┙ߟߩߪ Condon-Shortley ᵹߩߺߢ޽ࠆ㧚߽ߒ߽ Eyring-Walter-Kimball ᵹߩ phase ࠍ↪޿ࠆߥࠄ㧘ᑼ(5.4.9)㧔ߔߥࠊߜ㧘Eyring,

Walter, Kimball ߩ⪺ᦠߢߪ p.46 ߩᑼ(3.133)㧕ߪ૶߃ߥ޿ߩߢ޽ࠆ㧚ዋ‫ߥ߁ࠃ޿ߤߊޘ‬᳇߽ߔࠆ߇㧘㕖Ᏹߦࠃߊ޽ࠆ㑆㆑޿ߩ଀ߣߒߡේሶߩ 3 ߟߩ p 㑐ᢙߩߎߣࠍ໧㗴ߣߒࠃ߁㧚ේሶߩᵄേ㑐ᢙߦ߅޿ߡ A 1 ߩ 3 ߟߩᵄേ㑐ᢙߪ p ゠㆏ߣ⸒ࠊࠇࠆ㧚㧔ߎߩߎߣߦߟ޿ߡߪ Chapter 9 ߢ⹦ߒߊቇ⠌ߔࠆ㧚㧕ߎߩ 3 ߟߩ p ゠㆏ߣߒߡ㧘 Y1,1 T , I 㧘 Y1,0 T , I 㧘߅ࠃ߮ Y1,1 T , I ߩ 3 ߟ߇޽ࠆ߇ߎߩ߁ߜ Y1,1 T , I ߣ
Y1,1 T , I ߪⶄ⚛ᢙߣߥߞߡ޿ࠆ㧚
wavefunction ߇ⶄ⚛ᢙߢ߽৻ะߦߦᏅߒᡰ߃ߥ޿߇㧘ᚒ‫߫ߒ߫ߒߪޘ‬ታᢙߩ wavefunction ࠍ૶޿ߚ޿᳇ߦߥࠆ㧔ߘ߁ߔࠆߣ଀߃߫ࠣ࡜ࡈߦ
ឬߊᤨߥߤߦଢ೑ߢ޽ࠆ㧘ߥߤߩℂ↱ߢ㧕㧚ߘࠇߢ Y1,1 T , I ߣ Y1,1 T , I ߣߩ✢ᒻ⚿วࠍߣߞߡታᢙߩ wavefunction ࠍ૞ࠆߎߣ߇ࠃߊ޽ࠆ㧚ߎ߁ߒߡ૞ࠄࠇߚ wavefunctionߪ 1 ߟߪ㑐ᢙ x ߣ㧘߽߁ 1 ߟߪ㑐ᢙ y ߣหߓᒻߦߥࠆߩߢߘࠇࠄࠍ px ߅ࠃ߮ p y ߣฬઃߌࠆ㧚 Y1,0 T , Iߪ߽ߣ߽ߣታᢙߢ㑐ᢙ z ߣหߓᒻߥߩߢ pz ߣฬઃߌࠄࠇߡ޿ࠆ㧚

wavefunction ࠍ ታ ᢙ ߣ ߔ ࠆ ࠃ ߁ ߥ linear combination ߩ ข ࠅ ᣇ ߪ ਄ ㅀ ߒ ߚ
Condon-Shortley ᵹߣ Eyring-Walter-Kimball ᵹߢ⇣ߥࠅᰴߩࠃ߁ߦߥࠆߎߣߪኈᤃߦಽ
߆ࠆߢ޽ࠈ߁㧚Condon-Shortley ᵹ

Eyring-Walter-Kimball ᵹpx

1^Y1,1 T , I  Y1,1 T , I `2

1^Y1,1 T , I  Y1,1 T ,I `2

py1
i ^Y1,1 T , I  Y1,1 T , I `2

pz

Y1,0 T , I1
i ^Y1, 1 T , I  Y1, 1 T , I `2
Y1,0 T , Iߎߩࠃ߁ߦ phase ߩขࠅᣇߦࠃߞߡ linear combination ߩᒻ߇⇣ߥࠆࠊߌߢ޽ࠆ߇㧘ߎ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 25ߩߎߣߦ᳇߇ߟ߆ߕߦᖠ߻⎇ⓥ⠪߇ᴛጊ޿ࠆߩߢᵈᗧ߇ᔅⷐߢ޽ࠆ㧚ߎߩࠃ߁ߥ㑆㆑޿ߪ㧘Eyring ߩᢎ⑼ᦠએᄖߦ߽߆ߥࠅᴛጊ޽ࠆ㧚ᢎ⑼ᦠߩ㑆㆑޿ࠍ޽ߍߡ޽ࠄតߒࠍߔࠆ᳇ᜬߜߪᲫ㗡ߥ޿߇㧘㑆㆑ߞߚᑼߦၮߠ޿ߡ㧘ᓟߢ޿ࠈ޿ࠈߥ
⍦⋫ߦㆣㆄߒߡᖠ߻ቇ↢߿⎇ⓥ⠪߇ᄙ޿ߩߢ㧘ᵈᗧࠍ༐⿠ߔࠆᗧ๧ߢ㑆㆑޿ߩ޽ࠆᢎ
⑼ᦠࠍએਅߦ⟜೉ߒߡ߅ߊ㧚x

L. Pauling, E. G. Wilson, Jr., "Introduction to Quantum Mechanics With Application
to Chemistry", (1935), page 134.

x
x
x
x

W. Kauzman, "Quantum Chemistry" (1957), page 730.
A. K. Chandra, "Introduction to Quantum Chemistry" (1974), page 41.
L. I. Schiff, "Quantum Mechanics" 2nd edition (1955), page 73.
D. A. McQuarrie, "Quantum Chemistry" (1983), page 216.

৻ᣇ㧘ᰴߦ޽ߍࠆᢎ⑼ᦠߢߪ㧘ᱜߒ޿ phase factor ߇ขࠄࠇߡ޿ࠆߩߢ㧘቟ᔃߒߡ⚿ᨐࠍ↪޿ߡᄢਂᄦߣᕁ߁㧚x
x
x

E. U. Condon, G. H. Shortley "The Theory of Atomic Spectra" (1935), page 52.
A. S. Davydov, "Quantum Mechanics" (English Translation) (1965), page 22.
C. J. Ballhausen, "Introduction to Ligand Field Theory" (1962), page 19.

ᦨᓟߦᒁ↪ߒߚ Ballhausen ߩᧄߩ 19 㗁ߩ⣉ᵈߢߪ㧘․ߦ phase ߩขࠅᣇߦߟ޿ߡᵈᗧࠍߒߥߌࠇ߫޿ߌߥ޿㧘ߣᰴߩࠃ߁ߦㅀߴߡ޿ࠆ㧚A very important point is the choice of phase for the spherical harmonics
YA ,m T , I . We shall always use the Condon-Shortley "natural choice of phase"; afactor of 1 for positive odd values of m. Thus, YA , m T , I

1 YA , m T , I .
m

This is a point to be carefully checked every time a formula from a different bookis used; many hours of work have been spent in order to find mistakes in signs due
to such errors.
ోߊߎߩㅢࠅߢ޽ࠆ㧚⑳⥄り߽ Ballhausen ߣหߓߊ㧘ߎߩߎߣߢ∩޿⋡ߦ޽ߞߡ޿ࠆ㧚ߎߩࠃ߁ߥ⚻㛎ࠍᾖࠄߒߡߺࠆߣ㧘㑆㆑ߞߚ phase ࠍᐔ᳇ߢᢎ⑼ᦠߦᦠ޿ߡ޿ࠆੱߪ㧘߅ߘࠄߊ 1 ࿁߽⥄ಽߢ⸘▚ࠍߒߥ޿ߢ㧘ߤߞ߆ߩᧄ߆ࠄ౮ߒߡ᧪ߚߩߢ޽ࠈ߁ߣߒ߆
ᕁ߃ߥ޿㧚
Condon, Shortley ߽㧘߽ߜࠈࠎ phase ߩขࠅᣇߦࠃࠆᷙੂ߇޽ࠆߎߣߪ㊀‫ޘ‬ᛚ⍮ߒߡ޿ߡ㧘਄ㅀߩ⪺ᦠߩ 53 㗁ߩ⣉ᵈߢᰴߩࠃ߁ߦㅀߴߡ޿ࠆ㧚This question of phase choice has caused some confusion with regard to the䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 26

relative phases of the matrix components of angular momentum and ofelectrostatic interaction. See the discussion by Ufford and Shortley, Phys. Rev. Vol.
42, 167 (1932).
਄ߢᒁ↪ߒߚᢎ⑼ᦠએᄖߢ߽޿ࠈ޿ࠈߣ㑆㆑޿߇޽ࠆߩߢචಽߦ᳇ࠍઃߌߥߊߡߪ
޿ߌߥ޿㧚଀߃߫㧘P. W. Atkins ߩ"Molecular Quantum Mechanics"㧔second edition㧕ߢߪ page 66 ߩ⴫ߢߪ Condon-Shortley ߩᱜߒ޿ phase ߇ขࠄࠇߡ޿ࠆ㧚ߒ߆ߒ㧘ߔߋߘߩᓟߢ page 75 ߢ px ߣ p y ࠍቯ⟵ߔࠆ႐วߦߪ Eyring-Walter-Kimball ᵹߩ㑆㆑ߞߚphase ߦၮߠߊቯ⟵ᑼࠍ߽ߜ޿ߡ޿ࠆ㧚ᄙಽߎߩ⪺⠪ߪ⥄ಽߢ⸘▚ߔࠆߎߣߥߒߦ
⦡‫ࠄ߆ᧄߥޘ‬ነߖ㓸߼ߡ᧪ߚߩߢ޽ࠈ߁㧚ߘࠇߢߎࠎߥ߅߆ߒߥߎߣ߇⿠ߎߞߡߒ߹
߁ߩߢ޽ࠆ㧚ߎࠇࠄߩ଀߆ࠄಽ߆ࠆࠃ߁ߦ phase ߦ㑐ߒߡߪ߹ߕ㑆㆑ߞߡ޿ࠆߣᕁߞ
ߡ⥄ಽߢ⚊ᓧ޿ߊ߹ߢ⸘▚ߒߡߺࠆߎߣ߇ᄢಾߢ޽ࠆ㧚[Let’s play with Mathematica] ⃿㑐ᢙߩ Mathematica ߦࠃࠆ⴫⃻Mathematica ߦߪ⃿㑐ᢙ YA ,m T , I ߇ SphericalHarmonicY ߣ޿߁ౝㇱ㑐ᢙߣߒߡቯ⟵ߐࠇߡ޿ࠆߩߢଢ೑ߢ޽ࠆ㧚଀߃߫㧘 Y1,m T , Iࠍm1 ߆ࠄ m

1 ߹ߢߔߴߡࡊ࡝

ࡦ࠻ߒࠃ߁ߣ޿߁ᤨߦߪએਅߩࠃ߁ߦߔࠆ㧚

Do[Print[SphericalHarmonicY[1,m,theta,phi]],{m,-1,+1}]⋥ߜߦએਅߩࠃ߁ߥ⚿ᨐ߇ᓧࠄࠇࠆ㧚਄߆ࠄ m-I phiE

1, 0,  1 ߩ㗅ߢ޽ࠆ㧚3
Sqrt[----] Sin[theta]

2 Pi----------------------------2

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 27

3
Sqrt[--] Cos[theta]
Pi------------------2

I phi

3
-(E
Sqrt[----] Sin[theta])
2 Pi------------------------------2
਄ߩ㧘Mathematica ߇಴ജߒߚᑼߩᗧ๧ߪ᣿ࠄ߆ߣߪᕁ߁߇㧘ᔨߩߚ߼ߎߩ⚿ᨐࠍㅢᏱߩᑼߢ⴫ߔߣᰴߩࠃ߁ߦߥࠆ㧚e  iI

3
sin T
2S
2

3
cos T
S
2
 e iI

3
sin T
2S
2

ߎߩ⚿ᨐ߇ᱜߒ޿ߎߣߪ⴫ 5.5.2 ߣᲧセߔࠆ߹ߢ߽ߥߊ᣿ࠄ߆ߢ޽ࠆ㧚߁ࠇߒ޿ߎ
ߣߦ Condon-Shortley ߩ phase ࠍ↪޿ߡ޿ࠆ㧚਄ߩ଀߆ࠄ߽ಽ߆ࠆߣᕁ߁߇㧘Mathematica ߢߪ⥄ὼኻᢙߩᐩߩ e ࠍᄢᢥሼߩ‫ޟ‬E‫ޠ‬ߢ㧘߹ߚ㧘౞๟₸ߩ S ࠍ‫ޟ‬Pi‫߁޿ߣޠ‬ౝㇱቯᢙߣߒߡቯ⟵ߒߡ޿ࠆ㧚

5.6 ⷺㆇേ㊂ߩวᚑ೨▵ߢቇ⠌ߒߚⷺㆇേ㊂ߪ 1 ߟߩ☸ሶ㧔଀߃߫㔚ሶ㧕ߦߟ޿ߡ޽ࠆ୯ࠍᜬߞߡ޿ࠆ㧚߽ߒ♽߇ 2 ߟߩ☸ሶ߆ࠄᚑߞߡ޿ࠆߣߔࠆߣ㧘ߘࠇߙࠇߩ☸ሶߦߟ޿ߡⷺㆇേ㊂߇ቯ⟵ߐࠇࠆ߇㧘ߎߩ႐ว㧘♽ో૕ߣߒߡߩⷺㆇേ㊂ߪߤ߁ߥࠆߢ޽ࠈ߁߆㧚ߘࠇࠍ⍮ࠆ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 28ߦߪߘࠇߙࠇߩⷺㆇേ㊂ࠍวᚑߒߡߺࠇ߫޿޿㧚ⷺㆇേ㊂ߩวᚑߩ໧㗴ߪ㧘ᄙ㔚ሶ♽ߩේሶ߿ಽሶࠍᛒ߁႐ว㧘߹ߚߪ৻㔚ሶ♽ߩේሶߦ߅ߌࠆ゠㆏ⷺㆇേ㊂ߣࠬࡇࡦⷺㆇ
േ㊂㧔ᓟߢቇ⠌ߔࠆ㧕ߩ⋧੕૞↪ߥߤ޿ߚࠆᚲ↪޿ࠄࠇࠆ㧚ᧄ▵ߢߪ㧘2 ߟߩⷺㆇേ
㊂ߩวᚑߩ໧㗴ࠍቇ⠌ߔࠆ㧚

2 ߟߩⷺㆇേ㊂ࡌࠢ࠻࡞ L1 ߣ L2 ࠍ⠨߃ࠃ߁㧚 L1 ߦߟ޿ߡߪ L12 ߩ࿕᦭୯߇
A1 A1  1 = 2 㧘 L1z ߩ࿕᦭୯߇ m1= ߢ޽ࠆߣߔࠆ㧚 L2 ߦߟ޿ߡ߽ห᭽ߦ㧘 L2 2 ߩ࿕᦭୯߇
A 2 A 2  1 = 2 㧘 L2 z ߩ࿕᦭୯߇ m2 = ߢ޽ࠆߣߔࠆ㧚 L1 ߩᚑಽ㧘 L2 ߩᚑಽߘࠇߙࠇߦߟ޿

ߡߪ㧘ᑼ(5.1.3)ߩ੤឵㑐ଥ߇ᚑࠅ┙ߟߎߣߪᒰὼߢ޽ࠆ㧚ª¬ L1x , L1 y º¼ i=L1zª¬ L1 y , L1z º¼ i=L1x(5.6.1)> L1z , L1x @ i=L1 yª¬ L2 x , L2 y º¼ i=L2 zª¬ L2 y , L2 z º¼ i=L2 x(5.6.2)> L2 z , L2 x @ i=L2 y

ోⷺㆇേ㊂㧔total angular momentum㧕 L ࠍቯ⟵ߔࠆ㧚
L L1  L2(5.6.3)

ߎߎߢ L ߪᒰὼ㧘ᑼ(5.1.2)ߢቯ⟵ߒߚࠃ߁ߦ 3 ߟߩᚑಽߩวᚑ߆ࠄߥࠆࡌࠢ࠻࡞ߢ޽ࠆ㧚

L iLx  jLy  kLz(5.6.4)ᑼ(5.6.3)ߣᑼ(5.6.4)ࠃࠅ

Lx

L1x  L2 x

Ly

L1 y  L2 y

Lz

L1z  L2 z

(5.6.5)L2 ߦߟ޿ߡߪᑼ(5.6.4)ࠃࠅ
L2

L L
Lx 2  Ly 2  Lz 2৻ᣇ㧘ᑼ(5.6.3)ࠃࠅ

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 29(5.6.6)

L2

L1  L2  L1  L2

(5.6.7)L12  L2 2  L1 L2  L2 L1

L1 ߣ L2 ߪᒰὼ߅੕޿ߦ commutable ߢ޽ࠆ㧚㧔৻ᣇߩ☸ሶߦߟ޿ߡߩṶ▚߇ઁᣇߩ☸ሶߦߪ෸߫ߥ޿㧕ᓥߞߡ㧘ᑼ(5.6.7)ߪ

L2

L12  L2 2  2 L1 L2
L12  L2 2  2 L1x L2 x  L1 y L2 y  L1z L2 z

(5.6.8)એਅ㧘ోⷺㆇേ㊂ߩᚑಽ Lx 㧘 Ly 㧘 Lz ߦߟ޿ߡ߽㧘ᑼ(5.1.3)ߩࠃ߁ߥㅢᏱߩ੤឵㑐ଥ߇ᚑࠅ┙ߟߎߣࠍ␜ߘ߁㧚଀߃߫ Lx ߣ Ly ߦߟ޿ߡߪ
ª¬ Lx , Ly º¼ª¬ L1x  L2 x , L1 y  L2 y º¼ª¬ L1x , L1 y  L2 y ¼º  ª¬ L2 x , L1 y  L2 y º¼

(5.6.9)ª¬ L1x , L1 y º¼  ª¬ L1x , L2 y º¼  ª¬ L2 x , L1 y º¼  ¬ª L2 x , L2 y ¼ºߎߎߢ㧘ߔߴߡߩ L1 ߩᚑಽ߇ߔߴߡߩ L2 ߩᚑಽߣ੤឵น⢻ߢ޽ࠆ߆ࠄ㧘ᑼ(5.6.9)ߩ 3
ⴕ⋡ߩ╙ 2 㗄ߣ╙ 3 㗄ߪ࠯ࡠߢ޽ࠆ㧚ᓥߞߡª¬ Lx , Ly º¼ª¬ L1x , L1 y º¼  ª¬ L2 x , L2 y º¼
i=L1z  i=L2 z(5.6.10)i=Lzઁߩᚑಽߦߟ޿ߡ߽ోߊห᭽ߩ⸘▚߇ᚑࠅ┙ߟ㧚ߔߥࠊߜ㧘㧔ᑼ(5.6.10)ࠍ߽߹ߣ߼ߡ㧕ª¬ Lx , Ly º¼

i=Lzª¬ Ly , Lz º¼ i=Lx
> Lz , Lx @ i=Ly

(5.6.11)ߩࠃ߁ߦ㧘ᒻᑼ⊛ߦߪᑼ(5.1.3)ߣోߊหߓᑼ߇ోⷺㆇേ㊂ߦߟ޿ߡ߽ᚑࠅ┙ߟߎߣ߇
␜ߐࠇߚ㧚
ోߊห᭽ߦ

ª¬ L2 , Lx º¼ª¬ L2 , Ly º¼ª¬ L2 , Lz º¼

0(5.6.12)߇ᚑࠅ┙ߟ߇㧘⸽᣿ߪ⻉ำߩṶ⠌ߣߒߡ߅ߎ߁㧚㧔Problem 5.6.1㧕
2
ᑼ(5.6.12)߇ᚑࠅ┙ߟ߆ࠄ㧘 L ߣ㧘 L ߩ 1 ߟߩᚑಽ㧔଀߃߫ Lz 㧕ߩਔᣇߩṶ▚ሶߦ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 30ߟ޿ߡ౒ㅢߩ eigenfunction ߇ሽ࿷ߔࠆߎߣ߇ಽ߆ࠆ㧚㧔Section 4.4 ߩቯℂࠍෳ㧕Section

5.3 ߩ⼏⺰ߪ੤឵㑐ଥߩߺࠍ↪޿ߡน⢻ߛߞߚߎߣߦ⌕⋡ߔࠆߣోߊห᭽ߥᚻ⛯߈ߦࠃࠅ㧘 L2 ߣ Lz ߩ࿕᦭୯ࠍ᳿ቯߔࠆߎߣ߇಴᧪ࠆ㧚ᑼ(5.3.29)߅ࠃ߮ᑼ(5.3.30)ߢߪ 1 ߟߩ☸ሶߦߟ޿ߡߩ L2 ߩ࿕᦭୯ࠍ A A  1 = 2 㧘 Lz ߩ࿕᦭୯ࠍ m= ߣߒߚ߇㧘ߎߎߢߪ㧘․ߦోⷺㆇേ㊂ߣ޿߁ߎߣࠍߪߞ߈ࠅߣ␜ߔߚ߼ߦ㧘 A ߅ࠃ߮ m ߩઍࠅߦᄢᢥሼߩ L ߅
ࠃ߮ M ࠍ↪޿ࠆߎߣߦߔࠆ㧚ߔߥࠊߜ㧘Ṷ▚ሶ L2 ߦߟ޿ߡߩ࿕᦭୯ߪ L L  1 = 2 㧘Ṷ▚ሶ Lz ߦߟ޿ߡߩ࿕᦭୯ߪ M = ߣߔࠆߎߣߦߒࠃ߁㧚㧔หߓ L ߣ޿߁⸥ภ߇Ṷ▚ሶߦ૶ࠊࠇߚࠅ࿕᦭୯ࠍ⴫ߔ㊂ሶᢙߦ૶ࠊࠇߚࠅߒߡ޿ࠆ߇ᷙੂߒߥ޿ࠃ߁ᵈᗧߒࠃ߁㧚㧕ߐߡ㧘ᚒ‫ߢߎߎ߇ޘ‬᣿ࠄ߆ߦߒߚ޿ߎߣߪ 㧔1㧕ోⷺㆇേ㊂ߩ㊂ሶᢙ L ߿ M ߇㧘ฦ☸ሶߦߟ޿ߡߩ㊂ሶᢙ A1 㧘 m1 㧘 A 2 㧘 m2 ߣߤ߁޿߁㑐ଥߦ޽ࠆ߆㧘ߣ޿߁ߎߣ㧘߅ࠃ߮㧔2㧕ోⷺㆇേ㊂ߦߟ޿ߡߩ࿕᦭㑐ᢙ߇㧘ฦ☸ሶߦߟ޿ߡߩ࿕᦭㑐ᢙߣߤ߁޿߁
㑐ଥߦ޽ࠆ߆㧘ߣ޿߁ߎߣߩ 2 ὐߢ޽ࠆ㧚એਅ㧘notation ࠍ◲⇛ൻߔࠆߚ߼ߦ࿕᦭㑐ᢙࠍ⴫ߔߩߦ㧘bracket notation ࠍ↪޿ࠆߎߣߦߒࠃ߁㧚bracket notation ߢ eigenfunction ࠍ⴫ߔߦߪ㧘ߘߩᤨߘߩᤨߢㇺวߩࠃ޿⴫⸥ࠍ᳿߼ࠇ߫ࠃ޿㧚ߎߩ▵ߢߪએਅᰴߩࠃ߁ߥ⴫⸥ᴺࠍ↪޿ࠆߎߣߦߒࠃ߁㧚

operator L2 ߦߟ޿ߡߩ eigenvalue ߇ A A  1 = 2 㧘operator Lz ߦߟ޿ߡߩ eigenvalue ߇ m=㧔ߎߩߎߣࠍએਅᧄ▵ߢߪ◲නߩߚ߼ߦ㧘නߦ㧘㊂ሶᢙ߇ A 㧘 m ߢ޽ࠆߣ޿߁ߎߣߦ
ߔࠆ㧚2 ߟߩ㊂ሶᢙߩਗߴᣇߩ㗅ᐨߪ਄ߦㅀߴߚ㗅ᐨߢ޽ࠆߎߣߦ᳿߼ߡ߅ߊ㧕ߢ޽ࠆࠃ߁ߥ eigenfunction ࠍ A, m ߣ޿߁⸥ภߢ⴫ߒ㧘ߎࠇࠍ ket ߣ๭߱㧚ߘߒߡ㧘 A, m ߩcomplex conjugate ࠍ A, m ߣ⴫ߒ㧘ߎࠇࠍ bra ߣ๭߱㧚Section 5.3 ߢߎߩࠃ߁ߥ
wavefunction ߪ⃿㑐ᢙߣߒߡ⴫ߐࠇ㧘৻⥸ߦ YA ,m T , I ߣᦠߊߎߣࠍቇ⠌ߒߚ߇㧘ߎߎߢߪ wavefunction ߩౕ૕⊛ߥ㑐ᢙᒻߪ໧㗴ߣߒߥ޿㧚㧔ߎߩࠃ߁ߥ bracket notation ߩ೑ὐߪ㧘㊀ⷐߥ㊂ߢ޽ࠆ㊂ሶᢙ߇ subscript ߣߒߡߢߪߥߊ㧘ߪߞ߈ࠅ␜ߐࠇࠆߎߣߢ
޽ࠆ㧚ߒ߆ߒߥ߇ࠄ㧘㊂ሶᢙߩਗߴᣇߩ㗅ᐨߣ߆㑐ᢙᒻߥߤߦߟ޿ߡ㧘ߘߩㇺᐲ߈ߜࠎߣቯ⟵ߒߥ޿ߣ૗ߩߎߣ߆ಽ߆ࠄߥ޿ߣ޿߁ਇଢߐߪ޽ࠆ㧕ߎߩࠃ߁ߥ bracket notation ࠍ↪޿ࠆߣ㧘ฦ☸ሶߦߟ޿ߡߩ eigenfunction ߪᰴߩࠃ߁
ߦߥࠆ㧚

L12 ߅ࠃ߮ L1z ߩ eigenfunction㧦 A1 , m1

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 31

L2 2 ߅ࠃ߮ L2 z ߩ eigenfunction㧦 A 2 , m2᳞߻ࠆߴ߈ో♽ߦߟ޿ߡߩ eigenfunction㧦 L, Mߐߡ㧘ᰴߩࠃ߁ߥ੤឵㑐ଥ߇ኈᤃߦᓧࠄࠇࠆ㧚㧔Problem 5.6.2㧕

ª¬ L12 , Lx º¼ ª¬ L12 , Ly º¼ ª¬ L12 , Lz º¼ ¬ª L12 , L2 ¼º 0ª¬ L2 2 , Lx º¼ ¬ª L2 2 , Ly ¼º ¬ª L2 2 , Lz ¼º ¬ª L2 2 , L2 ¼º 0

(5.6.13)ߎߩ੤឵㑐ଥࠃࠅ㧘 L12 㧘 L2 2 㧘 L2 㧘߅ࠃ߮ Lz ߩ 4 ߟߩ operator ߔߴߡߦ౒ㅢߥ

eigenfunction ߇޽ࠆߎߣ߇ಽ߆ࠆ㧚ߎߩߎߣࠃࠅ eigenfunction L, M ࠍ㧘⴫␜ߔࠆ㊂
ሶᢙࠍ 2 ߟჇ߿ߒߡ A1 , A 2 , L, M ߩࠃ߁ߥᒻߢ⴫ߔߎߣߦߒࠃ߁㧚☸ሶ 1 ߦߟ޿ߡߩ
ቢో♽ A1 , m1 ߩߔߴߡߣ㧘☸ሶ 2 ߦߟ޿ߡߩቢో♽ A 2 , m2 ߩߔߴߡߣߩ޽ࠄࠁࠆน⢻ߥⓍ A1 , m1 A 2 , m2 ߪో♽㧔2 ߟߩ☸ሶ♽㧕ߦߟ޿ߡߩቢో♽ߣߥࠆࠊߌߢ޽ࠆ߆ࠄ㧘ో♽ߩ eigenfunction A1 , A 2 , L, M ࠍߎࠇࠄߩ complete set ߢዷ㐿ߔࠆߎߣ߇಴᧪ࠆ㧚A 1 , A 2 , L, M

¦ c A , A , L, M ; m , m
1

2

1

2

A1 , m1 A 2 , m2

(5.6.14)c A1 , A 2 , L, M ; m1 , m2 ߪዷ㐿ଥᢙߢ޽ࠆ㧚ᑼ(5.6.14)ߩᏀㄝߪ਄ㅀߒߚࠃ߁ߦ L12 ߦߟ޿ߡߩ eigenfunction ߢ eigenvalue ߪ A1 A1  1 = 2 㧘߹ߚหᤨߦ㧘 L2 2 ߦߟ޿ߡߩ

eigenfunction ߢ eigenvalue ߪ A 2 A 2  1 = 2 㧚ᓥߞߡ㧘ᑼ(5.6.14)ߩዷ㐿ߦ߅޿ߡߪ A1 㧘 A 2
ߪᏀㄝ㧘ฝㄝหߓ୯ߒ߆ขࠅᓧߥ޿㧚ᓥߞߡᑼ(5.6.4)ߩ✚๺ߪ m1 ߅ࠃ߮ m2 ߦߟ޿ߡߥ
ߐࠇࠆ㧚㧔ዷ㐿ଥᢙ c A1 , A 2 , L, M ; m1 , m2 ߩࠞ࠶ࠦߩਛߢ m1 ߣ m2 ߩߺࠍ࠮ࡒࠦࡠࡦߩฝ
஥ߦಽߌߡᦠ޿ߚߩߪߘߩࠃ߁ߥࠊߌߢ޽ࠆ㧕ᦝߦ㧘ᑼ(5.6.5)ߩ 3 ⇟⋡ߩᑼࠍ↪޿㧘
m1  m2 M(5.6.15)ߢ޽ࠆߎߣ߇ኈᤃߦ⸽᣿಴᧪ࠃ߁㧔Problem 5.6.3㧕
㧚ో♽ߦߟ޿ߡߩ

eigenfunction A1 , A 2 , L, M ࠍ᳞߼ࠆߦߪᑼ(5.6.14)ߩଥᢙ c A1 , A 2 , L, M ; m1 , m2 ߇ߔߴߡ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 32ᔅⷐߢ޽ࠆ㧚ߎߩଥᢙߪ㧘Clebsch-Gordan coefficient ߣ߆ Wigner coefficient ߥߤߣ๭߫
ࠇ㧔ߎߩਔ⠪ࠍᄙዋ⇣ߥߞߚᗧ๧ߢ↪޿ߡ޿ࠆ႐ว߽޽ࠆ㧕㧘ᄙߊߩᢎ⑼ᦠ߿ handbookߥߤߦ⴫߇ਈ߃ࠄࠇߡ޿ࠆ㧚ᄙߊߩߩᄢቇߩᄢဳ⸘▚ᯏ࠮ࡦ࠲࡯ߩ library program ߩਛߦ߽ߎߩଥᢙࠍ᳞߼ࠆࡊࡠࠣ࡜ࡓ߇⊓㍳ߐࠇߡ޿ࠆ㧚ታ↪ߩᲑ㓏ߢߪߎࠇࠄߩ⴫ࠍ
↪޿ࠆߩ߇ଢ೑ߢ޽ࠆ߇㧘ᢎ⢒ߩ႐ߢߪ㧘ߘࠇߙࠇߩ႐ว㧘႐วߦ⥄ಽߢ᳞߼ߡߺࠆ
ߎߣ߇ᔅⷐߢ޽ࠈ߁㧚ߚߛߒ㧘ߎߩଥᢙࠍ᳞߼ࠆߎߣߪߎߎߢߪⴕࠊߕ㧘߹ߕ㧘ߣࠅ޽߃ߕ L ߣ A1 㧘 A 2 ߣߩ㑐ଥࠍ᳞߼ࠆߎߣࠍ⠨߃ࠃ߁㧚৻⥸⺰ߦ౉ࠆ೨ߦ㧘Ṷ⠌ߣߒߡ㧘A1 1, A 2 2 ߣ޿߁ౕ૕଀ߦߟ޿ߡ⠨߃ߡߺࠃ߁㧚น⢻ߥ m1 ߩ୯ߪ m1 1, 0,  1 ߩ 3 ߟ㧘น⢻ߥ m2 ߩ୯ߪ m2 2,  1, 0,  1,  2 ߩ 5 ߟ㧚߽ߒ߽ో♽ߩ quantum number ࠍ A1 㧘 A 2 㧘 m1 㧘 m2 ߢ⴫ߔߣߔࠆߣ㧘น⢻ߥ⚵ߺวߖߪ3 u 5 15 ㅢࠅ޽ࠆߎߣߦߥࠆ㧚ߒ߆ߒ਄ㅀߒߚࠃ߁ߦ㧘ో♽ߩ quantum number ߪ A1 㧘

A 2 㧘 L 㧘 M ߢ߽ߞߡ⴫ߔߎߣߦߒߚࠊߌߢ޽ࠆ㧚ߎߩ⴫⸥ᴺߦ߅޿ߡ߽߿ߪࠅหߓ

15 ㅢࠅߩน⢻ߥ⚵ߺวߖ߇޽ࠆߎߣߦߥࠆ㧚ߎࠇࠄ 15 ㅢࠅߩน⢻ߥ⚵ߺวߖߦߟ޿
ߡᑼ(5.6.15)ߩ M ࠍ⴫ߦߒߡߺࠃ߁㧚

m2

m1

+2

+1

0

1

2

+1

+3

+2

+1

0

1

0

+2

+1

0

1

2

-1

+1

0

1

2

3

ߎߩ⴫߆ࠄ㧘ฦ M ߩ⴫ࠇࠆ㗫ᐲࠍ⴫ߦߔࠆߣᰴߩࠃ߁ߦߥࠆ㧚

M ߩ୯

+3

+2

+1

0

1

2

3

㗫ᐲ

1

2

3

3

3

2

1

ߎߩ⴫߆ࠄಽ߆ࠆࠃ߁ߦ㧘 M ߩᦨᄢ୯ߪ 3 ߢ޽ࠆ㧚 M ߪ L ߣ  L ߩ୯
M

L, L  1, L  2, ...,  L ࠍขࠆࠊߌߢ޽ࠆ߆ࠄ㧘L ߩᦨᄢ୯ߪ 3 ߣ޿߁ߎߣ߇ಽ߆ࠆ㧚

L 3 ߦߟ޿ߡߪ M

3, 2, 1, 0,  1,  2,  3 ߩ 7 ߟߩ୯ࠍขࠅᓧࠆ㧚਄⸥ߩ⴫߆ࠄ L 3 ߩ

7 ߟߩ⚵ࠍᒁ߈Ꮕࠆߣ㧘ᱷࠅߪᰴߩ⴫ߢ⴫ߐࠇࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 33

M ߩ୯

+2

+1

0

1

2

㗫ᐲ

1

2

2

2

1

ᱷߞߡ޿ࠆ M ߩਛߢᦨᄢ୯ߪ 2 ߢ޽ࠆ㧚ߎࠇߪ L 2 ߩᚑಽߢߥߌࠇ߫ߥࠄߥ޿㧚
L 2 ߦߟ޿ߡߪ M 2, 1, 0,  1,  2 ߩ 5 ߟߩᚑಽ߇޽ࠆߩߢ㧘ߎࠇࠄࠍౣ߮਄ߩ⴫߆
ࠄᏅߒᒁߊߣ㧘ᱷࠅߪᰴߩ⴫ߩࠃ߁ߦߥࠆ㧚

M ߩ୯

+1

0

1

㗫ᐲ

1

1

1

ߎࠇߪ᣿ࠄ߆ߦ㧘 L 1 ߩ 3 ߟߩᚑಽ M 1, 0,  1 ߦ⋧ᒰߔࠆ㧚એ਄ߩߎߣ߆ࠄ
A1 1, A 2 2 ߩวᚑߣߒߡ L 3, 2, 1 ߩ୯߇ᓧࠄࠇࠆߎߣ߇ಽ߆ߞߚ㧚એ਄ߩߎߣࠍ৻⥸⊛ߦขࠅᛒߞߡߺࠃ߁㧚 A1 ߣ A 2 ߩวᚑߢߪ 2A1  1 ㅢࠅߩ m1 ߣ
2A 2  1 ㅢࠅߩ m2 ߩ⚵ߺวߖߢ 2A1  1 2A 2  1 ㅢࠅߩน⢻ᕈ߇޽ࠆ㧚m1 ߩᦨᄢ୯ߪ A1 㧘

m2 ߩᦨᄢ୯ߪ A 2 ߢ޽ࠆ߆ࠄ㧘M ߩᦨᄢ୯ߪ A1  A 2 ߢ޽ࠆ㧚M ߪ L ߣ  L ߩ୯ࠍขࠆ ࠊߌߢ޽ࠆ߆ࠄ㧘 L ߩᦨᄢ୯ߪ߿ߪࠅ A1  A 2 ߢ޽ࠆ㧚2 ⇟⋡ߦᄢ߈ߥ M ߩ୯ߪ A1  A 2  1ߢ޽ࠆ߇㧘ߎࠇ߇⃻ࠇࠆ႐วߪ 2 ㅢࠅ޽ࠆ㧚ߔߥࠊߜ㧔a㧕m1 A1  1, m2 A 2 ߅ࠃ߮㧔b㧕
m1 A1 , m2 A 2  1 ߩน⢻ᕈ߇޽ࠆ㧚ᓥߞߡ㧘㧔a 㧕㧘㧔b㧕2 ߟߩ linear combination ߇

LA1  A 2 , M

A1  A 2  1 ߣ޿߁⁁ᘒߣ㧘LA1  A 2  1, M

A1  A 2  1 ߣ޿߁⁁ᘒࠍ૞ࠆߎߣߦߥࠆ㧚ߎߩࠃ߁ߥ⼏⺰ࠍ⛯ߌߡ޿ߊߣ㧘 L ߩน⢻ߥ୯ߪ A1  A 2 㧘 A1  A 2  1 㧘
A1  A 2  2 㧘...㧘 A1  A 2  b ߣ޿߁ߎߣߦߥࠆ㧚ߚߛߒߎߎߢ A1  A 2  b ߪ L ߩᦨዊ୯ࠍ

␜ߔ㧚b ࠍ᳿ቯߔࠆߚ߼ߦ㧘⁁ᘒߩᢙߪ 2A1  1 2A 2  1 ߢ޽ࠆߣ޿߁ߎߣߦᵈ⋡ߔࠆ㧚LA1  A 2  k ߣ޿߁⁁ᘒߦߟ޿ߡߪ 2 A1  A 2  k  1 ୘ߩน⢻ߥ M ߩ୯߇޽ࠆ㧚ᓥߞߡb¦ ^2 A  A  k  1`
1

2

2A1  1 2A 2  1

k 0

߇ᚑࠅ┙ߟ㧚ᓥߞߡ
b

b

k 0

k 0

¦ 2A1  2A 2  1  2¦ k

2A1  1 2A 2  1

b  1 2A1  2A 2  1  b b  1

2A1  1 2A 2  1䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 34ߔߥࠊߜ
b 2  2 A1  A 2 b  4A1A 2

0ߎߩੑᰴᣇ⒟ᑼߩ⸃ߪ
b 2A 1(5.6.16)(5.6.17a)

߅ࠃ߮
2A 2

b

(5.6.17b)ߢ޽ࠆ㧚ᑼ(5.6.17a)ߩᩮߦߟ޿ߡߪ L ߩᦨዊ୯ߪ
Lmin A 2  A1

(5.6.18a)߹ߚ㧘ᑼ(5.6.17b)ߩᩮߦߟ޿ߡߪ
Lmin A1  A 2
߽ߒ߽ A1A 2 ߥࠄ Lmin0 ߢ૗ߩਇㇺว߽ߥ޿㧚ߒ߆ߒ㧘߽ߒ A1 z A 2 ߩ႐ว㧘ᑼ(5.6.18)

ߩ 2 ߟߩᑼߩ߁ߜߩߤߜࠄ߆ߪ⽶ߩᢙߣߥࠅ㧘L ߩ୯ߣߒߡߪណ↪಴᧪ߥ޿㧚ᓥߞߡ㧘Lmin

A1  A 2(5.6.19)ߣ⠨߃ߑࠆࠍᓧߥ޿㧚⚿ዪ㧘਄ߩ⼏⺰ࠍ߹ߣ߼ࠆߣ㧘 A1 ߣ A 2 ߩ 2 ߟߩⷺㆇേ㊂ߩวᚑߢᓧࠄࠇࠆ L ߪ
LA1  A 2 , A1  A 2  1, A1  A 2  2, ..., A 1  A 2(5.6.20)ߣ޿߁ߎߣߦߥࠆ㧚ᦨᓟߩ⚿ᨐߪ㕖Ᏹߦ㊀ⷐߢ㧘ߎࠇ߆ࠄߚ߮ߚ߮↪޿ࠆߎߣߦߥࠆ
ߪߕߢ޽ࠆ㧚ዉ಴ߩᣇᴺߪᔓࠇߡߒ߹ߞߡ߽⚿ᨐߛߌߪⷡ߃ߡ߅ߊᔅⷐ߇޽ࠆ㧚[Coffee Break] bra ߣ ketbra ߅ࠃ߮ ket ߣ޿߁⸒⪲ߪ‫ޟ‬᜝ᒐ‫ࠍޠ‬ᗧ๧ߔࠆ bracket ߣ޿߁න⺆߆ࠄ૞ࠄࠇߚ߽
ߩߢ޽ࠆ㧚bra ߣ ket ࠍ↪޿ࠆ notation ߪ㧘ᓥߞߡ㧘"braket notation"ߢߪߥߊ"bracket
notation"ߢ޽ࠆߎߣߦᵈᗧߒࠃ߁㧚‫ڏ‬Exercise 5.6.1
A1 1, A 2 3 ߩวᚑߢᓧࠄࠇࠆߔߴߡߩ⁁ᘒࠍ ket ߢ␜ߖ㧚‫ޤ╵ޣ‬䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 35

4, 4 , 4,3 , 4, 2 , 4,1 , 4, 0 , 4, 1 , 4, 2 , 4, 3 , 4, 4 ,3,3 , 3, 2 , 3,1 , 3, 0 , 3, 1 , 3, 2 , 3, 3 ,2, 2 , 2,1 , 2, 0 , 2, 1 , 2, 2‫ڏ‬Problem 5.6.1
ᑼ(5.6.12)ࠍ⸽᣿ߖࠃ㧚‫ڏ‬Problem 5.6.2
ᑼ(5.6.13)ࠍ⸽᣿ߖࠃ㧚‫ڏ‬Problem 5.6.3
Lz L1z  L2 z ࠍᑼ(5.6.14)ߩਔㄝߦ૞↪ߔࠆߎߣߦࠃࠅᑼ(5.6.15)ࠍ⸽᣿ߖࠃ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 5: Page 36

CHAPTER 06ਗㅴㆇേߎߎ߆ࠄੑਃߩ┨ߦࠊߚߞߡ㧘޿ߊߟ߆ߩ◲නߥ♽ߦߟ޿ߡ Schrödinger equation§ =2 2·  V ¸\
¨
© 2m
¹

E\

ࠍ⸃޿ߡߺࠃ߁㧚ൻቇߩಽ㊁ߢ⃻ታߦㆣㆄߔࠆ⻉໧㗴ߪⶄ㔀ߢ޽ࠆ߆ࠄ㧘એਅߦㅀߴࠆࠃ߁ߥℂᗐൻߐࠇߚ◲නߥ♽ߦߟ޿ߡ Schrödinger equation ࠍ⸃޿ߡߺߚߣߎࠈߢ
ߒࠃ߁߇ߥ޿߆ߣᕁ߁ੱ߽޿ࠆ߆߽ߒࠇߥ޿㧚ߒ߆ߒ㧘৻⷗ⶄ㔀ߦ⷗߃ࠆൻቇߩ਎⇇߽㧘ᧄ⾰ߪන⚐ߥ߽ߩߢ޽ࠅ㧘ߎࠇ߆ࠄߩੑਃߩ┨ߢᛒ߁ၮᧄ⊛ߥ໧㗴ߪ㧘ߔߴߡ㧘ⶄ㔀ߥ⃻⽎ࠍන⚐ߥ♽ߩ⚵ߺวߖߣߒߡℂ⸃ߔࠆߩߦᄢᄌᓎߦ┙ߟ߽ߩߢ޽ࠆ㧚ਛߦߪ㧘ᄙዋᢙᑼ߇߿ߞ߆޿ߦߥࠆㇱಽ߇޽ࠆ߆߽ߒࠇߥ޿߇㧘᳇ࠍᛮ߆ߥ޿ߢቇ⠌ߒߡ߶ߒ޿㧚ᚻᆎ߼ߣߒߡ㧘ߎߩ┨ߢߪਗㅴㆇേࠍขࠅᛒ߁㧚

6.1 ৻ᰴరߩ⥄↱☸ሶ
1 ୘ߩ☸ሶ߇ᄖ⇇߆ࠄ૗ߩ⋧੕૞↪ࠍ߽ฃߌߕߦ৻ᰴరߩㆇേࠍߒߡ޿ࠆ♽ࠍ⠨߃ࠃ߁㧚ߎߩࠃ߁ߦ㧘potential ࠍฃߌߥ޿☸ሶߩߎߣࠍ⥄↱☸ሶ㧔free particle㧕ߣ޿߁⸒⪲ߢ⴫ߒߡ޿ࠆ㧚potential ߇޿ߚࠆᚲߢ࠯ࡠߢ޽ࠆ߆ࠄ㧘Schrödinger equation ߪ= 2 d 2\E\
(6.1.1)2m dx 2ߣߥࠆ㧚ߎߎߢ㧘৻ᰴరߩㆇേߩᣇะࠍ x ᣇะߣߒߚ㧚ߎߎߢ x ߩᄌၞߪ f d x d f ߢ޽ࠆ㧚ᑼ(6.1.1)ߩ৻⥸⸃ߪ\

Aeikx  Be  ikx

k2

2mE
=2

(6.1.2)ߣߥࠆߎߣߪ㧘ᑼ(6.1.2)ࠍᑼ(6.1.1)ߦઍ౉ߒߡߺࠇ߫ಽ߆ࠆ㧚৻ᣇ㧘eikx cos kx  i sin kx
ߦ⌕⋡ߔࠆߣ㧘ᑼ(6.1.2)ߩ⸃ߪᰴᑼߩࠃ߁ߦ⴫ߐࠇࠆ㧚\ C cos kx  D sin kx(6.1.3a)߹ߚߪ\

A 'sin kx  G䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 1(6.1.3b)ߣ߽⴫ߐࠇࠆ㧚޿ߕࠇߩᑼߢ߽ 2 ߟߩᧂ⍮ᢙ㧔 A ߣ B ߹ߚߪ C ߣ D ߹ߚߪ A' ߣ G 㧕ࠍ฽ߺ㧘ߎࠇࠄߪႺ⇇᧦ઙࠃࠅ᳿߼ࠆߎߣߦߥࠆ߇㧘ߘࠇߦߟ޿ߡߪߜࠂߞߣᓟ߹ࠊߒ
ߣߒ㧘߹ߕᑼ(6.1.2)߹ߚߪᑼ(6.1.3)ߩ⸃ࠍዋ‫߁ࠃߺߡ߼⌑ޘ‬㧚߹ߕ㧘࿕᦭୯ E ߪߤࠎߥ୯ߢ߽ขࠅᓧࠆߢ޽ࠈ߁߆㧫߽ߒ߽ E  0 ߢ޽ࠆߣ k ߪ
imaginary ߩᢙߣߥࠆ㧚ߘ߁ߔࠆߣᑼ(6.1.2)ߩ\ ߪ x rf ߢ⊒ᢔߒߡߒ߹߁㧚ߘߩࠃ߁ߥ㑐ᢙߪ⸃ߣߪߥࠅᓧߥ޿㧚ᓥߞߡ㧘energy E ߪ positive ߢߥߌࠇ߫޿ߌߥ޿ߎߣ߇
⚿⺰ߐࠇࠆ㧚ߚߛߒ㧘 x ߩ▸࿐ߦోὼ೙㒢߇ߥ޿ߚ߼ߦ㧘 E ߪ positive ߢ޽ࠆ㒢ࠅߤࠎߥ୯ߢ߽ขࠅᓧࠆߎߣߦᵈᗧߔߴ߈ߢ޽ࠆ㧚ߔߥࠊߜ㧘energy ߪ㊂ሶൻߒߡ޿ߥ޿㧚ᰴߦ㧘ฎౖജቇߣߩኻᔕߢㆇേ㊂ࠍ⠨߃ߡߺࠃ߁㧚ฎౖജቇߢߪ㧘⥄↱ߦㆇേߒߡ
޿ࠆ☸ሶߩㆇേ㊂ p ߪࠛࡀ࡞ࠡ࡯ E ߣ

1 2 p2mv
2
2mߣ޿߁㑐ଥߦ޽ࠆ㧚߹ߚ㧘ᑼ(6.1.2)ߦ␜ߒߚࠃ߁ߦ
E

k 2=2
2m
ߢ޽ࠆ㧚ߎࠇ߆ࠄ㧘ᑼ(6.1.2)߹ߚߪᑼ(6.1.3)ߩ wavefunction ߢ␜ߐࠇࠆ⥄↱☸ሶߩㆇേ㊂ߪ
p k=(6.1.4)Eߩࠃ߁ߦ⴫ߐࠇࠆߎߣ߇ಽ߆ࠆ㧚ߎߩߎߣࠍ߽߁ዋߒ⹦ߒߊ⷗ߡߺࠃ߁㧚߹ߕ㧘਄ߩ⼏⺰ࠍ෻ኻߩ┙႐߆ࠄㅴ߼ߡߺ
ࠆߣ㧘‫ޟ‬ㆇേ㊂ p ࠍ߽ߟ⥄↱☸ሶߩ wavefunction ߪᑼ(6.1.2)߹ߚߪᑼ(6.1.3)ߢ⴫ߐࠇࠆ‫ޠ‬ߣ⸒߁ߎߣ߇಴᧪ࠆ㧚ᰴߦ㧘ᑼ(6.1.2)߹ߚߪᑼ(6.1.3)ߩ wavefunction ߢ␜ߐࠇࠆ☸ሶߪ㧘޽ࠆ․ቯߩ‫ޟ‬ᵄ㐳‫ࠍޠ‬ᜬߞߡ޿ࠆ㧚ߎߩߎߣߪ㧘․ߦᑼ(6.1.3)ࠃࠅ᣿ࠄ߆ߢ޽ࠈ߁㧚৻⥸ߦ㧘ᵄ㐳 O ߩᵄߪ㧘 cos

2S x

O

߹ߚߪ sinᑼ(6.1.3)ߢ⴫ߐࠇࠆ☸ሶߩᵄ㐳ߪ O
ࠍᜬߟ☸ሶߩᵄ㐳ߪ O

p

2S

O

=

2S x

O

ߢ⴫ߐࠇࠆߎߣࠍᕁ޿ᶋ߆ߴࠆߣ㧘

2Sߣ޿߁ߎߣߦߥࠆ㧚ߔߥࠊߜ㧘ㆇേ㊂ p
k

k=

2S
ߢ޽ࠆ㧚ᓥߞߡㆇേ㊂ߣᵄ㐳ߣߪkhOߣ޿߁㑐ଥߦ޽ࠆߎߣ߇ಽ߆ࠆ㧚ߎࠇߪ㧘ߔߢߦቇ⠌ߒߚ de Broglie ߩ㑐ଥᑼߘߩ߽
ߩߢ޽ࠆ㧚ߢߪߎߎߢ㧘ᑼ (6.1.2) ߩଥᢙ A ߅ࠃ߮ B ߩᗧ๧ࠍ⠨߃ߡߺࠃ߁㧚઒ߦᑼ (6.1.2) ߢB0 ߣ߅޿ߚ߽ߩࠍ\  ߣ߅޿ߡ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 2

\

Aeikx

ߘࠇߦ㧘ㆇേ㊂ߩ operator

px

i=

d
dx

ࠍ૞↪ߒߡߺࠃ߁㧚

px\ 

i =

d\ 
dx

k =\ 

ߎߩࠃ߁ߦ㧘\  ߪ momentum operator px ߩ eigenfunction ߢ޽ࠅ㧘ߘߩ eigenvalue ߪ k =ߢ޽ࠆ㧚ᰴߦ㧘ᑼ(6.1.2)ߢ A 0 ߣߒߚ߽ߩࠍ\  ߣ߅߈㧘\Be  ikx

ߦหߓߊ momentum operator px ࠍ૞↪ߒߡߺࠆ㧚

px\ 

i=

d\ 
dx

 k =\ ߎߩࠃ߁ߦ㧘\  ߪ momentum operator ߩ eigenfunction ߢ޽ࠅ㧘ߘߩ eigenvalue ߪ  k =ߢ޽ࠆ㧚\  ߽\  ߽㧘ߤߜࠄ߽ momentum operator ߩ eigenfunction ߢ޽ࠆ߇㧘eigenvalue ߩ╓ภ߇⇣ߥࠆ㧚ߔߥࠊߜ㧘\  ߅ࠃ߮\  ߪߘࠇߙࠇหߓᄢ߈ߐߩㆇേ㊂ࠍᜬߞߡ߅੕޿
ߦㅒߩᣇะߦㆇേߒߡ޿ࠆ☸ሶߩ wavefunction ߦ⋧ᒰߔࠆ㧚ߎߩߎߣߪ㕖Ᏹߦ㊀ⷐߥߎߣߢ޽ࠆ㧚ߣ޿߁ߩߪ㧘ߎߩߎߣߦࠃߞߡᚒ‫ߪޘ‬㧘޽ࠆㆇേ㊂ࠍᜬߜ㧘޽ࠆᣇะߦ
ㆇേߒߡ޿ࠆ☸ሶߩ wavefunction ࠍ⋥ߜߦᦠ߈ਅߔߎߣ߇಴᧪ࠆ߆ࠄߢ޽ࠆ㧚ଥᢙ A ߅ࠃ߮ B ߩᗧ๧ߪߎࠇߢ߆ߥࠅߪߞ߈ࠅߒߚ㧚ߘࠇࠄߪ☸ሶ߇ᅤ૗ߦ૞ࠄࠇ
ࠆ߆㧘ߣ޿߁ߎߣߦࠃߞߡ᳿߹ࠆ㧚ߔߥࠊߜ㧘߽ߒ☸ሶ߇㋕⎔߆ࠄ x ゲߩᱜߩᣇะߦ
ᛂߜ಴ߐࠇࠇ߫ B 0 ߢ޽ࠆ㧚෻ኻߦ㧘x ゲߩ⽶ߩᣇะߦᛂߜ಴ߐࠇࠇ߫ A 0 ߢ޽ࠆ㧚
࿑ 6.1.1 ߦᮨᑼ⊛ߦ␜ߒߡ޽ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 3

\

A exp ikx

p

k=

\

B exp ikx

p

k =

࿑ 6.1.1
৻ቯᣇะߦㆇേߔࠆ☸ሶߩ wavefunction

A ߅ࠃ߮ B ߩౕ૕⊛ߥ୯ࠍ᳿ቯߔࠆߎߣ߇߹ߛᱷߞߡ޿ࠆ㧚ߎࠇߪ normalization
condition ߆ࠄ᳿߼ࠆߩߢ޽ࠆ߇㧘ታߪߎߎߦ໧㗴߇޽ࠆ㧚଀߃߫ A ࠍ᳿ቯߔࠆߚ߼ߦ
f³ \ *\ dx




f

f

A * A ³ dxfࠍ⸘▚ߒࠃ߁ߣߔࠆߣ㧘 x ߩᄌၞ߇ f d x d f ߢ޽ࠆߩߢⓍಽ߇⊒ᢔߒߡߒ߹߁㧚ߘࠇߢ A ࠍ᳿߼ࠆߎߣ߇಴᧪ߥ޿㧚ߘ߽ߘ߽㧘ߎ߁޿߁႐วߪ㧘㑐ᢙ߇ well behaved ߢߥ޿ߣߒߡ wavefunction ߦߪㆬ߫ߥ޿ߩ߇᥉ㅢߢ޽ࠆ߇㧘⥄↱☸ሶߩ႐วߦߪߘ߁߽
޿߆ߥ޿ߩߢ㧔ઁߦ wavefunction ߣߒߡㆬ߱૛࿾߇ߥ޿㧕޽ࠆ⒳ߩ trick ࠍ↪޿ߡߎߩ࿎㔍ࠍਸ਼ࠅ⿧߃ࠆ㧚ߎߎߢᚒ‫ߪޘ‬ᒻᑼ⊛ߦ㧘 x ᐳᮡ߇଀߃߫ x



L
߆ࠄ x
2



L
߹
2

ߢ㐳ߐ L ߩ▸࿐ߦ㒢ࠄࠇߡ޿ࠆߣ⠨߃ࠆ㧚ߎ߁ߒߡ normalization condition
L

A * A³ L 2 dx 1


2

ࠃࠅ
1
L
ߣߒߡ߅ߊ㧚㧔 B ߦߟ޿ߡ߽ห᭽ߦⴕ߁㧚㧕wavefunction ࠍ઒ߦߎ߁᳿߼ߡ߅޿ߡ㧘‛Aℂ㊂ࠍ⸘▚ߔࠆߣᄙߊߩ႐วߘࠇ߇ L ߦή㑐ଥߦ᳞߹ࠆ㧚ᐘ޿ߦߎ߁ߥߞߚ႐วߪ㧘x ߩᄌၞࠍ㒢ߞߚߎߣ߇ᒻᑼ⊛ߦߪ⴫ࠇߥ޿ߎߣߦߥࠆ㧚޿ߟ߽߁߹ߊ޿ߊࠊߌߢ߽

ߥ޿߇㧘߹޽ᄢ૕㧘ߎࠎߥ߰߁ߦߒߡ well behaved ߢߥ޿ߎߣߩ࿎㔍ࠍస᦯ߒߡ޿ࠆ㧚ߐߡ㧘ଥᢙ A ߿ B ߩᗧ๧ߪ᣿ࠄ߆ߣߥߞߚ߇㧘৻⥸⸃ߢ޽ࠆᑼ(6.1.2)ߪ૗ࠍᗧ๧ߔ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 4ࠆߩߢ޽ࠈ߁߆㧚wavefunction ߩᒻࠍߺࠇ߫㧘 x ߩᱜߩᣇะߩㆇേߣ⽶ߩᣇะߩㆇേ
ߣ߇ଥᢙ A ߣ B ߢ᳿߹ࠆഀวߢ㊀ߥߞߚ߽ߩߣ޿߁ߎߣ߇ಽ߆ࠆ㧚․ᱶߥ႐วߣߒߡ

A

B ߢ޽ࠇ߫

\A eikx  e ikx

2 A cos kxߣߥࠅ㧘ߎࠇߪᑼ(6.1.3)ߢ D 0 ߣ߅޿ߚ߽ߩߦ⋧ᒰߔࠆ㧚ฝะ߈ߩㆇേߣᏀะ߈ߩㆇ
േߣ߇หߓ㊀ߐߢ㊀ߨวߞߚ߽ߩ㧘ߔߥࠊߜቯᏱᵄߣ޿߁ߎߣߦߥࠆ㧚

6.2 ੗ᚭဳ potential ߩਛߩ⥄↱☸ሶ㧔৻ᰴరߩ▫ߩਛߩ☸ሶ㧕

Section 6.1 ߩᑧ㐳ߣߒߡ㧘੹ᐲߪ⾰㊂ m ߩ☸ሶ߇ x ゲ਄ x 0 ߣ x L ߩ㑆ߦ㒢ߞߡㆇേߒᓧࠆ႐วࠍ⠨߃ࠆ㧚potential ߪ 0 d x d L ߩ▸࿐ౝߢ࠯ࡠ㧘ߘߩ▸࿐ᄖߢߪ f ߢ޽ࠆߣ⠨߃ࠆ㧚ߔߥࠊߜ
V x

f

Lx

V x

0

0d x d L

V x

f

xd0

potential ߪ࿑ 6.2.1 ߦ␜ߔࠃ߁ߦ੗ᚭߩᒻࠍߒߡ޿ࠆߩߢ‫ޟ‬੗ᚭဳ potential‫ߣޠ‬๭߫ࠇࠆ㧚ߎߩ႐ว㧘ή㒢ߦᷓ޿੗ᚭࠍ⠨߃ߡ޿ࠆߎߣߦߥࠆ㧚෶㧘ߎߎߢᛒ߁໧㗴ߪ㧘એਅߩ▵ߢㅀߴࠆ▫ߩਛߩ☸ሶߩ໧㗴ߣૃߡ޿ࠆߎߣ߆ࠄ㧘ߒ߫ߒ߫‫৻ޟ‬ᰴరߩ▫ߩਛߩ☸ሶ‫޿⸒߁޿ߣޠ‬ᣇࠍߐࠇ㧘ߤࠎߥᢎ⑼ᦠߢ߽ᔅߕขࠅ਄ߍࠄࠇࠆ popular ߥ໧㗴ߢ޽ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 5࿑ 6.2.1 ੗ᚭဳ potential

0 d x d L ߩ▸࿐ౝߢߪ㧘Schrödinger equation ߪᒰὼᑼ(6.1.1)ߣหߓߊ

= 2 d 2\E\
2m dx 2ߣߥࠆ㧚⸃ߪᑼ(6.1.2)߅ࠃ߮ᑼ(6.1.3)ߣห᭽
\ Aeikx  Be ikx


(6.2.1)(6.2.2)߹ߚߪ\C cos kx  D sin kx(6.2.3a)\A 'sin kx  G(6.2.3b)߹ߚߪߎߎߢ k ߪᑼ(6.1.4)ߣห᭽

k2

2mE
=2

(6.2.4)ߣ␜ߐࠇࠆ㧚ߎߎ߹ߢߪ㧘 x ߩᄌၞߦ೙㒢߇ߥ޿ free particle ߣหߓߢ޽ࠆ߇㧘ߎߎ߆ࠄవ߇⇣ߥߞߡ᧪ࠆ㧚߹ߕ㧘Ⴚ⇇᧦ઙߣߒߡ㧘⸵ߐࠇࠆ㗔ၞߩ┵㧘ߔߥࠊߜ x 0 ߅ࠃ߮ xL ߢ\ xઍ౉ߔࠆߣ C0 ߢߥߌࠇ߫޿ߌߥ޿ߣ޿߁ⷐ⺧߇޽ࠆ㧚\ 00 ߇ᓧࠄࠇࠆ㧚ᰴߦ\ L0 ࠍઍ౉ߔࠆߣ

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 60 ࠍᑼ(6.2.3a)ߦ

\ L

D sin kLᑼ(6.2.5)ߩ 2 ߟߩ⸃㧘 D(6.2.5)00 ߹ߚߪ sin kLߢ޽ࠆ㧚㧔ߔߴߡߩ x ߦߟ޿ߡ\ x0 ߩ߁ߜ㧘 D

0 ߪ᣿ࠄ߆ߦ trivial solution

0 ߣߥߞߡߒ߹߁߆ࠄ㧚㧕ᓥߞߡ sin kL

0 ߢߥߌࠇ߫ߥࠄߥ޿㧚ߔߥࠊߜ
nS

kL
n

n 1, 2, 3, ...

(6.2.6)0 ߪหߓߊ trivial solution ࠍਈ߃ࠆߩߢ㒰޿ߡ޽ࠆ㧚ᑼ(6.2.6)ࠍᑼ(6.2.4)ߦઍ౉ߒߡ

h2n2(6.2.7)8mL2ࠍᓧࠆ㧚 n ߪᢛᢙߢ޽ࠆ߆ࠄ E ߪߣ߮ߣ߮ߩ୯ߒ߆ขࠅᓧߥ޿㧚ߔߥࠊߜ㧘energy ߪ㊂ሶൻߒߡ޿ࠆ㧔quantized㧕㧚 n ߪ㊂ሶᢙ㧔quantum number㧕ߣ⸒ࠊࠇࠆ㧚 E ߪ n ߩ㑐ᢙߢ޽ࠆߩߢ n ࠍ E ߩ subscript ߣߒߡߟߌߡ޿ࠆ㧚energy ߇㊂ሶൻߒߡ޿ࠆߣ޿߁
ߎߣߪ㧘ߘ߽ߘ߽ Planck ߇઒⺑ߣߒߡዉ౉ߒߚ߽ߩߢ޽ࠆ߇㧘ߎߎߢ⷗ߚࠃ߁ߦ㧘
Schrödinger equation ߦႺ⇇᧦ઙࠍട๧ߔࠆߎߣߦࠃࠅ⥄ὼߦ಴ߡ᧪ࠆߎߣߪ㕙⊕޿㧚߽ߞߣ߽㧘ߎߩߎߣߢ energy ߇㊂ሶൻߒߡ޿ࠆߣ޿߁ Planck ߩ઒⺑߇‫⸽ޟ‬᣿‫ࠇߐޠ‬ߚࠊߌߢߪߥ޿㧚 Schrödinger equation ߣ޿߁઒⺑ࠍណ↪ߔࠆߥࠄ߫ߘߎ߆ࠄ⥄ὼߦenergy ߩ㊂ሶൻ߇ዉ߆ࠇࠆ㧘ߣ⸒ߞߡ޿ࠆߛߌߢ޽ࠆ㧚1926 ᐕߩ⺰ᢥߢ Schrödinger⥄り㧘ᰴߩࠃ߁ߦㅀߴߡ޿ࠆ㧚"In this communication I wish to show that the usual rules of quantization can be
replaced by another postulate (the Schrödinger equation) in which there occurs no
mention of whole numbers. Instead, the introduction of integers arises in the same
natural way as, for example, in a vibrating string, for which the number of nodes
is integral. The new conception can be generalized, and I believe that it penetrates
deeply into the true nature of the quantum rules."Enߐߡ㧘eigenvalue En ߦ⋧ᒰߔࠆ wavefunction ߪ㧘

\n x

B sin kx

B sin

nS x
L

(6.2.8)ߢ޽ࠆ߇㧘 B ߪ normalization condition
L

³\

n

x *\ n x dx 10ࠃࠅ᳿ቯ಴᧪ࠆ㧚ߔߥࠊߜ㧘ᑼ(6.2.8)ࠍᑼ(6.2.9)ߦઍ౉ߔࠆߣ㧘䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 7(6.2.9)L

B * B ³ sin 20ቯⓍಽߩㇱಽߪ

B*B

nS x
dx 1
L

(6.2.10)

L
ߦߥࠆ߆ࠄ
2

2
L

ᓥߞߡ G ࠍછᗧߩታᢙߣߒߡ
2 iG
e
Lߢ޽ࠆ߇ߤ߁ㆬࠎߢ߽ࠃ޿ G ࠍ࠯ࡠߣߒߡ
B

2
Lߣߔࠆ㧚⚿ዪ㧘wavefunction ߪ
B

\n x

2
nSx
sin
,
L
L

(6.2.11)0d xd Ln 1, 2,3,...

(6.2.12)ߣ޿߁ߎߣߦߥࠆ㧚࿑ 6.2.2 ߦ En 㧘\ n 㧘߅ࠃ߮ \ n ࠍ n ߩዊߐ޿ᣇ߆ࠄ 4 ߟ߶ߤߦߟ
2޿ߡ࿑␜ߒߡ޿ࠆ㧚 n ߇ᄢ߈ߊߥࠆߦߟࠇߡ energy ߪ n ߩੑਸ਼ߦᲧ଀ߒߡᄢ߈ߊߥࠆ
ߎߣ㧘wavefunction ߪਔ┵ࠍ฽߼ߡ n  1 ୘ߩ node㧔▵㧕߇޽ࠆߎߣ߇ಽ߆ࠆߢ޽ࠈ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 8࿑ 6.2.2 ৻ᰴరߩ particle in a box ߦߟ޿ߡ\ n x ߅ࠃ߮\ n x䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 92ߎߎߢㅀߴߚ৻ᰴరߩ particle in a box ߩ໧㗴ߪ㧘✢ဳࡐ࡝ࠛࡦߩ㔚ሶࠬࡍࠢ࠻࡞ࠍ
⸃㉼ߔࠆߩߦߒ߫ߒ߫↪޿ࠄࠇࠆ㧚ߎߎߢ✢ဳࡐ࡝ࠛࡦߣߪᰴߦ␜ߔࠃ߁ߥࡐ࡝ࠛࡦ
ߢ޽ࠆ㧚ethylene

CH2=CH2

butadiene

CH2=CHCH=CH2

hexatriene

CH2=CHCH=CHCH=CH2

octatetraene

CH2=CHCH=CHCH=CHCH=CH2㊂ሶൻቇߩീᒝࠍᆎ߼ߚ߫߆ࠅߩ⃻Ბ㓏ߢ㧘ߎߩࠃ߁ߥᄢ߈ߥಽሶߩࠬࡍࠢ࠻࡞ࠍ㊂ሶൻቇ⊛ߦᛒ߁ߎߣߪ߶ߣࠎߤਇน⢻ߢ޽ࠆ㧚ࠬࡍࠢ࠻࡞ߣ޿ߞߡ߽᳓⚛ේሶߦߟ޿
ߡߩ Bohr model ࠍᛒߞߚߦߔ߉ߥ޿㧚㗅ᐨߣߒߡ㧘He ߆ࠄߪߓ߹ࠆᄙ㔚ሶ♽ߩේሶߦߟ޿ߡ߹ߕቇ⠌ߒ㧘ߘࠇ߆ࠄ㧘᳓⚛ಽሶߥߤߩዊߐߥಽሶࠍቇ⠌ߒ㧘ߘߩᓟߢ㧘਄ㅀߒߚࠃ߁ߥᄢ߈ߥಽሶ߳ߣᲑ㓏⊛ߦീᒝߒߡ޿߆ߥ޿ߎߣߦߪಽ߆ࠄߥ޿ߪߕߢ
޽ࠆ㧚ߒ߆ߒ㧘ߎߎߩᲑ㓏ߢ㧘ߤ߁ߒߡ߽৻⸒㧘਄ㅀߩࡐ࡝ࠛࡦ♽ߩࠬࡍࠢ࠻࡞ߦߟ
޿ߡㅀߴߡ߅߈ߚ޿㧚ᧄ⾰⊛ߥߎߣߢಽ߆ࠄߥ޿ߎߣ߇޽ߞߡ߽ߒ߫ࠄߊߪᚒᘟߒߡ
߽ࠄߞߡ㧘ߛ޿ߚ޿ߎࠎߥ߰߁ߦࠬࡍࠢ࠻࡞ߩ໧㗴ࠍขࠅᛒ߁ߩߛߥ㧘ߣ޿߁ᗵⷡࠍ
ߟ߆ࠎߢ߽ࠄ޿ߚ޿㧚਄ߦ␜ߒߚࠃ߁ߦ㧘ੑ㊀⚿ว߇ 1 ߟ߅߈ߦ޽ࠆࠃ߁ߥࡐ࡝ࠛࡦߢߪ㧘ੑ㊀⚿ว 1ߟߦߟ޿ߡ 2 ୘ߩ S 㔚ሶ߇ಽሶౝࠍ⥄↱ߦേ߈࿁ߞߡ޿ࠆߣ⠨߃ߡ߶߷Ꮕߒᡰ߃ߥ
޿㧚଀߃߫ butadiene ߢߪ C=CC=C ߩ㐳ߐߩ৻ᰴరߩ▫ߩਛࠍ 4 ୘ߩ S 㔚ሶ߇⥄↱ߦേ߈࿁ߞߡ޿ࠆߣ⠨߃ࠆ㧚ߎߩ 4 ୘ߩ S 㔚ሶߪ㧘energy ߩૐ޿⁁ᘒ㧔‫ޟ‬orbital,㧔゠㆏㧕‫ޠ‬ߣ⸒߁㧕߆ࠄ㧘Pauli ߩේℂߦᓥߞߡ 1 ߟߩ゠㆏ߦ 2 ୘ߕߟ෼ኈߐࠇࠆ㧚ߔߥࠊߜ㧘\ 1
ߦ 2 ୘㧘\ 2 ߦ 2 ୘౉ࠆ㧚ߎ߁޿߁⁁ᘒ߇ butadiene ߩ߽ߞߣ߽቟ቯߥ⁁ᘒߢ㧘‫ޟ‬ၮᐩ⁁
ᘒ㧔ground state㧕‫߁⸒ߣޠ‬㧚1 ୘ߩ㔚ሶ߇\ 2 ߆ࠄ\ 3 ߦ⒖ߞߚ㧘ߔߥࠊߜ‫ޟ‬ബ⿠‫ࠇߐޠ‬ߚ⁁ᘒ߇߽ߞߣ߽ energy ߩૐ޿ബ⿠⁁ᘒ㧔╙৻ബ⿠⁁ᘒ߹ߚߪᦨૐബ⿠⁁ᘒ㧕ߢ޽ࠆ㧚㧔࿑ 6.2.3㧕

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 10

\㪊

\㪊
\㪉

\㪉
\㪈

\㪈

㪜㫏㪺㫀㫋㪼㪻
㪪㫋㪸㫋㪼㪞㫉㫆㫌㫅㪻
㪪㫋㪸㫋㪼

࿑ 6.2.3 butadiene ߩ㔚ሶ㈩⟎㧚ਣߪฦ゠㆏ߦ෼ኈߐࠇߡ޿ࠆ S 㔚ሶࠍ␜ߔ㧚ߎߩ 2 ߟߩ⁁ᘒ㑆ߩࠛࡀ࡞ࠡ࡯Ꮕߦ⋧ᒰߔࠆ㔚⏛ᵄࠍ butadiene ߪๆ෼ߔࠆ㧚ߔߥࠊߜ㧘╙৻ๆ෼Ꮺߩࠛࡀ࡞ࠡ࡯ߪ㧘╙৻ബ⿠⁁ᘒߣၮᐩ⁁ᘒߣߩࠛࡀ࡞ࠡ࡯Ꮕߦ⋧ᒰߔࠆ㧚ߎࠇߪ㧘\ 3 ߣ\ 2 ߣߩࠛࡀ࡞ࠡ࡯Ꮕߦ╬ߒ޿ߣ⠨߃ߡࠃߐߘ߁ߢ޽ࠆ㧚☸ሶߩ⾰㊂ߣߒߡ㔚ሶߩ⾰㊂ me ࠍขࠆߣᑼ(6.2.7)ࠃࠅ

E3  E2h2
u 32  22
8me L2(6.2.13)ߢ޽ࠆ߇㧘ౕ૕⊛ߦ⸘▚ߒߡߺࠃ߁㧚ߎߎߢ L ߩขࠅᣇ߇໧㗴ߣߥࠆ㧚L ߪ C=C ߩ㐳ߐ 135 pm ߩ 2 ୚ߣ C㧙C ߩ㐳ߐ 154 pm ߣߩ๺ߣ⠨߃ߡ߽޿޿߇㧘ߒ߫ߒ߫ᦝߦਔ஥ߦߘࠇߙࠇ὇⚛ේሶߩඨᓘಽ㧔ߔߥࠊߜ C㧙C ৻㊀⚿วߩ㐳ߐ 154 pm
ߩඨಽ㧕ࠍട߃ߚ߽ߩࠍ L ߣߔࠆߣ޿߁ㄭૃ߇ขࠄࠇࠆ㧚ߔߥࠊߜ㧘ߎߩ⠨߃ߢߪ
L 135 pm u 2  154 pm  154 pm 578 pm

me ߅ࠃ߮ h ߩ୯ࠍ↪޿㧘ౕ૕⊛ߦ⸘▚ߔࠆߣ㧘
2E3  E2

6.626 u 10-34 Js u 5
8 u 9.110 u1031 kg u 578 u 1012 m

2

9.02 u 1019 Jᵄᢙߦ឵▚ߔࠆߣ
Q 4.54 u104 cm 1ߣߥࠆ㧚ታ㛎ߢߪ㧘butadiene ߩๆ෼ߪ 4.61u104 cm 1 ߦ⷗಴ߐࠇߡ޿ࠆ㧚ߎࠎߥ◲නߥࡕ࠺࡞ߢታ㛎୯߇߶߷ౣ⃻಴᧪ࠆߣߪ㛳ߊ߶ߤߢ޽ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 11‫ڏ‬Problem 6.2.1

hexatriene ߪ S 㔚ሶࠍ 6 ୘ᜬߟ㧚ᧄᢥਛߢᛒߞߚ 1 ᰴరߩ▫ߩਛߩ☸ሶߩࡕ࠺࡞ࠍ
↪޿ࠆߣ㧘hexatriene ╙ 1 ๆ෼Ꮺߪ 2.8 u104 cm 1 ߦ⃻ࠇࠆߎߣ߇ផቯߐࠇࠆߎߣࠍ␜
ߖ㧚‫ڏ‬Problem 6.2.2
1 ᰴరߩ particle in a box ߩฦ⁁ᘒߦߟ޿ߡ x ߩᐔဋ୯

x

³\

n

x * x\ n x dWࠍ᳞߼ࠃ㧚‫ڏ‬Problem 6.2.3
ᧄᢥߢߪ㧘a particle in a box ߩ wavefunction ࠍ㧘ᑼ(6.2.3a)ߦၮߠ޿ߡ᳞߼ߚ㧚ోߊหߓႺ⇇᧦ઙࠍᑼ(6.2.2)ߦㆡ↪ߔࠆߎߣߦࠃߞߡ wavefunction ࠍ᳞߼ࠃ㧚ᦝߦߘࠇ߇
ᑼ(6.2.8)ߣ৻⥌ߔࠆߎߣࠍ␜ߖ㧚‫ڏ‬Problem 6.2.4
ࡐ࡝ࠛࡦߪੑ㊀⚿วߩᢙ߇ᄙߊߥࠆߦߟࠇߡๆ෼ࠬࡍࠢ࠻࡞߇㐳ᵄ㐳ߦࠪࡈ࠻ߔ
ࠆߎߣ߇⍮ࠄࠇߡ޿ࠆ㧚ߎߩߎߣߪᣣᏱ↢ᵴߢߺ߆ߌࠆ߽ߩࠍᕁ޿಴ߒߡߺߡ߽ಽ߆
ࠆߢ޽ࠈ߁㧚଀߃߫⮎ዪߢᄁߞߡ޿ࠆࡆ࠲ࡒࡦ A ߪ㤛⦡޿⦡ࠍߒߡ޿ࠆ߇㧘ߘࠇߪᰴߦ␜ߔࠃ߁ߦ౒ᓎߒߡ޿ࠆੑ㊀⚿วࠍ 5 ୘ᜬߞߡ޿ࠆ㧚
%*

%*

%*

%*
%*1*

%*

߹ߚੱෳߩ⿒޿⦡ߩ߽ߣߢ޽ࠆEࠞࡠ࠹ࡦߪᰴ߉ߦ␜ߔࠃ߁ߦ౒ᓎߒߡ޿ࠆੑ㊀⚿วࠍ 11 ୘ᜬߞߡ޿ࠆ㧚
%*
%*

%*

%*

%*

%*
%*

%*

%*

౒ᓎߒߡ޿ࠆੑ㊀⚿วߩᢙ߇Ⴧ߃ߡ⦡߇㐳ᵄ㐳ߦࠪࡈ࠻ߒߡ޿ࠆࠊߌߢ޽ࠆ㧚ߎߩ଀ߩࠃ߁ߦ޿ࠈ޿ࠈߥ⟎឵ၮ߇ߟ޿ߡ޿ߥ޿⚐☴ߥࡐ࡝ࠛࡦߩ╙৻ๆ෼Ꮺߩ
ࠛࡀ࡞ࠡ࡯ࠍ⴫ 6.2.1 ߦ␜ߔ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 12⴫ 6.2.1

ࡐ࡝ࠛࡦߩ╙৻ๆ෼Ꮺߩㆫ⒖ࠛࡀ࡞ࠡ࡯

ੑ㊀⚿วߩᢙ

ㆫ⒖ࠛࡀ࡞ࠡ࡯

'E
㧔 cm 1 න૏㧕hc

161,5502

46,0803

39,7504

32,9005

29,9406

27,470725,6408

24,390ታ 㛎 ୯ ߪ H. H. Jaffe, M. Orchin, "Theory and Applications of Ultraviolet
Spectroscopy" ࠃࠅߎߩࠃ߁ߦ㧘ੑ㊀⚿วߩᢙ߇Ⴧ߃ࠆߦߟࠇߡๆ෼ߪ㐳ᵄ㐳ߦࠪࡈ࠻ߔࠆ⃻⽎ࠍ S 㔚ሶߩ⥄↱㔚ሶᮨဳߢ⠨߃ࠃ㧚ੑ㊀⚿วߩᢙࠍ N ߣߒߡ㧘ᰴߩ㗅ᐨߢ⠨߃ࠃ㧚(1) 㔚ሶߩᢙߪ૗୘߆㧚(2)න⚿วߪ૗୘㧘ੑ㊀⚿วߪ૗୘߆㧚(3)න⚿วߩ㐳ߐࠍ rC C 㧘ੑ㊀⚿วߩ㐳ߐࠍ rC C ߣߒߡ㧘౒ᓎ♽ߩ㐳ߐ L ࠍ⴫ߖ㧚ߎߩ㓙㧘਄ߩ଀ߢߦ␜ߒߚࠃ߁ߦᧃ┵ߩലᨐࠍ౉ࠇࠆ߽ߩߣߔࠆ㧚(4)ᦨૐࠛࡀ࡞ࠡ࡯ߩㆫ⒖ߪ㧘㊂ሶᢙ޿ߊߟ߆ࠄ޿ߊߟ߳ߩㆫ⒖߆㧚(5)

ㆫ⒖ࠛࡀ࡞ࠡ࡯ࠍ N 㧘 L ߅ࠃ߮㔚ሶߩ⾰㊂ me ࠍ↪޿ߡ⴫ߖ㧚(6)ๆ෼ߩᵄ㐳 O ࠍ㧘ห᭽ߦ⴫ߒ㧘 N ߣ O ߣߩ㑐ଥߦߟ޿ߡี๧ߖࠃ㧚⴫ 6.2.1 ߩታ㛎୯ߣ߽Ყセߒߡี๧ߖࠃ㧚㧔ߎߎߢ‫ีޟ‬๧ߖࠃ‫߇ߚ޿↪ࠍ⪲⸒߁޿ߣޠ‬૗ࠍߒߚࠄ޿޿߆චಽ⠨߃ߡߺࠃ㧚ታ㛎୯ߣ⸘▚୯ࠍߚߛਗߴߡߺߡ㧘ว߁ߣ߆วࠊߥ޿ߣ߆⸒ߞߡߺߚߣߎࠈߢ޽߹ࠅᗧ๧ߪߥ޿㧚⎇ⓥߩ⺰ᢥߢߪ

Discussion ߣ޿߁࠮࡚ࠢࠪࡦ߇޽ࠆ߇ߘ߁޿߁ߎߣࠍ߿ߞߡ᰼ߒ޿㧚ᒰὼ৻ੱ৻ੱ߇⇣ߥߞߚ discussion ࠍߔࠆߢ޽ࠈ߁㧚ߎߎߦ୘ᕈ߇⷗ࠄࠇࠆ߽ߩߢ޽
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 13ࠆ㧚ߤ߁޿߁ discussion ࠍߔࠆ߆ᭉߒߺߢ޽ࠆ㧚(7)એ਄ߩี๧ࠍ߰߹߃ߡ㧘 Nࠃ㧚10 ߩࡐ࡝ࠛࡦߩๆ෼ࠬࡍࠢ࠻࡞ߩᵄ㐳ࠍផቯߖ

6.3 ᦭㒢ߥᷓߐߩ੗ᚭဳ potential೨▵ߢߪή㒢ߩᷓߐߩ੗ᚭဳ potential ࠍ⠨߃ߚ߇㧘ߎߩ▵ߢߪ㧘ߘߩᑧ㐳ߣߒߡ᦭㒢ߩᷓߐߩ potential ࠍ⠨߃ߡߺࠆ㧚⷗ᣇࠍᄌ߃ࠆߣ㧘㓚ოߩ㜞ߐ߇ή㒢ߢߥߊ㧘᦭㒢ߩ㜞ߐߒ߆ߥ޿㧘ߣ޿߁ߎߣߢ޽ࠆ㧚Section 6.2 ߩ႐วߣߪߎࠇߛߌߩ㆑޿ߢ޽ࠆ߇㧘ᓧ ࠄ ࠇ ࠆ wavefunction ߦ ߪ ᄢ ߈ ߥ ㆑ ޿ ߇ ޽ ࠆ 㧚 ᦨ ߽ 㗼 ⪺ ߥ ߎ ߣ ߪ 㧘 ޿ ࠊ ࠁ ࠆ‫ޟ‬wavefunction ߩߒߺ಴ߒ‫ࠆ޽߇ޠ‬㧘ߣ޿߁ߎߣߢ޽ࠅ㧘ߎߩߎߣߩߚ߼ߦฎౖ⺰ߢ
ߪℂ⸃಴᧪ߥ޿⒳‫ࠆߥߦߣߎࠆࠇࠄ⷗߇⽎⃻ߩޘ‬㧚޿ࠊ߫㧘ߎߩ໧㗴ߪ㧘㊂ሶജቇߩᦨ߽㊂ሶജቇࠄߒ޿ᔕ↪ߩ৻ߟߢ޽ࠆߣ⸒߃ࠃ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 14

6.3.1 ᦭㒢ߩᷓߐߩ੗ᚭဳ potentialࠛࡀ࡞ࠡ࡯ߩၮḰߪߤߎߦߣߞߡ߽޿޿߆ࠄ㧘੗ᚭߩᐩࠍࠛࡀ࡞ࠡ࡯ߩ࠯ࡠߣߔ
ࠆ㧚ߘ߁ߔࠆߣ㧘potential ߩოߩ㜞ߐ߇᦭㒢ߣ޿߁ߎߣߦߥࠅ㧘ߘߩ potential ߩ㜞ߐࠍ V0 ߣߒࠃ߁㧚೨▵ߣห᭽㧘੗ᚭߪ 0 d x d L ߩ▸࿐ߦ޽ࠆߣߔࠆ㧚㧔࿑ 6.3.1㧕ߔߥࠊ
ߜ

V x

V0

Lx

V x

0

0d xd L

V x

V0

x0

ߎߩࠃ߁ߥ♽ߦߟ޿ߡߩ Schrödinger equation ࠍ⸃ߊߦ޽ߚߞߡ㧘ଢቱ਄㧘࿑ 6.3.1 ߦ␜ߒߚ 3 ߟߩ㗔ၞ A㧘B㧘C ߦಽߌߡ⠨߃ࠃ߁㧚ߘࠇߙࠇߩ㗔ၞߢߩ Schrödinger equationߪᰴߩࠃ߁ߦߥࠆ㧚
㗔ၞ A

xd0

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 15

d 2\ A
dx 2
k2
㗔ၞ C

k 2\ A

(6.3.1)2m E  V0=2(6.3.2)xtL

d 2\ Cdx 2
㗔ၞ B

 k 2\ C(6.3.3)0d xd Ld 2\ B
dx 2
k '2

k '2 \ B2mE
=2(6.3.4)(6.3.5)ߎࠇࠄߩ Schrödinger equation ࠍ⸃ߊߦ޽ߚߞߡએᓟᚒ‫ߪޘ‬㧘potential ߩᄢ߈ߐߦࠃߞ
ߡ㧔1㧕 E  V0 ߩ႐วߣ㧔2㧕 E ! V0 ߩ႐วߣߩ 2 ߟߩ႐วߦಽߌߡ⠨߃ࠆᔅⷐ߇޽ࠆ㧚

㧔1㧕 E  V0 ߩ႐วߎߩ႐ว㧘ᑼ(6.3.1)ߩ⸃ߪ

\A

Aeikx  A ' e ikx

ߢ޽ࠆ߇㧘ᑼ(6.3.2)ߩ k 2 ߇ negative ߢ޽ࠆ߆ࠄ k ߪ⚐⯯ᢙߣߥࠆ㧚ߘߎߢታᢙ N ࠍ↪
޿㧘 k iN ߣ⟎߈឵߃ࠆ㧚ߔߥࠊߜ\A

Ae N x  A ' eN x

ߣߥࠆ㧚ߎߎߢ x o f ߢ\ A ߇⊒ᢔߒߥ޿ߚ߼ߦߪ A 0 ߢ޽ࠆᔅⷐ߇޽ࠆ㧚ߔߥࠊߜ

\AA ' eN x(6.3.6)ห᭽ߦ㗔ၞ C ߢߩ⸃ߪ\C

Ce N x  C ' eN x

ߢ޽ࠆ߇㧘 x o f ߢ⊒ᢔߒߥ޿ߚ߼ߦߪ C ' 0 ߢߥߌࠇ߫ߥࠄߥ޿㧚ᦝߦ㧘એਅߩ⸘䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 16▚ߩଢ೑ߩߚ߼ߦ਄ᑼߩ C ࠍᡷ߼ߡ CeN L ߣ߅ߊ㧚㧔ߎ߁⟎߈឵߃ࠆߎߣߦࠃߞߡ৻⥸ᕈߪᄬࠊࠇߥ޿㧚㧕ߔߥࠊߜ\CCeN x  L(6.3.7)৻ᣇ㧘㗔ၞ B ߢߩ⸃ߪ㧘ᑼ(6.2.2)㧘ᑼ(6.2.3a)㧘ᑼ(6.2.3b)ߩ޿ߕࠇߩᒻߢ߽⴫ߐࠇࠆ߇
㧔ߎߎߢߪ E ! 0 ߩ႐วࠍ⠨߃ࠆߎߣߦߔࠆ㧚㧕㧘ߎߎߢߪᑼ(6.2.3a)ߩᒻࠍㆬ߮㧘\ B B sin k ' x  B 'cos k ' x(6.3.8)ߣߔࠆ㧚ߎߩᑼߪᑼ(6.2.3a)ߣ৻⷗หߓࠃ߁ߦ⷗߃ࠆ߆߽ߒࠇߥ޿߇㧘ᧄ⾰⊛ߥߎߣߢᄢ߈ߊ⇣ߥࠆ㧚ߔߥࠊߜ㧘ߎߩ▵ߩ໧㗴ߢߪ㧘\ ߪ੗ᚭߩ┵ x

0 ߅ࠃ߮ xL ߢ\ B0ߢ޽ࠆߣ޿߁᧦ઙߪ૗߽ߥ޿㧚\ A ߿\C ߇୯ࠍᜬߟߎߣ߆ࠄಽ߆ࠆࠃ߁ߦ㧘

wavefunction ߪ੗ᚭߩᄖ߳ߒߺ಴ߒߡ޿ࠆ㧚߻ߒࠈ㧘ᄢಾߥ᧦ઙߪ x 0 ߅ࠃ߮ x L ߢ
wavefunction ⥄り߅ࠃ߮ߘߩ൨㈩߇ㅪ⛯ߢߥߌࠇ߫޿ߌߥ޿㧘ߣ޿߁ߎߣߢ޽ࠆ㧚ߔߥࠊߜ㧘߹ߕ㧘 x 0 ߦߡ
\A \B
(6.3.9)
d\ A
dx

d\ B
dx

(6.3.10)߇ᚑࠅ┙ߟᔅⷐ߇޽ࠅ㧘ᦝߦ߹ߚ xL ߦߡ\ B \Cd\ B
dx(6.3.11)d\ C
dx(6.3.12)߇ᚑࠅ┙ߚߥߌࠇ߫ߥࠄߥ޿㧚ᑼ(6.3.9)ᑼ(6.3.12)ࠃࠅᰴᑼࠍᓧࠆ㧚A' B 'A ' N Bk 'C B sin k ' L  B 'cos k ' L
N C Bk 'cos k ' L  B ' k 'sin k ' L(6.3.13)(6.3.14)(6.3.15)(6.3.16)ᑼ(6.3.13)㧙ᑼ(6.3.16)ࠃࠅᰴᑼࠍᓧࠆ㧚tan k ' L

2N k '
N 2  k '2

(6.3.17)

߹ߚߪ
k 'L
2
k
'L
1  tan 2
2
2 tan

2

k'

N

§ k '·
1 ¨ ¸
©N ¹

2

ᢛℂߔࠆߣ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 17(6.3.18)

k ' L k ' ·§ k 'k 'L ·§
 ¸ ¨ tan
 1¸ 0¨ tan
2
2
N ¹© N
©
¹
⸃ߣߒߡ
tan

k 'L
2



k'

cot

k 'L
2

k'

(6.3.19)(6.3.20)

N(6.3.21)

N

෶ߪ
tan

2mE L
2=

cot

2mE L
2=



E
V0  E

(6.3.22)E
V0  E

(6.3.23)ߎߎߢએਅߩ⟎߈឵߃ࠍߔࠆ㧚

2mE L
2=

x

(6.3.24)ߔߥࠊߜE

2= 2 x 2
mL2(6.3.25)

ߎ߁ߔࠆߣ

x2
mL2V0
 x2
2
2=
ߣߥࠆߩߢ㧘ᑼ(6.3.22)߅ࠃ߮ᑼ(6.3.23)ߪᰴߩࠃ߁ߦߥࠆ㧚E
E  V0

tan x

cot x



x2
mL2V0
 x2
2
2=
x2
mL2V0
 x2
2= 2

(6.3.26)(6.3.27)(6.3.28)m㧘 L 㧘߅ࠃ߮ V0 ߪᣢ⍮ߢ޽ࠆ߆ࠄ㧘ߎࠇࠄߩᑼߦࠃߞߡ E ߪ޽ࠆ⸵ߐࠇߚ୯ߩߺࠍ
ߣࠆߎߣ߇ಽ߆ࠆ㧚ߔߥࠊߜࠛࡀ࡞ࠡ࡯ߪ㊂ሶൻߒߡ޿ࠆ㧚
ᑼ(6.3.27)߅ࠃ߮ᑼ(6.3.28)߆ࠄ x ߩ୯ࠍ⸃ᨆ⊛ߥᒻߢ಴ߔߎߣߪ಴᧪ߥ޿߇㧘ౕ૕⊛ߥ m㧘 L 㧘߅ࠃ߮ V0 ߩ୯߇ਈ߃ࠄࠇࠇ߫ᢙ୯⊛ߦ⸃ߊ߆㧘߹ߚߪࠣ࡜ࡈ߆ࠄ⸃ߊߎߣ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 18߇಴᧪ࠆ㧚
એਅ㧘଀ߣߒߡ

V0

20

h2
8mL2

(6.3.29)

h2
ߪή㒢
8mL2
ߩᷓߐߩ੗ᚭဳ potential ߩ႐วߩࠛࡀ࡞ࠡ࡯ߩන૏ߢ޽ࠆ㧚ߎߩ႐วߩ n 4 ߩࠛࡀߩ႐วߦߟ޿ߡ⸃ࠍ᳞߼ߡߺࠃ߁㧚ᑼ(6.2.7)߆ࠄಽ߆ࠆࠃ߁ߦ਄ᑼߩ߁ߜ

h2
h2
h2
n
5
㧘
ߩࠛࡀ࡞ࠡ࡯߇
25ߢ޽ࠆ߆ࠄ㧘ߎߎߢ⠨߃ߚ
20
ߣ
8mL2
8mL2
8mL2
޿߁୯ߪ n 4 ߣ n 5 ߩ 2 ߟߩࠛࡀ࡞ࠡ࡯Ḱ૏ߩ㑆ߩᷓߐߩ੗ᚭࠍ⠨߃ߡ޿ࠆࠊߌߢ
޽ࠆ㧔ᓟ಴ߩ࿑ 6.3.2 ࠍෳᾖ㧕㧚ᑼ(6.3.29)ࠍ↪޿ࠆߣᑼ(6.3.27)߅ࠃ߮ᑼ(6.3.28)ߪ
࡞ࠡ࡯߇ 16

tan x
cot x



x2
5S 2  x 2(6.3.30)x2
5S2  x 2

(6.3.31)ߣߥࠆ㧚⸃ࠍ᳞߼ࠆߚ߼ߦ㧘࿑ 6.3.1a ߢᑼ(6.3.30)ߩฝㄝ߅ࠃ߮Ꮐㄝࠍ x ߩ㑐ᢙߣߒߡࡊࡠ࠶
࠻ߒ㧘੤ὐࠍ᳞߼ߡ޿ࠆ㧚ห᭽ߦ࿑ 6.3.1b ߢߪᑼ(6.3.31)ߦߟ޿ߡߩࡊࡠ࠶࠻ࠍⴕ޿㧘੤ὐࠍ᳞߼ߡ޿ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 19

y

tan x

y

y

tan x

tan x

2
1.5
1
0.5
0
1

-0.5

2

3

4

5

6

7

8

-1
-1.5
-2

y
࿑ 6.3.1ax2
5S 2  x 2

ᑼ(6.3.30)ࠍࠣ࡜ࡈߢ⸃ߊ㧚ࠣ࡜ࡈߩ੤ὐࠃࠅ㧘⸃ߣߒߡ x

2.74 ߅ࠃ߮ x 5.41 ࠍᓧࠆ㧚

y
4
3

x2

y
y

cot x

y

5S 2  x 2

cot x

2

y

1

cot x

0
-1

x
1

2

3

4

5

6

7

8

-2
-3
࿑ 6.3.1b ᑼ(6.3.31)ࠍࠣ࡜ࡈߢ⸃ߊ㧚ࠣ࡜ࡈߩ੤ὐࠃࠅ㧘⸃ߣߒߡ x 1.37 㧘 x 4.09 ߅ࠃ߮ xࠣ࡜ࡈߩ੤ὐ߆ࠄ⸃ࠍ᳞߼ࠆ㧚߹ߕ㧘ᑼ(6.3.30)ߩ⸃ߪ

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 20

6.62 ࠍᓧࠆ㧚

x

2.74,

x 5.41,

E

3.04

h2
8mL2

E 11.86h2
8mL2

߹ߚ㧘ᑼ(6.3.31)ߩ⸃ߪ

x 1.37,

E

0.76

h2
8mL2

x

E

6.78

h2
8mL2

4.07,

h2
8mL2
એ਄ߩ⚿ᨐࠍ㧘ή㒢ߩᷓߐߩ potential ߩ႐วߣᲧセߒߡ࿑ 6.3.2 ߦ␜ߔ㧚
x

6.62,

E 17.76䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 21

࿑ 6.3.2

ή㒢ߥᷓߐߩ੗ᚭဳ potential ߣ᦭㒢ߥᷓߐߩ੗ᚭဳ potential

potential V ߅ࠃ߮࿕᦭୯ En ߪ

h2ߩන૏ߢ␜ߒߡ޽ࠆ㧚
8mL2

ᦨᓟߦ wavefunction ߦߟ޿ߡㅀߴࠆ㧚ᑼ(6.3.13)㧙ᑼ(6.3.16)ߩ᧦ઙࠃࠅ㧘ᦨ⚳⊛ߦ
ߪ wavefunction ߣߒߡᰴᑼ߇᳞߹ࠆ㧚\A

A ' eN x

xd0§N·0d xd LA ' ¨ sin k ' x  cos k ' x ¸(6.3.32)
©k'
¹
§k'
·
Ldx
\ C A ' ¨ sin k ' x  cos k ' x ¸ eN x  L©N¹
A ' ߪ x ߩᄌၞߔߴߡߦߟ޿ߡߩ normalization condition ߆ࠄ᳞߹ࠆ㧚਄ᑼ߆ࠄ᣿ࠄ߆ߥ

\B

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 22ࠃ߁ߦ㧘 x d 0 ߅ࠃ߮ x t L ߩ▸࿐ߢ߽ wavefunction ߪ࠯ࡠߢߪߥ޿㧚ߎߩߎߣߪߒ߫
ߒ߫‫ޟ‬wavefunction ߇ potential ߩᄖ߳ߒߺ಴ߒߡ޿ࠆ‫ࠆࠇߐ⃻⴫ߢ⪲⸒߁޿ߣޠ‬㧚ᑼ

(6.3.32)ࠍ↪޿ߡ㧘ታ㓙ߦ wavefunction ࠍឬ޿ߡߺߡ㧘߆ߟή㒢ߩᷓߐߩ potential ߩ႐ว㧔࿑ 6.3.2㧕ߣᲧセߒߡߺࠆߣ㧘ߎߩ‫ߺߒޟ‬಴ߒ‫ߊࠃߪߣߎߩޠ‬ℂ⸃಴᧪ࠆߪߕߢ޽
ࠆ㧚ߎࠇߪ⻉ำߩṶ⠌ߦᱷߒߡ߅ߎ߁㧚(Problem 6.3.1)
㧔2㧕 E ! V0 ߩ႐วߎߩ႐ว㧘ᑼ(6.3.2)ߩ k 2 ߪ positive ߥߩߢ k ߪታᢙߢ޽ࠆ㧚ᓥߞߡ\ A ߅ࠃ߮\ C ߪᰴߩࠃ߁ߦߥࠆ㧚\A\C

A sin kx  A 'cos kx
C sin kx  C 'cos kx(6.3.33)(6.3.34)\ B ߪᑼ(6.3.8)ߣหߓߢࠃ޿㧚ߎߎߢ㧘ଥᢙ A, A ' 㧘߅ࠃ߮ C , C ' ࠍߤߩࠃ߁ߦㆬࠎߢ߽
\ A ߅ࠃ߮\ C ߪ f d x d f ߩ▸࿐ߢ well behaved ߢ޽ࠆ㧚ᓥߞߡଥᢙߦ㑐ߔࠆ૗ߩⷐ
⺧߽ߥ޿㧚ࠛࡀ࡞ࠡ࡯ࠍ᳿߼ࠆ૗ߩⷐ⺧߽ߥ޿㧚ߔߥࠊߜ㧘ࠛࡀ࡞ࠡ࡯ߪ㊂ሶൻߒߡ޿ߥ޿㧚☸ሶߪ E ! V0 ߩ▸࿐ౝߩߤࠎߥࠛࡀ࡞ࠡ࡯ࠍ߽ขࠅᓧࠆߣ޿߁ߎߣߦߥࠆ㧚‫ڏ‬Problem 6.3.1
ᧄ▵ߢࠛࡀ࡞ࠡ࡯ࠍ᳞߼ߚౕ૕⊛ߥ႐วߦߟ޿ߡ㧘wavefunction ࠍ x ߩ㑐ᢙߣߒߡࡊࡠ࠶࠻ߖࠃ㧚ߘࠇࠍή㒢ߩᷓߐߩ႐วߩ wavefunction ߣᲧセߒ㧘࿑ 6.2.2 ߣหߓࠃ߁ߥᒻᑼߢឬߌ㧚㧔ߔߥࠊߜ㧘࿑ 6.3.2 ߩᏀฝ 2 ߟߩ࿑ߦ wavefunction ࠍᦠ߈౉ࠇࠆ㧚㧕

6.4 potential ߩოߣ⽾ㅢ

Section 6.2 ߅ࠃ߮ Section 6.3 ߩᑧ㐳ߣߒߡ㧘หߓ৻ᰴరߩਗㅴㆇേߢ޽ࠆ߇
0 d x d L ߩ▸࿐ߢߩߺ᦭㒢ߩ potential V0 ࠍᜬߜ㧘ߘߩઁߩ x ߩ▸࿐ߢߪ potential ࠍฃߌߥ޿႐วࠍ⠨߃ࠆ㧔࿑ 6.4.1㧕㧚ߔߥࠊߜ Section 6.3 ߢᛒߞߚ♽߇ potential ߩ‫ޟ‬੗ᚭ‫ޠ‬ߢ޽ࠆߩߦኻߒ㧘ߎߎߢᛒ߁ߩߪ potential ߩ‫ޟ‬ო‫ࠆ޽ߢޠ‬㧚ਔ⠪ߪ߶ߣࠎߤหߓขࠅ
ᛒ޿ࠍߔࠆࠊߌߢᡷ߼ߡㅀߴࠆ߶ߤߩߎߣ߽ߥ޿㧚ߒ߆ߒṶ⠌ߦ߽ߥࠆߢ޽ࠈ߁߆ࠄ
޽߃ߡขࠅ਄ߍࠆߎߣߦߔࠆ㧚ߚߛ㧘ዋ‫ࠍ⿰ޘ‬ᄌ߃ߡ㧘ߎߩ potential ߩოߦะ߆ߞߡ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 23☸ሶ߇ⴣ⓭ߔࠆߣ޿߁േ⊛ߥ໧㗴ߦߟ޿ߡ⠨߃ࠆ㧚

࿑ 6.4.1

potential ߩოߎࠇߪ㧘⭯⤑ߦ߱ߟ߆ࠆ☸ሶߩࡕ࠺࡞ߣ⠨߃ࠄࠇࠆ㧚ฎౖ⺰ߢ⠨߃ࠆߣ੐ᖱߪ᣿ࠄ
߆ߢ޽ࠆ㧚ߔߥࠊߜ㧘☸ሶ߇Ꮐ஥߆ࠄოߦ߱ߟߌࠄࠇߚᤨ㧘☸ሶ߇ర‫ޘ‬ᜬߞߡ޿ߚࠛ
ࡀ࡞ࠡ࡯߇ potential ࠛࡀ࡞ࠡ࡯ࠃࠅ߽ᄢ߈ߌࠇ߫☸ሶߪოߩਛߦ߽ߋࠅㄟ߻ߢ޽ࠈ
߁㧚ߘ߁ߢߥ޿႐วߪ☸ሶߪოߢߪߨ㄰ߐࠇࠆߛߌߢ޽ࠆ㧚ߢߪ㊂ሶ⺰ߢߪߤ߁ߥࠆ
ߢ޽ࠈ߁߆㧫
࿑ 6.4.1 ߦ␜ߒߚࠃ߁ߦ㧘ଢቱ਄ x ゲࠍ A㧘B㧘C ߩ 3 ߟߩ㗔ၞߦಽߌ㧘ߘࠇߙࠇߩ
㗔ၞߢ Schrödinger equation ࠍ┙ߡࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 24㗔ၞ A: H=2 d 22m dx 2x0㗔ၞ B: H

=2 d 2 V0
2m dx 20d xd L(6.4.1)=2 d 2Lx
2m dx 2ߎࠇࠄߩ⸃ߪ㧘㗔ၞ A ߣ C ߢߪ Section 6.1 ߢቇ⠌ߒߚ⥄↱☸ሶߣߥࠆ㧚㗔ၞ B ߢ߽㗔ၞ C: H⥄↱☸ሶߢ޽ࠅ㧘ߚߛ E ߩઍࠅߦ E  V0 ߣ߅޿ߚߎߣߛߌ߇⇣ߥࠆ㧚ߔߥࠊߜ㧘ᑼ

(6.1.2)ࠍෳᾖߒߡᰴߩࠃ߁ߦ᳞߹ࠆ㧚㗔ၞ A: \

Aeikx  A ' e  ikxk㗔ၞ B: \

Bei k ' x  B ' e  i k ' xk'㗔ၞ C: \

Ceikx  C ' e  ikx

k

2mE
=
2m E  V0(6.4.2)=2mE=ߎߎߢ߽㧘೨▵ߣห᭽ߦ E  V0 ߩ႐วߣ E ! V0 ߩ႐วߦಽߌߡ⠨߃ࠆᔅⷐ߇޽ࠆ㧚߹ߕ㧘 E  V0 ߩ႐วࠍ⠨߃ࠆߎߣߦߒࠃ߁㧚 E  V0  0 ߢ޽ࠆ߆ࠄ㧘 k ' ߪ imaginaryߣߥࠆ㧚ߘߎߢ
k ' iNߣߒߡ㧘ᣂߒ޿ real ߩᄌᢙ N ࠍዉ౉ߔࠆ㧚ߔߥࠊߜ㗔ၞ B: \Be N x  B ' eN x

2m V0  E

N

(6.4.3)=ߎࠇࠄߩᑼߩଥᢙ A, A ', B, B ', C , C ' ߪႺ⇇᧦ઙ߆ࠄ᳞߼ࠆߎߣ߇಴᧪ࠆ㧚ౕ૕⊛ߥ଀ߣߒߡ㧘☸ሶ߇㗔ၞ A ߩᏀ஥߆ࠄოߦⴣ⓭ߔࠆ႐วࠍ⠨߃ࠃ߁㧚޽ࠆㇱಽߪ〡ߨ㄰ࠅ৻ㇱߪოߩਛߦ߽ߋࠅㄟ߻㧚ߘࠇ߇߹ߚ㧘 x L ߢ〡ߨ㄰ߞߚࠅ㧘ო߆ࠄᛮߌ಴ߚࠅߔࠆ㧚ߒ߆ߒ㧘ߤ߁⠨߃ߡ߽㧘㗔ၞ C ߢᏀ஥ߦㅴⴕߔࠆ☸ሶߪ޽ࠅᓧ
ߥ޿㧚ᓥߞߡ C ' 0 ߢߥߌࠇ߫ߥࠄߥ޿㧚 C

2ߪოࠍ⽾ㅢߔࠆ☸ሶߩഀวࠍ␜ߔ㧚ห

2

᭽ߦ A ߪ㗔ၞ A ߢฝ஥ߦㅴⴕߔࠆ☸ሶߩഀวࠍ␜ߔ㧚ᓥߞߡ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 25

C
A

2
2

ߪ㧘ოߩᏀ஥߆ࠄ

ოߦ߱ߟ߆ߞߚ☸ሶ߇㧘ოࠍ⽾ㅢߔࠆ⏕₸㧔transmission probability㧕ࠍ␜ߔߎߣߦߥ
ࠆ㧚એਅߎࠇࠍ᳞߼ߡߺࠃ߁㧚߹ߕ㧘㗔ၞ A㧘B㧘C ߩႺ⇇㧔ߔߥࠊߜ x 0 ߅ࠃ߮ x L 㧕ߢ\ ߪㅪ⛯ߢߥߌࠇ߫ߥࠄߥ޿㧚A  A'B  B'

N L

Be  B ' eN L CeikL߹ߚ㧘ฦႺ⇇ߢߩ൨㈩߇ㅪ⛯ߢߥߌࠇ߫ߥࠄߥ޿㧚
ikA  ikA ' N B  N B '(6.4.4)(6.4.5)
N Be N L  N B ' eN L ikCeikLᧂ⍮ᢙߪ A, A ', B, B ', C ߩ 5 ߟߢ޽ࠅ㧘ᣇ⒟ᑼߪᑼ(6.4.4)ߣᑼ(6.4.5)วࠊߖߡ 4 ߟ޽ࠆ㧚ᓥߞߡ㧘ߘࠇߙࠇߩ୯ࠍ᳞߼ࠆߎߣߪ಴᧪ߥ޿߇㧘޽ࠆ 1 ߟߩᄌᢙߦኻߔࠆᲧߪ᳞߹
ࠆߎߣߦߥࠆ㧚਄⸥ߩㅪ┙ᣇ⒟ᑼࠍ⸃޿ߡ㧘C
ACߣߒߡᰴᑼࠍᓧࠆ㧚A

4iN ke

ikL^ N  ik e  N  ik e `NL2ߎࠇߩ complex conjugate

2

N L

(6.4.6)C*ࠍߣࠅ㧘ដߌวࠊߖߡ㧘transmission probability P ߣߒߡᰴ
A*

ᑼࠍᓧࠆ㧚
P

C
A

2

1

2

NL

1

e  e N L
16

2

(6.4.7)E§E·¨1  ¸
V0 © V0 ¹⚦߆޿ߎߣߪߣ߽߆ߊߣߒߡᄢಾߥߎߣߪ㧘ߚߣ߃ E  V0 ߢ޽ߞߡ߽ P ߪ࠯ࡠߢߥ޿
ߣ޿߁ߎߣߢ޽ࠆ㧚ߔߥࠊߜ㧘☸ሶ߇޽ࠆഀวߢოࠍ⽾ㅢߒᓧࠆߣ޿߁ߎߣߢ޽ࠆ㧚

V0 ! E ߢ޽ࠆ߆ࠄ㧘ฎౖ⺰ߢߪߘ߁޿߁ߎߣߪ⛘ኻߦ޽ࠅ߃ߥ޿㧚㊂ሶ⺰ߦࠃߞߡᆎ
߼ߡߎ߁޿߁ߎߣ߇⿠ࠆࠊߌߢ޽ࠆ߇㧘ߘߩේ࿃ߪ㧘potential ߩოߩᚲߢ wavefunction߇ᕆߦ࠯ࡠߦߥࠄߥ޿ߢ㧘޿ߊࠄ߆ߒߺ಴ߒߡ޿ࠆߣ޿߁㧘೨▵ߢ߽ᒝ⺞ߒߚߎߣߦ
޽ࠆ㧚ᰴߦ਄ߣߪ෻ኻߩ႐ว㧘ߔߥࠊߜ㧘E ! V0 ߩ႐วࠍ⠨߃ࠃ߁㧚ߎߩ႐วᑼ(6.4.2)ߩ k 'ߪᱜߢ޽ࠆ㧚\ ߇ㅪ⛯ߢ޽ࠆߣ޿߁᧦ઙ߆ࠄA  A' B  B 'Beik ' L  B ' e  ik ' L CeikL䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 26(6.4.8)߹ߚ㧘\ ߩᓸಽ߇ㅪ⛯ߢ޽ࠆߣ޿߁᧦ઙ߆ࠄ
ikA  ikA ' ik ' B  ik ' B '(6.4.9)ik ' Beik ' L  ik ' B ' e  ik ' L ikCeikL
ߎࠇࠄ 4 ߟߩ᧦ઙࠃࠅᰴᑼࠍᓧࠆ㧚
C
A

eikL

^

4kk '
k k

2

2

e  ik ' L  k  k ' e ik ' L(6.4.10)`ߎߩᑼߣߘߩ complex conjugate ࠍដߌࠆߎߣߦࠃࠅ㧘transmission probability P ߣߒߡᰴᑼࠍᓧࠆ㧚
P

C
A

2

1

2

1

e
16

 ik '

eik ' 2·E§E¨  1¸
V0 © V0 ¹

1
sin 2 k ' L
1·E§E
4 ¨  1¸
V0 © V0 ¹(6.4.11)

ᦨᓟߦ E V0 ߩ႐วࠍ⠨߃ࠃ߁㧚ߎߩ႐วߦߪ㧘ᑼ(6.4.7)߅ࠃ߮ᑼ(6.4.11)ߦ E V0 ࠍߘߩ߹߹ઍ౉ߔࠆࠊߌߦߪ޿߆ߥ޿㧚㕙ୟߢ߽ᆎ߼߆ࠄ E V0 ߩ႐วߦߟ޿ߡ᳞߼ࠆ
ᔅⷐ߇޽ࠆ㧚ߎࠇߪ◲නߢ޽ࠆ߆ࠄ⻉ำߩṶ⠌ߣߒߡ߅ߎ߁㧚(Problem 6.4.2)
ߎࠇࠄߩ⚿ᨐࠍ߹ߣ߼ߡ㧘㔚ሶߩⴣ⓭ߦߟ޿ߡ Lߣߒߡ࿑ 6.4.2 ߦࡊࡠ࠶࠻ߒߚ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 27

0.1 nm 㧘 V0

5 eV ߩ႐วࠍ଀

1.0

0.8

P

0.6

0.4

0.2

0.0

0

1

2

3

4

5

E/V0

࿑ 6.4.2㔚ሶߩ⽾ㅢ⏕₸㧔 V0

5 eV 㧘 L

0.1 nm 㧕‫ڏ‬Problem 6.4.1
ᑼ(6.4.7)߅ࠃ߮ᑼ(6.4.11)ࠍ⏕߆߼ࠃ㧚‫ڏ‬Problem 6.4.2

E V0 ߩ႐วߦߟ޿ߡ⽾ㅢߩ⏕₸ P ࠍ᳞߼ࠃ㧚‫ڏ‬Problem 6.4.3
ή㒢ߩෘߐࠍ߽ߜ㧘potential ߇ V0 ߢ޽ࠆࠃ߁ߥო߇޽ࠆ㧚⾰㊂ m ߩ☸ሶ߇㧘ߎߩოߩᏀ஥߆ࠄࠛࡀ࡞ࠡ࡯ E ߢოߦะ߆ߞߡ౉኿ߔࠆߎߣࠍ⠨߃ࠆ㧚ߎߩᤨ㧘ოߢߪߨ㄰
§E·
ߐࠇࠆ⏕₸㧔෻኿ଥᢙ㧕 R ¨ ¸ ࠍ㧔a㧕 E  V0 ߅ࠃ߮㧔b㧕 E ! V0 ߩ႐วߦߟ޿ߡ᳞߼
© V0 ¹ࠃ㧚⚿ᨐߪEߩ㑐ᢙߣߒߡ␜ߔߎߣ㧚V0䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 28ࡅࡦ࠻㧦࿑ 6.4.1 ߦ߅޿ߡ㧘A㧘B 2 ߟߩ㗔ၞߩߺࠍ⠨߃ࠇ߫ࠃ޿㧚෻኿ଥᢙߪᑼ(6.4.2)
2

A'
ߩ⸥ภߢ
ߢ⴫ߐࠇࠆ㧚
A

6.5 ᲑᏅߩ޽ࠆ੗ᚭဳ potential

Section 6.2 ߣ Section 6.3 ߢ㧘ᷓߐߩ⇣ߥࠆ੗ᚭဳ potential ߩ໧㗴ࠍᛒߞߚ߇㧘޿ߕࠇߩ႐ว߽੗ᚭߩᐩߪᐔߢ޽ߞߚ㧚ߎߩ੗ᚭߩᐩ߇ᐔߢߥ޿♽ߦߟ޿ߡ㧘 Schrödinger
equation ࠍ⸃ߊߎߣߪᩰᅢߥṶ⠌ߣߥࠆ㧚ߎߎߢߪ৻ߟߩ଀ߣߒߡ㧘એਅߦ␜ߔࠃ߁
ߥ㧘ᲑᏅߩ޽ࠆ potential ࠍขࠅ਄ߍߡߺࠃ߁㧚ᧄ⾰⊛ߦᣂߒ޿ߎߣߪ૗߽ߥ޿㧚Section
6.2 ߣหߓᣇᴺߢ⸃ߊߛߌߢ޽ࠆ߆ࠄ㔍ߥߊ⸃ߌࠆ໧㗴ߢ޽ࠆ㧚
potential ߪᰴߩࠃ߁ߥ႐วࠍ⠨߃ࠆ㧚



L
2

V x

f

L
d x d 0 㧔㗔ၞ A㧕 V x
20L㧔㗔ၞ B㧕
2

V x

V0

V x

f

f d x d 

0d xd

L
d xdf
2

potential ߩ᭎⇛ࠍ࿑ 6.5.1 ߦ␜ߔ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 29

࿑ 6.5.1

ᲑᏅߩ޽ࠆ potential

Schrödinger equation ߪ E  V0 ߩ႐วߣ E ! V0 ߩ႐วߣߦಽߌߡ⸃߆ߥߌࠇ߫ߥࠄߥ޿
ߎߣߪ੹߹ߢቇ⠌ߒߚㅢࠅߢ޽ࠆ㧚એਅ㧘E ! V0 ߩ႐วߩߺࠍᛒ߁ߎߣߦߔࠆ㧚E  V0ߩ႐วߪ⻉ำ⥄ࠄ⹜ߺߡ᰼ߒ޿㧚
Section 6.2 ߅ࠃ߮ Section 6.3 ࠍቇ⠌ߒߚ੹ߣߥߞߡߪ㧘㗔ၞ A ߅ࠃ߮ B ߦ߅޿ߡ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 30

ߪ㧘wavefunction ߪᰴߩࠃ߁ߦߥࠆߎߣߪ᣿ࠄ߆ߢ޽ࠈ߁㧚
㗔ၞ A:

\A x

A eikx  e  ik L  x(6.5.1)

㗔ၞ B:

\B x

B eik ' x  e  ik ' L  x(6.5.2)

ߎߎߢ k ߅ࠃ߮ k ' ߪᰴᑼߢਈ߃ࠄࠇࠆ㧚

2mE
=

k

(6.5.3)2m E  V0k'(6.5.4)=L
ߢ\ A x2

A 1  e  ikL

B 1  eik ' L(6.5.5)

kA 1  e  ikL

B ' 1  ei k ' L(6.5.6)k 'L
2(6.5.7)ᑼ(6.5.1)ߢߪ x

0 ߣ޿߁᧦ઙࠃࠅᰴᑼࠍᓧࠆ㧚ߎࠇࠍᢛℂߒߡ㧘

k cot

kL
2

k 'cotࠍᓧࠆ㧚ߔߥࠊߜ㧘

2mE
2mE L
cot
=
2=



2m E  V0=

cot

2m E  V0 L
2=ߎࠇ߇㊂ሶൻߒߚࠛࡀ࡞ࠡ࡯ E ࠍਈ߃ࠆᑼߢ޽ࠆ㧚߽ߒ߽ L ߣ V0 ߇ਈ߃ࠄࠇࠇ߫㧘⸃ᨆ⊛ߦ㧘߹ߚߪࠣ࡜ࡈ⊛ߦ⸃ߊߎߣ߇น⢻ߢ޽ࠆ㧚ߎߎߢ

2mE L
2=
ߣ߅ߊߣ㧘ᑼ(6.5.8)ߪᰴߩࠃ߁ߦߥࠆ㧚x

x cot x x2 

L2 mV0L2 mV0
2
cot
x

2= 2
2= 2

(6.5.9)(6.5.10)ౕ૕⊛ߥ଀ߣߒߡ

2= 2
h2 4(6.5.11)
L2 m 8mL2 S2ߩ႐วࠍߣࠅ㧘ᑼ(6.5.10)ࠍࠣ࡜ࡈ߆ࠄ⸃޿ߡߺࠃ߁㧚ߎߩ႐ว㧘ᑼ(6.5.10)ߪ
V0䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 31

x cot x

 x 2  1 cot x 2  1(6.5.12)߹ߚߪ
tan x 2  1
tan x
ߣߥࠆ㧚

x2 1
x



(6.5.13)

y
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2

x
2

4

6

8

y

y
࿑ 6.5.2x2  1x

tan x 2  1
tan x
ᑼ(6.5.13)ߩࠣ࡜ࡈߦࠃࠆ⸃ᴺ

࿑ 6.5.2 ߦߎߩᑼߩฝㄝ߅ࠃ߮Ꮐㄝࠍ x ߦߟ޿ߡࡊࡠ࠶࠻ߒߡ޿ࠆ㧚ࠣ࡜ࡈߩ੤ὐ
߆ࠄ㧘⸃ߪዊᢙὐએਅ 3 ᩴ߹ߢߩ♖ᐲߢ.
x 1715x 3.223x 4.765
x 6.323x 7.887...

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 32ߣ᳞߹ࠆ㧚ᑼ(6.5.9)ࠍ↪޿ࠆߣ㧘

4 x 2 h2
S 2 8mL2
ߢ޽ࠆ߆ࠄ㧘࿕᦭୯ߣߒߡᰴߩ⚿ᨐ߇ᓧࠄࠇࠆ㧚E

E 1.19

h2
8mL2

E

4.21

h2
8mL2

E

9.20

h2
8mL2

E 16.20E(6.5.14)

h2
8mL2

h2
25.21
8mL2

...
੗ᚭߩᐩ߇ V0 ߢᐔߢ޽ࠆ႐วߩ࿕᦭୯ߪ n 2ࠃࠅ߶ࠎߩࠊߕ߆㧘ߔߥࠊߜ⚂ 0.2

h2ߢ޽ߞߚߎߣࠍᕁ޿಴ߔߣ㧘ߘࠇ
8mL2

h2ߛߌᄢ߈ߊߥߞߡ޿ࠆߎߣ߇ಽ߆ࠆ㧚࿑ 6.5.3
8mL2

ߦਔ⠪ࠍᲧセߒߡ߅ߊ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 33

࿑ 6.5.3

ᐩߩᐔࠄߥ੗ᚭဳ potential ߣᐩߦᲑᏅߩ޽ࠆ੗ᚭဳ potential䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 34

ᑼ(6.4.7)ߪห᭽ߥ໧㗴ࠍᛒߞߚ Atkins ߩ"Molecular Quantum Mechanics"ߩ 43 㗁ߩᑼ(3.2.5)ߣ߶ࠎߩዋߒ⇣ߥߞߡ޿ࠆ㧚ᧄ lecture note ߩᣇ߇ᱜߒ޿ߣ⑳ߪାߓߡ޿ࠆ㧚߹ߚ㧘߿ߪࠅ Atkins ߩᢎ⑼ᦠߩ 44 㗁ߦ proton ߣ deuteron ߦߟ޿ߡߩ⽾ㅢ⏕₸ߩ࿑߇ਈ߃ࠄࠇߡ޿ࠆ߇㧘ᧄ lecture note ߩ࿑ 6.4.2 ߦ␜ߒߚ㔚ሶߦᲧߴߡ 1000 ୚߽㊀޿ේሶߦߟ޿ߡߪ E  V0 ߩ႐วߦߪ⽾ㅢߩ⏕₸ߪ㒢ࠅߥߊ࠯ࡠߦㄭ޿୯ߢ޽ࠅ㧘Atkins ߩ⸘▚㑆㆑޿ߩࠃ߁ߢ޽ࠆ㧚

6.6 ਃᰴరߩ▫ߩਛߩ☸ሶᧄ┨ߢߪ㧘Section 6.1 ߆ࠄ Section 6.5 ߹ߢߔߴߡ㧘৻ᰴరߩ Schrödinger equation ߦ⹤ࠍ㒢ߞߡ᧪ߚ߇㧘ᦨᓟߦ㧘ਃᰴరߩ▫ߩਛߩ⥄↱☸ሶߩ໧㗴ࠍቇ⠌ߒߡ߅߈ߚ޿㧚ߎࠇߪ Section 6.2 ߢቇ⠌ߒߚ৻ᰴరߩ▫ߩਛߩ⥄↱☸ሶߩන⚐ߥᑧ㐳ߢߪ޽ࠆ߇㧘ൻቇߦ߅ߌࠆᄙߊߩ໧㗴㧔଀߃߫㧘⛔⸘ജቇ⊛ߦ᳇૕ߩ⁁ᘒᣇ⒟ᑼࠍዉ಴ߒߚࠅ㧘ਗㅴㆇേߦ㑐ߔࠆᾲኈ㊂ࠍዉ಴ߒߚࠅߔࠆ໧㗴㧕ߦᔕ↪ߐࠇ㧘㊀ⷐߥ໧㗴ߢ޽ࠆ㧚Section
6.2 ߩᓳ⠌ࠍ౗ߨߡቇ⠌ߔࠆߩ߇ࠃ߆ࠈ߁㧚ㄝߩ㐳ߐ߇ߘࠇߙࠇ a 㧘 b 㧘߅ࠃ߮ c ߢ޽ࠆ⋥ᣇ૕ࠍ⠨߃㧘ߘߩ⋥ᣇ૕ߩਛߢߪ

potential ߪ࠯ࡠ㧘ߘߩოࠍ฽߼ߡᄖ஥ߢߪ potential ߇ f ߢ޽ࠆߣ⠨߃ࠆ㧚ߔߥࠊߜV x, y , z

0 for 0 d x d a, 0 d y d b, 0 d z d c

V x, y , z

f otherwise(6.6.1)▫ߩౝ஥ߢߪ Schrödinger equation ߪᰴߩࠃ߁ߦ⴫ࠊߐࠇࠆ㧚


w2
w2 ·
=2 § w 2¨ 2  2  2 ¸\ x, y, z
2m © wx wy wz ¹

E\ x, y, z

(6.6.2)ߎߩਃᄌᢙߩ஍ᓸಽᣇ⒟ᑼߢ޽ࠆ Schrödinger equation ࠍ⸃ߊߦ޽ߚࠅ㧘߹ߕ㧘eigenfunction ߪ x ߩߺߩ㑐ᢙ㧘y ߩߺߩ㑐ᢙ㧘߅ࠃ߮ z ߩߺߩ㑐ᢙߩⓍߣߒߡ⴫ࠊߐࠇ
ࠆߣ઒ቯߔࠆ㧚ߔߥࠊߜ\ x, y , z

X x Y y Z z(6.6.3)ߎߩࠃ߁ߦ઒ቯߔࠆߣ㧘ᑼ(6.6.3)ߩࠃ߁ߥᒻߢ⴫ࠊߐࠇߥ޿⸃ࠍ⷗⪭ߣߒߡߒ߹߁ߩ
ߢߪߥ޿߆ߣᔃ㈩ߔࠆੱ߽޿ࠃ߁㧚ߒ߆ߒ㧘߽ߒᑼ(6.6.3)ߩᒻߩ⸃߇ቯ߼ࠄࠇߚႺ⇇᧦ઙࠍḩ⿷ߔࠆߥࠄ߫หߓႺ⇇᧦ઙࠍḩߚߔઁߩᒻߩ⸃ߪሽ࿷ߒߥ޿ߎߣ߇ᢙቇ⊛ߦ␜ߐࠇߡ޿ࠆ㧚ߎࠇ߇޿ࠊࠁࠆᄌᢙಽ㔌ᴺߣ⸒ࠊࠇࠆᣇᴺߢᧄ lecture note ߢ߽ߔߢߦ૗࿁߆↪޿ߡ޿ࠆ㧚ᑼ(6.6.3)ࠍᑼ(6.6.2)ߦઍ౉ߒ㧘ో૕ࠍ X x Y y Z z ߢഀࠆߎ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 35ߣߦࠃࠅᰴᑼࠍᓧࠆ㧚22
w2 X x
1 wY y1 w Z z 2mE


 2
X x
wx 2
Y y wy 2
Z z
wz 2
=

1

0

(6.6.4)ߔߥࠊߜ
w2 X xX x
wx 2
1

1 wY y1 w Z z 2mE


 2
2
Y y wy
Z z
wz 2
=
2

2

(6.6.5)ߎߩᑼߩᏀㄝߪ x ߩߺߩ㑐ᢙߢ㧘৻ᣇฝㄝߪ y ߣ z ߩߺߩ㑐ᢙߢ޽ࠆ㧚ߘߩਔ⠪߇╬ߒ޿ߣ޿߁ߎߣߪਔㄝߣ߽޽ࠆቯᢙߦ╬ߒ޿ߣ޿߁ߎߣࠍ␜ߒߡ޿ࠆ㧚ߎߩቯᢙࠍ㧘଀߃߫ kx ߣߢ߽߅ߌ߫޿޿ߩߢ޽ࠈ߁߇㨇଀߃߫ᑼ(3.1.5)߅ࠃ߮ᑼ(3.1.6)ࠍෳᾖ㨉ߎߎߢߪ㧘ᑼ(6.6.4)ߩᒻࠍ⠨߃㧘ߎߩቯᢙࠍ 
d2X x
X x
dx 2
1



2mEx
ߣ߅ߊߎߣߦߒࠃ߁㧚
=2

2mEx
=2

(6.6.6)

ߎߎߢߪ৻ᄌᢙߩߺߢ޽ࠆ߆ࠄ஍ᓸಽᣇ⒟ᑼࠍᏱᓸಽᣇ⒟ᑼߣߒߡᦠ߈ᦧ߃ߡ޽ࠆ㧚
ᑼ(6.6.4)ߦ߅޿ߡ㧘x, y, z ߪኻ⒓⊛ߦ⃻ࠊࠇࠆߩߢోߊหߓ⼏⺰ࠍᒰߡߪ߼ࠆߎߣߦࠃ
ࠅ㧘ᰴߩ 2 ߟߩᑼࠍᓧࠆ㧚2
1 d Y y
Y y
dy 2



2
1 d Z z
Z z
dz 2



2mE y
=2

2mEz
=2

(6.6.7)(6.6.8)ᑼ(6.6.6)-ᑼ(6.6.8)ࠍᑼ(6.6.4)ߦઍ౉ߔࠆߎߣߦࠃࠅ eigenvalue E ߪ

EEx  E y  Ez

(6.6.9)ߩࠃ߁ߦ⴫ࠊߐࠇࠆߎߣ߇ಽ߆ࠆ㧚⚿ዪ 3 ᄌᢙߩ஍ᓸಽᣇ⒟ᑼ(6.6.2)߇৻ᄌᢙߩᏱᓸ
ಽᣇ⒟ᑼ 3 ߟߣߒߡᦠ߈⴫ߔߎߣ߇಴᧪ߚࠊߌߢ޽ࠆ㧚
ᑼ(6.6.6)-ᑼ(6.6.8)ߪ㧘ߔߢߦ⸃޿ߚᑼ(6.2.1)ߣోߊหߓᒻࠍߒߡ޿ࠆߩߢ㧘⸃߽ోߊหߓᒻߣߒߡ᳞߹ࠆ㧚ߔߥࠊߜ㧘eigenfunction ߦߟ޿ߡߪᑼ(6.2.12)ߣห᭽ߦ

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 36

nS
2
sin x
a
a
n yS
2
sin
b
b

X x
Y x

nx

1, 2, ...

ny

1, 2, ...

(6.6.10)nS
2
nz 1, 2, ...
sin z
c
c
eigenvalue ߦߟ޿ߡߪᑼ(6.2.7)ߣห᭽ߦ
Z x

Ex
Ey
Ez

h 2 nx 28ma 2
h 2 nx 2
8mb 2
h 2 nx 2
8mc 2

nx

1, 2, ...

ny

1, 2, ...

nz

1, 2, ...

(6.6.11)એ਄ࠍ߹ߣ߼㧘ᑼ(6.6.2)ߩ⸃ߣߒߡᦨ⚳⊛ߦᰴߩ⚿ᨐ߇ᓧࠄࠇࠆ㧚\ x, y , z
E

8
nS x
nS y
nS z
sin
sin
sin
abc
a
b
c

2
nz 2 ·
h 2 § nx 2 n y¨¸
8m ¨© a 2 b 2 c 2 ¸¹nx , n y , nz䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 6: Page 37(6.6.12)1, 2, ...

CHAPTER 07⺞๺ᝄേሶߎߩ┨ߢߪ㧘㊂ሶജቇߦ߅޿ߡ߽ߞߣ߽㊀ⷐߥ໧㗴ߩ 1 ߟߢ޽ࠆ⺞๺ᝄേሶ㧔harmonic oscillator㧕ࠍขࠅᛒ߁㧚harmonic oscillator ߪ㧘♽߇ߘߩᐔⴧ⁁ᘒ߆ࠄߕࠇ
ߚᤨߦ߽ߣߦᚯࠈ߁ߣߔࠆജ߇㧘ᐔⴧ⁁ᘒ߆ࠄߩߕࠇߦᲧ଀ߒߡ޿ࠆ႐วߦ⃻ࠇࠆ㧚ᝄࠅሶ߿߫ߨߩᝄേߪฎౖജቇߩቇ⠌ߢ߽ࠃߊ಴ߡ᧪ࠆ଀ߢ޽ࠆ㧚ൻቇߩಽ㊁ߢߩ଀ߪ㧘ੑේሶಽሶߩේሶߩᝄേ߿㧘࿕૕ਛߩ᭴ᚑේሶߩᝄേߥߤ߇޽ࠆ㧚ߎߩ႐ว߶ߤ
᣿ࠄ߆ߢߪߥ޿߆߽ߒࠇߥ޿߇㧘㔚⏛ᵄߩ႐߽ harmonic oscillator ߩ㓸วߣߒߡขࠅᛒ
߁ߎߣ߇಴᧪ࠆ㧚ᦝߦ㧘harmonic oscillator ࠍᛒ߁ᢙቇ⊛ᚻᴺߪ㊂ሶജቇߩ޿ࠈ޿ࠈߥ
ಽ㊁ߢ➅ࠅ㄰ߒ಴ߡ᧪ࠆߩߢ㧘ߘ߁޿߁ᗧ๧߆ࠄ߽ᄢᄌၮᧄ⊛ߢ㊀ⷐߥ໧㗴ߢ޽ࠆ㧚

7.1 Hooke ߩᴺೣߦᓥ߁♽ߦߟ޿ߡߩ Schrödinger ᣇ⒟ᑼ
৻┵߇ოߦ࿕ቯߐࠇߚ߫ߨࠍ⠨߃ࠆ㧚⥄ὼߩ㐳ߐ߆ࠄ x ߛߌ߫ߨࠍિ߫ߒߚᤨ㧘߹ߚߪ x ߛߌ᛼ߒ❗߼ߚᤨ㧘ߘߩᄌ૏ߦᲧ଀ߒߚജߢ߽ߣߦᚯࠈ߁ߣߔࠆ㧚ߎࠇߪࠃߊ
⍮ࠄࠇߚ Hooke ߩᴺೣߢ޽ࠆ㧚ߎߩ႐วߩᲧ଀ቯᢙ k ߪജߩቯᢙ㧔force constant㧕ߣ
๭߫ࠇࠆ㧚

Zゲ
࿑ 7.1.1 ฎౖ⊛ߥࡃࡀ߹ߕฎౖ⺰ߢ⠨߃ߡߺࠃ߁㧚߫ߨߩవ┵ߦߟ޿ߡ޿ࠆ‛૕ߦ߆߆ࠆജߪ  kx ߢ㧘potential V ߪ

1 2
kx
2
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 1
V x(7.1.1)ߢ⴫ߐࠇࠆ㧚㧔ᵈ㧦potential ࠍᓸಽߒߡ╓ภࠍᄌ߃ߚ߽ߩ߇ജ㧘ߔߥࠊߜ㧘F



dV x
dx

ߢ޽ࠆ㧚㧕ฎౖ⺰ߢߪ㧘Newton ߩㆇേᣇ⒟ᑼ㧔╙ੑᴺೣ㧕
d 2x
dt 2
ߩ⸃ߣߒߡ
m

kx

(7.1.2)x

A cos 2SQ t  t0

(7.1.3)߇ᓧࠄࠇࠆ㧚ߎߎߢ A ߪᝄ᏷㧘 2SQ t0 ߪ૏⋧Ꮕࠍ⴫ߔ㧚Q ߪᝄേᢙߢ
1 k
2S mㆇേߩࠛࡀ࡞ࠡ࡯ T ߪQT

1 § dx ·m¨ ¸
2 © dt ¹(7.1.4)2

2mS 2Q 2 A2 sin 2 2SQ t  t0

potential V ߪ

V

1 2
kx
2

2mS 2Q 2 A2 cos 2 2SQ t  t0ᓥߞߡ㧘ోࠛࡀ࡞ࠡ࡯ E ߪ

E

2mS 2Q 2 A2

1 2
kA
2

(7.1.5)ߣߥࠆ㧚ߔߥࠊߜ㧘ฎౖ⺰ߢߪ E ߩ୯ߦ․ߦ೙㒢ߪߥ޿㧚
ᰴߦ㊂ሶ⺰ߢ⠨߃ࠃ߁㧚Schrödinger equation ߪ=2 w \ x 1 2(7.1.6) kx \ x E\ x
2m wx 22ߣߥࠆ㧚એᓟߎߩᣇ⒟ᑼࠍ⸃ߊߎߣࠍ⠨߃ࠆ߇㧘ߘߩ೨ߦߜࠂߞߣ potential ࠍ⷗ߡߺࠃ߁㧚2䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 2

Vx࿑ 7.1.2
⺞๺ᝄേߦ߅ߌࠆ potential㧔2 ᰴ㑐ᢙ㧕potential ߪ᡼‛✢ߢ⴫ߐࠇࠆߩߢ࿑ 7.1.2 ߩࠃ߁ߦߥࠆ߇㧘ߎࠇࠍ࿑ 6.3.2 ߩ੗ᚭဳpotential ߣᲧߴߡߺࠆ㧚੗ᚭဳ potential ߩ႐วߦߪ੗ᚭߩ┵ߢ V⋥ߦ┙ߜ਄߇ߞߡ޿ࠆߩߦ෻ߒ㧘᡼‛✢ߩ potential ߩ႐วߦߪ㧘V

0 ߆ࠄ Vf ߹ߢု

0 ߆ࠄ Vf ߹ߢߩჇട߇᡼‛✢ߢ޽ࠆ߆ࠄ㧘V ߇ή㒢ᄢߦߥࠆߩ߇ࠁߞߊࠅߣߥߞߡ޿ࠆ㧚ߘߩઁߪࠃߊૃߡ޿ࠆߣ⸒߃ࠃ߁㧚ߘ߁޿߁ࠊߌߢ wavefunction ߽ૃߡ޿ࠆߩߢߪߥ޿߆ߣ੍
ᗐߐࠇࠆ㧚ߚߛ㧘੹࿁ߪ㧘potential ߩჇട߇ࠁߞߊࠅߥߩߢ wavefunction ߇ߎߩ potentialߩᄖ஥ߦ಴ࠆ‫ߺߒޟ‬಴ߒ‫ߣࠆ޽߇ޠ‬⠨߃ࠄࠇࠆ㧚ߎߩ੍ᗐ߇ߤߩ⒟ᐲᱜߒ޿߆㧘ߘࠇ
ߪ Schrödinger equation ࠍ⸃޿ߡ߆ࠄߩ߅ᭉߒߺߣߒࠃ߁㧚

7.2 Hermite ߩᓸಽᣇ⒟ᑼ
Schrödinger equation (7.1.6)ߪߘ߁◲නߦ⸃ߌߘ߁߽ߥ޿ߩߢ㧘⷗ㅢߒࠍࠃߊߔࠆߚ

߼ߦᄌᢙߩ⟎߈឵߃ࠍߒߡߺࠃ߁㧚ᑼ(7.1.6)ߪ
d 2\ x § 2mE mkx 2 · ¨ 2  2 ¸\ x
= ¹
dx 2
© =

0

(7.2.1)ߣߥࠆ߇㧘╙ 2 㗄ߩ᜝ᒐߩਛߩ 2 ߟߩ㗄ࠍߘࠇߙࠇ D ߅ࠃ߮ E 2 x 2 ߣ߅ߊߎߣߦߒࠃ߁㧚ߔߥࠊߜ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 3

2mE
=2

D
E

2

mk
=2

§ 2S mQ ·
¨
¸
© = ¹

(7.2.2)2ߎߩ⟎߈឵߃ߢ Schrödinger equation ߪ

d 2\ x D  E 2 x2 \ x 0
dx 2ߣߥࠆ߇㧘ߎߎߢᦝߦήᰴరߩᄌᢙ [ ࠍዉ౉ߒߡ
km
x=2ߣ߅ߊߣ㧘ᑼ(7.2.3)ߪ

[Ex(7.2.4)4d 2\ [ § D· ¨  [ 2 ¸\ [
2
d[
©E
¹

(7.2.3)0(7.2.5)ߣߥࠆ㧚ߎߩᒻߦߒߡߺߚߣߎࠈߢ⸃߇ߤ߁ߥࠆ߆ᔅߕߒ߽ߪߞ߈ࠅߒߥ޿㧚ߘߎߢ߹ߕ [ ߇ᄢ߈޿ᭂ㒢ߢߤ߁ߥࠆ߆ࠍ⠨߃ߡߺࠃ߁㧚ߎߩ႐วߪ [ 2 ߦኻߒߡD
ࠍήⷞ಴᧪ࠆ߆ࠄᑼ(7.2.5)ߪ
E
d 2\ [
 [ 2\ [02d[ߣߥࠆ㧚ߎࠇࠍ⸃ߊߚ߼ߦ㧘઒ߦ

\ [

[ of

exp a[ 2(7.2.6)(7.2.7)

ߣ߅޿ߡߺࠃ߁㧚 a ߪ parameter ߢᑼ(7.2.6)߇ᚑࠅ┙ߟࠃ߁ߦ᳿߼ࠆ㧚ᑼ(7.2.7)ࠍᑼ
(7.2.6)ߦઍ౉ߔࠆߣ

2a  4a 2[ 2  [ 2 exp a[ 2

0

ߔߥࠊߜ

[ 2 4a 2  1  2a 0

(7.2.8)[ o f ߩᭂ㒢ࠍ⠨߃ߡ޿ࠆߩߢ㧘╙ 2 㗄⋡ߪήⷞ಴᧪ࠆ㧚ᓥߞߡ㧘 a r

1
ߣߥࠆ㧚
2

§ [2 ·§ [2 ·
exp ¨  ¸ ߽ exp ¨  ¸ ߽ߤߜࠄ߽ᑼ(7.2.6)ࠍḩ⿷ߔࠆ㧚ߒ߆ߒ㧘Chapter 04 ߢቇ⠌ߒ
© 2 ¹
© 2 ¹

ߚ‫ ߪ [ \ޟ‬wavefunction ߢ޽ࠅ well behaved ߢߥߌࠇ߫޿ߌߥ޿‫ߩ⺧ⷐ߁޿ߣޠ‬ὑߦ㧘䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 4೨⠪ߪណ↪಴᧪ߥ޿㧚ᓥߞߡ\ [ ߩ asymptotic solution ߣߒߡ\ [§ [2 ·
exp ¨  ¸ 0
© 2 ¹

[ of

(7.2.9)ࠍᓧࠆ㧚ߎߩࠃ߁ߦ [ o f ߩ႐วߩ asymptotic solution ߇᳞߹ߞߚߣߎࠈߢ㧘޿ࠃ޿ࠃᑼ(7.2.5)ߩ Schrödinger equation ࠍ⠨߃ࠆߩߢ޽ࠆ߇㧘ߎߎߢ㧘ᑼ(7.2.5)ߩ⸃ߪ

\ [§ [2 ·
H [ exp ¨  ¸© 2 ¹ߩᒻߢ⴫ߐࠇࠆߣ઒ቯߔࠆ㧚H [(7.2.10)ߪ [ ߩ㑐ᢙߢ㧘ᓟߢ᳿ቯߔࠆߎߣߦߔࠆ㧚ᑼ(7.2.10)

ࠍᑼ(7.2.5)ߦઍ౉ߔࠆߣᰴᑼࠍᓧࠆ㧚d 2H [dH [ § D · 2[
 ¨  1¸ H [
2
d[
d[
©E
¹

0

(7.2.11)

ߎࠇߪ Hermite's differential equation ߣߒߡ⍮ࠄࠇߡ޿ࠆᓸಽᣇ⒟ᑼߢ޽ࠆ㧚ࠃߊ⍮ࠄࠇߡ޿ࠆᓸಽᣇ⒟ᑼߢ㧘ᢙቇߩᢎ⑼ᦠߦ⸃߇ਈ߃ࠄࠇߡ޿ࠆ߆ࠄ㧘ߘࠇࠍߘߩ߹߹↪
޿ࠆߣ޿߁ᚻ߽ߥ޿ࠊߌߢߪߥ޿㧚ߒ߆ߒ㧘ߎߎߢߪ㧘ᢙቇߩᧄߦ㗬ࠄߕߦ H [ ߩᒻࠍ᳞߼ࠆߎߣࠍ⠨߃ࠃ߁㧚

7.3 ౹⚖ᢙᴺߦࠃࠆ Hermite ߩᓸಽᣇ⒟ᑼߩ⸃
ߎߎߢߪ㧘Hermite ߩᓸಽᣇ⒟ᑼ(7.2.11)ࠍ౹⚖ᢙᴺߢ⸃߈㧘߆ߟ Hermite ߩᄙ㗄ᑼߣ⸒ࠊࠇࠆ⸃ߩᕈ⾰ࠍ⺞ߴߡߺࠃ߁㧚߹ߕ H [ ߪ [ ߩ power series ߢዷ㐿಴᧪ࠆߣ઒ቯߒߡߺࠃ߁㧚H [a0  a1[  a2[ 2  ...

ߎࠇࠃࠅᰴᑼࠍᓧࠆ㧚dH [a1  2a2[  3a3[ 2  ...
d[d 2H [d[ 21u 2a2  2 u 3a3[  3 u 4a4[ 2  ...

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 5(7.3.1)(7.3.2)(7.3.3)ߎࠇࠄߩᑼࠍᑼ(7.2.11)ߦઍ౉ߒߡ㧘 [ ߩ᣹౹ߦਗߴࠆ㧚ª§D· º«1u 2a2  ¨  1¸ a0 »
©E
¹ ¼
¬
ª
§D
· º
 « 2 u 3a3  ¨  1  2 u 1¸ a1 » [
©E
¹ ¼
¬
ª
§D
· º
 «3 u 4a4  ¨  1  2 u 2 ¸ a2 » [ 2
©E
¹ ¼
¬

(7.3.4)ª§D· º « 4 u 5a5  ¨  1  2 u 3 ¸ a3 » [ 3
©E
¹ ¼
¬
...

0

਄ᑼ߇޿߆ߥࠆ [ ߩ୯ߢ߽ᚑࠅ┙ߚߥߌࠇ߫޿ߌߥ޿ߣ޿߁ߎߣ߆ࠄ㧘ᰴߩฦᑼ߇ᓧ
ࠄࠇࠆ㧚
§D
·
1u 2a2  ¨  1¸ a0 0
©E
¹
§D
·
2 u 3a3  ¨  1  2 u 1¸ a1 0
©E
¹
§D
·
3 u 4 a4  ¨  1  2 u 2 ¸ a2 0
©E
¹
§D
·
4 u 5a5  ¨  1  2 u 3 ¸ a3 0
©E
¹
...
৻⥸ᑼߪ
§D
·
m  1 m  2 am  2  ¨  1  2m ¸ am
©E¹

(7.3.5)0m 0, 1, 2, ...

(7.3.6)

ߎࠇ߆ࠄಽ߆ࠆࠃ߁ߦ㧘߽ߒ߽ a0 ߇ਈ߃ࠄࠇࠇ߫㧘a2 㧘a4 㧘a6 㧘...ߩ஧ᢙ㗄߇᳿߹ࠆ㧚ห᭽ߦ a1 ߇ਈ߃ࠄࠇࠇ߫ a3 㧘 a5 㧘 a7 㧘...ߩᄸᢙ㗄߇᳿߹ࠆߎߣߦߥࠆ㧚଀ߣߒߡᦨೋ
ߩᢙ㗄ࠍ᳞߼ߡߺࠃ߁㧚m ߇஧ᢙߩ႐วߦߪ
a2



·
1 §D
¨  1¸ a0
2 u1 © E¹

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 6(7.3.7)

a4



·
1 §D
¨  1  2 u 2 ¸ a2
3u 4 © E¹(7.3.8)

·§ D
·
1 §D
¨  1¸¨  1  2 u 2 ¸ a0
4! © E¹© E¹

߹ߚ m ߇ᄸᢙߩ႐วߦߪ
a3



·
1 §D
¨  1  2 u 1¸ a1
2u3 © E
¹

a5



·
1 §D
¨  1  2 u 3 ¸ a3
4u5 © E¹

(7.3.9)(7.3.10)

·§ D
·
1 §D
¨  1  2 ¸¨  1  2 u 3 ¸ a1
5! © E¹© E¹

ߎߩࠃ߁ߥ power series ߣߒߡ᳞߹ࠆ H [ ߇ߤߩࠃ߁ߥᕈ⾰ࠍᜬߞߡ޿ࠆ߆ m ߩᄢ߈޿ߣߎࠈࠍี๧ߒߡߺࠃ߁㧚৻⥸ߦ㧘power series ߇෼᧤ߔࠆ߆ߤ߁߆ߪ㧘ㅪ⛯ߒߚa౹ߩଥᢙߩᲧࠍߣߞߡߺࠇ߫ಽ߆ࠆߣ޿߁ߎߣ߆ࠄ㧘 n  2 ࠍ⺞ߴߡߺࠃ߁㧚 [ ߇ᄢan߈޿ߣߎࠈߢߪ n ߇ᄢ߈޿ߣߎࠈߩߺࠍ⠨߃ࠇ߫ࠃ޿߆ࠄ㧘am  2
2m 2
o 2
mof
am
m
m

(7.3.11)ߎߎߢ exp [ 2 ߩዷ㐿ࠍᕁ޿⿠ߎߘ߁㧚exp [ 2

1 [ 2 

[4
2!



[6
3!

 ... 

[m
§m·
¨ ¸!
©2¹

 ...

(7.3.12)ߎߎߢ [ m ߩଥᢙࠍ bm ߣ߅ߊߣbm  2
bm

§m·
¨ ¸!
©2¹
§m ·
¨  1¸ !
©2¹

2
m(7.3.13)ߣߥߞߡᑼ(7.3.11)ߣหߓ㑐ଥߦ޽ࠆߎߣ߇ಽ߆ࠆ㧚ߔߥࠊߜ㧘[ ߇ᄢ߈޿ߣߎࠈߢߪ
H [ ߪ߶߷ exp [ 2 ߩࠃ߁ߥᒻߦߥࠆߎߣ߇ಽ߆ࠆ㧚H [߇ exp [ 2 ߩࠃ߁ߦᝄࠆ

⥰߁ߣ޿߁ߎߣ߆ࠄ㧘ᑼ(7.2.10)ࠍ⠨ᘦߔࠆߣ㧘Schrödinger equation ߩ⸃ߣߒߡߩ\ [䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 7

߇ exp [ 2 ߩࠃ߁ߦᝄࠆ⥰߁ߣ޿߁ߎߣ߇಴᧪ࠆ㧚ߎߩࠃ߁ߥ㑐ᢙߪ [ ߇ᄢ߈޿ᚲߢ
⊒ᢔߒߡߒ߹޿㧘᣿ࠄ߆ߦ㧘wavefunction ߣߒߡߪ⸵ߐࠇߥ޿㑐ᢙߢ޽ࠆ㧚ߎߩਇㇺวࠍㆱߌࠆߦߪ㧘ᑼ(7.3.1)ߩዷ㐿߇㧘޽ࠆ᦭㒢ߩ౹ߢᱛ߹ࠇ߫ࠃ޿㧚ߘߩ႐วߦߪ㧘
ᑼ(7.2.10)ߩ exp [ 2 ߩᷫዋ߇ߤࠎߥᄙ㗄ᑼߩჇടࠃࠅ߽ߪࠆ߆ߦᕆỗߢ޽ࠆߚ߼ߦ㧘D[ o f ߢ\ [ o 0 ߣߥࠆ㧚ߟ߹ࠅ㧘ᑼ(7.3.6)ߦ߅޿ߡ㧘  1 ߇ 2n ߦ╬ߒߊ㧘m t n  2Eߩߔߴߡߩ am ߇࠯ࡠߦߥࠇ߫ࠃ޿ߩߢ޽ࠆ㧚D
 1 2nEߦ D ߅ࠃ߮ E ߩቯ⟵ᑼ(7.2.2)ࠍઍ౉ߒ㧘߆ߟᑼ(7.1.4)ࠍ↪޿ࠆߣ

(7.3.14)1·§hQ ¨ n  ¸ n 0, 1, 2, ...
(7.3.15)
2¹©
߇ᓧࠄࠇࠆ㧚ߚߛߒ㧘 E ߇ n ߩ㑐ᢙߣߒߡਈ߃ࠄࠇߡ޿ࠆߩߢ㧘 n ࠍ E ߩ subscript ߣ
En

=

k §
1·
¨n ¸
m©
2¹

ߒߡߟߌߡ޿ࠆ㧚
ᑼ(7.3.15)߆ࠄಽ߆ࠆࠃ߁ߦ energy ߪ㊂ሶൻߒߡ޿ࠆ㧚ߎߩ㊂ሶൻߪ㧘\ [[ o f ߢ⊒ᢔߒߡߪߥࠄߥ޿㧘ߔߥࠊߜ㧘\ [߇

߇ well-behaved ߢߥߌࠇ߫ߥࠄߥ޿

ߣ޿߁᧦ઙ߆ࠄ಴ߡ᧪ࠆ㧚energy ߇᳞߹ߞߚߩߢ㧘ᰴߦ wavefunction ࠍ᳞߼ߡߺࠃ߁㧚
ᑼ(7.3.14)ࠍᑼ(7.3.7)߅ࠃ߮ᑼ(7.3.8)ߦઍ౉ߔࠆߣ

a2



1
2n a0
2 u1

(7.3.16)a4

12n 2n  4 a0
4!

(7.3.17)߇ᓧࠄࠇࠆ㧚ߎߩࠃ߁ߥᠲ૞ࠍ➅ࠅ㄰ߒߡ㧘 H n [ࠍ᳞߼ࠆߎߣߪ◲නߥṶ⠌ߢ޽ࠆ㧚㧔Problem 7.3.1㧕㧚⚿ᨐߪᰴߩࠃ߁ߦߥࠆ㧚

Hn [

ª 2n
º
2 2 n n  2 4 23 n n  2 n  4 6[ [  ...»
a0 «1  [ 2 
4!
6!
¬ 2!
¼
(for even n)

ߎߎߢ n ࠍ H [ ߩ subscript ߣߒߡߟߌߚ߇㧘ߎࠇߪ㧘 H [䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 8(7.3.18)ߩᄙ㗄ᑼ߇ [ n ߩ㗄ߢಾࠇࠆߎߣࠍ␜ߔߎߣߣߔࠆ㧚ోߊหߓࠃ߁ߦ㧘ᑼ(7.3.14)ࠍᑼ(7.3.9)߅ࠃ߮ᑼ(7.3.10)ߥߤߦઍ౉ߔࠆߎߣߦࠃࠅ n߇ᄸᢙߢ޽ࠆ႐วߩᑼ߽᳞߹ࠆ㧚

Hn [

ª
º
2 n  1 3 22 n  1 n  3 5[ [  ...»
a1 «[ 3!
5!
¬¼
(for odd n)(7.3.19)ߎࠇࠄߩᑼߢ a0 ߣ a1 ߦኻߒߡߪ૗ߩ೙㒢߽ߥ޿ߩߢൎᚻߦ᳿߼ߡࠃ޿㧚৻⥸ߦߪ㧘
Hn [ߦ߅޿ߡ [ ߩ߽ߞߣ߽㜞ᰴߩ㗄ߔߥࠊߜ㧘 [ n ߩଥᢙ߇ 2n ߦߥࠆࠃ߁ߦ a0 ߣ

a1 ࠍ᳿߼ࠆߩ߇᥉ㅢߢ޽ࠆ㧚ߎߩࠃ߁ߦ᳿߼ࠆߣ H n [ ߣߒߡᦨ⚳⊛ߦᰴߩᑼ߇ᓧࠄ

ࠇࠆ㧚
Hn [

1

n/2

n!

§n·
¨ ¸!
©2¹

ª 2 n 2 2 2 n n  2 4 23 n n  2 n  4 6ºu «1  [ [ [  ...»
4!
6!
¬ 2!
¼
(for even n)

Hn [

1

n 1 / 2(7.3.20)2 n!

§ n 1 ·
¨
¸!
© 2 ¹

ª
º
2 n  1 3 22 n  1 n  3 5[ [  ...»u «[ 
3!
5!
¬¼(for odd n)ᦨೋߩᢙ㗄ߦߟ޿ߡߩౕ૕⊛ߥᒻࠍ⴫ 7.3.1 ߦ߹ߣ߼ߡ߅ߊ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 9(7.3.21)

⴫ 7.3.1

Hermite ߩᄙ㗄ᑼߩ޿ߊߟ߆
H0 [

1

H1 [

2[

H2 [

4[ 2  2

H3 [8[ 3  12[

H4 [
H5 [
H6 [
H7 [

H10 [
H11 [

16[ 4  48[ 2  12
32[ 5  160[ 3  120[

64[ 6  480[ 4  720[ 2  120
128[ 7  1344[ 5  3360[ 3  1680[

H8 [256[ 8  3584[ 6  13440[ 4  13440[ 2  1680

H9 [

512[ 9  9216[ 7  48384[ 5  80640[ 3  30240[

1024[ 10  23040[ 8  161280[ 6  403200[ 4  302400[ 2  30240
2048[ 11  56320[ 9  506880[ 7  1774080[ 5  2217600[ 3  665280[‫ڏ‬Problem 7.3.1
ᑼ(7.3.18)߅ࠃ߮ᑼ(7.3.19)ࠍዉ಴ߖࠃ㧚‫ڏ‬Problem 7.3.2
ߎߩ section ߢߪ Hermite ߩᓸಽᣇ⒟ᑼࠍ౹⚖ᢙᴺߢ⸃޿ߚ߇㧘ߎߩᣇᴺࠍ↪޿ߡ㧘
ᓸಽᣇ⒟ᑼ

d2 f x
 k2 f x 0
dx 2ࠍ⸃޿ߡߺࠃ߁㧚߽ߜࠈࠎ㧘ߎߩ⸃ߪ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 10(7.3.22)

A cos kx  B sin kx

f x(7.3.23)ߣߥࠆߎߣߪಽ߆ߞߡ޿ࠆ߇㧘ߎߎߢߪߘߩ⸃ࠍ⍮ࠄߕߦ㧘౹⚖ᢙᴺߢ⸃ߊߎߣࠍ⠨߃ߡߺࠃ߁㧚
߹ߕ㧘 f xf x

ࠍ x ߩ౹ߢዷ㐿ߔࠆ㧚
a0  a1 x  a2 x 2  ...

(7.3.24)ߎߩᑼࠍᑼ(7.3.22)ߦઍ౉ߒ㧘 x ߩ౹ਸ਼ߩ㗄ࠍ㓸߼ࠆߎߣߦࠃࠅ㧘ᰴߩᑼࠍዉߌ㧚n n  1 an  k 2 an  2

0

for n t 2

(7.3.25)ߎߩᑼߦࠃࠅ㧘߽ߒ߽ a0 ߇ਈ߃ࠄࠇࠇ߫㧘 a2 㧘 a4 㧘...ߩ߇ߔߴߡߩ஧ᢙ㗄߇᳞߹ࠆ㧚ห᭽ߦ a1 ߇ਈ߃ࠄࠇࠇ߫㧘 a3 㧘 a5 㧘...ߩ߇ߔߴߡߩᄸᢙ㗄߇᳞߹ࠆ㧚ߔߥࠊߜ㧘 n t 2
ߩ an ߪߔߴߡ a0 ߣ a1 ߢ߽ߞߡ⴫ߐࠇࠆ㧚ߘߩ⚿ᨐࠍᑼ (7.3.24) ߦઍ౉ߒ㧘⚿ᨐߪᑼ
(7.3.23)ߣ╬ߒߊߥࠆߎߣࠍ␜ߖ㧚ߎߩ႐วᑼ(7.3.23)ߩ A ߅ࠃ߮ B ߪߤ߁⴫ߐࠇࠆ߆㧚[Let’s play with Mathematica] Hermite polynomial
Hermite ߩᄙ㗄ᑼߩౕ૕⊛ߥ㑐ᢙᒻߪᰴᢙ߇ 0 ߆ࠄ 5 ૏߹ߢߪ޿ࠈ޿ࠈߥᢎ⑼ᦠߦ

タߞߡ޿ࠆߩߢᔅⷐ߇޽ࠇ߫޿ߟߢ߽⺞ߴࠆߎߣ߇಴᧪ࠆ㧚ߒ߆ߒ㧘ᰴᢙ߇ᄢ߈ߊߥ
ࠆߣ㧔଀߃߫ n 10 એ਄㧕㑐ᢙᒻࠍ⷗ߟߌࠆߎߣߪ⥋㔍ߩࠊߑߢ޽ࠆ㧚ᐘ޿ߦ߽㧘
Mathematica ߢߪ Hermite ߩᄙ㗄ᑼ߇ HermiteH ߣ޿߁ౝㇱߢቯ⟵ߐࠇߡ޿ࠆߩߢᔅⷐ

ߦᔕߓߡߤࠎߥ n ߦߟ޿ߡߩ㑐ᢙᒻ߽⋥ߜߦᓧࠄࠇࠆ㧚଀߃߫㧘 n 0 ߆ࠄ n 11 ߹ߢߩ㑐ᢙᒻࠍߔߴߡࡊ࡝ࡦ࠻ߒߚߌࠇ߫એਅߦ␜ߔࠃ
߁ߦߔࠇ߫ࠃ޿㧚
Do[Print["H(", n, ", x) = ", HermiteH[n,x]], {n,0,11}]Mathematica ߪ⋥ߜߦએਅߩࠃ߁ߥ⚿ᨐࠍਈ߃ߡߊࠇࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 11* Z 
* Z Z

* Z  Z

* Z Z Z

* Z Z Z

* Z ZZ Z

* Z  ZZ Z

* Z Z ZZ Z

* Z Z ZZ Z

* Z ZZ ZZ Z

* Z  ZZ ZZ Z

* Z Z ZZ ZZ Z

7.4 Hermite ߩᄙ㗄ᑼߦ㑐ߔࠆੑਃߩᕈ⾰
Hermite ߩᄙ㗄ᑼߪᑼ(7.3.20)߅ࠃ߮ᑼ(7.3.21)ߢਈ߃ࠄࠇߚ߇㧘ᰴߩᑼߢ߽ቯ⟵ߐࠇ

ࠆ㧚
Hn [

n

1 exp [2

d n exp [ 2d[ n‫ڏ‬Exercise 7.4.1㧦
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 12(7.4.1)ᑼ(7.4.1)ࠍ↪޿ߡ H 0 [ 㧘 H1 [ 㧘߅ࠃ߮ H 2 [ ࠍ᳞߼ࠃ㧚㧔Problem 7.4.1 ߽ෳ㧕‫ޤ⸃ޣ‬㧦
ᑼ(7.4.1)ߢ n

H0 [

0 ߣ߅ߊ

exp [ 2 exp [ 21n 1 ߣ߅ߊߣ

H1 [1 exp [ 21 exp [ 2d exp [ 2d[^2[ exp [ `22[n2 ߣ߅ߊߣ
H2 [

exp [

2

d 2 exp [ 2d[ 2exp [ 2

2  4[ 2 exp [ 24[ 2  2ߐߡ㧘ᑼ(7.4.1)ࠍ [ ߦߟ޿ߡᓸಽߔࠆ㧚dH n [d[1 2[ exp [ 2nd n exp [ 2d[ n

 1 exp [ 2nd n 1 exp [ 2d [ n 1

2[ H n [  H n 1 [(7.4.2)ߎߩᑼࠍ 2 ࿁↪޿ߡ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 13

d 2Hn [d[ 22 H n [  2[ ^2[ H n [  H n 1 [ `  ^2[ H n 1 [  H n  2 [ `2 1  2[ 2 H n [  4[ H n 1 [  H n  2 [(7.4.3)ߎࠇࠄߩ 2 ߟߩ㑐ଥᑼࠍᑼ(7.2.11)ߦઍ౉ߒ㧘߆ߟH n  2 [  2[ H n 1 [  2 n  1 H n [D 1 ࠍ 2n ߣ⟎߈឵߃ࠆߣ
E0߹ߚߪ n  1 ࠍ n ߣ⟎߈឵߃ߡH n 1 [  2[ H n [  2nH n 1 [0

㧔7.4.4㧕ߣ޿߁㕖Ᏹߦ᦭↪ߥ recursion formula ߇ᓧࠄࠇࠆ㧚߹ߚ㧘ߎߩᑼࠍᑼ(7.4.2)ߦઍ౉ߔ
ࠆߣ㧘ᰴߩ᦭↪ߥᑼ߇ᓧࠄࠇࠆ㧚dH n [
d[2nH n 1 [(7.4.5)ᰴߦ㧘ᑼ(7.2.10)ߔߥࠊߜ

\n [§ [2 ·
H n [ exp ¨  ¸© 2 ¹(7.4.6)߇⋥੤♽ࠍߥߔߎߣࠍ␜ߒ㧘߆ߟ㧘ⷙᩰൻቯᢙࠍ᳞߼ࠆߎߣࠍ⹜ߺࠃ߁㧚ଢቱ਄ m d nߣ⠨߃ࠆ㧚ᑼ(7.4.1)ࠍ↪޿ߡ㧘䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 14

f

³H

m

[ H n [ exp [ 2 d [

f

f

³H

m

[ 1

n

d n exp [ 2
d[ n

f

n

1 H m [d n 1 exp [ 2d [ n 1d [ 㧔ㇱಽⓍಽࠍߒߡ㧕
f

 1

n

f

³

n 1
2
dH m [ d exp [

d[

f

f

d [ n 1d[㧔╙ 1 㗄ߪ࠯ࡠ㧚╙ 2 㗄ߦᑼ(7.4.5)ࠍ↪޿ߡ㧕
1

n 1

f

2m ³ H m 1 [d n 1 exp [ 2d [ n 1fd[ߎߩᠲ૞ࠍ➅ࠅ㄰ߒߡߟ޿ߦᰴᑼࠍᓧࠆ㧚
f

³H

m

[ H n [ exp [

2

d[

1

nm

f

f

2 m! ³ H 0 [m

d n  m exp [ 2

f

1
0

nm

m

2 m!

d[ nmd n  m 1 exp [ 2d[ n  m 1d[

f

(7.4.8)f

for n ! mߎࠇߢ H n [ ߇⋥੤♽ࠍߥߒߡ޿ࠆߎߣ߇⸽᣿ߐࠇߚ㧚ᰴߦⷙᩰൻ࿃ሶࠍ᳞߼ࠃ߁㧚nm ߩ႐วߦߪ㧘ᑼ(7.4.8)ߩ 1 ⇟⋡ߩᑼࠃࠅf2³ H m [ H n [ exp [ d[

1

2n

f

f

2n n ! ³ exp [ 2 d [f(7.4.9)2n n ! S

߆ߊߒߡ

\n [

§ [2 ·
1
exp
H
[
¨ ¸
n
2n n ! S© 2 ¹

n0, 1, 2, ...

(7.4.10)ߪ [ ᐳᮡߦ߅޿ߡ orthonormal set ࠍ૞ࠆߎߣ߇⸽᣿ߐࠇߚ㧚ᑼ(7.4.10)ߩ\ n [ ߪ㧘ᑼ(7.4.9)ߦ␜ߒߚࠃ߁ߦ㧘ήᰴరᐳᮡߩ [ ߦߟ޿ߡⷙᩰൻߐࠇߡ޿ࠆߎߣߦᵈᗧߔߴ߈ߢ޽ࠆ㧚⃻ታߩ⺞๺ᝄേሶߩ⸃ߣߒߡߪ㧘㐳ߐߩᰴరࠍᜬߞߚ x ᐳᮡߦߟ޿ߡⷙᩰൻ
ߒߚᣇ߇޿޿ߣ޿߁⠨߃ᣇ߽޽ࠆ㧚ߎߩ႐วߦߪᑼ(7.2.4)ߦ⌕⋡ߒߡ㧘ᑼ(7.4.10)ߩ 4
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 15

1E୚ߦߒߥ޿ߣ޿ߌߥ޿㧚ߔߥࠊߜ㧘
f

³H [
n

2

exp [ 2 dx 1

(7.4.11)

f

ࠃࠅ

\n [

§ [2 ·
H n [ exp ¨  ¸S© 2 ¹2n n !
1n(7.4.12)0, 1, 2, ...

E߇㧘x ᐳᮡߦ߅޿ߡ orthonormal set ࠍ૞ࠆ㧚ᑼ(7.4.10)ߣᑼ(7.4.12)ߪ㧘නߦ㧘normalizationߦ߅ߌࠆᐳᮡ߇⇣ߥࠆߚ߼ߦ㧘normalization factor ߛߌ߇⇣ߥࠆ߽ߩߢ޽ࠅ㧘߹߉ࠄࠊߒ޿ߩߢᵈᗧࠍⷐߔࠆ㧚ߎߩࠃ߁ߦ 2 ߟߩ⇣ߥߞߚ normalization ߩᣇᴺ߇޽ࠆߣ޿߁ߎߣߪታߦා੺ߥ⃻ታߢ޽ࠆ㧚ߚ߆߇ normalization ߩ㆑޿ߦߔ߉ߥ޿ߣߪ޿ߞߡ߽㧘⃻ታߦ㧘‛ℂ㊂ߩ⸘▚ߥߤߢᄙߊߩੱ߇㑆㆑ߞߡߒ߹߁㧚ᑼ(7.4.10)ߣᑼ(7.4.12)ߣ 4

1

Eߩ࿃ሶߛߌ⇣ߥࠆ߽ߩࠍߤߜࠄ߽\ n [ ߣ޿߁หߓ⸥ภߢ⴫ߔߣ޿߁ߎߣߪᦸ߹ߒ޿ߎߣߢߪߥ޿㧚ߘߎߢ㧘ᧄ lecture note ߢߪ㧘એਅߦㅀߴࠆࠃ߁ߥ⚂᧤ࠍߒߡ normalization ߩᏅࠍߪߞ߈
ࠅߐߖࠆߎߣߦߔࠆ߇㧘৻⥸ߦណ↪ߐࠇߡ޿ࠆᣇᴺߢߪߥ޿㧚ᧄ lecture note ߢߪ [ ߦߟ޿ߡⷙᩰൻߐࠇߡ޿ࠆ㑐ᢙߪᑼ(7.4.10)ߣหߓߊ\ n [ ߣ⴫ߒ㧘৻ᣇ㧘 x ߦߟ޿ߡⷙ
ᩰൻߐࠇߡ޿ࠆ㑐ᢙߪᑼ(7.4.12)ߩઍࠅߦ\ n x ߣ⴫ߔߎߣߦߔࠆ㧚ߔߥࠊߜ\n x§ [2 ·
H n [ exp ¨  ¸S© 2 ¹2n n !
1n0, 1, 2, ...

(7.4.12')Eߣߔࠆ㧚ᑼ(7.4.12)߽ᑼ(7.4.12')߽ߘ߽ߘ߽หߓ㑐ᢙߢ޽ࠅ㧘[ ߩ㑐ᢙߢ޽ࠆߣ౒ߦ x ߩ㑐ᢙߢߢ߽޽ࠆߎߣߢ޽ࠆ߆ࠄ㧘ߘࠇࠍ\ n [ ߣ⴫ߘ߁߇\ n x ߣ⴫ߘ߁߇ߤߜࠄߢ
߽ࠃ޿ࠊߌߢ޽ࠆ㧚ߒ߆ߒ㧘ᧄ lecture note ߦ㒢ࠅ㧘 x ߦߟ޿ߡⷙᩰൻߒߚߎߣࠍߪߞ
߈ࠅߐߖࠆߚ߼ߦ޽߃ߡᑼ(7.4.12')ߩࠃ߁ߦ⴫ߔߎߣߦߔࠆ㧚એ਄ߩ⚂᧤ࠍᔨߩߚ߼ᰴߩᑼߢ⴫ߒߡ߅ߊ㧚f³ \ [ \ [ d[ 1
n

n

f

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 16

f

³\

n

x \ n x dx 1fߥ߅㧘ߊߤ޿ࠃ߁ߢ޽ࠆ߇㧘਄ߦㅀߴߚࠃ߁ߦ㧘\ n [ߣ\ n x ࠍߎߩࠃ߁ߦ૶

޿ಽߌࠆߎߣߪᧄ lecture note ߦ㒢ߞߡߩ⚂᧤ߣᔃᓧߡ߶ߒ޿㧚৻⥸ߦ⹺߼ࠄࠇߡ޿ࠆ
ᣇᴺߢߪߥ޿㧚଀߃߫㧘Eyring, Walter, Kimball ߩ"Quantum Chemistry"ߩ page 77 ߩ 5.50
ᑼߩ\ n [ ߪ x ߦߟ޿ߡⷙᩰߐࠇߚ߽ߩࠍ␜ߔ㧚ߔߥࠊߜᧄ lecture note ߩᑼ(7.4.12')ߩᗧ๧ߢ↪޿ࠄࠇߡ޿ࠆ㧚߹ߚ㧘Atkins ߩ"Molecular Quantum Mechanics"㧘2nd edition
ߩ page 52 ߩᑼ(3.5.5)ߪ㧘ᧄ lecture note ߩᑼ(7.4.12')ߣหߓᒻߢ⴫ߐࠇߡ޿ࠆ߇㧘ߘࠇߪ [㧔Atkins ߩᧄߢߪ y ߣ޿߁ᄌᢙߦߥߞߡ޿ࠆ߇㧕ߦߟ޿ߡⷙᩰൻߐࠇߚ㧘ᧄ lecturenote ߩᑼ(7.4.10)ߩᗧ๧ߢ↪޿ࠄࠇߡ޿ࠆ㧚ߎߩ 2 ߟߩߺࠍᲧߴࠆߣ㧘ᒻᑼ⊛ߦߪᧄ
lecture note ߣㅒߦߥߞߡ޿ࠆ߇㧘߹ߤࠊߐࠇߥ޿ࠃ߁ߦᵈᗧߒߡ߶ߒ޿㧚એ਄㧘ᄢಽ㐳ߊ߆߆ߞߚ߇㧘harmonic oscillator ߦ㑐ߔࠆ Schrödinger equation (7.2.5)
ߦኻߔࠆ eigenvalue (7.3.15)ߣ eigenfunction (7.4.10)߇᳞߹ߞߚࠊߌߢ޽ࠆ㧚ዉ಴ߦᚻ㑆ขߞߚߩߢ㧘ᔨߩߚ߼߹ߣ߼ߡ߅ߎ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 17⺞๺ᝄേሶߦ㑐ߔࠆ㊀ⷐ౏ᑼ߹ߣ߼㧦
Schrödinger equation ߪ=2 w \ x 1 2 kx \ x
2m wx 22ߢ޽ࠆ߇㧘ߎߎߢ
2mE
D
=2
2

E

§ 2S mQ ·
¨
¸
© = ¹

mk
=2

2

E\ x

(7.1.6)(7.2.2)2km
x=2ߩ⟎߈឵߃ࠍߔࠆߣ㧘Schrödinger equation ߪ

[

Ex

4

d 2\ [ § D· ¨  [ 2 ¸\ [
2
d[
©E
¹

0

(7.2.4)(7.2.5)ߣߥࠆ㧚ߎߩ Schrödinger equation ߩ eigenvalue ߪ
k §
1·
¨n ¸
m©
2¹
ߢ޽ࠆ㧚ߚߛߒ㧘ߎߎߢ
En

=

Q

1
2S

1·
§
hQ ¨ n  ¸2¹©

n0, 1, 2, ...

km(7.3.15)(7.1.4)ߢ޽ࠆ㧚
eigenfunction ߪ [ ᐳᮡߦߟ޿ߡⷙᩰൻߐࠇߡ޿ࠆ߽ߩߪ

§ [2 ·
1
H
[
exp
¨ ¸
n
2n n ! S
© 2 ¹

\n [
f

2

³ \ [ d[ 1(7.4.10)

n 0, 1, 2, ...

nfߢ޽ࠅ㧘৻ᣇ㧘 x ᐳᮡߦߟ޿ߡⷙᩰൻߐࠇߡ޿ࠆ߽ߩߪ

1

\n x

2n n !
f³\

2

n

x dx 1SE

Hn§ E x2 ·¸© 2 ¹

E x exp ¨ (7.4.12')n 0, 1, 2, ...

f

ߣߥࠆ㧚
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 18

\ n [ ߩౕ૕⊛ߥ㑐ᢙᒻࠍ㧘⴫ 7.4.1 ߦ޿ߊߟ߆ߩ n ߦߟ޿ߡ߹ߣ߼ߡ߅ߊ

⴫ 7.4.1\ n [ ߩ㑐ᢙᒻ㧔 n 0  10 㧕
§ 1 ·exp ¨  [ 2 ¸
S
© 2 ¹
1
§ 1 ·
exp ¨  [ 2 ¸ 2[
4
S 2
© 2 ¹

\0 [
\1 [
\2 [
\3 [
\4 [
\5 [
\6 [
\7 [
\8 [
\9 [
\ 10 [

4

4

4

4

4

4

4

4

4

1

4

1
§ 1 ·
exp ¨  [ 2 ¸ 4[ 2  2
S2 2
© 2 ¹

1
§ 1 ·
exp ¨  [ 2 ¸ 8[ 3  12[
S4 3
© 2 ¹

1
§ 1 ·
exp ¨  [ 2 ¸ 16[ 4  48[ 2  12
S8 6
© 2 ¹

1
§ 1 ·
exp ¨  [ 2 ¸ 32[ 5  160[ 3  120[
S 16 15
© 2 ¹

1
§ 1 ·
exp ¨  [ 2 ¸ 64[ 6  480[ 4  720[ 2  120
S 96 5
© 2 ¹

1
§ 1 ·
exp ¨  [ 2 ¸ 128[ 7  1344[ 5  3360[ 3  1680[
S 96 70
© 2 ¹

1
§ 1 ·
exp ¨  [ 2 ¸ 256[ 8  3584[ 6  13440[ 4  13440[ 2  1680
S 384 70
© 2 ¹

1
§ 1 ·
exp ¨  [ 2 ¸ 512[ 9  9216[ 7  48384[ 5  80640[ 3  30240[
S 2304 35
© 2 ¹

1
§ 1 ·
exp ¨  [ 2 ¸ 1024[ 10  23040[ 8  161280[ 6  403200[ 4  302400[ 2  30240
S 23040 7© 2 ¹

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 19࿑ 7.4.1a ߅ࠃ߮࿑ 7.4.1b ߦ harmonic oscillator ߩ wavefunction \ ߅ࠃ߮ \ ࠍ n20߆ࠄ n 5 ߹ߢࠍ potential ߣ౒ߦ␜ߒߡ޽ࠆ㧚࿑ 7.4.1aharmonic oscillator ߩ potential ߅ࠃ߮ wavefunction \ n 㧚\ n ߪ㧘ߘࠇߙࠇߩࠛࡀ࡞ࠡ࡯ಽߛߌ਄ߦࠪࡈ࠻ߒߡឬ޿ߡ޿ࠆ㧚❑ゲߪ hQ න૏ߩࠛࡀ࡞ࠡ࡯㧚ߚߛߒ㧘ᵄേ㑐ᢙߩᒻߦߟ޿ߡߪ❑ゲߪᝄ᏷ߢࠛࡀ࡞ࠡ࡯ߢߪ
ߥ޿㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 20

࿑ 7.4.1b
harmonic oscillator ߩ potential ߅ࠃ߮ wavefunction \ n ߩੑਸ਼㧘\ n 2 㧚\ n 2 ߪ㧘ߘࠇߙࠇߩࠛࡀ࡞ࠡ࡯ಽߛߌ਄ߦࠪࡈ࠻ߒߡឬ޿ߡ޿ࠆ㧚❑ゲߪ hQ න૏ߩࠛࡀ࡞ࠡ࡯㧚ߚߛߒ㧘ᵄേ㑐ᢙߩᒻߦߟ޿ߡߪ❑ゲߪᝄ᏷ߢࠛࡀ࡞ࠡ࡯ߢߪ
ߥ޿㧚ߎߩ࿑ࠍ⌑߼ߡߺࠆߣ㧘ᆎ߼ߦ੍ᗐߒߚࠃ߁ߦ㧘࿑ 7.3.2 ߩ੗ᚭဳ potential ߩ႐วߣ߆ߥࠅૃߡ޿ࠆߎߣߦ᳇߇ߟߊ㧚ߒ߆ߒ߹ߚ㧘޿ࠈ޿ࠈߥὐߢ⇣ߥߞߡ޿ࠆ㧚੗ᚭဳ potential ߩ႐วߦߪ nߪn

0 ߇ߥߊ n 1 ߆ࠄᆎ߹ߞߡ޿ߚߩߦ㧘harmonic oscillator ߢ

0 ߆ࠄᆎ߹ࠆ㧚߹ߚ㧘 n

0 ߩ႐วߢ߽ E ߪ࠯ࡠߢߥߊ޽ࠆ୯㧔zero point energyߣ⸒ࠊࠇࠆ㧕ࠍᜬߞߡ޿ࠆߎߣߦ߽ᵈᗧߔࠆᔅⷐ߇޽ࠆ㧚߹ߚ㧘ᆎ߼ߦ੍ᗐߒߚࠃ߁
ߦ㧘harmonic oscillator ߩ႐วߦߪ wavefunction ߇ potential ߩᄖ஥߳߆ߥࠅߒߺ಴ߡ޿
ࠆ㧚ߎߩࠃ߁ߥߎߣࠍ tunneling ߣ޿߁߇㧘ߎߩߎߣߦߟ޿ߡߪ㧘ᓟߢ߽߁ዋߒߊࠊߒ
ߊี๧ߒࠃ߁㧚ᝄേᵄേ㑐ᢙ\ n [ ߩᒻࠍ࿑ 7.4.1a ߦ n

0 ߆ࠄ n 11 ߹ߢ␜ߔ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 21

0.7
0.6
0.5
0.4
0.3
0.2
0.1

4

4

2

2

0.6
0.4
0.2

4

2

2

2

4

2

4

2

4

 0.6

4

\0 [

\1 [

0.6

0.6

0.4

0.4

0.2

0.2

 0.2

2

 0.4

2

4

4

2

 0.2
 0.4

 0.4

4

 0.2

 0.6

\2 [

\3 [

0.4

0.4

0.2

0.2
2

4

4

2

 0.2

 0.2
 0.4

 0.4
\4 [

\5 [
࿑ 7.4.2aᝄേᵄേ㑐ᢙ\ n [ , n䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 22

05

4

2

0.4

0.4

0.2

0.2
2

4

4

2

 0.2

\6 [

\7 [

0.4

0.4

0.2

0.2

2

2

4

4

2

 0.2

4

2

4

 0.2

\8 [

\9 [

0.4

0.4

0.2

0.2

2

2

 0.4

 0.4

4

4

 0.4

 0.4

4

 0.2

2

2

4

4

2

 0.2

 0.2
 0.4

 0.4
\ 10 [\ 11 [
࿑ 7.4.2bᝄേᵄേ㑐ᢙ\ n [ , n䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 23

6  11ߎࠇ߹ߢߩ⸃ᴺࠍቇ⠌ߒߡ߈ߡ㧘߆ߥࠅ㕙ୟߛߥߣᕁߞߚߎߣߢ޽ࠈ߁㧚Hermite
ߩᓸಽᣇ⒟ᑼߦߟ޿ߡ೨߽ߞߡീᒝߒߡ޿ࠆੱߥࠄ޿ߑ⍮ࠄߕ㧘ᆎ߼ߡߩੱߦߣߞߡ
ߪ㧘㕙ୟߛߣ޿߁ߩ߇޿ߟࠊࠄߑࠆᗵߓߢ޽ࠆߣᕁ߁㧚Schrödinger equation ߩ⸃ࠍ᳞߼ࠆߛߌߥࠄ߫㧘߽ߞߣ◲නߥᣇᴺ߇޽ࠆ㧚ߘࠇߢ߽޽߃ߡ㧘߿߿ߎߒ޿ߎߣࠍ㐳‫ޘ‬ߣㅀߴߚߩߪ㧘ߎߩᣇᴺ߇ᄙߊߩ standard ߥᢎ⑼ᦠߩ߿ࠅᣇߢ޽ࠆߒ㧘⸃ߊㆊ⒟ߦ޿
ߊߟ߆ߩᄢಾߥ᭎ᔨ߿ᚻᴺ߇฽߹ࠇߡ޿ࠆ߆ࠄߢ޽ࠆ㧚ᰴߩ▵ߢߪ㧘ߎࠇ߹ߢߣߪోߊ⇣ߥߞߚᣇᴺߢ Schrödinger equation ߩ⸃ࠍ᳞߼ߡߺ
ࠃ߁㧚↢ᚑṶ▚ሶ߿ᶖṌṶ▚ሶࠍ↪޿ࠆߘߩᣇᴺߪ㧘੹߹ߢߩൻቇߩᢎ⑼ᦠߦߪ޽߹
ࠅ಴ߡ᧪ߥ߆ߞߚߎߣߪ੐ታߢߪ޽ࠆ߇㧘㕖Ᏹߦ㊀ⷐߢ㧘߆ߟ㧘ᓎߦ┙ߟᢙቇ⊛ᚻᴺ
ߢ޽ࠆ㧚ߔߢߦ᳞߼ߚ⚿ᨐࠍ㧘ઁߩᣇᴺߢ಴ߔߛߌ߆߽ߒࠇߥ޿߇㧘ߎ߁޿߁ᣇᴺ⺰ߩଢ೑ߐࠍߥࠎߣߥߊᗵߓࠆߛߌߢ߽ᗧ๧ߩ޽ࠆߎߣߢ޽ࠆߣᕁ߁㧚‫ڏ‬Problem 7.4.1
ᑼ(7.4.1)ࠃࠅ H 3 [ 㧘 H 4 [ 㧘߅ࠃ߮ H 5 [ ࠍ᳞߼ߡ⴫ 7.3.1 ߩ⚿ᨐߣᲧセߖࠃ㧚‫ڏ‬Problem 7.4.2
ᑼ(7.4.4)߆ࠄᰴߩ㑐ଥᑼࠍዉ಴ߖࠃ㧚nn 1\ n 1 [ \ n 1 [2
2[\ n [‫ڏ‬Problem 7.4.3

ᑼ(7.4.5)ࠍ↪޿ߡ㧘

d\ n [d[

ࠍ\ n 1 [ ߣ \ n 1 [ ߣߢ⴫ߖ㧚‫ڏ‬Problem 7.4.4
ᑼ(7.4.9)ߩ normalization ߢᔅⷐߥⓍಽ
f

I0 a

³ exp ax dx
2(7.4.13)fࠍᰴߩࠃ߁ߦߒߡ᳞߼ߡߺࠃ㧚߹ߕ
f

I0 a

2 ³ exp  ax 2 dx
0

I 0 a ߩੑਸ਼ߪᰴᑼߢ␜ߐࠇࠆߎߣࠍ␜ߖ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 24(7.4.14)

I0 a

2

f f

^

`

4 ³ ³ exp a x 2  y 2 dxdy00ߎߎߢᭂᐳᮡ߳ߩᄌ឵ࠍⴕ߁㧚r 2 x2  y 2
dxdy rdrdT(7.4.15)(7.4.16)(7.4.17)ᑼ(7.4.15)ࠍ r ߅ࠃ߮ T ߦ㑐ߔࠆቯⓍಽߩᑼߦ⟎߈឵߃㧘߆ߟⓍಽࠍታⴕߖࠃ㧚ᓟߢ૶߃ࠆࠃ߁ߦ╵߃ߪਅߦᦠ޿ߡ߅ߎ߁㧚f³ exp ax dx
2

I0 a(7.4.18)

f‫ڏ‬Problem 7.4.5
Problem 7.4.6 ߩᑧ㐳ߣߒߡ㧘f³ x exp ax dx
2n

In a

2

(7.4.19)fࠍᰴߩࠃ߁ߦߒߡ᳞߼ࠃ㧚߹ߕ㧘ᰴᑼࠍ⸽᣿ߖࠃ㧚d n I0 ada nn

1 I n aߎߩᑼࠍ↪޿ߡ㧘 I1 a

(7.4.20)

߅ࠃ߮, I 2 a ࠍ᳞߼ࠃ㧚ߎߩ⚿ᨐߪ㧘ߎࠇ߆ࠄ߽ߒ߫ߒ߫૶

߁ߎߣߦߥࠈ߁߆ࠄਅߦᦠ߈౉ࠇߡ߅ߊߣࠃ޿㧚
f

I1 a

³ x exp ax dx(7.4.21)³ x exp ax dx(7.4.22)

2

2

f
f

I2 a

4

2

f

[Let’s play with Mathematica] ⺞๺ᝄേሶߩ Mathematica ߦࠃࠆ⴫⃻ᑼ㧔7.4.10㧕ߢቯ⟵ߐࠇߚ⺞๺ᝄേሶߩᵄേ㑐ᢙ\ n [ ߅ࠃ߮㑐ㅪߒߚ࿑߿ᑼߪ㧘
Mathematica ߦࠃߞߡ◲නߦ⴫⃻಴᧪ࠆ㧚߹ߕ㧘ᑼ㧔7.4.10㧕ࠍ◲නߥᑼߢቯ⟵ߒߡ߅ߊߣଢ೑ߢ޽ࠆ㧚ߎߎߢ n ߪᝄേ㊂ሶᢙߢ޽ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 25

f[n_,x_]:=HermiteH[n,x] Exp[-x^2/2]/Sqrt[2^n Factorial[n] Sqrt[Pi]]৻ᐲߎߩࠃ߁ߦቯ⟵ߒߡ߅ߊߣ㧘ᓟߪ㧘f[n,x]ࠍ↪޿ߡ㧘◲ẖߦ⴫⃻಴᧪ࠆ㧚଀߃߫㧘n0 ߆ࠄ n 11 ߹ߢߩ\ n [ ߩ [ ߦኻߔࠆࡊࡠ࠶࠻ߪ㧘ᰴߩࠃ߁ߦ◲නߦ᳞߹ࠆ㧚࿑

7.4.2 ߪߎߩࠃ߁ߦߒߡឬ޿ߚ߽ߩߢ޽ࠆ㧚
Table[Plot[f[n,x],{x,-5,5},{n,0,11}]7.5 ↢ᚑṶ▚ሶߣᶖṌṶ▚ሶࠍ↪޿ߚ⺞๺ᝄേሶߩ⸃
ౣ߮ᑼ(7.2.5)ߢਈ߃ߚήᰴరᐳᮡߢ⴫ߒߚ⺞๺ᝄേሶߦ㑐ߔࠆ Schrödinger equationd 2\ [ § D· ¨  [ 2 ¸\ [
2
d[
©E
¹

0

ࠍ⠨߃ࠆ㧚ߎߎߢᑼ(7.2.2)ࠃࠅ
D 2E
E hQ

(7.5.1)(7.5.2)ߢ޽ࠆ߇㧘ᡷ߼ߡ

OE
hQ(7.5.3)ߢቯ⟵ߐࠇࠆήᰴరߩ㊂ O ࠍዉ౉ߔࠆ㧚ߘ߁ߔࠆߣᑼ(7.5.1)ߪ
1 § 2 d2 ·¨ [  2 ¸\ O [2©
d[ ¹O\ O [(7.5.4)1§d2 ·ߣߥࠆ㧚ߔߥࠊߜ㧘operator ¨ [ 2  2 ¸ ߩ eigenvalue ߇ O 㧘eigenfunction ߇\ O [ ߣ2©
d[ ¹޿߁ߎߣߦߥࠆ㧚ߎߎߢ\ O [ ߩ subscript ߣߒߡߘߩ eigenvalue O ࠍߟߌࠆߎߣߦߔ
ࠆ㧚ᑼ(7.5.4)ߪήᰴరᐳᮡ [ ߦ㑐ߔࠆ⺞๺ᝄേሶߩ Schrödinger equation ߢ޽ࠅ㧘ߘߩ
eigenvalue ߪήᰴరߢ޽ࠆߎߣߦ⇐ᗧߔࠆᔅⷐ߇޽ࠆ㧚ߐߡߎߎߢ㧘ᣂߒߊ↢ᚑṶ▚ሶ b  ߅ࠃ߮ᶖṌṶ▚ሶ b ࠍᰴߩࠃ߁ߦቯ⟵ߔࠆ㧚㧔‫↢ޟ‬ᚑ‫ޟ߮ࠃ߅ޠ‬ᶖṌ‫ߩޠ‬ᗧ๧ߪ߅޿߅޿᣿ࠄ߆ߦߥࠆ㧕䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 26

b

1§
d ·
¨[ 
¸
d[ ¹
2©

(7.5.5)

b

1§
d ·
¨[ 
¸
d[ ¹
2©

(7.5.6)

ߎࠇ߆ࠄߒ߫ࠄߊߩ㑆㧘ᣂߒߊቯ⟵ߒߚ↢ᚑṶ▚ሶ b  ߅ࠃ߮ᶖṌṶ▚ሶ b ߦߟ޿ߡ޿ࠈ޿ࠈߥⷺᐲ߆ࠄี๧ߔࠆߎߣߦߥࠆ㧚߹ߕߎࠇࠄ 2 ߟߩṶ▚ሶߩⓍࠍ᳞߼ߡߺࠃ
߁㧚
d ·§
d\ ·
1§
¨[ 
¸ ¨ [\ ¸
d[ ¹ ©
d[ ¹2©

b  b\·
1 § 2 d2¨ [  2  1¸\
2©
d[
¹
d ·§
d\ ·
1§
¨[ 
¸ ¨ [\ ¸
d[ ¹ ©
d[ ¹
2©

bb \

·
1 § 2 d2¨ [  2  1¸\
2©
d[
¹

ߔߥࠊߜ
bb

·
1 § 2 d2¨ [  2  1¸d[2©
¹

(7.5.7)bb ·
1 § 2 d2¨ [  2  1¸d[2©
¹

(7.5.8)ߎߩᑼࠃࠅª¬b, b  º¼

bb   b  b 1(7.5.9)ߣ޿߁੤឵㑐ଥ߇ᓧࠄࠇࠆ㧚ߎߩ੤឵㑐ଥߪ㕖Ᏹߦ㊀ⷐߢߎࠇ߆ࠄవ➅ࠅ㄰ߒ૶ࠊࠇ
ࠆߎߣߦߥࠆ㧚ᑼ(7.5.9)ߩࠃ߁ߦ commutator ߇ 1 ߦߥࠆࠃ߁ߥ੤឵㑐ଥߪ Bose ߩ੤឵㑐ଥߣ⸒ࠊࠇߡ޿ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 27

ᑼ(7.5.6)ߩ operator ࠍ b 㧘߹ߚ㧘ᑼ(7.5.5)ߩ operator ߪߘࠇߦ  ߩ superscript ࠍߟߌߡ b  ߣ߅޿ߚ߇㧘ߘࠇߪߎߩ 2 ߟߩ operator ߇߅੕޿ߦ Hermitian conjugate ߢ޽ࠆߣ޿߁㑐ଥߦ޽ࠆ߆ࠄߢ޽ࠆ㧚એਅߎߩߎߣࠍ⺑᣿ߒࠃ߁㧚 b ߅ࠃ߮ b  ߪ޿ߕࠇ߽㧘 [d߅ࠃ߮ߣ޿߁ 2 ߟߩ operator ࠍ฽߻㧚ߎߩ߁ߜ [ ߪ Hermitian ߢ޽ࠆ㧚ߔߥࠊߜ㧘d[[ o rf ߢ 0 ߦ෼᧤ߔࠆછᗧߩ wavefunction\ ߅ࠃ߮ M ߦߟ޿ߡ³\ * [M d[³ M * [\ d[ *dߪ anti-Hermitian ߢ޽ࠆ㧚ߔߥࠊߜd[ߣߥࠆ㧚߹ߚ㧘Problem 4.2.1 ߢቇ⠌ߒߚࠃ߁ߦ*§·d ¨ ³ M * \ d[ ¸
d[
©
¹

d
³\ * d[ M d[ߣߥࠆ㧚2 ߟߩᑼࠍㄝ‫ޘ‬ᒁ޿ߡ§d ·³\ * ¨© [  d[ ¸¹ M d[߹ߚߪ㧘ਔㄝߦ

*

§
·
§
d ·
¨ ³M *¨[ ¸\ d[ ¸d[ ¹©
©
¹

(7.5.10)1ࠍ߆ߌߡ
2³\ * b M d[

³ M * b\ d[³ Mb\ * d[

*(7.5.11)2 ߟ⋡ߩ╬ภߪ㧘operator b ߇ real ߢ޽ࠆߣ޿߁ߎߣ߆ࠄᓧࠄࠇࠆ㧚ᑼ(7.5.11)߆ࠄ㧘bߣ b  ߣߪ੕޿ߦ Hermitian conjugate㧔ࠛ࡞ࡒ࡯࠻౒ᓎ㧕ߢ޽ࠆߎߣ߇ಽ߆ࠆ㧚‫ޤ⸼ޣ‬㧦
৻⥸ߦ㧘✢ᒻṶ▚ሶ C ߦߟ޿ߡ³ M *C\ dW*³ C\ M dW*

߇ W o rf ߢ 0 ߦߥࠆછᗧߩ wavefunction\ ߅ࠃ߮ M ߦߟ޿ߡ³ M * C\ dW*³\ * C I dW

(7.5.12)߇ᚑࠅ┙ߟṶ▚ሶ C  ߇޽ࠇ߫ C ߣ C  ߪ߅੕޿ߦ Hermitian conjugate ߢ޽ࠆߣ޿߁㧚ߐߡ㧘 b  ߿ b ߩṶ▚ሶߩᕈ⾰߇ߪߞ߈ࠅߒߡ᧪ߚߩߢ㧘޿ࠃ޿ࠃ Schrödinger
equation ࠍ⸃ߊߎߣࠍ⠨߃ࠃ߁㧚ᑼ(7.5.7)ࠍ↪޿ࠆߣ Schrödinger equation ߪᰴߩࠃ߁
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 28ߦᦠߌࠆ㧚

1·§ ¨ b b  ¸\ O O\ O
2¹©
ߎߩᑼߩਔㄝߦ b ࠍ૞↪ߔࠆ㧚1·§b ¨ b  b  ¸\ O Ob\ O
2¹©
ߣߥࠆ߇㧘Ꮐㄝߩ╙ 1 㗄⋡ߦᑼ(7.5.9)ߩ੤឵㑐ଥࠍ↪޿ࠆߣ1·§ ¨ b b  ¸ b\ O2¹©

O  1 b\ O

(7.5.13)(7.5.14)(7.5.15)৻ᣇ㧘ᑼ(7.5.13)ࠃࠅ eigenvalue ߇ O  1 ࠍᜬߟ eigenfunction \ O 1 ߪ

1·§ ¨ b b  ¸\ O 1
2¹©

O  1 \ O 1

(7.5.16)ߢਈ߃ࠄࠇࠆ㧚ߎߩ 2 ߟߩᑼࠍᲧߴࠆߣ b\ O ߪ\ O 1 ߩቯᢙ୚ߣ޿߁ߎߣ߇ಽ߆ࠆ㧚ߔߥࠊߜ㧘operator b ߪ wavefunction \ O ߦ૞↪ߒߡ㧘 O ߇ 1 ߟዊߐ޿\ O 1 ࠍ૞ࠆ㧚ߘ߁޿߁ᗧ๧ߢ㧘operator b ߪߒ߫ߒ߫‫ޟ‬ᶖṌṶ▚ሶ‫(ޠ‬annihilation operator)ߣ⸒ࠊࠇࠆ㧚‫ޤ⸼ޣ‬㧦"annihilation"ࠍ‫⊒ߣޠࡦ࡚ࠪ࡯࡟ࡅ࠾ࠕޟ‬㖸ߔࠆቇ↢⻉ำ߇ࠃߊ޿ࠆ㧚ㄉᦠߢ⊒㖸ࠍ⺞ߴߡ߅ߊߎߣ㧚ห ߓ ࠃ ߁ ߥ ߎ ߣ ࠍ b  ߦ ߟ ޿ ߡ ߽ ⴕ ߞ ߡ ߺ ࠃ ߁ 㧚 ߹ ߕ ᑼ (7.5.8) ߦ ⌕ ⋡ ߔ ࠆ ߣ
Schrödinger equation ߪ§  1·¨ bb  ¸\ O O\ O
2¹©
ߣᦠߌࠆ㧚ߎߩᑼߩਔㄝߦ b  ࠍ૞↪ߔࠆ㧚§  1· ¨ b b  ¸ b \ O Ob \ O
2¹©

(7.5.17)ਔㄝߦ b \ O ࠍട߃ߡ

1· 
§ 
¨b b  ¸b \O
2¹©
৻ᣇ㧘ᑼ(7.5.13)ࠃࠅ
1· § ¨ b b  ¸ b \ O 1
2¹©

O  1 b \ OO  1 b \ O 1䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 29(7.5.18)ߎߩ 2 ߟߩᑼࠍᲧߴߡߺࠆߣ㧘 b \ O ߪ\ O 1 ߩቯᢙ୚ߦߥࠆߎߣ߇ಽ߆ࠆ㧚ߔߥࠊߜ㧘
operator b  ߪ wavefunction \ O ߦ૞↪ߒߡ㧘O ߇ 1 ߟᄢ߈޿\ O 1 ࠍ૞ࠆ㧚ߘߩᗧ๧ߢ b ߪߒ߫ߒ߫‫↢ޟ‬ᚑṶ▚ሶ‫(ޠ‬creation operator)ߣ⸒ࠊࠇࠆ㧚ߐߡ㧘annihilation operator b ߦ⌕⋡ߒࠃ߁㧚 b ࠍ wavefunction ߦ 1 ࿁૞↪ߔࠆߏߣߦ eigenvalue ߇ 1 ߟߕߟዊߐߊߥࠆ㧚ߒ߆ߒ㧘ߎߩᠲ૞ࠍή㒢࿁➅ࠅ㄰ߔߎߣߪ಴᧪
ߥ޿㧚ߥߗߥࠄ߫㧘harmonic oscillator ߩ energy ߪ kinetic energy ߣ potential energy ߣߩ๺ߢᔅߕᱜߢߥߌࠇ߫ߥࠄߥ޿㧔ᑼ(7.5.3)ߢ E ! 0 ߢ޽ࠆ߆ࠄ O ! 0 㧕㧚ᓥߞߡ㧘 b ࠍ૞↪ߔࠆߎߣߦࠃߞߡ O ࠍ 1 ߟߕߟᷫࠄߒߡ޿ߞߡ߽ O ߪ⽶ߦߪߥࠄߥ޿㧚ᒰὼ O ߩᦨዊ୯߇޽ࠆߪߕߢ㧘ߘߩ O ߩᦨዊ୯ࠍ Omin ߣ߅ߎ߁㧚
ᑼ(7.5.15)ߪ O

Omin ߢ߽ᚑࠅ┙ߟߪߕߢ޽ࠆ߆ࠄ

1·§ ¨ b b  ¸ b\ Omin
2¹©

Omin  1 b\ Omin(7.5.19)1·§ߎ ߩ ߎ ߣ ߆ ࠄ 㧘 (a) b\ Omin ߪ Hamiltonian ¨ b  b  ¸ ߩ eigenfunction ߢ ޽ ࠅ 㧘
2¹©
eigenvalue Omin  1 ࠍᜬߟ㧘(b) b\ Omin ߪ࠯ࡠߢ޽ࠆ㧘ߩ޿ߕࠇ߆ߢ޽ࠆߣ⚿⺰ߐࠇࠆ㧚(a)ߪ Omin ߇ᦨዊߩ eigenvalue ߢ޽ࠆߣ޿߁઒ቯߦ෻ߔࠆ㧚ᓥߞߡ(b)߇ᚑࠅ┙ߚߥߌࠇ߫
ߥࠄߥ޿㧚ߔߥࠊߜ

b\ Omin

0(7.5.20)ਔㄝߦ b  ࠍ૞↪ߔࠆ㧚b  b\ Omin01ਔㄝߦ \ Omin ࠍട߃ࠆ㧚
2
1·
1
§ 
Omin
¨ b b  ¸\ Omin
2¹
2©
ߎࠇࠍ Schrödinger equation (7.5.13)
1·§ ¨ b b  ¸\ O
2¹
©
ߩO

O\ O

Omin ߩ႐วߩᑼ

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 30(7.5.21)1·§ ¨ b b  ¸\ Omin
2¹©
ߣᲧߴࠆߎߣߦࠃࠅ

Omin\ O

min

1
2

Omin

(7.5.22)ߣ޿߁⚿ᨐ߇ᓧࠄࠇࠆ㧚਄ㅀߩ⼏⺰ߢ᣿ࠄ߆ߥࠃ߁ߦ O ߪ߅੕޿ߦᢛᢙ୯ߛߌ⇣ߥࠆߩߢ㧘 n ࠍᱜߩᢛᢙߣߒߡOn  Ominn1
2

n 0, 1, 2, ...

(7.5.23)ߣ⴫ߐࠇࠆߎߣߦߥࠆ㧚੹߹ߢ㧘eigenvalue O ࠍᜬߟ eigenfunction ࠍ\ O ߣ߅޿ߡ᧪ߚ߇㧘ᑼ(7.5.23)ߦ⌕⋡1
ߒߡ㧘੹ᓟ㧘eigenvalue n  ࠍᜬߟ eigenfunction ࠍ\ n ߣ߅ߊߎߣߦߒࠃ߁㧚 n ࠍ㊂ሶ
2ᢙ㧔quantum number㧕ߣ޿߁㧚ߎߩ notation ࠍ↪޿ࠆߣ Schrödinger equation (7.5.13)ߪ

1·
1·
§ 
§
(7.5.24)¨ b b  ¸\ n ¨ n  ¸\ n
2¹
2¹
©
©
ߣ⴫⃻ߐࠇࠆ㧚operator b  ߅ࠃ߮ b ߪߘࠇߙࠇ eigenvalue ࠍ 1 ߟ਄ߍ߹ߚߪਅߍࠆ૞
↪ࠍᜬߟࠊߌߢ㧘\ n ⴫␜ߢߪb \ nJ\ n 1(7.5.25)

b\ n

G\ n 1(7.5.26)
ߣߥࠆ㧚 J ߅ࠃ߮ G ߪ੹ߩߣߎࠈಽ߆ࠄߥ޿Ყ଀ቯᢙߢ޽ࠆ㧚ᑼ(7.5.26)ߩᭂ㒢ߩᑼ
(7.5.20)ߪ
b\ 0

0(7.5.27)

ߣ޿߁ߎߣߦߥࠆ㧚ߐߡ㧘਄ߢ eigenvalue ߇᳞߹ߞߚߩߢᰴߦ eigenfunction ߩౕ૕⊛ߥᒻࠍ᳞߼ࠆߎߣ
ࠍ⹜ߺࠆ㧚߹ߕᑼ(7.5.27)ߦ⌕⋡ߔࠆ㧚ᑼ(7.5.6)ࠍ↪޿ߡ
§
d ·
¨[ 
¸\ 0
d[ ¹
©

0

߹ߚߪ

d\ 0

\0

[ d [

ߎࠇࠍ⸃޿ߡ

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 31

ln\ 0[22

 constantߔߥࠊߜ§ [2 ·
N 0 exp ¨  ¸
© 2 ¹

\0

ቯᢙ㗄ߩ N 0 ߪ normalization condition ࠃࠅ᳞߼ࠆ㧚ߔߥࠊߜ㧘
f

³\ [
0

2

d[

1

f

ࠃࠅ
1

N0

4

S

ࠍᓧࠆ㧚ߔߥࠊߜ㧘normalize ߐࠇߚၮᐩ⁁ᘒ㧔ߔߥࠊߜ energy ߇ᦨૐߩ⁁ᘒ㧕ߩ
wavefunction ߣߒߡ\0 [§ [2 ·
exp
¨ ¸
4
S
© 2 ¹
1

(7.5.28)߇᳞߹ࠆ㧚ߎ߁ߒߡ\ 0 [ ߇᳞߹ࠇ߫㧘ߘࠇߦ㗅ᰴ b  ࠍ૞↪ߒߡ޿ߌ߫\ 1 [ 㧘\ 2 [ 㧘\ 3 [ 㧘㧚㧚㧚ߣછᗧߩ\ n [ ߩᒻ߇᳞߹ࠆߪߕߢ޽ࠆ㧚ߚߛߒᑼ(7.5.25)ߣᑼ(7.5.26)ߦ
߅ߌࠆᲧ଀ቯᢙ J ߅ࠃ߮ G ߪᧂ⍮ߢ޽ࠆ㧚਄ߢⴕߞߚࠃ߁ߦ㧘Ფᐲ normalization
condition ࠃࠅቯᢙㇱಽࠍ᳿߼ߡ߽ࠃ޿߇㧘಴᧪ࠆߎߣߥࠄᆎ߼߆ࠄᑼ (7.5.25) ߿ᑼ(7.5.26)ࠍ㧘ቯᢙ㗄ࠍ߽฽߼ߡ߈ߜࠎߣ᳞߼ߡ߅߈ߚ޿㧚એਅߘߩߎߣࠍ⹜ߺࠆ㧚੹㧘઒ߦ\ n [ ߪߔߢߦⷙᩰൻߐࠇߡ޿ࠆߣߔࠆ㧚ߎࠇߦ b  ࠍ૞↪ߒߡ\ n 1

Nb \ nߣߒߡ\ n1 ߇ⷙᩰൻߐࠇࠆࠃ߁ߦ N ࠍ᳞߼ࠃ߁㧚

³\

n 1

*\ n 1d[*

N 2 ³ b \ n b \ n d[*

N 2 ³\ n b b \ n d [' b ߪHermitianߛ߆ࠄ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 32

N 2 ³\ n bb \ n d[

' b, b  ,[ߪߔߴߡreal ߛ߆ࠄ

N 2 ³\ n b  b  1 \ n d [N 2 n 1'੤឵㑐ଥ(7.5.9)ࠃࠅ

'ᑼ (7.5.24)ࠃࠅ 㧔ᑼ(7.5.24)ࠃࠅ㧕

1
ߔߥࠊߜ

1
n 1

N
⚿ዪ

1 
b \n
n 1

\ n 1(n0, 1, 2, ...)

(7.5.29)ߣ޿߁㑐ଥ߇ᓧࠄࠇࠆ㧚ߎߩ㑐ଥࠍ↪޿ߡ㧘\ 1 [ 㧘\ 2 [ 㧘\ 3 [ 㧘㧚㧚㧚ߣ㗅‫ߦޘ‬છᗧߩ\ n [ ࠍ᳞߼ࠆߎߣ߇಴᧪ࠆ㧚‫ڏ‬Exercise 7.5.1
ᑼ(7.5.29)ࠍ↪޿ߡ\ 1 [ ࠍ᳞߼ࠃ㧚‫ޤ⸃ޣ‬㧦
ᑼ(7.5.29)ࠃࠅ

\ 1 b \ 0
1§
d ·
¨[ 
¸\ 0
2©
d[ ¹
=

( ' (7.5.28) )§ [2 ·
1 1 §
d ·
exp
[

 ¸
¨
¨
¸
2 4S ©
d[ ¹© 2 ¹§ [2 ·
1 1
exp
[

[
¨ ¸
2 4S
© 2 ¹
24

1

S

§ [2 ·¸© 2 ¹[ exp ¨ ߹ߚ㧘ᑼ(7.5.29)ߢ n ࠍ n  1 ߣ⟎߈឵߃㧘ਔㄝߦ b ࠍ૞↪ߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 33

1 
bb \ n 1
n

b\ n

1 
b b  1 \ n 1nn\ n 1'ᑼ (7.5.24)ࠍ↪޿ߡߎࠇ߽߹ߚ᦭↪ߥ㑐ଥᑼߢ޽ࠆ㧚ᑼ(7.5.29)ࠍᦠ߈⋥ߒߚ߽ߩߣ߹ߣ߼ߡ㧘એਅߦ␜ߒߡ߅ߊ㧚b \ nn  1\ n 1

(7.5.30)b\ n

n\ n 1(7.5.31)ߎߩ 2 ߟߩ㑐ଥᑼߪ޿ࠈ޿ࠈߥὐߢ㊀ⷐߢ޽ࠆ㧚଀߃߫ᑼ(7.5.5)ߣᑼ(7.5.6)ࠃࠅ

1b  b
2[dd[(7.5.32)1b  b
2ߢ޽ࠆߩߢ recursion formula ߩ [\ n ߅ࠃ߮(7.5.33)d\ n߇ Section 7.4 ߢቇ⠌ߒߚࠃࠅߪ◲නߦd[᳞߹ࠆ㧚㧔Problem 7.4.2,߅ࠃ߮ Problem 7.4.3㧕ߔߥࠊߜ[\ n [
d\ n [d[nn 1\ n 1 [ \ n 1 [2
2(7.5.34)nn 1\ n 1 [ \ n 1 [2
2(7.5.35)

7.6 ⺞๺ᝄേሶߦ㑐ߔࠆੑਃߩ‛ℂ㊂ߩᐔဋ୯harmonic oscillator wavefunction ߇ਈ߃ࠄࠇߚ߆ࠄߦߪ㧘޿ࠈ޿ࠈߩ‛ℂ㊂ߦ㑐ߔࠆᐔဋ୯㧔߹ߚߪᦼᓙ୯㧕ࠍ᳞߼ࠆߎߣߪ◲නߥṶ⠌໧㗴ߢ޽ࠆ㧚ߎߩ▵ߢߪੑਃߩ଀
ࠍ޽ߍߡߺࠃ߁㧚ߚߛߎߎߢ㧘ᐳᮡߩขࠅᣇߦߟ޿ߡߜࠂߞߣ remark ߒߡ߅ߊᔅⷐ
߇޽ࠆ㧚ᑼ(7.4.10)߅ࠃ߮ᑼ(7.4.12')ߢ␜ߒߚࠃ߁ߦ㧘harmonic oscillator ߩ wavefunction䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 34

\ [E x ߪ㧘޽ࠆ႐วߦߪήᰴరߩ [ ᐳᮡߦ㑐ߒߡⷙᩰൻߒ㧘෶㧘޽ࠆ႐วߦ

\ߪ㐳ߐߩᰴరࠍ߽ߞߚ x ᐳᮡߦ㑐ߒߡⷙᩰൻߒߚࠅߔࠆ㧚ᒰὼߩߎߣߥ߇ࠄ㧘ਔ⠪ߩ㑆ߦߪ 4 E ߛߌߩⷙᩰൻቯᢙߩ㆑޿߇↢ߕࠆࠊߌߢ޽ࠆ㧚޽ࠆ‛ℂ㊂ߦߟ޿ߡߩᐔဋ
୯ߦߒߡ߽㧘 [ ߢⓍಽߔࠆ߆߹ߚߪ x ߢⓍಽߔࠆ߆ߢ୯ߪ⇣ߥߞߡ᧪ࠆ㧚એ਄ߩߎߣ
ߦߪߊࠇߋࠇ߽ᵈᗧߒߡ߶ߒ޿㧚ᢙቇ⊛ߦߪ㧘 [ ߦ㑐ߔࠆⓍಽ߇◲නߢ޽ࠈ߁㧚ߘ߽ߘ߽㧘ߘߩߚ߼ߦߎߘ㧘 [ ߣ޿߁ᄌᢙࠍዉ౉ߒߚࠊߌߢ޽ࠆ㧚ߘߎߢ㧘ᧄ▵ߦ߅޿ߡߪ㧘ේೣߣߒߡήᰴరߩ [ ᐳᮡߦ㑐ߒߡᐔဋ୯ࠍ᳞߼㧘႐วߦࠃߞߡߪ x ᐳᮡߦ㑐ߔࠆᐔဋ୯ߦߟ޿ߡ߽⸒෸ߔࠆߎߣߦߔࠆ㧚߹ߕ㧘૏⟎ߦ㑐ߔࠆᐔဋ୯ [[f³ \ [ [\ [ d[
n

n

f

ࠍ᳞߼ߡߺࠃ߁㧚ߎߎߢᑼ(7.5.34)ࠍ↪޿ࠇ߫ eigenfunction ߩ⋥੤ᕈࠃࠅⓍಽߪ࠯ࡠߦ
ߥࠆߎߣ߇ಽ߆ࠆ߇㧘ߘߩߎߣߪ㧘߽ߞߣ◲නߦ㧘ᄸ㑐ᢙߩⓍಽ߇࠯ࡠߦߥࠆߣ޿߁
ߎߣ߆ࠄ߽᣿ࠄ߆ߢ޽ࠆ㧚ߔߥࠊߜ㧘[f³ \ [ [\ [ d[nn0(7.6.1)fߥ߅㧘ᒰὼߩߎߣߥ߇ࠄ
1x[ 0(7.6.2)Eߢ޽ࠆ㧚ㆇേ㊂ߩᐔဋ p[ ߪ
f

p[

§

³ \ [ ¨© i=n

fd\ n [ ·
¸ d[
d[ ¹

0

(7.6.3)ߩࠃ߁ߦ࠯ࡠߦߥࠆߎߣߪ㧘ᑼ(7.5.35)߆ࠄ߽᣿ࠄ߆ߢ޽ࠆ㧚ౣ߮ᒰὼߩߎߣߥ߇ࠄ

px

E p[

0

(7.6.4)૏⟎߿ㆇേ㊂ߩᐔဋ୯߇࠯ࡠߢ޽ࠆߎߣ߇ಽ߆ߞߚߩߢ㧘ᰴߦߎࠇࠄ‛ℂ㊂ߩੑਸ਼ᐔဋࠍ᳞߼ߡߺࠃ߁㧚߹ߕ૏⟎ߦ㑐ߒߡߪ㧘ᑼ(7.5.34)ࠃࠅ

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 35

nn 1[\ n 1 [ [\ n 1 [2
2[ 2\ n [½°
n ­° n  1
n\ n2 [  \ n [ ¾®
2 ¯° 22¿°(7.6.5)½°
n  1 ­° n  1n2\n [ 
\ n2 [ ¾®
2 ¯° 22¿°ߣ޿߁㑐ଥᑼࠍ↪޿ࠆߣ㧘[2f³ \ [ [ \ [ d[
2

n

n

f

n n 1

2
2
1
n
2

(7.6.6)

ߣߥࠆߎߣ߇ಽ߆ࠆ㧚ᓥߞߡ

x2

1

E

[2

n

E

1
2

(7.6.7)‫ޣ‬ᵈᗧ‫ޤ‬㧦ߎߩࠃ߁ߥ⸘▚ࠍߔࠆᤨ㧘Section 7.4 ߩᦨᓟߦㅀߴߚ㧘normalization factor ߩ⇣ߥࠆ 2 ߟߩᑼ㧘\ n [ߢ޽ࠆ㧚 x 2

ߣ\ n x ߩߤߜࠄࠍ૶ߞߡ߽ࠃ޿߇㧘ߘߩ૶޿ᣇߦߪᵈᗧ߇ᔅⷐ

ߪ
f

x2

³ \ [ x \ [ d[2nn(7.6.7a)nx x 2\ n x dx

(7.6.7b)

f

߅ࠃ߮
f

x2

³\

f

ߩ޿ߕࠇߩᑼࠍ↪޿ߡ߽ࠃ޿ߎߣߪᒰὼߢ޽ࠆ㧚ߚߛߒ㧘ᑼ(7.6.7b)ࠍ↪޿ࠆ႐วߦߪ㧘x\ n xߦߟ޿ߡߩ recursion formula ߇ᔅⷐߢ޽ࠆ㧚ߎߩ႐ว㧘x 2\ n x

1E[ 2\ n xߦ⇐ᗧߔࠇ߫(7.6.7)߇ᓧࠄࠇࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 36ᰴߦㆇേ㊂ߦߟ޿ߡߩੑਸ਼ᐔဋࠍ᳞߼ࠆ㧚ㆇേ㊂ࠍ⴫ߔ operator ߦߟ޿ߡߪ

p[

§
w ·
¨ i = ¸
w[ ¹
©

2

2

= 2

w2
w[ 2

ߢ޽ࠆ߆ࠄ㧘ᑼ(7.5.35)ࠍ 2 ࿁↪޿ߡd 2\ n [= ³ \ n [
d[
d[ 2
f
f

p[

2

2

1·
§
=2 ¨ n  ¸
2¹©

(7.6.8)߹ߚߪ

1·§=2 E ¨ n  ¸(7.6.8a)2¹©
ߐߡ㧘ᑼ(5.4.13)ߩቯ⟵߆ࠄ㧘૏⟎ߩ root mean square deviation ߪᰴߩࠃ߁ߦߥࠆ㧚px 2

p[ 2 EG[[2  [

2

E

1
2(7.6.9)1·§= ¨n ¸2¹©

(7.6.10)nห᭽ߦ㧘ㆇേ㊂ߦߟ޿ߡߪ

G p[

p[ 2  p[

G[G p[

1·§=¨n  ¸
2¹
©

2

ߣߥࠅ

ߣ޿߁㑐ଥ߇ᓧࠄࠇࠆ㧚 n 

G[G p[ t(7.6.11)

1 1
t ߢ޽ࠆ߆ࠄ㧘
2 2

1
2(7.6.11')x ᐳᮡߦߟ޿ߡ߽ߎߩ⚿ᨐߪหߓߢ

G xG px1·§=¨n  ¸2¹©

(7.6.12)

G xG px t

12(7.6.12')ࠃࠅߣߥࠆ㧚ߔߥࠊߜ㧘Chapter 04 ߢቇ⠌ߒߚ Heisenberg uncertainty principle ߩᑼ (4.5.16)

1
2

G xG px t =߇⺞๺ᝄേሶߦߟ޿ߡ߽ᚑࠅ┙ߞߡ޿ࠆߎߣߪᒰὼߣ޿߃߫ᒰὼߢ޽ࠆ㧚
ᑼ(7.6.7)߅ࠃ߮ᑼ(7.6.8)߆ࠄ㧘⋥ߜߦㆇേࠛࡀ࡞ࠡ࡯߅ࠃ߮ࡐ࠹ࡦࠪࡖ࡞ࠛࡀ࡞ࠡ

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 37࡯ߩᐔဋ୯߇᳞߹ࠆ㧚ߔߥࠊߜ㧘kinetic energy ߦߟ޿ߡߪ

px 2
2m

1· 2 E§¨n ¸=
2 ¹ 2m©

1 k §1·=¨n  ¸
2 m ©
2¹

1·
§
hQ ¨ n  ¸
2¹
©
2
En
2

(7.6.13)߹ߚ㧘potential energy ߦߟ޿ߡߪ

1 2
kx
2

1·
§
k ¨n ¸
2¹
©
2E

1 k §
1·
=¨n  ¸
2 m ©
2¹

1·
§
hQ ¨ n  ¸
2¹
©
2
En
2

(7.6.14)ߔߥࠊߜ㧘ᐔဋߣߒߡ㧘ߤߩ㊂ሶᢙߦ߅޿ߡ߽ోࠛࡀ࡞ࠡ࡯ߪㆇേࠛࡀ࡞ࠡ࡯ߣࡐ࠹
ࡦࠪࡖ࡞ࠛࡀ࡞ࠡ࡯ߣߦ╬ߒߊ㈩ಽߐࠇߡ޿ࠆߎߣߦߥࠆ㧚

7.7 ⺞๺ᝄേሶࡕ࠺࡞ߣੑේሶಽሶߦ߅ߌࠆᝄേ
ൻቇߦ߅޿ߡ⺞๺ᝄേሶ߇㕖Ᏹߦ㊀ⷐߢ޽ࠆߎߣߩ 1 ߟߩℂ↱ߪ㧘ߎߩ┨ߩ౨㗡ߢ߽ㅀߴߚࠃ߁ߦಽሶࠍ᭴ᚑߒߡ޿ࠆේሶߩᝄേࠍ⠨߃ࠆࡕ࠺࡞ߣߒߡ⺞๺ᝄേሶࡕ
࠺࡞߇૶ࠊࠇࠆߎߣ߇ᄙ޿߆ࠄߢ޽ࠆ㧚ߎߩ▵ߢߪ㧘harmonic oscillator ߩᔕ↪ߩ৻଀ߣߒߡੑේሶಽሶߦ߅ߌࠆේሶߩᝄേߩ໧㗴ࠍ⚫੺ߔࠆ㧚ੑේሶಽሶࠍ᭴ᚑߒߡ޿ࠆ 2 ߟߩේሶߩ૏⟎ߪ㧘ߘߩᐔⴧ૏⟎ߦ߇ߞߜࠅߣ࿕ቯߐ
ࠇߡ޿ࠆࠊߌߢߪߥߊ㧘ᐔⴧ૏⟎ߩ߹ࠊࠅߢᝄേߒߡ޿ࠆ㧚ᄢ㔀ᛠߦ⠨߃ࠆߣ㧘޽ߚ
߆߽࿑ 7.7.1 ߦ␜ߔࠃ߁ߦ 2 ߟߩේሶ߇߫ߨߢ⚿߫ࠇߡ޿ࠆߺߚ޿ߥ߽ߩߣߺߥߔߎ
ߣ߇಴᧪ࠃ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 38

X=X1
࿑ 7.7.1

X=X2

2 ේሶಽሶߩᝄേࠍࡃࡀߣߒߡ⠨߃ࠆ㧚⃻ታߩੑේሶಽሶ߇ᧄᒰߦ Hooke ߩᴺೣߦᓥ߁߆ߤ߁߆⇼໧ߪ޽ࠆ߇㧘ߘ߁޿߁⹤ߪ
ᓟ࿁ߒߦߒߡ㧘ߣߦ߆ߊ࿑ 7.7.1 ߩࠃ߁ߦ㧘⥄ὼ㐳 L ߩ߫ߨߢ⾰㊂ m1 ߣ m2 ߩ 2 ߟߩ‛૕߇⚿߫ࠇߡ޿ࠆ♽ࠍ⠨߃ࠃ߁㧚ߒ߆ߒ㧘ߎߩ♽ߪ㧘࿑ 7.1.1 ߣߪ᣿ࠄ߆ߦ⇣ߥߞߡ޿ߡ㧘޿ߞߚ޿㧘੹߹ߢቇ⠌ߒߚߎߣ߇ㆡ↪಴᧪ࠆߩ߆ߤ߁߆ಽ߆ࠄߥ޿㧚ߘߎߢ߹ߕ࿑ 7.7.1 ߩ♽ߦߟ޿ߡ Newton ߩㆇേᣇ⒟ᑼࠍߚߡߡߺࠃ߁㧚m1

d 2 X1
dt 2

d2X2
dt 2
ߎߩ 2 ᑼࠃࠅ
m2

k X 2  X1  L(7.7.1)

k X 2  X 1  L(7.7.2)d 2 X 2 d 2 X1

dt 2
dt 2



d 2 X 2  X1
dt 2

§ 1
1 ·
k ¨ 
¸ X 2  X1  L© m1 m2 ¹kk
X 2  X1  L 
X 2  X1  L
m2
m1

߹ߚߪ

ߎߎߢ

1

P

1
1

m1 m2

ߢቯ⟵ߐࠇࠆ reduced mass P ࠍዉ౉ߒ㧘߆ߟ x(7.7.3)X 2  X 1  L ߣ⟎߈឵߃ࠆߣ㧘Newton

ߩㆇേᣇ⒟ᑼߪ

w2 x
 kx 0(7.7.4)wt 2
ߣߥࠆ㧚ߎߩᑼߪᑼ(7.1.2)ߩ m ࠍ P ߢ⟎߈឵߃ߚᒻߣߥߞߡ޿ࠆ㧚ߔߥࠊߜ㧘࿑ 7.7.1

Pߦ␜ߒߚੑ૕໧㗴ߪ㧘឵▚⾰㊂ࠍዉ౉ߔࠆߎߣߦࠃߞߡ࿑ 7.1.1 ߦ␜ߒߚ৻૕໧㗴ߣหߓߦᛒ߁ߎߣ߇಴᧪ࠆ㧚ᓥߞߡᧄ┨ߢ੹߹ߢዉ಴ߒߚోߡߩᑼ߇㧘m o P ߩ⟎߈឵䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 39߃ߢߘߩ߹߹ੑ૕໧㗴ߦㆡ↪಴᧪ࠆߎߣߦߥࠆ㧚ߢߪᰴߦ㧘⃻ታߩੑේሶಽሶ߇㧘ᧄᒰߦ Hooke ߩᴺೣߦᓥ߁߆ߤ߁߆⠨߃ߡߺࠆ㧚ੑේሶಽሶߥࠄߕߣ߽㧘࿑ 7.7.1 ߩ߫ߨߦߒߚߣߎࠈߢ㧘ᄌ૏ߩᄢ߈޿ߣߎࠈߢ Hookeߩᴺೣ߇ᚑࠅ┙ߚߥ޿ߎߣߪ᣿ࠄ߆ߢ޽ࠆ㧚߹ߕ߫ߨࠍߤࠎߤࠎ᛼ߒ❗߼ߡ޿ߌ߫㧘ߘߩ߁ߜߦ 2 ߟߩ⃿ߪ߱ߟ߆ࠅ㧘ߘࠇએ਄〒㔌ߪ❗߹ࠄߥ޿㧚potential ߪߘߎߢή㒢ᄢߦ⊒ᢔߔࠆߎߣߦߥࠆ㧚৻ᣇ㧘෻ኻߦ߫ߨࠍߤࠎߤࠎિ߫ߒߡ޿ߊߣ㧘ߘߩ߁ߜߦ
߫ߨߪಾࠇߡߒ߹߁ߢ޽ࠈ߁㧚⃻ታߩੑේሶಽሶߢ߽ૃߚࠃ߁ߥߎߣ߇⠨߃ࠄࠇࠆ㧚޽ࠆ〒㔌એਅߦߥࠆߣේሶ㑆ߩᢺജߩߚ߼ߦ potential ߪ᡼‛✢ࠃࠅߪߕߞߣᕆߦ਄᣹
ߔࠆߢ޽ࠈ߁㧚߹ߚ㧘ේሶ㑆〒㔌ࠍᄢ߈ߊߒߡ޿ߊߣ㧘⸃㔌ߒߡ 2 ୘ߩේሶߦߥߞߡ
ߒ߹߁ߩߢߘࠇએ਄ߪ potential ߪჇടߒߥ޿㧚ߘ߁޿߁ߎߣࠍ⠨߃ࠆߣ㧘potential ߪ߶߷࿑ 7.7.2 ߩὐ✢ߩࠃ߁ߥᒻߦߥࠆߎߣ߇⠨߃ࠄࠇࠆ㧚Vx
࿑ 7.7.2

࿑ߪ xharmonic potential㧔ታ✢㧕߅ࠃ߮ࠃࠅ⃻ታߦㄭ޿ potential㧔⎕✢㧕

X 2  X 1  L ࠍᮮゲߦߣߞߡ޿ࠆߚ߼ߦ x

0 ߩߣߎࠈ߇ᐔⴧ૏⟎ߦ⋧ᒰߔࠆ㧚

potential V x ߪ x ߦߟ޿ߡⶄ㔀ߥ㑐ᢙߣߥࠈ߁߇㧘 V x㐿ߒߡߺࠃ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 40

ࠍx0 ߩㄭㄝߢ Taylor ዷ23§ dV x ·1 §d V x ·1 §d V x ·23
V 0 ¨
x
x


¨
¸
¨
¸ x  ...
¸
2
3
2! © dx ¹ x 0
3! © dx ¹ x 0© dx ¹ x 0

V x(7.7.5)ߎߎߢ x§ dV x ·0 ߢ V x ߪᭂ୯ࠍߣࠆ߆ࠄ ¨¸ ߪ࠯ࡠߢ޽ࠆ㧚ᦝߦ㧘x© dx ¹ x 0⠨߃ࠆ㒢ࠅ x 3 ߿ x 4 ߥߤߩ㜞ᰴߩ㗄ߪήⷞߒߡ߽ࠃߐߘ߁ߢ޽ࠆ㧚 V 0

0 ߩㄭㄝࠍ

ࠍ potentialߩၮḰߣߒߡ࠯ࡠߣ߅ߊߣ potential ߪ

1 §d V x ·
2
¨
¸ x
2
2! © dx ¹ x 0
2

V x(7.7.6)§ d 2V x ·ߣ⴫ߔߎߣ߇಴᧪ࠆ㧚 ¨
ࠍ k ߣ߅ߌ߫¸2© dx ¹ x 0
1 2
kx
2

V x

(7.7.7)ߣ㧘ࠃߊ⍮ࠄࠇߚ᡼‛✢߇ᓧࠄࠇࠆ㧚࿑ 7.7.2 ߢߪߎߩ potential ࠍታ✢ߢ⴫ߒߡ޿ࠆ㧚ߔߥࠊߜ㧘⃻ታߩੑේሶಽሶߩᝄേ߇ Hooke ߩᴺೣߦᓥߞߡ޿ࠆߣߪᕁ߃ߥ޿߇㧘ᐔⴧ૏⟎ߩㄭㄝߢߪㄭૃ⊛ߦߘ߁߅ߌࠆߣ⠨߃ࠄࠇࠆ㧚⃻ታߩੑේሶಽሶ߇ㄭૃ⊛ߦ⺞๺ᝄേሶߢ⴫ߐࠇࠆߣߔࠆߣ㧘ㅢᏱߩ⁁ᘒ㧔଀߃߫Ᏹ᷷㧕ߢ㧘ᄢㇱಽߩಽሶߪ߽ߞߣ߽቟ቯߥ n 0 ߩ⁁ᘒ㧔ၮᐩ⁁ᘒ㧕ߦ޽ࠆ㧚ߘߩಽሶߪ㧘 n 1 ߣ n 0 ߣߩࠛࡀ࡞ࠡ࡯Ꮕ㧘ߔߥࠊߜ E1  E2 hQ ߩࠛࡀ࡞ࠡ࡯ࠍᜬߞߚ㔚⏛ᵄࠍๆ෼ߔࠆ㧚Q ߪಽሶߦ࿕᦭ߩ୯ߢ޽ࠅ㧘ၮḰᝄേᢙ㧔fundamental vibrationalfrequency㧕ߣ⸒ࠊࠇࠆ㧚Q ߪ㧘ᵄᢙQ ߦߒߡ߶߷ 103 cm 1 ߩࠝ࡯࠳࡯ߢ޽ࠆ㧚ߎߩ⒟ᐲߩࠛࡀ࡞ࠡ࡯ࠍᜬߟ㔚⏛ᵄߪ⿒ᄖ✢ߢ޽ࠅ㧘⿒ᄖ✢㗔ၞߩๆ෼ࠬࡍࠢ࠻࡞߆ࠄಽሶ
ߩᝄേ⁁ᘒߦ㑐ߔࠆ⍮⷗߇ᓧࠄࠇࠆ㧚ߔߥࠊߜᑼ(7.1.4)ࠃࠅ

Q

1
2S

k

P

߹ߚߪᵄᢙන૏ߢ

Q

1k

2S c Pߢ޽ࠆ߆ࠄ㧘Q ߹ߚߪQ ߩ᷹ቯ୯߆ࠄ㧘force constant k ࠍ᳞߼ࠆߎߣ߇಴᧪ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 41‫ڏ‬Exercise 7.7.1
H35Cl ߪQ 2886 cm 1 ߣ᷹ⷰߐࠇߡ޿ࠆ㧚force constant ࠍ᳞߼ࠃ㧚‫ޤ⸃ޣ‬㧦k

2S cQ P235
u 1.66 u 10 kg^2S u 3.00 u10 m/s u 2.8886 u10 m ` u 36
8

5

1

2

2

4.82 u 102 N/m޿ߊߟ߆ߩੑේሶಽሶߦߟ޿ߡߩၮḰᝄേᢙߣജߩቯᢙࠍ⴫ 7.7.1 ߦ␜ߔ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 42

⴫ 7.7.1

޿ߊߟ߆ߩੑේሶಽሶߦߟ޿ߡߩၮḰᝄേᢙߣജߩቯᢙ

ಽሶQ /cm 1

k / Nm 1

H2

4159

520

D2

2990

530

H35Cl

2886

482

H79Br

2559

385

H127I

2230

293

35Cl35Cl

556

320

79Br79Br

321

240

127I127I

213

170

16O16O

1556

1140

14N14N

2331

2260

12C16O

2143

1870

14N16O

1876

1550

23Na23Na

158

170

23Na35Cl

378

120

39K35Cl

278

80ߎߩ⴫ߢߪၮḰᝄേᢙ㧔fundamental vibrational frequency㧕ߩ୯ߣߒߡQ ࠍਈ߃ߡ޿ࠆ㧚ᝄേᢙߣ޿߁㒢ࠅ㧘 s 1 ߩᰴరࠍᜬߟQ ߢ޽ࠆߴ߈ߢ޽ࠅ㧘ᵄᢙQ ߢߪߥ޿ߪߕߢ޽ࠆ
߇㧘㐳޿㑆ߩ⠌ᘠߢ㧘ߒ߫ߒ߫ᝄേᢙQ ࠍ㧘

QcQ ߩᵄᢙ⴫␜ߢ⴫ߔ㧚න૏ߩᷙੂߪ

ᅢ߹ߒ޿ߎߣߢߪߥ޿߇㧘⃻ታߦߪߒ߫ߒ߫ߎ߁޿߁⴫ߒᣇࠍߔࠆߩߢᧄⓂߢ߽޽߃ߡߘߩ⠌ᘠߦߒߚ߇ߞߚ㧚⴫ 7.7.1 ߢಽ߆ࠆࠃ߁ߦ㧘ੑේሶಽሶߦ߅ߌࠆജߩቯᢙߪ߶߷ 102 N/m ߩᄢ߈ߐߢ

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 43޽ࠆ㧚ߎࠇࠍ⃻ታߦሽ࿷ߔࠆ߫ߨ⸘ࠅߣᲧߴߡߺࠆߩ߽㕙⊕߆ࠈ߁㧚1 kg㊀ 9.80 N ߢ޽ࠆߎߣࠍ⠨ᘦߔࠆߣ㧘 102 N/m ߣ޿߁ജߩቯᢙࠍᜬߟ߫ߨߪ㧘1 kg ߩ߅߽ࠅࠍษࠆ
ߒߚᤨߦ 10 cm િ߮ࠆ߫ߨ⸘ࠅߦ߶߷⋧ᒰߔࠆ㧚⻉ำߪߎߩ⚿ᨐࠍߺߡ㧘ಽሶߩ߫ߨ
ߪᕁߞߚࠃࠅᒝ޿ߣ⠨߃ࠆߢ޽ࠈ߁߆㧘߹ߚߪᒙ޿ߣ⠨߃ࠆߢ޽ࠈ߁߆㧫‫ڏ‬Problem 7.7.1
H79Br ߩၮḰᝄേᢙߪᵄᢙන૏ߢQ

2559 cm 1 ߣ᷹ⷰߐࠇߡ޿ࠆ㧚force constant ࠍ᳞߼ࠃ㧚ᦝߦᝄേߩ๟ᦼࠍ߽᳞߼ࠃ㧚‫ڏ‬Problem 7.7.2
79Br79Br ߩ force constant ߪ 240 Nm 1 ߢ޽ࠆ㧚ၮḰᝄേᢙࠍᵄᢙන૏ߢ᳞߼ࠃ㧚ᦝߦ࠯ࡠὐᝄേࠛࡀ࡞ࠡ࡯ࠍ߽᳞߼ࠃ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 7: Page 44

CHAPTER 08
࿁ォㆇേ
Schrödinger equation ߩ⸃ߩ଀ߣߒߡᧄ┨ߢߪ㧘޽ࠆቯὐߩ๟ࠅߩ࿁ォㆇേࠍขࠅ਄
ߍࠆ㧚ኈᤃߦᗐ௝಴᧪ࠆࠃ߁ߦ㧘ߎߩขࠅᛒ޿ߪ㧘ߔߢߦቇ⠌ߒߚⷺㆇേ㊂ߣኒធߥ߆߆ࠊࠅࠍᜬߟ㧚

8.1 ౞๟਄ࠍ⥄↱ㆇേߒߡ޿ࠆ☸ሶ߹ߕᆎ߼ߦ㧘߽ߞߣ߽◲නߥ଀ߣߒߡ㧘⾰㊂ m ߩ☸ሶ߇ඨᓘ r ߩ౞๟਄ࠍ㧘․ߦ
potential ࠍฃߌࠆߎߣߥߊ㧘⥄↱ㆇേࠍߒߡ޿ࠆ♽ࠍ⠨߃ࠆ㧚౞゠㆏ࠍ xy 㕙ౝߦߣߞߡ߽৻⥸ᕈߪᄬࠊࠇߥ޿㧚㧔࿑ 8.1.1㧕

zr

࿑ 8.1.1
౞๟਄ߩ☸ሶߩㆇേ

ㆇേߪ xy 㕙ౝߦ㒢ࠄࠇߡ޿ࠆߩߢ Hamiltonian ߪH=2 § w 2
w2 ·¨¸2m © wx 2 wy 2 ¹(8.1.1)ߣߥࠆ㧚ߎߩ႐วߩࠃ߁ߦ㧘 xy ᐔ㕙ౝߩ౞ㆇേࠍ⸥ㅀߔࠆߦߪ⋥ⷺᐳᮡࠃࠅ߽㧘ᐔ㕙ౝߩ polar coordinate ߇ࠃࠅଢ೑ߢ޽ࠆ㧚cylindrical coordinate ߢ߽ࠃ޿㧚෶㧘ߔߢߦ
Section 5.2 ߢ↪޿ߚ spherical polar coordinate ߢ TS2㧘r ৻ቯߣ⠨߃ߡ߽หߓߎߣߢ޽

ࠆ㧚ᑼ(5.2.6)㧙(5.2.8)ߦߎࠇࠄߩ᧦ઙࠍ౉ࠇࠆߎߣߦࠃࠅ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 1

sin I w
r wI

w
wx



w
wy

cos I w
r wI

(8.1.2)ࠍᓧࠆ㧚ᓥߞߡ
H=2 w 2
2mr 2 wI 2

(8.1.3)ߣߥࠆ㧚ߎߎߢ࿁ォߩᘠᕈ⢻₸㧔moment of inertia㧕ߣߒߡ

I mr 2ࠍቯ⟵ߔࠆ㧚ߘ߁ߔࠆߣ Hamiltonian ߪH=2 w 22 I wI 2(8.1.4)(8.1.5)ߣߥࠆ㧚ߔߥࠊߜ㧘Schrödinger equation ߪ
w 2\
wI 2



2IE
\
=2

ߣߥࠆ㧚ߎߩ⸃ߪᰴߩࠃ߁ߦߥࠆ㧚\ AeimI  Be imI(8.1.6)(8.1.7)

ߎߎߢ m ߪήᰴరߩ㊂ߢ޽ࠆ㧚㧔⸼㧦ᑼ(8.1.1)㧙ᑼ(8.1.4)ߢߪ m ߪ☸ሶߩ⾰㊂ࠍ␜ߒߚ߇㧘ᑼ(8.1.7)ߢߪ m ࠍ㊂ሶᢙߣߒߡ޿ࠆ㧚ߎߩࠃ߁ߦోߊ⇣ߥߞߚᗧ๧ߢ↪޿ࠄࠇߚ߇㧘ᄙಽᷙੂߩ߅ߘࠇߪߥ޿ߎߣߣᕁ߁㧚㧕੹ߩߣߎࠈ㧘 m ߩขࠅᓧࠆ୯ߦ૗ߩ೙㒢߽ߥ޿㧚ᰴߦႺ⇇᧦ઙࠍ⠨߃ࠃ߁㧚☸ሶ߇౞๟਄ࠍㆇേߒߡ޿ࠆ㒢ࠅ I ᐳᮡߦߪ೙㒢ߪߥ޿ߩߢ㧘ߤߎ߆ߢ wavefunction ߇࠯ࡠߦߥࠄߥߊߡߪ޿ߌߥ޿ߣ޿ߞߚ᧦ઙߪ૗߽ߥ޿㧚ߚߛߒ㧘\ ߇ well behaved ߢ޽ࠆߚ߼ߦ 1 ଔߢߥߌࠇ߫ߥࠄߥ޿ߣ޿߁᧦ઙ߇޽ࠆ㧚
ߔߥࠊߜ

\ I

\ I  2S

(8.1.8)1(8.1.9)m 0, r 1, r 2, ....

(8.1.10)ߎߩ᧦ઙࠃࠅ

e 2S im߹ߚߪߩࠃ߁ߦ೙㒢ߐࠇࠆ㧚ᓥߞߡ⸵ߐࠇࠆ energy ߪ

m2 =2
m 0, r 1, r 2, ....
2I
ߣߥࠆ㧚ߔߥࠊߜ energy ߪ㊂ሶൻߒߡ޿ࠆ㧚E䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 2(8.1.11)ᑼ(8.1.7)ߩ╙ 1 㗄⋡ߣ╙ 2 㗄⋡ߩᗧ๧ߪ㧘Section 6.1 ߢቇ⠌ߒߚਗㅴㆇേߩ႐วߣ
หߓࠃ߁ߦߒߡ⠨߃ࠆߎߣ߇಴᧪ࠆ㧚ᑼ(8.1.7)ߢ B

0 ߣ߅޿ߚ߽ߩࠍ\  㧘߹ߚ㧘A 0ߣ߅޿ߚ߽ߩࠍ\  ߣߒࠃ߁㧚

\

AeimI

(8.1.12)\

Be  imI(8.1.13)ߐߡ㧘 z ゲ๟ࠅߩⷺㆇേ㊂Ṷ▚ሶߪ§ ww ·Lz i= ¨ x  y ¸
wx ¹
© wy
ߢਈ߃ࠄࠇࠆ߇㧘ߎߩᑼߦᑼ(8.1.2)ࠍઍ౉ߔࠆߎߣߦࠃࠅ

Lz

§ w ·
i= ¨ ¸
© wI ¹(8.1.14)ߣߥࠆ㧚ߎߩ operator ࠍ\  ߅ࠃ߮\  ߦ૞↪ߔࠆ㧚

Lz\ 

m=\ 

Lz\ 

 m=\ (8.1.15)ߔߥࠊߜ㧘\  ߅ࠃ߮\  ߪ㧘ߘࠇߙࠇ㧘 z ゲ๟ࠅߩⷺㆇേ㊂߇ m= ߅ࠃ߮  m= ߩ⁁ᘒߦ⋧ᒰߔࠆ㧚⛘ኻ୯߇หߓߢ╓ภߩߺ߇෻ኻߢ޽ࠆ߆ࠄ㧘߅੕޿ߦㅒะ߈ߩ࿁ォߢ޽ࠆߣ⚿⺰ߐࠇࠆ㧚ߔߥࠊߜ\  ߇ฝ࿁ࠅߩ࿁ォߣߔࠆߣ\  ߪᏀ࿁ࠅߩ࿁ォߣ޿߁ߎߣߦ
ߥࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 3

AeimI\\

Be  imI

࿑ 8.1.2
ฝ࿁ࠅߩ࿁ォ߅ࠃ߮Ꮐ࿁ࠅߩ࿁ォᱷߐࠇߚ໧㗴ߪ normalization factor ߢ޽ࠆ㧚ߒ߆ߒ㧘ߎߩ႐วߦ߽㧘Section 6.1 ߩ⥄↱☸ሶߩ႐วߣหߓࠃ߁ߥ࿎㔍߇޽ࠆ㧚ߔߥࠊߜ㧘 I ߩขࠅᓧࠆ୯ߦߪᧄ᧪೙㒢ߪߥ޿ߪߕߥߩߢⓍಽߩ਄㒢ਅ㒢ࠍ᳿߼ࠆߎߣ߇಴᧪ߥ޿ߩߢ޽ࠆ㧚ߘߎߢᄙߊߩ႐
ว㧘ᒻᑼ⊛ߦ㧘\ ߪ 0 ߆ࠄ 2S ߹ߢߩ㑆ߢⷙᩰൻߐࠇߡ޿ࠆߣ⠨߃ࠆ㧚ߔߥࠊߜ㧘
2S³ \  *\  dI
0

2S

³ \ *\ dI


0



2S

A2 ³ d I

2S A2

1

2S B 21

0

2S

B ³ dI
2

0

ࠃࠅ

\
\

1 imI
(m 0, 1, 2, ...)e2S
1  imI(m 1, 2, ...)
e2S

(8.1.16)ߣ޿߁⚿ᨐ߇ᓧࠄࠇࠆ㧚m 0 એᄖߪ\  ߣ\  ߣߪหߓ energy ࠍᜬߟ㧘ߔߥࠊߜ㧘❗ㅌߒߡ޿ࠆ㧚ࠃߊ⍮ࠄࠇߡ޿ࠆࠃ߁ߦ㧘❗ㅌߒߡ޿ࠆ♽ߩ wavefunction ߩขࠅᣇߦߪછᗧᕈ߇޽ࠆࠊߌߢ㧘଀߃߫㧘ᑼ(8.1.16)ߩઍࠅߦ

\ cos

1
\  \ 
2

\ sin

1
 i \  \ 2

1

S

cos mIm 0, 1, 2, ...
(8.1.17)

1S

sin mIm䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 4

1, 2, ...

ߩ⚵ࠍߣߞߡ߽ࠃ޿㧚ᑼ(8.1.17)ߪ㧘 m z 0 ߩ❗ㅌߒߡ޿ࠆ 1 ⚵ߩ wavefunction หᔒ߇⋥੤ߒߡ޿ࠆߎߣߣ㧘౒ߦ real ߩ㑐ᢙߢ޽ࠆߣ޿߁ߎߣߩߚ߼ߦ㧘ℂ⸃ߒ߿ߔ޿⚵ߣ⸒޿ᓧࠆߢ޽ࠈ߁㧚wavefunction ߣ energy level ࠍ࿑ 8.1.3 ߦ␜ߔ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 5ᮮゲߪⷺᐲߢ I

࿑ 8.1.3 ࿁ォߩᵄേ㑐ᢙ
0 ߆ࠄ I 2S ߹ߢ㧚❑ゲߪ㧘ᵄേ㑐ᢙߩᒻߦߟ޿ߡߪᝄ᏷ߢ޽

ࠆ߇㧘หᤨߦߘࠇߙࠇߩࠛࡀ࡞ࠡ࡯ߦᲧ଀ߔࠆࠃ߁ߦࠪࡈ࠻ߐߖߡ޿ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 6ߎߩ࿑ࠍ㧘੗ᚭဳࡐ࠹ࡦࠪࡖ࡞ߩ႐วߩ࿑ 6.3.2 ߣᲧߴߡߺࠆߩ߽㕙⊕߆ࠈ߁㧚ߤߜ
ࠄ߽ energy ߪ㊂ሶᢙߩੑਸ਼ߦᲧ଀ߒߡ޿ࠆ㧚ߒ߆ߒᄢ߈ߥ㆑޿߇ 2 ߟ޽ࠆ㧚1 ߟߪ㧘੗ᚭဳࡐ࠹ࡦࠪࡖ࡞ߩ႐วߦߪ n

0 ߪߥߊ n 1 ߆ࠄᆎ߹ߞߚߩߦኻߒ㧘౞ㆇേߩߣ

߈ߦߪ m 0 ߆ࠄᆎ߹ߞߡ޿ࠆ㧚߽߁ 1 ߟߪ㧘੗ᚭဳࡐ࠹ࡦࠪࡖ࡞ߩ႐วߦߪ❗ㅌ߇ߥ߆ߞߚߩߦኻߒ㧘౞ㆇേߩ႐วߦߪ m 0 ߩ⁁ᘒࠍ㒰޿ߡߔߴߡ❗ㅌߒߡ޿ࠆ㧚‫ޤ⿷⵬ޣ‬
㧦ߎߎߢቇ⠌ߒߚ౞ㆇേߦ㑐ߔࠆ Schrödinger equation ߩ⸃ߪታߪ Section 6.1 ߩ৻ᰴరߩ⥄↱☸ሶߩᑧ㐳ߣߒߡ߽᳞߼ࠆߎߣ߇಴᧪ࠆ㧚ߔߥࠊߜ㧘Section 6.1 ߢⴕߞߚ৻ᰴ
రߩ x ᐳᮡࠍ㧘౞ㆇേߩᣇะߢ޽ࠆߣ⠨߃ࠆ㧚ᑼ(6.1.2)ߩࠃ߁ߦ

\ x

Aeikx  Be ikx

k2

2mE
=2

(8.1.18)

߇ᓧࠄࠇࠆ߇㧘ߎߎߢ\ x ߇ 1 ଔߣ޿߁᧦ઙ㧘ߔߥࠊߜ\ x\ x  2S r

(8.1.19)ߣ޿߁๟ᦼ᧦ઙࠍዉ౉ߔࠆ㧚ߎࠇ߆ࠄeik 2S r 1
ߔߥࠊߜ
kr integer(8.1.20)(8.1.21)ߣߥࠆ㧚(8.1.18)ߩ k ߩ⴫ᑼࠍઍ౉ߔࠆߣ

2 IE
=2

(integer) 2(8.1.22)ߣ㧘ᑼ(8.1.11)ߣหߓᑼ߇ᓧࠄࠇࠆ㧚ߎߎߢⴕߞߚࠃ߁ߥ㧘৻ᰴరߩᐳᮡߦ๟ᦼ᧦ઙࠍዉ౉ߔࠆߎߣߪ㧘⚿᥏ߩขࠅᛒ޿
ߥߤ㧘ᄙߊߩ႐วߦࠃߊߥߐࠇࠆℂ⺰⊛ߥขࠅᛒ޿ߢ޽ࠆ㧚‫ڏ‬Problem 8.1.1
ᑼ(8.1.11)ߢߪߤ߁ߒߡ m 0 ߇฽߹ࠇߡ޿ࠆ߆㧚෶ߤ߁ߒߡ m  0 ߩ୯߇฽߹ࠇߡ
޿ࠆ߆㧚㊂ሶᢙߦ㑐ߒߡᑼ(6.2.7)ߣߩ㆑޿ߪߤ߁ߒߡ⿠ࠆ߆㧚‫ڏ‬Problem 8.1.2
ᑼ(8.1.16)߅ࠃ߮ᑼ(8.1.17)ߩ m ߩขࠅᓧࠆ୯߇㧘ᑼ(8.1.11)ߣ⇣ߥࠆߩߪߤ߁޿߁ࠊ
ߌ߆㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 78.2 ⃿ߩ⴫㕙ࠍ⥄↱ㆇേߒߡ޿ࠆ☸ሶ೨▵ߢቇ⠌ߒߚੑᰴరߩᐔ㕙ౝㆇേࠍਃᰴర߳᜛ᒛߒߡߺࠃ߁㧚ߎߎߢߪ㧘⾰㊂ mߩ☸ሶ߇ඨᓘ r ߩ⃿ߩ⴫㕙ࠍ⥄↱ㆇേߒߡ޿ࠆ႐วࠍ⠨߃ࠆ㧚Hamiltonian ߪᒰὼߩߎ
ߣߥ߇ࠄ㧘
H



w2
w2 ·
=2 § w 2¨ 2 2 2¸
2m © wx wy wz ¹



=2 2

2m

ߢ޽ࠆ㧚ౣ߮ spherical polar coordinate ࠍ↪޿ࠃ߁㧚ᑼ(5.2.6)㧙(5.2.8)ߢ
᳞߼ߚ߇㧘ߎࠇ߆ࠄ

(8.2.1)w
w
w,,
ࠍwx wy wz

w2
w2
w2
,
,
ࠍ㧘ߘߒߡᦝߦߘߩ๺ࠍߣߞߡ  2 ࠍ᳞߼ࠆߎߣߪ㧘wx 2 wy 2 wz 2

tedious ߢߪ޽ࠈ߁߇㔍ߒ޿ߎߣߢߪߥ޿㧚⚿ᨐߪᰴߩࠃ߁ߦߥࠆ㧚

1 w § 2 w ·
1
w §
w ·
1
w2
¨r
¸
¨ sin T¸
wT ¹ r 2 sin 2 T wI 2
r 2 wr © wr ¹ r 2 sin T wT ©

2

(8.2.2)ᑼ(8.2.2)ߪ spherical polar coordinate ߦ߅ߌࠆ  2 ߩ৻⥸ᑼࠍ␜ߒߚ߽ߩߢ㧘ߎࠇ߆ࠄవ㧘
➅ࠅ㄰ߒ↪޿ࠆߎߣߦߥࠈ߁㧚⃻࿷ᛒߞߡ޿ࠆ໧㗴ߩ႐วߦߪ㧘r ߇৻ቯߢ޽ࠆ߆ࠄ Hamiltonian (8.2.1)ߪᰴߩࠃ߁
ߦߥࠆ㧚
H=2 1 w §w ·1 w2
¨ sin T¸
2 I sin T wT ©
wT ¹ sin 2 T wI 2

(8.2.3)ߚߛߒ㧘ᑼ(8.1.4)ߣห᭽ߎߎߢ߽ᘠᕈ⢻₸I mr 2(8.2.4)ࠍዉ౉ߒߚ㧚ᑼ(8.2.3)ߩ Hamiltonian ߪ㧘ታߪᚒ‫ࠆ޽ߢߩ߽ߩߺߓߥ߅ߪߡߞߣߦޘ‬㧚
ᑼ(5.2.10)ࠍෳᾖߔࠆߣ
L2(8.2.5)2Iߣ޿߁ߎߣ߇ಽ߆ࠆ㧚ᚒ‫ ߦߢߔߪޘ‬5 ┨ߢ㧘 L2 ߦኻߔࠆ eigenfunction ߣ eigenvalue ࠍ᳞߼ߡ޿ߚࠊߌߢ޽ࠆ㧚ߔߥࠊߜ㧘ᑼ(5.3.29)ߢ␜ߒߚࠃ߁ߦ

H

L2Yl ,ml l  1 = 2Yl ,mߢ޽ࠆ㧚ߎࠇ߆ࠄ⋥ߜߦL2=2
Yl ,m l l  1
Yl ,m(8.2.6)2I
2I߇ᓧࠄࠇࠆ㧚ߔߥࠊߜ㧘⃿ߩ⴫㕙਄ߩ⥄↱☸ሶߦߟ޿ߡߩ eigenfunction ߪ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 8

Yl , m T , I

4l , m T ) I
4l , m T

1 imI
e
2S

(8.2.7)ߢ޽ࠅ㧘eigenvalue ߪ=2
l l 1
l 0, 1, 2, ...
(8.2.8)2Iߢ޽ࠆ㧚eigenvalue ߪ A ߩߺߩ㑐ᢙߢ m ߦߪࠃࠄߥ޿㧚1 ߟߩ A ߦߟ޿ߡ m l ߆ࠄElml ߹ߢ 2l  1 ୘ߩ eigenfunction Yl , m ߇޽ࠅ㧘ߘࠇࠄ߇ߔߴߡหߓ energy ࠍᜬߞߡ޿

ࠆ㧚ߔߥࠊߜ Yl ,m ߪ 2l  1 ㊀ߦ❗ㅌߒߡ޿ࠆ㧚‫⵬ޣ‬ㆮ‫ޤ‬㧦  2 ߩᭂᐳᮡ⴫␜

(8.2.2)ߩዉ಴ߦߟ޿ߡߪ‫ޟ‬tedious ߢߪ޽ࠈ߁߇㔍ߒ޿ߎߣߢߪߥ޿‫߇ߚ޿ᦠߣޠ‬㧘ߤ߁ߒߚࠄ޿޿߆ಽ߆ࠄߥ޿ߣ޿߁⾰໧ࠍߒ߫ߒ߫ฃߌࠆߩߢ㧘એਅߦߘߩዉ಴ࠍ⹦
ww
ߒߊ⸥ߒߡ߅ߊ㧚 ߅ࠃ߮ ߪ㧘ᑼ(5.2.6)߅ࠃ߮ᑼ(5.2.7)߆ࠄಽ߆ࠆࠃ߁ߦ 3 ߟߩ㗄wxwyߩ๺㧘߹ߚ㧘

w2w
ߪᑼ(5.2.8)߆ࠄಽ߆ࠆࠃ߁ߦ 2 ߟߩ㗄ߩ๺ߣߥߞߡ޿ࠆ㧚ᓥߞߡ 2
wx
wz

w2
w2
߅ࠃ߮ 2 ߪ 33 ߩ 9 ߟߩ㗄ߩ๺ߢ㧘 2 ߪ 22 ߩ 4 ߟߩ㗄ߩ๺ߢ⴫ࠊߐࠇࠆ㧚એwywzਅߩᑼߢߪ㧘ಽ߆ࠅ߿ߔ޿ࠃ߁ߦ㧘ߎࠇࠄ 9 ߟ޽ࠆ޿ߪ 4 ߟߩ㗄ࠍߘࠇߙࠇⴕࠍᄌ߃
ߡ␜ߒߡ޿ࠆ㧚w2wߥ߅㧘଀߃߫㧘 2 ߪ ߩੑਸ਼ߢߪߥ޿ߎߣߪ൩⺰ߢ޽ࠆ㧚޿ߟ߽ⴕߞߡ޿ࠆࠃ߁ߦ㧘
wx
wx
w\
w
w 2\
ߦ ࠍ૞↪ߔࠆߎߣߦࠃߞߡ 2 ࠍ᳞߼ࠆߩ߇ᏱᅃᚻᲑߢ޽ࠆ㧚
wx
wx
wx

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 9

w 2\
wx 2

sin T cos I

w ª
w\ º
sin T cos I
«
wr ¬
wr »¼

 sin T cos I

w ª1
w\ º
cos T cos I«
wr ¬ rwT »¼

 sin T cos I

w ª 1 sin I w\ ºwr «¬ r sin T wI »¼

w ª
w\ º
1
 cos T cos I
sin T cos I
«
r
wT ¬
wr »¼
1
w ª1
w\ º
cos T cos I cos T cos I«rwT ¬ rwT »¼
1
w ª 1 sin I w\ º
 cos T cos IrwT «¬ r sin T wI »¼

w\ º
1 sin I w ª
sin T cos I«r sin T wI ¬
wr »¼



w\ º
1 sin I w ª 1
cos T cos I«r sin T wI ¬ r
wT »¼



1 sin I w ª 1 sin I w\ ºr sin T wI «¬ r sin T wI »¼
sin 2 T cos 2 I

w 2\
wr 2

 sin T cos T cos 2 I sin I cos I

w § 1 w\ ·¨¸
wr © r wT ¹

w § 1 w\ ·¨¸
wr © r wI ¹

1
w §
w\ ·
 cos T cos 2 I¨ sin T
¸
r
wT ©
wr ¹


1
w §
w\ ·
cos T cos 2 I¨ cos T
¸
2
r
wT ©
wT ¹

1
w § 1 w\ ·
 2 cos T cos I sin I¨¸rwT © sin T wI ¹

1 sin I
w §
w\ ·
sin T
¨ cos I¸r sin T
wI ©
wr ¹

w §
w\ ·
1 sin Icos T
 2
¨ cos I¸
wI ©
wT ¹r sin T


1 sin I 1 w §w\ ·¨ sin I¸
2r sin T sin T wI ©
wI ¹䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 10

w2
\
wr 2
§ 1 w 1 w2 · sin T cos T cos 2 I ¨  2
¸\© r wT r wr wT ¹
sin 2 T cos 2 I§ 1 w 1 w2 · sin T cos I ¨  2
¸\© r wI r wr wI ¹
§
1
w
w2 ·
 cos T cos 2 I ¨ cos T  sin T¸\rwr
wr wT ¹
©


1
w
w2 ·
2 §
cos
T
cos
I
sin
T
cos
T


¨
¸\
r2
wT
wT 2 ¹©

§ cos T w
11
w2 ·
 2 cos T cos I sin I ¨  2

¸\
I
w
T
I
T
T
sin
sin
r
w
w
©
¹


§
1 sin Iw
w2 ·sin T ¨  sin I  cos I¸\r sin T
wr
wr wI ¹©

§
1 sin Iw
w2 ·cos T ¨  sin I
 2
 cos I¸\r sin T
wT
wTwI ¹©


1 sin 2 I w 2\
r 2 sin 2 T wI 2

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 11

w 2\
wy 2

sin T sin I

w ª
w\ º
sin T sin I
«
wr ¬
wr »¼

 sin T sin I

w ª1
w\ º
cos T sin I«
wr ¬ rwT »¼

 sin T sin I

w ª 1 cos I w\ ºwr «¬ r sin T wI »¼

1
w ª
w\ º
sin
sin
T
I
 cos T sin I
r
wT «¬
wr »¼
1
w ª1
w\ º
cos T sin I cos T sin I«rwT ¬ rwT »¼
1
w ª 1 cos I w\ º
 cos T sin IrwT «¬ r sin T wI »¼


1 cos I w ª
w\ º
sin
T
sin
I
r sin T wI «¬
wr »¼



1 cos I w ª 1
w\ º
cos T sin I«r sin T wI ¬ r
wT »¼



1 cos I w ª 1 cos I w\ ºr sin T wI «¬ r sin T wI »¼
w 2\
sin T sin I 2
wr22

 sin T cos T sin 2 I sin I cos I

w ª 1 w\ ºwr «¬ r wT »¼

w ª 1 w\ ºwr «¬ r wI »¼

1
w ª
w\ º
sin T
 cos T sin 2 I
«
r
wT ¬
wr »¼


1
w ª
w\ º
cos T sin 2 I
cos T
2
«
r
wT ¬
wT »¼



1
w ª 1 w\ ºcos T sin I cos I
2rwT «¬ sin T wI »¼



1 cos I
w ª
w\ º
sin T
sin I
«
wI ¬
wr »¼
r sin T



w ª
w\ º
1 cos I
cos T
sin I
2
«
wI ¬
wT »¼
r sin T



1 1
w §
w\ ·
cos I
¨ cos I¸2
2
r sin TwI ©
wI ¹䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 12

w2
\
wr 2
ª 1 w 1 w2 º sin T cos T sin 2 I «  2
»\¬ r wT r wr wT ¼
sin 2 T sin 2 Iª 1 w 1 w2 º sin I cos I «  2
»\¬ r wI r wr wI ¼
ª
1
w
w2 º
\
 cos T sin 2 I « cos T  sin Trwr
wr wT »¼
¬


ª
1
w
w2 º
2
cos
T
sin
I
sin
T
cos
T


«
»\
r2
wT
wT 2 ¼¬

ª cos T w
1
1
w2 º
cos
T
sin
I
cos
I


« sin 2 T wI sin T wTwI »\
r2
¬
¼



ª
1 cos Iw
w2 º
sin T «cos I  sin T\
wr
wr wI »¼
r sin T¬ª
1 cos Iw
w2 º 2 sin Icos T «cos I\r sin T
wT
wTwI »¼
¬


w 2\
wz 2

w2
1 1
2
cos
I
\
wI 2
r 2 sin 2 T

cos T

w §
w\ ·
¨ cos T
¸
wr ©
wr ¹

 cos Tw §1w\ ·¨ sin T¸
wr © r
wT ¹

1
w §
w\ ·
 sin T¨ cos T
¸
r
wT ©
wr ¹
1
w §1
w\ ·
 sin T¨ sin T¸rwT © r
wT ¹
w 2\
cos
wr 2
2

 sin T cos Tw § 1 w\ ·¨¸
wr © r wT ¹

1
w §
w\ ·
 sin T¨ cos T
¸
r
wT ©
wr ¹


1
w §
w\ ·
sin T
¨ sin T
¸
2
r
wT ©
wT ¹

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 13

w2
\
wr 2
§ 1 w 1 w2 · sin T cos T ¨  2
¸\© r wT r wr wT ¹
cos 2 T§1w
w2 ·
 sin T ¨  sin T  cos T¸\rwr
wr wT ¹
©
§
1
w
w2 ·
 2 sin T ¨ cos T sin T 2 ¸\
r
wT
wT ¹
©
એ਄ࠃࠅ‫ޔ‬

w2
w2
w2


wx 2 wy 2 wz 2

w2
1 w2
1 1 w 2 2 w 1 cos T wwr 2 r 2 wT 2 r 2 sin 2 T wI 2 r wr r 2 sin T wT
1 w § 2 w ·
1 §
1
w ·
w2
r

sin
T

¨
¸
¨
¸
r 2 wr © wr ¹ r 2 sin T ©
wT ¹ r 2 sin 2 T wI 2

8.3 ೰૕࿁ォሶ೰૕࿁ォሶߦߟ޿ߡߩ Schrödinger equation ߩ⸃ߪੑේሶಽሶߩࠬࡍࠢ࠻࡞ߩ⸃㉼ߦ㊀ⷐߢ޽ࠆ㧚ߎߎߢ޿߁೰૕࿁ォሶߣߪ㧘2 ߟߩ⃿߇િ߮❗ߺߒߥ޿᫔ߢ⚿߫ࠇߚ㧘
㊀㊂᜼ߍߩㆬᚻ߇↪޿ࠆࡃ࡯ࡌ࡞ߺߚ޿ߥ߽ߩߢ޽ࠆ㧚ߎࠇ߇ੑේሶಽሶߩࠃ޿ࡕ࠺
࡞ߦߥߞߡ޿ࠆ߆ߤ߁߆ߪ໧㗴ߢ޽ࠆ㧚೨ߩ┨ߢߪ㧘ੑේሶಽሶࠍ㧘࿑ 7.7.1 ߩࠃ߁ߦ 2 ߟߩේሶ߇߫ߨߢ⚿߫ࠇߚ߽ߩߣ⠨߃ߚ߇㧘ߘ߁ߔࠆߣ㧘࿁ォߔࠆߎߣߦࠃߞߡ
㆙ᔃജߦࠃࠅ߫ߨߪᄙዋિ߮ࠆࠊߌߢ޽ࠆ㧚ߒ߆ߒ㧘ߘߩિ߮ߪේሶ㑆〒㔌ߦᲧߴߡዊߐ޿ߣ⠨߃ࠄࠇࠆߩߢߎߎߢߪ㧘ߣࠅ޽߃ߕ㧘2 ߟߩේሶ㑆ߩ〒㔌ߪ࿁ォߦࠃߞߡᄌࠄߥ޿ߣߒߡขࠅᛒ߁㧚ዋߥߊߣ߽╙৻ㄭૃߣߒߡណ↪ߔࠆࡕ࠺࡞ߣߒߡߪᗧ๧ߩ
ߥ޿ߎߣߢߪ޽ࠆ߹޿㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 14

㊀ᔃ

㫉㪉
㫄㪉

㫉㪈
㫄㪈

࿑ 8.3.1 2 ේሶಽሶߩ࿁ォේሶ 1 ߩ⾰㊂ࠍ m1 㧘ߘߩᐳᮡࠍ x1 , y1 , z1 ߣߒࠃ߁㧚ห᭽ߦේሶ 2 ߩ⾰㊂ࠍ m2 㧘ᐳᮡࠍ x2 , y2 , z2 ߣߔࠆ㧚੹㧘potential ߪ⠨߃ߡ޿ߥ޿߆ࠄ㧘Hamiltonian ߪᒰὼ

H



w2
w 2 · =2 § w 2
w2
w2 ·=2 § w 2¨ 2  2  2 ¸
¨ 2¸
2m1 © wx1 wy1 wz1 ¹ 2m2 © wx2 wy2 2 wz2 2 ¹(8.3.1)ߣߥࠆ㧚ߎߎߢ㧘㊀ᔃߩᐳᮡࠍ X , Y , Z ߣߒ㧘߹ߚ㧘ේሶ 2 ߆ࠄ⷗ߚේሶ 1 ߩ⋧ኻᐳᮡࠍ x, y, z ߣߔࠆ㧚એᓟ㧘ᑼ(8.3.1)ߩ Hamiltonian ࠍ X , Y , Z ߅ࠃ߮ x, y, z ࠍ↪޿ߡ⴫ߔߎߣࠍ⹜ߺࠆ㧚 x1 , y1 , z1 ; x2 , y2 , z2

ᐳᮡߣ X , Y , Z ; x, y, z ᐳᮡߩ㑆ߦߪᰴߩ㑐ଥ߇޽

ࠆ㧚

X
Y
Z

m1
m
x1  2 x2mm
m1m
y1  2 y2mm
m1m
z1  2 z2mm(8.3.2)

m1  m2(8.3.3)ߚߛߒm䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 15ߢ޽ࠆ㧚ᦝߦ
x x1  x2

yy1  y2zz1  z2(8.3.4)ߎࠇࠄߩ㑐ଥࠃࠅ
w
wX w
wx w

wx1 wx1 wX wx1 wx

w
wx2

wX w
wx w

wx2 wX wx2 wx

m1 w
w

m wX wx(8.3.5)m2 w
w
m wX wx(8.3.6)

ห᭽ߥᑼ߇ y ߅ࠃ߮ z ᚑಽߦߟ޿ߡ߽ᓧࠄࠇࠆ㧚ᓥߞߡᑼ(8.3.1)ߩ x ᚑಽߦߟ޿ߡߪ
=2 w 2=2 w 2
2m1 wx12 2m2 wx2 22

= 2 § m1 w
= 2 § m2 w
w ·
w ·



¨
¸
¨
¸
2m1 © m wx wX ¹ 2m2 © m wx wX ¹



=2 w 2=2 § 1
1 · w2¨¸
2m wX 2 2 © m1 m2 ¹ wx 2

2

(8.3.7)

ߣߥࠆ㧚 y ߅ࠃ߮ z ᚑಽߦߟ޿ߡ߽ห᭽ߥ㑐ଥᑼ߇ᓧࠄࠇࠆ㧚ᓥߞߡ㧘reduced mass
1 1
1
(8.3.8)

P m1 m2ࠍዉ౉ߔࠆߣ㧘Hamiltonian(8.3.1)ߪᰴߩࠃ߁ߦߥࠆ㧚H

=2 § w 2
w2
w 2 · =2 § w 2
w2
w2 ·¨ 2  2  2 ¸
¨ 2 2 2¸
2m © wX
wY
wZ ¹ 2 P © wx wy wz ¹

(8.3.9)

ߎߩ Hamiltonian ߩ╙ 1 㗄⋡ߪ㧘㊀ᔃߩㆇേࠍ⴫ߔ㗄ߢ޽ࠆ㧚⃻࿷㧘ᚒ‫ߪޘ‬ੑේሶಽሶో૕ߩਗㅴㆇേߦߪ⥝๧߇ߥ޿ߩߢ㧘㊀ᔃߪⓨ㑆਄޽ࠆὐߦ࿕ቯߐࠇߡ޿ߡേ߆ߥ
޿ߣ⠨߃ࠃ߁㧚ߘ߁ߔࠆߣ Hamiltonian ߪH=2 § w 2
w2
w2 ·¨ 2 2 2¸
2 P © wx wy wz ¹

(8.3.10)ߣߥࠆ㧚⚿ዪ㧘㊀ᔃߩ๟ࠅࠍ࿁ߞߡ޿ࠆ 2 ߟߩ☸ሶߩ໧㗴ߪ㧘឵▚⾰㊂ P ࠍᜬߞߚ 1୘ߩ☸ሶߩ࿁ォߩ໧㗴ߣߒߡขࠅᛒ߁ߎߣ߇಴᧪ࠆ㧚ߎߩࠃ߁ߥℂ↱ߢ㧘೰૕࿁ォ૕ߩ໧㗴ߪ㧘Section 8.2 ߣోߊห╬ߦᛒ߁ߎߣ߇಴᧪ࠆࠊߌߢ޽ࠆ㧚ߔߥࠊߜ㧘ᘠᕈ⢻₸ߩ⸘▚ߦ↪޿ࠆ⾰㊂߇឵▚⾰㊂ߢ޽ࠆߣ޿߁ὐߛߌࠍ⠨ᘦߒߡ㧘eigenfunction,
eigenvalue ߪ㧘ᑼ(8.2.7)߅ࠃ߮ᑼ(8.2.8)ߢਈ߃ࠄࠇࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 16‫ ڏ‬Problem 8.3.1

R
mA

mB

࿑ߩࠃ߁ߥ 2 ේሶಽሶߩᘠᕈࡕ࡯ࡔࡦ࠻ߪᰴߩࠃ߁ߥᑼߢ⴫ߐࠇࠆߎߣࠍ␜ߖ㧚
mA mB
I
R2
mA  mB‫ڏ‬Problem 8.3.2
R'

R
mA

mB

mC

࿑ߩࠃ߁ߥ⋥✢ 3 ේሶಽሶߩᘠᕈࡕ࡯ࡔࡦ࠻ߪᰴߩࠃ߁ߥᑼߢ⴫ߐࠇࠆߎߣࠍ␜
ߖI

m R  mC R 'mA R  mC R '  A
mA  mB  mC
2

2

2

8.4 ੑේሶಽሶߩ࿁ォࠬࡍࠢ࠻࡞
೨▵ߢቇ⠌ߒߚߎߣࠍ߽ߣߦ㧘ੑේሶಽሶߩ࿁ォࠬࡍࠢ࠻࡞ߦߟ޿ߡ㧘߶ࠎߩዋߒ
ߛߌㅀߴߡ߅ߊ㧚೰૕࿁ォ૕ߩ energy level ߪ=2l l 1(8.4.1)
2Iߢਈ߃ࠄࠇࠆߎߣߪߔߢߦቇ⠌ߒߚߣ߅ࠅߢ޽ࠆ㧚ࠬࡍࠢ࠻࡞ߩߎߣߪ੹߹ߢోὼቇ
⠌ߒߡ޿ߥ޿߇㧘 l ߇ 1 ߟ⇣ߥࠆ⁁ᘒ㑆ߩㆫ⒖߇น⢻ߢ޽ࠆߣ޿߁ߎߣࠍ޿ߕࠇቇ⠌ߔࠆߎߣߦߥࠆ㧚ߎߎߢߪߘߩ⚿⺰ߩߺࠍవขࠅߒߡ⠨߃ࠆ㧚㔚⏛ᵄߩๆ෼ߩ႐วߦ
ߪ㊂ሶᢙ߇ l ߩ⁁ᘒ߆ࠄ l  1 ߩ⁁ᘒ߳ߩㆫ⒖߇⿠ࠆ㧚ߎߩᤨߩㆫ⒖ࠛࡀ࡞ࠡ࡯ߪEA'E

EA 1  EA

=2
^ l 1 l  2  l l 1 `
2I
=2
l 1
I

(8.4.2)ੑේሶಽሶߩ឵▚⾰㊂ߪ߶߷ 1025  1026 kg 㧘⚿ว〒㔌ߪ߶߷ 1010 m ߢ޽ࠆ㧚ᓥߞߡ㧘ᘠᕈ⢻₸ I ߪ߶߷ 1045  1046 kgm 2 ߣߥࠆ㧚ߎߩ୯ࠍ↪޿ࠆߣ㧘ᑼ(8.4.2)ߩㆫ⒖ࠛࡀ࡞䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 17ࠡ࡯ߪࡑࠗࠢࡠᵄߩࠛࡀ࡞ࠡ࡯ߦ⋧ᒰߔࠆߎߣ߇ಽ߆ࠆ㧚ߘߎߢ㧘ࡑࠗࠢࡠᵄ㗔ၞߢߩಽሶߩ࿁ォࠬࡍࠢ࠻࡞ߪߒ߫ߒ߫ࡑࠗࠢࡠᵄಽశߣ⸒ࠊࠇࠆ㧚⃻࿷㧘ಽሶߩ᭴ㅧࠍ
᳿ቯߔࠆ߽ߞߣ߽ᱜ⏕ߥᣇᴺߩ 1 ߟߢ޽ࠆ㧚ᑼ(8.4.2)ߩ 'E ࠍ㧘ๆ෼ߔࠆ㔚⏛ᵄߩᝄേᢙQ ߢ⴫ߔߣ

Q

'E
h

h

4S 2 Il 1(8.4.3)ߣߥࠆ㧚ߒ߫ߒ߫⠌ᘠߣߒߡ㧘

B

h
8S2 I

(8.4.4)ߣ߅߈㧘ᑼ(8.4.3)ࠍ

Q

2B l  1(8.4.5)ߩࠃ߁ߦ⴫ߔ㧚ᑼ(8.4.4)ߩ B ߪ rotational constant ߣ๭߫ࠇࠆ㧚Ᏹ᷷ㄭㄝߩ᷷ᐲߢߪ㧘ಽሶߪ l 0 㧘 l 1 㧘 l 2 㧘.㧚
㧚ߣ߆ߥࠅᐢ޿ l ߩ▸࿐ߦࠊߚ§ E ·ߞߡሽ࿷ߔࠆ㧚㧔Boltzmann ಽᏓೣߦࠃࠅ㧘 l ߩ⁁ᘒߩಽᏓߪ 2l  1 exp ¨  l ¸ ߦᲧ଀© kT ¹ߔࠆ㧕ߎ߁޿߁⁁ᘒߢࡑࠗࠢࡠᵄߩๆ෼ࠬࡍࠢ࠻࡞ࠍ᷹ቯߔࠆߣ㧘 Q 2 B, 4 B, 6 B, ...
ߣ㧘 2B ߩ㑆㓒ߢๆ෼✢߇ߺࠄࠇࠆ㧚ߘߩ޽ࠅߐ߹ࠍ࿑ 8.4.1 ߦᮨᑼ⊛ߦ␜ߔ㧚ߎߩࠃ߁ߥࠬࡍࠢ࠻࡞㧔࿁ォ⁁ᘒ㑆ߩࠬࡍࠢ࠻࡞ߢ޽ࠆ߆ࠄ࿁ォࠬࡍࠢ࠻࡞ߣ⸒ࠊࠇࠆ㧚㧕ࠍ⸃ᨆߒߡ㧘ಽሶߩ⚿ว〒㔌ࠍ᳞߼ࠆߎߣ߇಴᧪ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 18l =5'E=10B
l =4
'E=8B
l =3
'E=6B
l =2

'E=4B

l =1
l =0

'E=2B
࿑ 8.4.1 ࿁ォࠬࡍࠢ࠻࡞

‫ڏ‬Exercise 8.4.1
H35Cl ߪ 6.26 u1011 Hz ߩ╬㑆㓒ߢࡑࠗࠢࡠᵄߩๆ෼߇᷹ⷰߐࠇߡ޿ࠆ㧚H ߣ Cl ߣ
ߩ〒㔌ࠍ᳞߼ࠃ㧚
‫ޤ⸃ޣ‬
㧦

2B

h
4S 2 I6.626 u 1011 Hzߎࠇ߆ࠄ I ࠍ᳞߼ࠆ㧚I

6.626 u1034 Js
4S2 u 6.26 u 1011 s 1

2.68 u 1047 kg m 2

H35Cl ߩ reduced mass ߪ
1.00 u 35.00
u1.66 u 1027 kg
P
36.00
ᓥߞߡ

I

Pr 2

ࠃࠅ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 19

r2.68 u1047 kg m 2
1.61u1027 kg

1.29 u1010 m 129 pm‫ڏ‬Problem 8.4.1
H79Br ߩ㆙⿒ᄖ✢ࠬࡍࠢ࠻࡞ߦ߅޿ߡ㧘ᵄᢙන૏ߢ 16.72 cm 1 ߩ╬㑆㓒ߩๆ෼ߩ೉
߇޽ࠆ㧚moment of inertia ߅ࠃ߮⚿ว〒㔌ࠍ᳞߼ࠃ㧚‫ڏ‬Problem 8.4.2
12C16O ߩ l 0 ߆ࠄ l 1 ߳ߩㆫ⒖ߪ 1.153 u105 MHz ߦ޽ࠆ㧚CO ߩ⚿ว〒㔌ࠍ᳞߼ࠃ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟㩷

Lecture Note on Quantum Chemistry by T. Azumi
Chapter 8: Page 20

CHAPTER 09᳓⚛ේሶߎߩ┨ߢߪ㧘߽ߞߣ߽◲නߥൻቇ⒳ߢ޽ࠆ᳓⚛ේሶߦߟ޿ߡ Schrödinger equationࠍ⸃޿ߡߺࠆ㧚⸒߁߹ߢ߽ߥ޿ߎߣߥ߇ࠄ㧘᳓⚛ේሶߪේሶᩭ 1 ୘ߣ㔚ሶ 1 ୘ߩߺ߆ࠄ᭴ᚑߐࠇ㧘ߎࠇࠃࠅ߽◲නߥේሶߪሽ࿷ߒߥ޿㧚ಽሶ߿ේሶߩᕈ⾰ߦᩰ೎ߩ㑐ᔃࠍᜬߟൻቇ⠪ߦߣߞߡ㧘ߘߩ╙৻ࠬ࠹࠶ࡊߣߒߡ᳓⚛ේሶࠍℂ⸃ߒߚ޿ߣᕁ߁ߩߪᒰὼ
ߢ޽ࠆ㧚ߒ߆ߒ㧘ᚒ‫ߡߞߣߦޘ‬᳓⚛ේሶߩ Schrödinger equation ࠍ⸃ߊߎߣ߇㊀ⷐߥߩߪ㧘᳿ߒߡ㧘╙৻ࠬ࠹࠶ࡊߣߒߡᗧ๧߇޽ࠆߣ޿߁ߎߣߢߪߥ޿㧚ታߩߣߎࠈ㧘Schrödinger
equation ࠍᱜ⏕ߦ⸃ߊߎߣ߇಴᧪ࠆൻቇ⒳ߪේሶᩭ 1 ୘ߣ㔚ሶ 1 ୘ߩߺ߆ࠄ᭴ᚑߐࠇ
ࠆ᳓⚛ේሶߒ߆ߥ޿ߩߢ޽ࠆ㧚㔚ሶ߇ 1 ୘Ⴧ߃ߡࡋ࡝࠙ࡓේሶߦߥࠆߣ߽߁ᱜ⏕ߦߪ⸃ߌߥ޿㧚ߘߎߢ㧘෩ኒߦ⸃ߊߎߣ߇಴᧪ߥ޿᳓⚛એᄖߩේሶ߿ಽሶࠍ㧘ᅤ૗ߦㄭૃ
⊛ߦ⸃ߊ߆߇໧㗴ߣߥࠆ㧚ߎߩ႐ว㧘᳓⚛ේሶࠍ಴⊒ߩࡕ࠺࡞ߣߒߡ↪޿㧘ߘߎߦߥࠎࠄ߆ߩㄭૃࠍขࠅ౉ࠇߡ⠨߃ࠆ㧚ࡕ࠺࡞‛⾰ߢ޽ࠆ߆ࠄ㧘᳓⚛ේሶߩߎߣߪቢోߦ
ℂ⸃ߒߡ߅ߊᔅⷐ߇޽ࠆࠊߌߢ޽ࠆ㧚ߥ߅㧘߅޿߅޿᣿ࠄ߆ߦߥࠆࠃ߁ߦ㧘േᓘㇱಽߩ wavefunction ߪ౹⚖ᢙᴺߦࠃߞߡ⸃ߊߩ߇㧘⃻ሽߔࠆᢎ⑼ᦠߩᄢㇱಽ߇ណ↪ߒߡ޿ࠆᣇᴺߢ޽ࠆ㧚ߒ߆ߒ㧘harmonic
oscillator ߩ႐วߦቇ⠌ߒߚࠃ߁ߥṶ▚ሶᴺࠍ↪޿ࠆߣ㧘߆ߥࠅ◲නߦ⸃߇᳞߹ࠆ㧚ᧄ
Ⓜߦ߅޿ߡߪ㧘ᓟ⠪ߩขࠅᛒ޿ࠍߒࠃ߁ߣ߽⠨߃ߚ߇㧘90%એ਄ߩᢎ⑼ᦠߢᛒࠊࠇߡ
޿ࠆᣇᴺࠍㅀߴߥ޿ߢ㧘ઁߩᣇᴺߛߌࠍㅀߴࠆߎߣߩ᳿ᔃ߇ߟ߈߆ߨߚ㧚elegant ߥᣇᴺࠍቇ⠌ߔࠆߩ߽޿޿߇㧘conventional ߢ standard ߥᣇᴺࠍోߊീᒝߒߥ޿ߣ޿߁ߩ
߽Ꮕߒ㓚ࠅ߇޽ࠆ߆߽ߒࠇߥ޿㧚ߘ߁ᕁߞߡ㧘ߎߎߢߪ㧘ߘߩਔᣇࠍ⸥ㅀߔࠆߎߣߦ
ߔࠆ㧚಴᧪ࠆߛߌ߇ࠎ߫ߞߡਔᣇߩᣇᴺࠍቇ⠌ߒߡ߶ߒ޿㧚

9.1 ᳓⚛ේሶߦ㑐ߔࠆ Schrödinger ᣇ⒟ᑼ
ේሶᩭߩ⾰㊂ࠍ M 㧘ߘߩᐳᮡࠍ xN , yN , z N ߣߒ㧘߹ߚ㧘㔚ሶߩ⾰㊂ࠍ me 㧘ߘߩᐳᮡࠍ xe , ye , ze ߣߔࠆ㧚potential ߪේሶᩭߣ㔚ሶߣߩ㑆ߩ Coulomb ⋧੕૞↪ߛߌߢ޽ࠆ㧔Problem 9.1.1㧕ߩߢ㧘Hamiltonian ߪᰴߩࠃ߁ߦ⴫ߐࠇࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 1

H




=2 § w 2
w2
w 2 · =2 § w 2
w2
w2 ·





¨
¸
¨
¸
2 M © wX N 2 wYN 2 wZ N 2 ¹ 2me © wxe 2 wye 2 wze 2 ¹
e2

1
4SH 0

X N  xe2

 YN  ye2

 Z N  ze

(9.1.1)

2

ߎߎߢ Section 8.3 ߢⴕߞߚࠃ߁ߦ㧘᳓⚛ේሶߩ㊀ᔃߩᐳᮡ X , Y , Z ߅ࠃ߮ේሶᩭ߆ࠄ
ߺߚ㔚ሶߩᐳᮡ x, y, z ࠍዉ౉ߔࠆ㧚ᑼ(8.3.7)ࠍᓧߚߩߣหߓࠃ߁ߦߒߡ㧘Hamiltonianߣߒߡᰴᑼࠍᓧࠆ㧚
H




§ w2
=2
w2
w 2 · =2 § w 2
w2
w2 ·¨ 2  2  2 ¸
¨ 2 2 2¸
2 M  me © wX
wY
wZ ¹ 2 P © wx wy wz ¹
1

e2

4SH 0

x2  y2  z 2(9.1.2)

ߎߎߢ P ߪ឵▚⾰㊂ߢ㧘
1

P

1 1

M m

(9.1.3)ߣቯ⟵ߐࠇࠆ㧚ᑼ(9.1.2)ߩ╙ 1 㗄⋡ߪ᳓⚛ේሶో૕ߩਗㅴㆇേࠍ⴫ߔ߽ߩߢ޽ࠆ߇㧘ߘࠇߪߔߢߦ╙ 6 ┨ߢᛒߞߚ߽ߩߢ㧘ߎߎߢߪ⥝๧߇ߥ޿㧚ᓥߞߡ㧘㊀ᔃߪⓨ㑆ਛߦ࿕ቯߐࠇߡ޿ࠆߣ⠨߃ߡ㧘Hamiltonian ߪ

H

1
w2
w2 ·
=2 § w 2¨ 2  2  2 ¸
2 P © wx wy wz ¹ 4SH 0e2
x2  y2  z 2(9.1.4)ߣߒߡࠃ޿㧚ᦝߦ㧘ᑼ(8.2.2)ࠍ↪޿ߡ㧘ߎࠇࠍ spherical polar coordinate ߳ᄌ឵ߔࠆ㧚ߘߩ⚿ᨐ㧘

­1 w § 2 w ·
1
w §
w ·½
° r 2 wr ¨ r wr ¸  r 2 sin T wT ¨ sin T wT ¸ °
e2
= °
©
¹
©
¹°
H 

®
¾
2P °
1
w2
° 4SH 0 r
 2 2
2
°¯
°¿
r sin T wIߣߥࠆ㧚ߔߥࠊߜ㧘 Schrödinger equation ߪᰴߩࠃ߁ߦ⴫ߐࠇࠆ㧚
2䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 2(9.1.5)w §w\ ·1 w § 2 w\ ·
1
¨r
¸ 2
¨ sin T¸
2
wT ¹
r wr © w r ¹ r sin T wT ©
w 2\ 2 P §
e2 ·
1
 2 2


E
¨
¸\
r sin T wI 2 = 2 ©
4SH 0 r ¹

(9.1.6)0ߎߩᑼߪ r , T , I ߣ޿߁ 3 ߟߩᄌᢙࠍ฽ࠎߢ޿ߡߥ߆ߥ߆ⶄ㔀ߢ޽ࠆ㧚ߘߎߢ㧘\ r ,T , I

R r Y T ,Iߩࠃ߁ߦ㧘\ r , T , I(9.1.7)

ࠍ r ߛߌߩ㑐ᢙ R r ߣ T , I ߛߌߩ㑐ᢙ Y T , I ߣߩⓍߢ⴫ߐࠇࠆ

ߣ⠨߃ࠆ㧚ᑼ(9.1.7)ࠍᑼ(9.1.6)ߦઍ౉ߔࠆߎߣߦࠃࠅ

1 w § 2 wR r · 2P §e2 · 2


r
E
¨
¸
¨
¸r
4SH 0 r ¹
R r wr ©
wr ¹ = 2 ©
wY T , I ·w 2Y T , I
1
1
w §
sin
T


¨
¸
2
Y T , I sin T wT ©
wT
wI 2
¹ Y T , I sin T(9.1.8)ࠍᓧࠆ㧚 r ߛߌߩ㑐ᢙߣ T , I ߛߌߩ㑐ᢙߣ߇߅੕޿ߦ╬ߒ޿ߣ޿߁ߎߣߪ㧘ߘࠇߙࠇ߇޽ࠆቯᢙߦ╬ߒ޿ߎߣࠍ␜ߒߡ޿ࠆ㧚ߎߩቯᢙࠍ O ߣ߅ߊߣᰴߩ 2 ߟߩᑼ߇ᓧࠄࠇࠆ㧚
2
wY T , I ·1 w §1 w Y T ,I¨ sin T
¸ 2
wT
wI 2
sin T wT ©
¹ sin T

1 w § 2 wR r · ­° 2P §e 2 · O ½°


r
E
¨
¸ ® ¨
¸ ¾R r
4SH 0 r ¹ r 2 °¿r2 w r ©
wr ¹ ¯° = 2 ©

OY T , I(9.1.9)0(9.1.10)ߎߩ߁ߜ╙ 1 ⇟⋡ߩᑼ(9.1.9)ߪᚒ‫ߩߺߓߥ߅ߦߢߔߡߞߣߦޘ‬ᑼߢ޽ࠆ㧚ߔߥࠊߜ㧘
ᑼ(5.2.10)ࠍ⠨ᘦߔࠆߣᑼ(9.1.9)ߪ
L2Y T , I

O = 2Y T , I(9.1.11)ߢ޽ࠆߎߣ߇ಽ߆ࠆ㧚ᑼ(5.3.29)ࠍᡷ߼ߡෳᾖߔࠆ߹ߢ߽ߥߊ㧘ߎߩᑼߪ߹ߐߒߊ㧘Y T ,I߇⃿㑐ᢙߢ޽ࠆߎߣࠍ␜ߒߡ޿ࠆ㧚ߔߥࠊߜ㧘ᑼ(9.1.9)ߩ⸃ߪ
Y T ,IO

YA ,m T , I

A d m d A

A A 1䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 3(9.1.12)(9.1.13)ߣߥࠆ㧚ᱷࠅߩᑼ(9.1.10)ߪᑼ(9.1.13)ࠍઍ౉ߔࠆߎߣߦࠃࠅ

1 w § 2 wR r · ­° 2 P §e 2 · A A  1 ½°


r
E
¾R r
¨
¸ ® ¨
¸
4SH 0 r ¹
r 2 wr ©
r 2 °¿
wr ¹ °¯ = 2 ©

0(9.1.14)0(9.1.15)╙ 1 㗄⋡ߩṶ▚ࠍⴕ޿

w 2 R r 2 wR r ­° 2P §e2 · A A  1 ½°



E
® 2 ¨
¾R r
¸
4SH 0 r ¹
r wr
r 2 °¿
wr 2°¯ = ©

ࠍᓧࠆ㧚㑐ᢙ R r ߪ r ߛߌߩ㑐ᢙߢ޽ࠅ㧘േᓘᵄേ㑐ᢙ㧔radial wavefunction㧕ߣ⸒ࠊࠇࠆ㧚ᑼ(9.1.15)ߪേᓘㇱಽߩ Schrödinger equation ߢ޽ࠆߣ޿߁ߎߣ߽಴᧪ࠆ㧚ߎߩᓸ
ಽᣇ⒟ᑼߩ⸃ߪ㧘ߘ߁◲නߦᓧࠄࠇߘ߁ߦ߽ߥ޿ߩߢ▵ࠍᡷ߼ߡ⠨߃ࠆߎߣߦߒࠃ
߁㧚‫ڏ‬Problem 9.1.1
ᄥ㓁ߣ࿾⃿ߣߩ⋧੕૞↪ߪਁ᦭ᒁജߩߺߦࠃߞߡਈ߃ࠄࠇࠆ㧚৻⷗ૃߚ♽ߩේሶᩭߣ㔚ሶߩ♽ߦߟ޿ߡ㧘ਁ᦭ᒁജࠍ⸘▚ߒ㧘Coulomb ജߣᲧセߖࠃ㧚ේሶᩭߣ㔚ሶߣߩ〒㔌ߪᄢ૕ 1010 m ߩࠝ࡯࠳࡯ߣ⠨߃ࠃ㧚ਁ᦭ᒁജቯᢙߘߩઁᔅⷐߥቯᢙߪ⥄ಽߢ⺞ߴߡߺࠃ㧚

9.2 േᓘᵄേ㑐ᢙߩቯᕈ⊛ี๧േᓘᵄേ㑐ᢙ R r ߦ㑐ߔࠆ Schrödinger equation (9.1.15)ࠍ⸃ߊߎߣࠍ⠨߃ࠆ㧚ᣂߒ
ߊ
P r

R r r

(9.2.1)ߢቯ⟵ߐࠇࠆ P r ߣ޿߁㑐ᢙࠍዉ౉ߔࠆߣᑼ(9.1.15)ߪ

d 2 P r ­° 2P §e 2 · A A  1 ½°


E
®
¾P r
¨
¸
2
4SH 0 r ¹
dr 2
r 2 ¿°¯° = ©

0(9.2.2)ߩࠃ߁ߦߥࠆ㧚ߎߩᓸಽᣇ⒟ᑼࠍᢙቇ⊛ߦ⸃ߊ೨ߦ㧘ߎߩᣇ⒟ᑼࠍቯᕈ⊛ߦี๧ߒߡ㧘⸃ߦߟ޿ߡߩ߅߅߹߆ߥ޽ࠅߐ߹ࠍᛠីߒߡ߅ߎ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 4߽ߒ߽ᑼ(9.2.2)ߦ߅޿ߡ Coulomb potential ߩઍࠅߦ effective potential Veff ߣߒߡ

Veff



e2
4SH 0 r

A A  1 =2
2P r 2(9.2.3)ࠍ⠨߃ࠆߣ㧘ᑼ(9.2.2)ߪ

=2 d P r(9.2.4)
 Veff P r EP r
2 P dr 2ߣ޿߁৻ᰴరߩ Schrödinger equation ߣߺߥߔߎߣ߇಴᧪ࠆ㧚ᑼ(9.2.4)ߩ effective
potential ߩ‛ℂ⊛ᗧ๧ࠍ⠨߃ߡߺࠃ߁㧚╙ 1 㗄⋡ߪ Coulomb ߩ potential ߢ޽ࠅᒁജߢ
޽ࠆ㧚╙ 2 㗄⋡ߪⷺㆇേ㊂ߦଐሽߔࠆ෻⊒ജߢ㆙ᔃജߦ⋧ᒰߔࠆ㧚
2



‫ޤ⸼ޣ‬
㧦
momentumฎౖ⊛ߦߪ㆙ᔃജߪPr3
momentumߩ߇ potential ߢ
2P r 22ߢਈ߃ࠄࠇࠆ㧚ߎࠇࠍⓍಽߒߡ╓ภࠍᄌ߃ߚ߽

2ߣߥࠆ㧚 momentum

2

A A  1 = 2 ࠍઍ౉ߔࠆߣ㧘A A  1 =2߇ᓧࠄࠇࠆ㧚
2P r 2A

0 ߩ႐วߦߪ㔚ሶߪⷺㆇേ㊂ࠍᜬߚߕ㧘㔚ሶߦߪᩭ߆ࠄㅏߍࠃ߁ߣߔࠆജߪ௛

߆ߕᒁജߩߺ߇௛ߊ㧚ߎߩ႐วߩ potential ߪ࿑ 9.2.1㧔a㧕ߦ␜ߔࠃ߁ߦ޿ߚࠆߣߎࠈ
⽶ߢ޽ࠆ㧚৻ᣇ A z 0 ߩ႐วߦߪ㧘㔚ሶߪⷺㆇേ㊂ࠍᜬߜ㧘ߘࠇߦࠃߞߡਛᔃ߆ࠄㅏࠇ
ࠃ߁ߣߔࠆജ߇௛ߊ㧚ᓥߞߡ㧘ߎߩലᨐߣ㧘ᒁജߩㇱಽߣߩ┹੎ߣߥࠆ㧚 r ߇ዊߐ޿ߣߎࠈߢߪ potential ߩ 11ߩ㗄߇  ߩ㗄ࠃࠅ߽ࠃߊല߈㧘 r ߇ᄢ߈޿ߣߎࠈߢߪㅒ
2rr

1
ߦ  ߩ㗄߇ఝ൓ߣߥࠆ㧚A 1 ߩ႐วߩ Veff ࠍ࿑ 9.2.1㧔b㧕ߦ㧘߹ߚ㧘Ar
ࠍ࿑ 9.2.1㧔c㧕ߦ␜ߔ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 5

2 ߩ႐วߩ Veff‫ޣ‬ᵈ‫ޤ‬㧦effective potential㧚❑ゲߪ Veff ࠍ atomic unit㧔 1 atomic unitߪේሶᩭ߆ࠄ㔚ሶ߹ߢߩ〒㔌ࠍ atomic unit㧔 1 atomic unitme 4
8h 2H 0 2

H 0h2
S me4

13.6 eV 㧕㧘ᮮゲ

52.9 pm 㧕ߢ⴫ࠊߔ

2 A A 1ߣ⴫ߐࠇࠆ㧚࿑ 9.2.1 ߪߎߩන૏ߢ⋡⋓ߞߡ
 
r
r2

ߣ㧘effective potential ߪ Veff
޽ࠆ㧚

㪋

㪭

㪉

㪇

㪄㪉

㪄㪋
㪇

㪈

㪉

㪊

㪋

㫉

࿑ 9.2.1(a)A

0 ߩ႐วߩ effective potential❑ゲ㧘ᮮゲ౒ߦ atomic unit ߢ⴫␜ߒߡ޿ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 6

㪌

㪋

㪭

㪉

㪇

㪄㪉

㪄㪋

㪇

㪈

㪉

㪊

㪋

㪌

㫉
࿑ 9.2.1(b)A 1 ߩ႐วߩ effective potential❑ゲ㧘ᮮゲ౒ߦ atomic unit ߢ⴫␜ߒߡ޿ࠆ㧚ὐ✢ߪᒁജ㧘ᢺജߘࠇߙࠇߩ㗄ࠍ␜ߔ㧚ߎߩ 2 ߟߩ㗄ߩ๺߇ታ✢ߢ␜ߒߚ effective potential ߣߥࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 7

㪋

㪭

㪉

㪇

㪄㪉

㪄㪋

㪇

㪈

㪉

㪊

㪋

㪌

㫉
࿑ 9.2.1(c)A 2 ߩ႐วߩ effective potential❑ゲ㧘ᮮゲ౒ߦ atomic unit ߢ⴫␜ߒߡ޿ࠆ㧚ὐ✢ߪᒁജ㧘ᢺജߘࠇߙࠇߩ㗄ࠍ␜ߔ㧚ߎߩ 2 ߟߩ㗄ߩ๺߇ታ✢ߢ␜ߒߚ effective potential ߣߥࠆ㧚߹ߕ A

0 ߩ႐วࠍี๧ߔࠆ㧚 r o 0 ߩᭂ㒢ߢߪ potential energy

1 e2
߇ E ࠃࠅ߽
4SH 0 rߪࠆ߆ߦᄢ߈ߊߥࠅ E ࠍήⷞ಴᧪ࠆ㧚ᓥߞߡᑼ(9.2.2)ߪ

d 2 P r 2P e2 P r
 2
dr 2= 4SH 0 r

0A

0, r o 0(9.2.5)ߣㄭૃ಴᧪ࠆ㧚ߎߩ⸃ߪP r | Ar 

P e2
Ar 2
2
= 4SH 0(9.2.6)

ߩᒻߦߥࠆߎߣߪ㧘ᑼ(9.2.6)ࠍᑼ(9.2.5)ߦઍ౉ߒߡ r o 0 ߩᭂ㒢ࠍߣߞߡߺࠇ߫ಽ߆
ࠆ㧚ᓥߞߡ㧘radial wavefunction ⥄૕ߪ r o 0 ߩᭂ㒢ߢ R r䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 8

P r| A ߣߥࠆ㧚ߔߥࠊr

ߜ r o 0 ߢ㧘 R r
Aߪ࠯ࡠߢߥ޿޽ࠆ୯ࠍขࠆߎߣ߇ಽ߆ࠆ㧚⸒⪲ࠍ឵߃ߡ⸒߁ߣ㧘

0 ߩ႐วߦߪ㧘㔚ሶࠍᩭߩ૏⟎ߦ⷗಴ߔ⏕₸߇࠯ࡠߢߪߥ޿ߣ޿߁ߎߣߦߥࠆ㧚A z 0 ߩ႐วߦߪ㧘ᄢ߈ߥ෻⊒ജߩߚ߼ߦ㔚ሶࠍᩭߩ૏⟎ߦ⷗಴ߔ⏕₸ߪ࠯ࡠߣߥ

ࠆ㧚߹ߚ㧘 A z 0 㧘 r o 0 ߩᭂ㒢ߢߪᑼ(9.2.2)ߪ

d 2P r A A 1
P r |0dr 2
r2
ߣߥࠆ㧚ߎߩ⸃ߪ

A z 0, r o 0

P r | Ar A 1  Br  A(9.2.7)(9.2.8)ߣ޿߁ᒻߦߥࠆߎߣߪઍ౉ߒߡߺࠇ߫᣿ࠄ߆ߢ޽ࠆ㧚ߒ߆ߒ㧘਄ㅀߒߚࠃ߁ߦ r o 0 ߩᭂ㒢ߢ P r o 0 ߢ޽ࠆ߆ࠄ㧘ᑼ(9.2.8)ߢ B

P r| Ar Ar

R r

0 ߢߥߌࠇ߫ߥࠄߥ޿㧚ᓥߞߡ㧘r o0

ߣ޿߁ asymptotic solution ߇ᓧࠄࠇࠆ㧚߆ߊߒߡ㧘R r(9.2.9)ߩ߅߅ࠃߘߩ޽ࠅߐ߹߇ಽ߆

ߞߚߩߢ㧘ᰴߩ▵ߢ㧘ᱜ⏕ߥ㑐ᢙᒻࠍ᳞߼ߡߺࠃ߁㧚

9.3 േᓘᵄേ㑐ᢙߩ౹⚖ᢙᴺߦࠃࠆ⸃߹ߕ E ߩ╓ภࠍ⠨߃ߡߺࠆ㧚㔚ሶߣᩭߣ߇ή㒢ߦ㔌ࠇߡ޿ࠆ႐วࠍ energy ߩ࠯ࡠߣߒߚࠊߌߢ޽ࠆ߆ࠄ㧘㔚ሶ߇ᩭ߆ࠄ᧤❈ߐࠇߡ޿ࠆ㒢ࠅ㧔ߎ߁޿߁⁁ᘒࠍ᧤❈⁁ᘒ㧔bound state㧕ߣ⸒߁㧕 E  0 ߢ޽ࠆ㧚㔚ሶ߇ߎߩ᧤❈ࠍ⣕ߒߚ⁁ᘒߢߪ E ! 0 ߢ޽ࠆ㧚એਅ㧘 E ߇⽶ߩ႐วߣ E ߇ᱜߩ႐วࠍ೎‫ߦޘ‬ขࠅᛒ߁㧚E ! 0 ߩ႐วߪᓟ⒟ Section 9.4 ߢ⠨߃ࠆߎߣߦߒ㧘ߎߎߢߪ߹ߕ㧘bound state㧘ߔߥࠊߜ㧘 E  0 ߩ႐วࠍ⠨߃ࠃ߁㧚ߎߎߢᰴᑼߢቯ⟵ߐࠇࠆᣂߒ޿ήᰴరߩ parameter Qࠍዉ౉ߔࠆ㧚

Q

e2

P

4SH 0 = 2 E

(9.3.1)

ᦝߦ r ߩઍࠅߦᣂߒ޿ήᰴరߩᄌᢙ x ࠍዉ౉ߔࠆ㧚x

8 P Er=䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 9(9.3.2)એ਄ߩᄌᢙߩ⟎߈឵߃ࠍⴕ߁ߣ㧘⸃ߊߴ߈ R ߦߟ޿ߡߩᓸಽᣇ⒟ᑼ(9.1.15)ߪᰴߩࠃ߁
ߦߥࠆ㧚
d 2 R 2 dR ­ 1 Q A A  1 ½ ®  ¾R
dx 2 x dx ¯ 4 xx2 ¿(9.3.3)0߹ߕߎߩᑼࠍ x o f ߩ㗔ၞߢี๧ߒߡߺࠃ߁㧚 x o f ߢߪᑼ(9.3.3)ߪ

d 2R 1Rdx 2 4ߣߥࠆ㧚ߎࠇࠍ⸃޿ߡ

(9.3.4)§ x·§x·
A exp ¨  ¸  B exp ¨ ¸© 2¹©2¹߇ᓧࠄࠇࠆ߇㧘 x o f ߢ R ߇⊒ᢔߒߥ޿ߚ߼ߦߪ B
R

R

Ae



x
2

(9.3.5)0 ߢߥߌࠇ߫ߥࠄߥ޿㧚ᓥߞߡ

(9.3.6)x o f ߢߎߩࠃ߁ߥ asymptotic solution ߇ᓧࠄࠇࠆߣ޿߁ߎߣ߆ࠄ㧘ᑼ(9.3.3)ߩ⸃ߣߒ

ߡ

R x



x

K x e 2(9.3.7)

ߣ߅޿ߡࠃߐߘ߁ߢ޽ࠆߣ⚿⺰ߐࠇࠆ㧚ߎߎߢ K x ߪᑼ(9.3.3)ࠍḩ⿷ߔࠆࠃ߁ߦ᳿߼
ࠄࠇࠆࠊߌߢ޽ࠆ߇㧘ዋߥߊߣ߽㧘 R ߇ x o f ߢ⊒ᢔߒߥ޿ߚ߼ߦߪ㧘 x o f ߦߥࠆ§ x·
႐วߩ K x ߩჇട߇ exp ¨ ¸ ߩჇടࠃࠅ߽ࠁߞߊࠅߢߥߌࠇ߫ߥࠄߥ޿ߎߣߪ⏕߆
©2¹ߢ޽ࠆ㧚ᑼ(9.3.7)ࠍᑼ(9.3.3)ߦઍ౉ߒߡᰴᑼࠍᓧࠆ㧚d 2KdK
 x 2 x ^ Q 1 x  A A 1 ` K 02dx
dxߎߩᑼߪ x 0 ߢ߽ᚑࠅ┙ߚߥ޿ߣ޿ߌߥ޿㧚ᓥߞߡ
x2

A A 1 K 0(9.3.9)0ߔߥࠊߜ㧘 A z 0 ߩ႐วߦߪ K 0(9.3.8)0 ߢ޽ࠆ㧚

K x ߩᒻࠍ౹⚖ᢙᴺߢ᳞߼ࠃ߁㧚ᑼ(9.3.9)߆ࠄ㧘A z 0 ߩ႐วߦߪ K x ߩዷ㐿ߢ x 0ߩ㗄ࠍ฽߹ߥ޿ߎߣ߇ಽ߆ࠆ㧚ߎߩߎߣ߆ࠄ㧘 K x

K x

x s ¦ ak x k
k 0䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 10ࠍᰴߩࠃ߁ߦ⴫ߔߎߣߦߔࠆ㧚(9.3.10)ߎߎߢ s ߪ A z 0 ߩ႐วߪᱜߩᢛᢙߢ޽ࠆ㧚ᑼ(9.3.10)ࠍᑼ(9.3.8)ߦઍ౉ߔࠆ㧚¦^ s  k s  k 1  2 s  k  A A  1 ` a x

sk

k

k 0

 ¦ Q  s  k  1 ak x s  k 1

(9.3.11)0

k 0ߎࠇ߇ߔߴߡߩ x ߦߟ޿ߡᚑࠅ┙ߚߥߌࠇ߫޿ߌߥ޿߆ࠄ㧘 x ߩฦ౹ߩଥᢙ߇࠯ࡠߢ
ߥߌࠇ߫ߥࠄߥ޿㧚 x s ߩଥᢙߦ⌕⋡ߔࠆߣs s  1  2s  A A  10߹ߚߪ
s s 1ߎߩ⸃ߪ㧘 sA A 1A ߹ߚߪ s(9.3.12)
 A  1 ߩ޿ߕࠇ߆ߢ޽ࠆ㧚 s ߇ negative ߢ޽ࠆߣᑼ(9.3.10)

ߪ x o 0 ߢ⊒ᢔߒߡߒ߹޿㧘ᑼ(9.3.9)ߣ⍦⋫ߔࠆ㧚ᓥߞߡ sዪ㧘ᑼ(9.3.10)ߪ

x A ¦ ak x k

K x

A ߇໑৻ߩ⸃ߢ޽ࠆ㧚⚿(9.3.13)k 0ߣߥࠆ㧚ߘߒߡᑼ(9.3.11)ࠃࠅଥᢙ ak ߣ ak 1 ߩ㑐ଥࠍ᳞߼ࠆ㧚ak 1

A  k  1 Qakk  1 k  2  2A

(9.3.14)k ߩᄢ߈޿ߣߎࠈߢߪ

ak 1

ak
k

(9.3.15)ߣߥࠆ㧚৻ᣇ㧘 e x ߩዷ㐿
e

x

xk
¦
k 0 k!

(9.3.16)ߦ߅޿ߡ x k ߩଥᢙࠍ bk ߣ⴫ߔߣ㧘

bk 1

bk
k

ߣᑼ(9.3.15)ߣหߓ㑐ଥ߇ᓧࠄࠇࠆ㧚ߔߥࠊߜ㧘 K xหߓࠃ߁ߦᝄ⥰߁㧚ᑼ(9.3.7)ࠍ⠨ᘦߔࠆߣ㧘䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 11(9.3.17)
ߪ x ߩᄢ߈޿ߣߎࠈߢ e x ߣ߶߷

x

x

exe 2

e2



R x

xof

(9.3.18)

ߣߥߞߡ well behaved ߥ㑐ᢙߣߥࠄߥ޿㧚ߎߩࠃ߁ߦߥࠄߥ޿ߚ߼ߦߪ㧘ᑼ(9.3.13)
ߩዷ㐿߇޽ࠆ᦭㒢ߩߣߎࠈߢᱛ߹ࠆᔅⷐ߇޽ࠆ㧚ߎߩߚ߼ߦߪᑼ(9.3.14)ߦ߅޿ߡ k ߇
޽ࠆ୯ n ' n ' 0, 1, 2, ... ࠃࠅ߽ᄢ߈޿ᚲߢߔߴߡߩ ak ߇࠯ࡠߦߥࠇ߫ࠃ޿㧚ߔߥࠊߜ

Q A  n' 1ߣ޿߁᧦ઙ߇ᓧࠄࠇࠆ㧚ߎߎߢᣂߒߊ
n(9.3.19)A  n ' 1

(9.3.20)

ߣ n ࠍቯ⟵ߔࠆ㧚ᑼ(9.3.19)ࠃࠅ㧘n t A 1

A

(9.3.21)0, 1, 2, ...

ߢ޽ࠆ߆ࠄ
n t1
ߢ޽ࠆ㧚ᑼ(9.3.1)ࠍ↪޿ࠆߣᑼ(9.3.20)߆ࠄ

En



P e4

1
4SH 0

2

n 1, 2, ...

2n 2 = 2(9.3.22)߇ᓧࠄࠇࠆ㧚E ߪ n ߦࠃߞߡ᳿߹ࠆߩߢ E ߩ subscript ߣߒߡ n ࠍߟߌߚ㧚ߎߩࠃ߁ߦ㧘E  0 ߩ႐วߦߪ energy ߇㊂ሶൻߒߡ޿ࠆߎߣ߇ಽ߆ࠆ㧚ߥ߅㧘ߎߩ⚿ᨐߪ㧘Bohr ߩ㊂ሶ᧦ઙ߆ࠄ಴ߒߚᑼ(1.6.11)ߣቢోߦหߓߦߥߞߡ޿ࠆ㧚㧔ߒ߆ߒ㧘ߎߩߎߣߪ Bohr
ߩ㊂ሶ᧦ઙ߇ᱜᒰൻߐࠇࠆߎߣࠍᗧ๧ߒߡ޿ࠆߩߢߪߥ޿㧚㧕ᑼ(9.3.2)ࠍ↪޿ࠆߣr1
na0 ' x2

(9.3.23)ߣߥࠆ㧚ߚߛߒ a0 ' ߪ㐳ߐߩ dimension ࠍ߽ߜa0 '4SH 0 = 2P e2ߢቯ⟵ߐࠇࠆ㊂ߢ㧘Bohr radius a0 ߦ

(9.3.24)

me

P

1

me
ࠍដߌࠆߎߣߦࠃߞߡ reduced massMߦߟ޿ߡߩ⵬ᱜࠍߒߚ߽ߩߢ޽ࠆ㧚
ᑼ(9.3.21)߆ࠄಽ߆ࠆࠃ߁ߦ㧘1 ߟߩ n ߦߟ޿ߡ A ߪ 0 ߆ࠄ n  1 ߹ߢ n ୘ߩ୯ࠍขࠅ
ᓧࠆ㧚ߣߎࠈ߇㧘energy ߪᑼ(9.3.22)߆ࠄ᣿ࠄ߆ߥࠃ߁ߦ n ߩߺߦࠃߞߡ᳿߹ࠅ A ߦߪࠃࠄߥ޿㧚ߔߥࠊߜ㧘 A ߦ㑐ߒߡߪ❗ㅌߒߡ޿ࠆߎߣߦߥࠆ㧚ಽశቇߩಽ㊁ߢߪ㧘
A 0, 1, 2, 3 ߦኻᔕߒߡߘࠇߙࠇ s 㧘p 㧘d 㧘f ߣ޿߁ฬ೨ߢ๭ࠎߢ޿ࠆ㧚଀߃߫ n 1 㧘䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 12A

0 ߩ⁁ᘒߪ 1s 㧘 nࠇࠆ㧚ห᭽ߦ n 3 ߢ A

2ߢA

0 ߣ A 1 ߩ⁁ᘒߪߘࠇߙࠇ 2s 㧘 2 p ߣ޿ߞߚ๭߮ᣇ߇ߐ

0 㧘A 1 㧘A

2 ߩฦ⁁ᘒߪߘࠇߙࠇ 3s 㧘3 p 㧘3d ߣ๭߫ࠇࠆ㧚energy ࠍ࿑ 9.3.1 ߦᮨᑼ⊛ߦ␜ߔ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 13

0

4s
3s
2s

4p
3p

4d
3d
486.4 nm
2p 456.2 nm-4-6

102.6 nm121.5 nm

-8

-10

-12
1s

࿑ 9.3.1 ᳓⚛ේሶߩࠛࡀ࡞ࠡ࡯Ḱ૏࿑Ḱ૏㑆ࠍ⚿ࠎߢ޿ࠆ⋥✢ߪ᷹ⷰߐࠇࠆࠬࡍࠢ࠻࡞ߣᵄ㐳䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 14એ਄㧘radial wavefunction ߦߟ޿ߡ energy ߇᳞߹ߞߚߩߢ㧘ᰴߦ wavefunction ߩᒻߦߟ޿ߡี๧ߒࠃ߁㧚߹ߕ㧘੹߹ߢಽ߆ߞߚߎߣࠍᓳ⠌ߒߡ߅ߎ߁㧚ᑼ(9.3,7)㧘ᑼ(9.3.13)߅ࠃ߮ᑼ(9.3.20)ࠃࠅ

§ x · n 1A
exp ¨  ¸ ¦ ak x A  k© 2¹ k 0

R x(9.3.25)§ x·ߢ޽ࠆߎߣ߇ಽ߆ߞߡ޿ࠆ㧚ߔߥࠊߜ㧘 exp ¨  ¸ ߦ߆߆ࠆᄙ㗄ᑼߪ x A ߩ㗄߆ࠄᆎ߹© 2¹
ࠅ x n 1 ߩ㗄ߢ⚳ࠆߎߣߦߥࠆ㧚଀߃߫㧘1s ߩ㑐ᢙߢߪቯᢙ㗄ߩߺ㧘2s ߪቯᢙ㗄ߣ x1 ߩ§ x·
㗄㧘 2 p ߪ x1 ߩ㗄ߩߺ߆ࠄߥࠆᄙ㗄ᑼߦ exp ¨  ¸ ࠍដߌߚᒻߣߥߞߡ޿ࠆ㧚ߘࠇߙࠇ© 2¹ߩ႐วߦߟ޿ߡ㧘ᄙ㗄ᑼߩଥᢙ ak ߪᑼ(9.3.14)ߔߥࠊߜ
A  k 1 nakk  1 k  2  2A

ak 1

(9.3.26)ࠃࠅ㧘ᦨೋߩ ak 㧘ߔߥࠊߜ x A ߩଥᢙ㧘aA ࠍਈ߃ࠇ߫ᰴ‫ߣޘ‬᳞߼ࠆߎߣ߇಴᧪ࠆ㧚aA ߪ

wavefunction ߇ normalize ߐࠇࠆࠃ߁ߦ᳿߼ࠇ߫ࠃ޿㧚ᆎ߼߆ࠄߘߩࠃ߁ߥ aA ࠍ᳿߼ࠆߎߣߪ߻ߕ߆ߒ޿ߩߢ㧘ߣࠅ޽߃ߕ 1 ߣߒߡ߅޿ߡ㧘ᓟߢ normalization factor ࠍ⠨߃ࠇ߫ࠃ߆ࠈ߁㧚
‫ڏ‬Exercise 9.3.1
1s ߩ radial wavefunction ࠍ᳞߼ࠃ㧚

‫ޤ⸃ޣ‬
㧦
n 1㧘A

0 ߢ޽ࠆ߆ࠄ㧘ᄙ㗄ᑼߪቯᢙ㗄ߩߺߣ޿߁ߎߣߦߥࠆ㧚ߘࠇࠍ 1 ߣ⟎޿

ߡ

R1s x



x

e 2

ᑼ(9.3.23)ࠃࠅ
2r
x
a0 '
ߢ޽ࠆ߆ࠄ normalization factor ࠍ N ߣߒߡ


R1s r

Ne

r
a0 '

ߣߥࠆ㧚normalization factor ߪ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 15

f§ 2r · 2³0 exp ¨©  a0 ' ¸¹r dr

1
N2

a0 '3
4

ߣߥࠅ㧘⚿ዪ§ r ·exp ¨ ¸
a0 '3
© a0 ' ¹
2

R1s r
ߣ᳞߹ࠆ㧚

‫ڏ‬Exercise 9.3.2
2 s ߅ࠃ߮ 2 p ߩ radial wavefunction ࠍ᳞߼ࠃ㧚‫ޤ⸃ޣ‬:
߹ߕ 2 s ࠍ⠨߃ࠆ㧚n 2 㧘A 0 ߢ޽ࠆ߆ࠄ㧘ᄙ㗄ᑼߩㇱಽߪ x 0 ߣ x1 ߩ 2 ߟߩ㗄ࠃࠅߥࠆ㧚 x 0 ߩଥᢙࠍ 1 ߣߔࠆ㧚 x1 ߩଥᢙߪᑼ(9.3.26)ߦ n 2 㧘 A 0 㧘 k 0 ࠍઍ౉ߒ㧘

a1



1
2

ߣߥࠆ㧚ᓥߞߡ

R2s x

§ 1 ·
§ x·
¨1  x ¸ exp ¨  ¸
© 2 ¹
© 2¹

ᑼ(9.3.23)ࠃࠅ
rx
a0 'ߢ޽ࠆ߆ࠄ㧘normalization factor ࠍ N 2 s ߣߒߡ

R2s r

§
§ r ·
1 ·
N 2s ¨1 r ¸ exp ¨ ¸© 2a0 ¹
© 2a0 ¹ߣߥࠆ㧚ᰴߦ 2 p ࠍ⠨߃ࠆ㧚 n

2 㧘A 1 ߢ޽ࠆ߆ࠄ㧘ᄙ㗄ᑼߩㇱಽߪ x1 ߩ㗄ߒ߆ߥ޿㧚ߘߩ

ଥᢙࠍ 1 ߣ߅޿ߡ㧘

R2p x

§ x·
x exp ¨  ¸© 2¹ߣߥࠆ㧚ᓥߞߡ㧘normalization factor ࠍ N 2p ߣߒߡ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 16

R2p r

N 2p

§
r
r ·
exp ¨ ¸a0 '© 2a0 ' ¹ߣߥࠆ㧚

2 ߟߩ normalization factor N 2s ߅ࠃ߮ N 2p ࠍ᳞߼ࠆߎߣߪ⻉ำߩṶ⠌ߣߒࠃ߁㧚਄ߩ exercise ߢⴕߞߚࠃ߁ߦ㧘ౕ૕⊛ߥ n ߣ A ߣ߇ਈ߃ࠄࠇߚߣ߈㧘radial
wavefunction ࠍ᳞߼ࠆߎߣߪ㕖Ᏹߦ◲නߥߎߣߢ޽ࠆ㧚ߚߛ㧘৻⥸ᑼߣߒߡ㑐ᢙᒻ߇ਈ߃ࠄࠇߡ޿ࠇ߫⴫ߒᣇߣߔࠇ߫߈ࠇ޿ߢ޽ࠆߒ㧘෶㧘૗߆ߣଢ೑ߥߎߣߪ޽ࠆ㧚ߘ߁޿߁ࠊߌߢ㧘ᦨᓟߦ৻⸒ߎߩߎߣߦߟ޿ߡㅀߴߡ߅ߎ߁㧚ߚߛߒ㧘ߎߎߢߪ㧘޽߹ࠅᢙቇ⊛ߦ෩ኒߦᛒ߁ߎߣࠍㆱߌߡ㧘ᢎ⑼ᦠߩ⚿ᨐࠍ୫ࠅࠆߎߣߦߒࠃ߁㧚ᑼ(9.3.13)
ߦ⴫ߐࠇࠆᄙ㗄ᑼߪ㧘ታߪ㧘Laguerre ߩᄙ㗄ᑼߣߒߡ 19 ਎♿ߩᤨઍ߆ࠄ⍮ࠄࠇߡ޿ߚ㑐ᢙߣኒធߥ㑐ଥ߇޽ࠆ㧚৻⥸ߦ㧘dD D xx e(9.3.27)dxDߢቯ⟵ߐࠇࠆ㑐ᢙߪ㧘Laguerre ߩᄙ㗄ᑼ㧔Laguerre polynomial㧕ߣ⸒ࠊࠇࠆ㧚ߎߩ㑐ᢙࠍ E ࿁ᓸಽߒߚ߽ߩ
LD x

ex

d E LD x
L x(9.3.28)dx E
ߪ Laguerre ߩห઻ᄙ㗄ᑼ㧔associated Laguerre polynomial㧕ߣ⸒߁㧚㧔Laguerre ߩ㒳ᄙ㗄ᑼߣ߽⸒ࠊࠇࠆ㧚㧕ED

߹ߚᑼ(9.3.28)ߢቯ⟵ߒߚ LDE x ߪᓸಽᣇ⒟ᑼ

d2 E
d E
L x  E 1 x
LD x  D  E LDE x
2 D
dx
dx
ߩ⸃ߢ߽޽ࠆ㧚x0(9.3.29)ߐߡ㧘ߎߎߢߪ⹦ߒ޿ߎߣߪ⋭⇛ߔࠆ߇㧘ᑼ(9.3.13)ߩᄙ㗄ᑼ K x ߪ
K x

Nx A L2nAA1 x(9.3.30)ߣߥࠆߎߣ߇⸽᣿ߐࠇࠆ㧚 N ߪછᗧߩቯᢙߢ㧘normalization condition ࠃࠅ᳿߼ࠆߎߣ
߇಴᧪ࠆ㧚ᓥߞߡ㧘ᑼ(9.3.7)ࠃࠅ㧘radial wavefunction R x ߪ

R x

§ x·
N exp ¨  ¸ x A L2nAA1 x© 2¹

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 17(9.3.31)ߣਈ߃ࠄࠇࠆ㧚normalization ߪ㧘 x ߦߟ޿ߡⓍಽߔࠆߩߢߪߥߊ㧘ㅢᏱߪ r ߦߟ޿ߡ
ߩⓍಽ߆ࠄ᳿ቯߔࠆ㧚ߔߥࠊߜ㧘f³R r r dr 1
2

(9.3.32)0ࠃࠅ㧘normalization factor N ࠍ᳿ቯߔࠆ㧚ߎ߁ߒߡ᳿ቯߐࠇߚ radial wavefunction ߩ৻⥸ᑼߪ㧘ᄙߊߩᢎ⑼ᦠߦਈ߃ࠄࠇߡ޿ࠆ㧔଀߃߫㧘Eyring, Walter, Kimball, "Quantum
Chemistry" page 84㧕߇㧘৻ᔕ޽ߍߡ߅ߎ߁㧚3

Rn ,A rAn  A  1 ! § 2r ·§ 2 ·§r · 2 A 1 § 2r · ¨¸
¸ exp ¨ 
¸ Ln  A ¨
¸
3 ¨
na
'
na
'
na
'
2
n
n

!
A
© 0 ¹
^
` © 0 ¹ © 0 ¹ © na0 ' ¹(9.3.33)ߎߎߢᦨೋߩࡑࠗ࠽ࠬߩ╓ภߪ㑐ᢙో૕ߣߒߡᱜߦߥࠆࠃ߁ߦߔࠆߚ߼ߦߟߌߚ߽
ߩߢ޽ࠆ㧚߹ߚ㧘ߎߩᑼ߆ࠄ᣿ࠄ߆ߥࠃ߁ߦ R r ߪ㊂ሶᢙ n ߣ A ߩ㑐ᢙߢ޽ࠆߩߢߎ
ࠇࠄߩ㊂ሶᢙࠍ subscript ߣߒߡߟߌߡ޽ࠆ㧚
Rn ,A r ߩౕ૕⊛ߥ㑐ᢙᒻࠍ n 1, 2, 3 ߩ႐วߦߟ޿ߡ⴫ 9.3.1 ߦ␜ߔ㧚ߥ߅㧘ߎߩ⴫ߢߪ㧘ࠃࠅ৻⥸⊛ߥ႐วߣߒߡේሶᩭߩ㔚⩄߇ Ze ߢ޽ࠆ႐วߩ㑐ᢙᒻࠍ␜ߒߡ޿ࠆ
ߩߢ㧘᳓⚛ߩ႐วߦߪ Z

1 㧘ߘߒߡᒰὼߩߎߣߥ߇ࠄ a0 ' a0 ߢ޽ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 18

⴫ 9.3.1 ᳓⚛㘃ૃේሶߩേᓘᵄേ㑐ᢙ

Z
r
a0 '

U
a0 '

4SH 0

=2
P e2

§ m ·
a0 ¨1  e ¸© M¹

A d n 1

3

n 1

n

n

2

2

A

A

0

0

A 1

R1s

§ Z · U
2 ¨
¸e
a
'
© 0 ¹

R2 s

U

§ Z ·
2

2
U
e
¨
¸
'
a
2 2 © 0 ¹

R2 p

U

§ Z ·
2
¨
¸ Ue
2 6 © a0 ' ¹

R3 s

U

§ Z ·
2
2
3
¨
¸ 27  18 U  2 U e
81 3 © a0 ' ¹

R3 p

U

§ Z ·
4
2
3

6
U
U
e
¨
¸
81 6 © a0 ' ¹

R3d§ Z · 2  U3
4
¨
¸ U e
81 30 © a0 ' ¹

1

1

3

3

3

n

3

A

0

3

n

3

A 1

3

n

3

A

2

[Coffee Break] Bohr ߩᖺᤊࡕ࠺࡞㧘቟ࠄ߆ߦ߅⌁ࠅߊߛߐ޿
Schrödinger equation ߩ⸃ߣߒߡᓧࠄࠇߚ࿕᦭୯ᑼ(9.3.22)ᑼߪ㧘Bohr ߩ㊂ሶ᧦ઙ߆ࠄ಴ߒߚᑼ㨇(1.6.11)ᑼ㨉ߣ৻⥌ߒߡ޿ࠆ㧚ߎߩ৻⥌ߩߚ߼ߦ㧘Bohr ߩࡕ࠺࡞㧔㔚ሶ߇ᩭߩ๟ࠅࠍ࿁ߞߡ޿ࠆߣ޿߁ࡕ࠺࡞ߢ㧘ᖺᤊ߇ᄥ㓁ߩ๟ࠅࠍ࿁ߞߡ޿ࠆߩߣหߓࡕ
࠺࡞ߥߩߢߒ߫ߒ߫‫ޟ‬Bohr ߩᖺᤊࡕ࠺࡞‫ࠆࠇࠊ⸒ߣޠ‬㧕߇ታߪᱜߒ߆ߞߚߩߛߣᕁࠊ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 19ࠇ߇ߜߢ޽ࠆ㧚ߒ߆ߒ㧘‫ߪߩ߁޿ߣޠߚߒ⥌৻ޟ‬㧘ࠛࡀ࡞ࠡ࡯߇ߚ߹ߚ߹৻⥌ߒߚߣ޿߁ߎߣߦߔ߉ߕ㧘Bohr ߩᖺᤊࡕ࠺࡞ߪ޿ࠈ޿ࠈߥὐߢ㊂ሶജቇߩ⸃ߣߪ⇣ߥߞߡ޿ࠆ㧚߹ߕ㧘ߤ߁޿߁ߎߣ߆ࠄࠛࡀ࡞ࠡ࡯ߩ㊂ሶൻ߇ዉ߆ࠇߚߩߢ޽ࠈ߁߆㧚Bohr ࡕ࠺࡞ߢߪ㧘ⷺㆇേ㊂߇ = ߩᢛᢙ୚ߢ޽ࠆߣ޿߁઒ቯ߆ࠄ㊂ሶൻ߇↢ߓߚ㧚ߘࠇߦ෻ߒ㧘㊂ሶജቇߢߪ㧘wavefunction ߇ well behaved ߢߥߌࠇ߫ߥࠄߥ޿㧘ߔߥࠊߜᩭ߆ࠄή
㒢㆙ߩ࿾ὐߢ wavefunction ߇⊒ᢔߒߥ޿ߣ޿߁᧦ઙ߆ࠄ㊂ሶൻ߇↢ߓߚ㧚㊂ሶജቇߢߪ㧘Bohr ߩ⠨߃ߚࠊߌߩಽ߆ࠄߥ޿઒ቯࠍዉ౉ߔࠆᔅⷐ߇ߥߊ㧘ߏߊ⥄ὼߦ㊂ሶൻ߇ዉ߆ࠇߚࠊߌߢ޽ࠆ㧚ⷺㆇേ㊂ߩᄢ߈ߐ߽㧘Bohr ࡕ࠺࡞ߣ㊂ሶജቇߢߪᄢ߈ߊ⇣ߥߞߡ޿ࠆ㧚Bohr ࡕ࠺࡞ߢߪ㧘਄ㅀߒߚࠃ߁ߦⷺㆇേ㊂ߪ = ߩᢛᢙ୚ߢ޽ࠆ㧚㊂ሶജቇߢߪⷺㆇേ㊂⥄૕ߪobservable ߣߪ⸒߃ߥ޿㧚ⷺㆇേ㊂ߩੑਸ਼ߩߺ߇ observable ߢ޽ࠅ㧘ߘߩ୯ߪ A A  1 = 2ߢ޽ࠆ㧚ߎࠇ߇ᱜ⏕ߥ⸒޿ᣇߢ޽ࠆ߇㧘ߒ߫ߒ߫㧘㊂ሶജቇߢߪ‫ⷺޟ‬ㆇേ㊂ߪ A A  1 =ߢ޽ࠆ‫޿⸒߁޿ߣޠ‬ᣇ߇ߐࠇࠆ㧚ዋ‫߇ࠆ޽ߢࠎߍ߆޿޿ޘ‬㧘ߎߩࠃ߁ߥ⴫ߒᣇߢ 2 ߟ
ߩᣇᴺߢߩⷺㆇേ㊂ࠍᲧセߔࠆߣ⴫ 9.3.2 ߩࠃ߁ߦߥࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 20

⴫ 9.3.2 ⷺㆇേ㊂ߩᄢ߈ߐߦ㑐ߒߡ Bohr ࡕ࠺࡞ߣ㊂ሶജቇߣߩᲧセ⁁ᘒBohr ࡕ࠺࡞

㊂ሶജቇ

1s

=

0

2s

2=

0

2p

2=

2=

3s

3=

0

3p

3=

2=

3d

3=

6=

4s

4=

0

4p

4=

2=

4d

4=

6=

4f

4=

12=

㆑޿ߪ߹ߛ߹ߛ޽ࠆ㧚଀߃߫㧘Bohr ࡕ࠺࡞ߢߪ㔚ሶߩሽ࿷⏕₸ߪේሶᩭ߆ࠄ޽ࠆ
᳿߹ߞߚ〒㔌ߦ޽ࠆ゠㆏਄ߦ㒢ࠄࠇߡ޿ࠆ㧚ᓥߞߡ㧘ᒰὼߩߎߣߥ߇ࠄ㧘㔚ሶ߇ේሶᩭߩ૏⟎ߦ⷗಴ߐࠇࠆ⏕₸ߪቢోߦ࠯ࡠߢ޽ࠆ㧚ߒ߆ߒ㧘㊂ሶജቇߢߪ㧘 s 㔚ሶߩ႐ว㧘㔚ሶ߇ේሶᩭߩ૏⟎ߦ⷗಴ߐࠇࠆ⏕₸ߪ߆ߥࠅ޽ࠆ㧚㧔Section 9.8 ࠍෳᾖ㧕ߘߒߡߎߩߎߣߪ㧘᳓⚛ߦ߅ߌࠆ㔚ሶࠬࡇࡦߣᩭࠬࡇࡦߣߩ⋧੕૞↪㧘޿ࠊࠁࠆ⿥ᓸ⚦⋧੕૞↪㧔hyperfine coupling㧕߇㧘๟ᵄᢙන૏ߢ 1420 MHz ߣ޿߁߆ߥࠅᄢ߈ߥ୯ߣߒ
ߡ᷹ⷰߐࠇߡ޿ࠆߣ޿߁ታ㛎੐ታ߆ࠄ߽ⵣઃߌߐࠇߡ޿ࠆ㧚
Bohr ߩᖺᤊࡕ࠺࡞ߪ㧘㊂ሶജቇߩ⊒ዷㅜ਄ߦߪ↹ᦼ⊛ߥᬺ❣ߢ޽ߞߚߣᕁࠊࠇࠆ㧚ߒ߆ߒ㧘⃻࿷ߩ਎ߩਛߢߪ㧘ోߊ㧘ߣ޿ߞߡ޿޿߶ߤᗧ๧ߩߥ޿㧘ㆊ෰ߩㆮ‛ߢߒ߆ߥ޿㧚ߎߩࠃ߁ߦ㧘޽ࠆᤨᦼߦߪ↹ᦼ⊛ߥᬺ❣ߢ޽ߞߚ߽ߩ߇ᓟߦߥߞߡోߊ㘈ߺࠄࠇࠆߎߣ߇ߥߊߥߞߚℂ⺰ߪᢙᄙߊ޽ࠆߪߕߢ޽ࠆ߇㧘ߤ߁޿߁ࠊߌ߆ Bohr ࡕ࠺࡞ߛߌߪ⃻࿷ᄙߊߩᢎ⑼ᦠߢขࠅ਄ߍࠄࠇߡ޿ࠆ㧚ߎߩߎߣߪ㧘Bohr ߦߣߞߡߪ߅ߘࠄ
ߊᐘㆇߥߎߣߢ޽ࠆߣߪᕁ߁߇㧘ߘߩߚ߼ߦ㧘᳓⚛ේሶߩ㊂ሶജቇࠍ⺋⸃ߒߡߒ߹߁
ቇ↢⻉ำ߇ߌߞߎ߁޿ࠆߎߣߪ Bohr ⥄り߽ᦸࠎߢߪ޿ߥ޿ߢ޽ࠈ߁㧚Pimentel ߣSpratley ߩ⪺ᦠߦߪฎ߷ߌߚ Bohr ࡕ࠺࡞ߩ߅Ⴤߩ਄ߦࠞ࡜ࠬ߇ߣ߹ߞߡ޿ߡᶡࠍᵹߒ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 21ߥ߇ࠄ౵⑔ࠍ␨ߞߡ޿ࠆẂ↹߇ߩߞߡ޿ࠆ㧚⪺૞ᮭߩ໧㗴߇޽ࠆߩߢ㧘ߘߩẂ↹ߩࠦࡇ࡯ࠍߎߎߦઃߌࠆࠊߌߦߪ޿߆ߥ޿߇㧘㕙⊕޿ߩߢ㧘ᤚ㕖ේ⪺ࠍ⷗ߡ߽ࠄ޿ߚ޿㧚ᚒ‫߽ޘ‬㧘౵⑔ࠍ␨ߞߡ㧘ᓟߪ Bohr ߩᖺᤊࡕ࠺࡞ߩߎߣࠍᔓࠇߡߒ߹߅߁ߢߪߥ޿߆㧚[Let’s play with Mathematica] Laguerre polynomialᑼ(9.3.28)߅ࠃ߮ᑼ(9.3.29)ߢቯ⟵ߒߚ associated Laguerre polynomial ߪ㧘޽ࠅ߇ߚ޿
ߎߣߦ Mathematica ߢߪ LaguerreL ߣ޿߁ౝㇱ㑐ᢙߣߒߡߔߢߦቯ⟵ߒߡ޽ࠆ㧚ߎߩౝㇱ㑐ᢙࠍ↪޿ߡᑼ(9.3.33)ߩ radial function Rn ,A r ࠍ R[n, L, r]ߣቯ⟵ߔࠆߣR[n_,L_,r_]:=Sqrt[4 (n-L-1)! n^(-4) (n+L)!^(-1)]*(2r/n)^L Exp[-r/n]LaguerreL[n-L-1,2L+1,2r/n];
ߩࠃ߁ߦߥࠆ㧚৻ᣤߎ߁߿ߞߡቯ⟵ߒߡ߅ߌ߫ Rn ,A r ࠍ฽߻⒳‫ߩޘ‬㑐ᢙߩ⸘▚߿ࡊࡠ࠶࠻ࠍ㕖Ᏹߦ◲නߦⴕ߁ߎߣ߇಴᧪ߡଢ೑ߢ޽ࠆ㧚

9.4 ᳓⚛ߩㅪ⛯⁁ᘒ೨▵ߢߪ E  0 ߩ bound state ߩߺߦߟ޿ߡ⠨߃㧘E ! 0 ߩ႐วࠍ㒰ᄖߒߡ᧪ߚ㧚ߘߎߢ㧘એਅ㧘 E ! 0 ߩ႐วࠍ⠨߃ࠆ㧚ߎߩ႐วߪ㧘ᑼ(9.3.1)㧘ᑼ(9.3.2)ߩઍࠅߦ
k

e2

P

(9.4.1)

4SH 0 = 2 E

8P E
r
=
ߣ߅ߊ㧚ߘ߁ߔࠆߣᑼ(9.1.15)ߪ

x

d 2 R 2 dR ­ 1 k A A  1 ½®  
¾R
dx 2 x dx ¯ 4 xx2 ¿(9.4.2)0䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 22(9.4.3)ߣߥࠆ㧚 x ߩᄢ߈޿ߣߎࠈߢߪ

d 2R 1
 R 0
dx 2 4ߣߥࠆ߇㧘ߎߩ⸃ߪ
R

i

x
2

Ae  Be

i

xof(9.4.4)x2(9.4.5)ߣߥࠅ㧘 x ߩᄢ߈޿ߣߎࠈߢ߽⊒ᢔߒߥ޿㧚wavefunction ߩᒻߪ⥄↱☸ሶߣหߓߢ޽
ࠆ㧚ߎߩࠃ߁ߦ㧘 E ! 0 ߩ႐วߦߪ E ߦ૗ߩ᧦ઙ߽ߥߊ㧘㊂ሶൻߔࠆߎߣߥߊㅪ⛯⊛ߥࠛࡀ࡞ࠡ࡯ࠍขࠅᓧࠆ㧚

9.5 Ṷ▚ሶᴺߦࠃࠆേᓘᵄേᣇ⒟ᑼߩ⸃
േᓘᵄേ㑐ᢙ R r ࠍ Section 9.3 ߢߪ౹⚖ᢙᴺߢ⸃޿ߚ㧚ߎߩᣇᴺߪ㧘Laguerre

polynomial ߦߟ޿ߡ೨߽ߞߡቇ⠌ߒߚߎߣ߇޽ࠆੱߥࠄ޿ߑ⍮ࠄߕ㧘ᆎ߼ߡߩੱߦߣ
ߞߡߪ㧘߻ߕ߆ߒߊߪߥ߆ߞߚࠈ߁߇㧘㕙ୟߢߪ޽ߞߚߣᕁ߁㧚ߘߩ㕙ୟߥᣇᴺࠍ޽
߃ߡߎߎߢ߽ขࠅᛒߞߚ߇㧘ߘࠇߪ⃻ሽߔࠆ㊂ሶജቇߩᢎ⑼ᦠߩᄢㇱಽ߇ߎߩᣇᴺࠍ
ߣߞߡ޿ࠆ߆ࠄߢ޽ࠆ㧚ߣߎࠈ߇㧘harmonic oscillator ߩ໧㗴ࠍ㧘Ṷ▚ሶᴺߢᲧセ⊛◲නߦ⸃ߊߎߣ߇಴᧪ߚߩߣห᭽㧘᳓⚛ේሶߩ radial wavefunction ߽㧘Ṷ▚ሶᴺߦࠃߞ
ߡ߆ߥࠅ◲නߦ⸃ߊߎߣ߇಴᧪ࠆ㧚Section 9.3 ߢ㐳‫ߡߞ߆߆ߣޘ‬ዉ಴ߒߚᑼߣోߊห
ߓ߽ߩ߇ᓧࠄࠇࠆߦߔ߉ߥ޿߇㧘ߎߩࠃ߁ߥᢙቇ⊛ᚻᴺߦᘠࠇߡ߅ߊߩߪ᳿ߒߡή㚝ߥߎߣߢߪ޽ࠆ߹޿㧚ߘ߁޿߁ࠊߌߢએਅ㧘Ṷ▚ሶᴺࠍ⚫੺ߔࠆ㧚ߥ߅㧘એਅߩขࠅ
ᛒ޿ߪ㧘ᄢ╭ߦ߅޿ߡ㧘ਛ᎑⽵㓶‫ޟ‬㊂ሶജቇ‫ޠ‬㧔ጤᵄᦠᐫ㧕ߦ⽶ߞߡ޿ࠆ߇ᄙዋᚻ⋥ߒࠍߒߡ޿ࠆ㧚ߎߎߢߪ R r ߦ㑐ߔࠆᓸಽᣇ⒟ᑼ(9.1.15)ߢߥߊ
P r

R r r

(9.5.1)ߣ߅޿ߡᓧࠄࠇࠆᑼ(9.2.2)㧘ߔߥࠊߜd 2 P r °­ 2P §e 2 · A A  1 °½


E
® 2 ¨
¾P r
¸
4SH 0 r ¹
dr 2
r 2 °¿°¯ = ©

0(9.5.2)߆ࠄ಴⊒ߔࠆ㧚ߎߎߢᣂߒߊήᰴరߩ㐳ߐ U ߣήᰴరߩ energy H ࠍᰴߩࠃ߁ߦቯ⟵ߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 23

U

r
a0 '

H

(9.5.3)

4SH 0

2

2= 2

P e4

E

(9.5.4)

ߎߎߢ a0 ' ߪߔߢߦቯ⟵ߒߚࠃ߁ߦ Bohr radius ߦ reduced mass ߩ⵬ᱜࠍ౉ࠇߚ߽ߩߢ
޽ࠆ㧚
m
=2
§ m ·
a0 e a0 ¨1  e ¸(9.5.5)2
PeP© M¹
Section 9.1 ߢቯ⟵ߒߚࠃ߁ߦ M ߪේሶᩭߩ⾰㊂ߢ޽ࠆ㧚ᣂߒ޿ᄌᢙ U ߣ H ࠍ↪޿ࠆߣ
a0 '

4SH 0

ᑼ(9.5.2)ߪᰴߩࠃ߁ߦ⴫ߐࠇࠆ㧚­ d2
2 A A 1 ½
 
®
¾P
2
U
U 2 ¿ H ,A¯ dU

H PH ,A(9.5.6)ߎߎߢ eigenvalue H ߅ࠃ߮Ṷ▚ሶߩਛߦ฽߹ࠇߡ޿ࠆ A ࠍ eigenfunction P ߩ subscriptߣߒߡߟߌߡ޿ࠆ㧚ߎߩ࿕᦭୯ᣇ⒟ᑼࠍ⸃ߊߦ޽ߚࠅ㧘߹ߕ㧘੕޿ߦ Hermitian conjugate ߢ޽ࠆ 2 ߟߩṶ▚ሶ aA ߣ aA ࠍቯ⟵ߒࠃ߁㧚㧔Problem 9.5.1㧕

d
A 1
 
dU U A

aA 



aA

d
A 1
 
dU U A

(9.5.7)(9.5.8)ߎࠇࠃࠅaA  aA

2 A A 1
1
d2
 
 2
2
2
dU
A
U
U

(9.5.9)aA aA 



2 A A 1
1
d2
 
 2
2
2
dU
A
U
U

(9.5.10)ᑼ(9.5.9)ࠍ↪޿ࠆߣᑼ(9.5.6)ߪ

1·§¨ H  2 ¸ PH ,A
A ¹©
ߣߥࠆ㧚߹ߚᑼ(9.5.10)ߦ⌕⋡ߔࠆߣᑼ(9.5.6)ߪ

aA  aA PH ,AaA 1aA 1 PH ,A

§
1 ·
¨H 
¸ PH ,A
2
¨
¸
1
A

©
¹

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 24(9.5.11)(9.5.12)ߣ߽ᦠߊߎߣ߇಴᧪ࠆ㧚ߐߡ㧘ᑼ(9.5.11)ߦᏀ߆ࠄ aA ࠍ૞↪ߔࠆ㧚

1·§¨ H  2 ¸ aA PH ,A
A ¹©
ߎߩᑼߣ㧘ᑼ(9.5.12)ߩ A ࠍ A  1 ߣ⟎߈឵߃ߚᑼ

aA aA  aA PH ,A1·§¨ H  2 ¸ PH ,A 1
A ¹©

aA aA  PH ,A 1(9.5.13)(9.5.14)ߣᲧセߔࠆߣ㧘aA PH ,A ߪ PH ,A ߣหߓ࿕᦭୯ H ࠍᜬߜ㧘A ߇ 1 ߟዊߐ޿ PH ,A 1 ߩቯᢙ୚ߢ޽
ࠆ߆߹ߚߪᕡ╬⊛ߦ࠯ࡠߢ޽ࠆ߆ߩ޿ߕࠇ߆ߢ޽ࠆߎߣ߇ಽ߆ࠆ㧚ห᭽ߦ㧘ᑼ(9.5.12)
ߦᏀ߆ࠄ aA 1 ࠍ૞↪ߐߖࠆ㧚aA 1 aA 1aA 1 PH ,A

§
1 · 
¨H 
¸ aA 1 PH ,A
2
¨
¸
1
A

©
¹

(9.5.15)ߎߩᑼߣ㧘ᑼ(9.5.11)ߢ A ࠍ A  1 ߦ⟎߈឵߃ߚᑼ

aA 1 aA 1 PH ,A 1

§
1 ·
¨H 
¸ PH ,A 1
2
¨
¸
1
A

©
¹

(9.5.16)ߣᲧߴࠆߣ㧘 aA 1 PH ,A ߪ㧘 PH ,A ߣหߓ࿕᦭୯ H ࠍ߽ߜ㧘 A ߇ 1 ߟᄢ߈޿ PH ,A 1 ߩቯᢙ୚ߢ
޽ࠆ߆㧘ᕡ╬⊛ߦ࠯ࡠߢ޽ࠆ߆ߩ޿ߕࠇ߆ߢ޽ࠆߎߣ߇ಽ߆ࠆ㧚࿕᦭୯ H ࠍᜬߟ⁁ᘒ߇޿ߊߟ߆޽ߞߚߣߒߡ߽㧘ߔߥࠊߜ❗ㅌߒߡ޿ߚߣߒߡ߽㧘ߘߩ❗ㅌߪ᦭㒢୘ߦ㒢ࠄࠇࠆߣ⠨߃ࠄࠇࠆ㧚ᓥߞߡ㧘 aA 1 ࠍ➅ࠅ㄰ߒߡ૞↪ߐߖߡ A ࠍ 1 ߟߕߟჇ߿ߒߡ޿
ߊߣ㧘ߟ޿ߦߪ㧘৻ቯߩ࿕᦭୯ H ߩ߽ߣߢ⸵ߐࠇࠆ A ߩᦨᄢ୯ߦ㆐ߔࠆߪߕߢ޽ࠆ㧚ߎߩ႐วߩ A ߩᦨᄢ୯ࠍ A max ߣ߅ߎ߁㧚ߘ߁ߔࠆߣ㧘

aA max 1 PH ,A max

0(9.5.17)ߣߥࠆ㧚ߎߩ A max ࠍᡷ߼ߡ n  1 ߣ߅ߊߣ㧘ᑼ(9.5.17)ߪ


an PH ,n 10ߣ⴫ߔߎߣ߇಴᧪ࠆ㧚ߎߩᑼࠍ㧘ᑼ(9.5.12)ߢ A ࠍ n  1 ߣ⟎߈឵߃ߚᑼ

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 25(9.5.18)

1 ·§¨ H  2 ¸ PH , n 1
n ¹©
ߣᲧߴࠆߎߣߦࠃࠅ㧘an an  PH , n 1

H



(9.5.19)1
n2(9.5.20)ࠍᓧࠆ㧚ߎࠇࠍᑼ(9.5.4)ߦઍ౉ߔࠆߣ㧘En

P e4 1

1



4SH 0

2

(9.5.21)2= 2 n 2ߎߎߢ㧘࿕᦭୯ E ߪ n ߦࠃߞߡ᳿߹ࠆ߆ࠄ㧘 n ࠍ E ߩ subscript ߣߒߡߟߌߚ㧚ߎߩ

subscript ߩዉ౉ߦࠃߞߡ㧘੹߹ߢߩ subscript H ࠍߔߴߡ n ߦ⟎߈឵߃ࠆߎߣ߇಴᧪ࠆ㧚ߔߥࠊߜ㧘ᑼ(9.5.11)ߩ Schrödinger equation ߪaA aA Pn ,A1·§¨ En  2 ¸ Pn ,A
A ¹
©

A

0, 1, 2, ..., n  1(9.5.22)ߣߥࠆ㧚ࠛࡀ࡞ࠡ࡯ En ߪ n ߩߺߦࠃߞߡ᳿߹ࠅ㧘 A ߦߪଐሽߒߥ޿㧚ߔߥࠊߜ㧘1 ߟ
ߩ n ߦߟ޿ߡ㧘 A 0, 1, 2, ..., n  1 ߦዻߔࠆ n ୘ߩ Pn ,A 㑐ᢙߪ❗ㅌߒߡ޿ࠆ㧚એ਄ߩࠃ߁ߦߒߡ㧘Ṷ▚ሶᴺࠍ↪޿ࠆߎߣߦࠃߞߡ㕖Ᏹߦ◲නߦ eigenvalue㧔ߔߥࠊߜ energy㧕ࠍ᳞߼ࠆߎߣ߇಴᧪ߚ㧚ߘߎߢᰴߦ eigenfunction ߩᒻࠍ᳿߼ࠆߎߣࠍ⠨߃ࠃ߁㧚ᑼ(9.5.18)ߔߥࠊߜ㧘

an Pn ,n 1

0(9.5.23)

ߩ⸃ߪPn ,n 1

U

N U ne n(9.5.24)ߢ޽ࠆߎߣߪ਄ᑼߦ
dn 1an   
dU U n(9.5.25)ࠍ૞↪ߒߡߺࠇ߫ಽ߆ࠆ㧚 N ߪ normalization factor ߢ޽ࠆ㧚ᑼ(9.5.24)ߦ➅ࠅ㄰ߒ aA ࠍ૞↪ߒߡ޿ߌ߫㧘ߔߴߡߩ Pn ,A㧔 A

n  1, n  2, ..., 0 㧕ࠍ㗅‫ߦޘ‬᳿ቯߔࠆߎߣ߇಴᧪ࠆ㧚Pn ,A ߇᳞߹ࠇ߫㧘ᑼ(9.5.1)߆ࠄ R r ߪ⋥ߜߦ᳞߹ࠆ㧚normalization factor ߪߘߩㇺᐲ᳿߼ߡ߿ࠇ߫ࠃ߆ࠈ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 26‫ڏ‬Exercise 9.5.1
2 s ߩ radial wavefunction R2s r ࠍ᳞߼ࠃ㧚‫ޤ⸃ޣ‬㧦
ᑼ(9.5.24)ࠃࠅ㧘normalization factor ࠍ㒰޿ߡ㧘

P21 Uߎࠇߦ㧘 a1U

U 2e 2d1
  1 ࠍ૞↪ߔࠆ㧚
dU U

P20 U

a1 P21 UU §U·
3U e 2 ¨ 1  ¸© 2¹ᓥߞߡᑼ(9.5.3)ࠍ↪޿㧘normalization factor ࠍ N 2s ߣߒߡ

R2s

§
§
r ·
r ·
N 2s ¨1 ¸ exp ¨ ¸
© 2a0 ' ¹
© 2a0 ' ¹N 2s ࠍ᳞߼ࠆߎߣߪ⻉ำߩṶ⠌ߣߒߡᱷߒߡ߅ߎ߁㧚‫ڏ‬Problem 9.5.1
ᑼ(9.5.7),ᑼ(9.5.8)ߢቯ⟵ߒߚ 2 ߟߩṶ▚ሶ aA ߣ aA ߪ੕޿ߦ Hermitian conjugate ߢ޽ࠆߎߣࠍ␜ߖ㧚‫ޣ‬Hint‫ޤ‬㧦ߎࠇࠄߩṶ▚ሶߪ㧘Section 7.5 ߢዉ౉ߒߚ b  㧘b ߩ႐วߣห᭽㧘anti-Hermitian operator
ߣ Hermitian operator ߩ๺߅ࠃ߮Ꮕߣߒߡቯ⟵ߐࠇߡ޿ࠆ㧚b  㧘b ߣหߓࠃ߁ߦ⠨߃ࠇ߫ࠃ޿㧚

9.6 ᳓⚛㘃ૃේሶ

He+㧘Li2+㧘Be3+ߣ޿ߞߚࠗࠝࡦߪ㧘᳓⚛ේሶߣหߓߊේሶᩭ 1 ୘ߣ㔚ሶ 1 ୘ࠃࠅߥࠆ 2 ☸ሶ♽ߢ޽ࠆ߆ࠄ㧘᳓⚛ේሶߩ႐วߣหߓࠃ߁ߦขࠅᛒ߁ߎߣ߇಴᧪ࠆߪߕߢ
޽ࠆ㧚ߚߛේሶᩭߩ㔚⩄߇ e ߢߥߊ Ze 㧔 Z ߪේሶ⇟ภ㧕ߢ޽ࠆߎߣߣ㧘ේሶᩭߩ⾰㊂䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 27߇ᄢ߈޿ߎߣ߇᳓⚛ේሶߩ႐วߣ⇣ߥࠆὐߢ޽ࠆ㧚೨⠪ߩ㔚⩄ߦߟ޿ߡߪ㧘ᩭߣ㔚ሶ

e2

Ze 2
ߣߩ෻⊒ࠍ
ߩઍࠅߦ
ߣߔࠆߎߣߢ⸃᳿ߐࠇࠆ㧚ᓟ⠪ߩ⾰㊂ߦߟ޿ߡߪ㧘
4SH 0 r
4SH 0 rᑼ (9.1.3) ߩ឵▚⾰㊂ࠍߣࠆߎߣߦࠃߞߡ⥄േ⊛ߦ⸃᳿ߐࠇࠆ㧚⚿ዪ㧘ᑼ (9.1.4) ߩ

Hamiltonian ߩ e 2 ߩઍࠅߦ Ze2 ߣ߅ߌ߫㧘᳓⚛㘃ૃේሶߩ Hamiltonian ߇ᓧࠄࠇࠆ㧚
Section 9.1 ߆ࠄ㧘㐳‫ߡߞ߆߆ߣޘ‬㧘ᑼ(9.1.4)ߩ Hamiltonian ߩ⸃ࠍ᳞߼ߚࠊߌߢ޽ࠆ߇㧘ታߪ㧘ᆎ߼߆ࠄ㧘᳓⚛㘃ૃේሶߦߟ޿ߡߩ Hamiltonian ߦߟ޿ߡߩ⸃ࠍ᳞߼ࠇ߫ࠃࠅ
৻⥸⊛ߢ޽ߞߚߪߕߢ޽ࠆ㧚ߚߛ㧘ᧄⓂߦ߅޿ߡߪ㧘Section 9.1 ߆ࠄ Section 9.5 ߹ߢߪ㧘޽߃ߡ㧘 Z 1 ߣ◲නൻߒߚ♽ߩߺߦߟ޿ߡᑼߩዉ಴ࠍⴕߞߚ㧚หߓࠃ߁ߥߎߣ
ࠍ᳓⚛㘃ૃේሶߦߟ޿ߡⴕ߁ߣ㧘ㅜਛߪ⋭⇛ߔࠆ߇㧘eigenvalue ߦߟ޿ߡߪ㧘ᑼ(9.3.22)
ߩઍࠅߦ

En



PZ Z 2e4

1
4SH 0

2

2n 2 = 2

n 1, 2, ...

(9.6.1)ߣߥࠆ㧚ߚߛߒ㧘឵▚⾰㊂ߪ㧘ේሶ⇟ภ Z ߩ㑐ᢙߢ޽ࠆߎߣࠍᒝ⺞ߔࠆߚ߼ߦ P ߩઍࠅߦ P Z ߣ߅޿ߚ㧚

wavefunction ߦߟ޿ߡߪ㧘᳓⚛ߣ⇣ߥࠆߩߪേᓘㇱಽ R r ߩߺߢ޽ࠆ㧚ߎࠇߩౕ૕
⊛ߥᒻߪ㧘ᑼߩዉ಴ߪ⋭⇛ߒߚ߇㧘ߔߢߦ⴫ 9.3.1 ߦ␜ߒߚ㧚ⷺᐲಽᏓ YA ,m T , I ߦߟ޿ߡߪ㧘᳓⚛ේሶߩ႐วߣోߊหߓߢ޽ࠆ㧚᳓⚛㘃ૃේሶߦߟ޿ߡ㧘▵ࠍᡷ߼ߡ߹ߢㅀߴߥߌࠇ߫޿ߌߥ޿ℂ↱ߪߤߎߦ޽ࠆߩ
ߢ޽ࠈ߁߆㧚He+߇ߤߩ૏቟ቯߦሽ࿷ߔࠆ߆ߪ೎໧㗴ߣߒߡ㧘He+ࠍ⍮ࠆߎߣ߇ߘࠇ
߶ߤ㊀ⷐߢ޽ࠆࠃ߁ߦߪᕁ߃ߥ޿ߢ޽ࠈ߁㧚߹ߒߡ߿ Li2+ߣ߆ Be3+ߣ޿ߞߚਇ቟ቯ
ࠗࠝࡦߦߤࠇߛߌ⥝๧߇޽ࠆߣ޿߁ߩߢ޽ࠈ߁߆㧫ታߩߣߎࠈ㧘᳓⚛㘃ૃේሶࠍ⠨߃
ࠆߎߣߩ㊀ⷐᕈߪ㧘᳓⚛ߣห᭽ߦේሶᩭ 1 ୘ߣ㔚ሶ 1 ୘ߩߺ߆ࠄߥࠆ 2 ☸ሶ♽ߦߟ޿ߡߩ⥝๧ࠃࠅ߽㧘߻ߒࠈ㧘ᄙ㔚ሶේሶߦ޽ࠆࠃ߁ߢ޽ࠆ㧚଀߃߫㧘He ߪ㔚ሶ߇ 2 ୘޽ࠅ㧘ߎ߁ߥࠆߣ߽߁ Schrödinger equation ࠍ෩ኒߦ⸃ߊߎߣߪ಴᧪ߥ޿㧚ߒ߆ߒ㧘╙

1 ࠬ࠹࠶ࡊߣߒߡ㧘߹ߕ㧘2 ୘ߩ㔚ሶߘࠇߙࠇ߇+2 ߩ㔚⩄ࠍᜬߟේሶᩭ߆ࠄߩ potentialࠍฃߌߡ޿ࠆ㧘ߣ޿߁ࡕ࠺࡞߆ࠄ಴⊒ߔࠆ㧚㔚ሶߣ㔚ሶߣߩ෻⊒ߪᓟ߆ࠄ⠨߃ࠇ߫ࠃ߆ࠈ߁㧚ߘ߁޿߁⠨߃ᣇ߇ߤࠇ߶ߤᚑഞߔࠆ߆ߪ⇼໧ߢ޽ࠆ߇㧘߹޽㧘ߣࠅ޽߃ߕߘ
߁޿߁┙႐ߢ⠨߃ߡߺࠆߎߣߪᗧ๧ߩߥ޿ߎߣߢߪ޽ࠆ߹޿㧚ߎ߁޿߁႐ว㧘᳓⚛㘃ૃේሶߩᵄേ㑐ᢙ߿ energy ߇ᔅⷐߣߥࠆࠊߌߢ޽ࠆ㧚ࠕ࡞ࠞ࡝㊄ዻේሶࠍ᳓⚛㘃ૃේሶߣߒߡᛒ߁ߎߣߪ㧘਄ߩ଀ࠃࠅߪዋ‫޿ߥࠇߒ߽߆ߒ߹ޘ‬㧚଀߃߫ Li ߩ႐ว㧘2 ߟߩ㔚ሶߪ K Ზߦ޽ࠅ㧘L ᩭߦ޽ࠆ 1 ୘ߩ㔚ሶ߆ࠄ⷗ࠇ߫ߎߩ K ᩭߦ޽ࠆ 2 㔚ሶࠍේሶ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 28ᩭߣ޿ߞߒࠂߦߒߡ⠨߃ࠆߎߣߪ╙ 1 ߩㄭૃߣߒߡ㧘ᖡߊߪߥߐߘ߁ߢ޽ࠆ㧚ߣߦ߆ߊ㧘㔚ሶࠍ 2 ୘એ਄ᜬߟේሶߦߟ޿ߡߪ㧘 Schrödinger equation ߩᱜ⏕ߥ⸃߇ᓧࠄࠇ
ߥ޿ߚ߼ߦ㧘ᓟߢቇ⠌ߔࠆࠃ߁ߥߐ߹ߑ߹ߥㄭૃᴺࠍ㚟૶ߒߡ⠨߃ࠆࠊߌߢ޽ࠆ߇㧘ߘߩ႐ว㧘᳓⚛㘃ૃේሶߦߟ޿ߡߩ⸃߇ᔅⷐߣߥࠆߎߣ߇ᄙ޿ߩߢ޽ࠆ㧚

9.7 ේሶ゠㆏㑐ᢙᚒ‫ߣߞ߿ߪޘ‬㧘᳓⚛ේሶߦߟ޿ߡ㧘ቢోߥᒻߩᵄേ㑐ᢙࠍ⺰ߕࠆߎߣ߇಴᧪ࠆᲑ㓏ߣߥߞߚ㧚ේሶߦߟ޿ߡߩ wavefunction ࠍ․ߦ atomic orbital ߣ޿߁⸒⪲ߢ⴫ߔ㧚orbitalߣ޿߁⸒⪲ߪ㧘૗ࠍᗧ๧ߒߡ޿ࠆ߆ࠃߊಽ߆ࠄߥ޿޽޿߹޿ߥ⸒⪲ߦ⡞ߎ߃ࠆ㧚orbitߣ޿߁ߩߪ゠㆏ߢ޽ࠅ㧘ߘߩ⸒⪲ߩᗧ๧ߪ㧘଀߃߫㧘‫ޟ‬ᄥ㓁♽ߦ߅޿ߡ࿾⃿ߩㅢࠆ゠㆏‫ࠆ޿ߡߒࠅ߈ߞߪߦ߁ࠃ߁޿ߣޠ‬㧚orbital ߣ޿߁⸒⪲ߪ㧘ᱧผ⊛ߦߪ㧘㔚ሶ߇ේሶᩭߩ๟ࠅߩ㧘޽ࠆ‫゠ޟ‬㆏‫ࠍޠ‬࿁ߞߡ޿ࠆ㧘ߣ޿߁ Bohr ߩ model ߆ࠄ⠨߃ࠄࠇߚ⸒⪲߆߽ߒࠇߥ޿㧚ߒ߆ߒ㧘ߎߩ┨ߢቇ⠌ߒߚࠃ߁ߦ㧘 Schrödinger equation ߩ⸃ߪ㧘㔚ሶ߇ⓨ㑆਄ߩ޽ࠆ႐ᚲߦ⷗಴ߐࠇࠆ⏕₸ࠍ␜ߒߡ޿ࠆࠊߌߢ㧘߽ߪ߿㧘‫゠ޟ‬㆏ࠍ࿁ߞ
ߡ޿ࠆ‫߁޿ߣޠ‬᭎ᔨߪߥ޿㧚ߒ߆ߒ㧘ߣߦ߆ߊ㧘ේሶߩ wavefunction ߪ atomic orbitalߣ޿߁⸒⪲ߢ⃻ታߦ↪޿ࠄࠇߡ޿ࠆ㧚ߎߩ orbital ߣ޿߁⸒⪲ߪ㧘2 㔚ሶએ਄ߩ㔚ሶࠍᜬߟේሶࠍ⠨߃ࠆᲑ㓏ߢߪ㧘ࠃࠅߪߞ߈ࠅߣߒߚᗧ๧ߠߌࠍߒߥߌࠇ߫ߥࠄߥ޿߇㧘ߎߎߢߪߣࠅ޽߃ߕ㧘‫ޟ‬wavefunction ߔߥࠊߜ orbital‫ߦ޿߹޿޽ߣޠ‬ℂ⸃ߒߡ߅ߊߎߣ
ߦߒࠃ߁㧚᳓⚛ේሶ߅ࠃ߮᳓⚛㘃ૃේሶߩ atomic orbital ߪ㧘ᑼ(9.1.7)\ r,T , I

R r Y T ,Iߢ⴫ࠊߒߚࠊߌߢ޽ࠆ㧚േᓘㇱಽ R r ߪ n ߣ A ߣ޿߁ 2 ߟߩ㊂ሶᢙߢⷙቯߐࠇߚ㧚ⷺᐲㇱಽ Y T , I ߪ A ߣ m ߩ 2 ߟߩ㊂ሶᢙߢⷙቯߐࠇߚ㧚ߘ߁޿߁ߎߣߢ㧘ోᵄേ㑐ᢙ\ r ,T , I ߪ㧘 n 㧘 A 㧘߅ࠃ߮ m ߣ޿߁ 3 ߟߩ㊂ሶᢙߢⷙቯߐࠇࠆߎߣߦߥࠆ㧚ߔߥࠊߜ\ n ,A , m r , T , I

Rn ,A r YA ,m T , I
Rn ,A r 4A ,m T

1 imI
e
2S

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 29(9.7.1)

Rn ,A r ߦߟ޿ߡߪౕ૕⊛ߥ㑐ᢙᒻࠍ⴫ 9.3.1 ߦ␜ߒߚ㧚߹ߚ 4A ,m T ߦߟ޿ߡߩౕ૕⊛ߥᒻߪߔߢߦ⴫ 5.5.1 ߢ␜ߒߚ㧚ᓥߞߡ㧘ోᵄേ㑐ᢙߪᑼ(9.7.1)ߦᓥߞߡ㧘නߦㆡᒰߥ
㗄ߩⓍࠍߣࠆߛߌߢ޽ࠅ㧘૗ߩ໧㗴߽ߥ޿㧚ߒ߆ߒ㧘ᓟߢߩଢቱ߽⠨߃ߡ㧘৻ᔕ n d 3
ߩฦ orbital ࠍ⴫ 9.7.1 ߦ߹ߣ߼ߡ߅ߊߎߣߦߒࠃ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 30

⴫ 9.7.1 ᳓⚛㘃ૃේሶߩේሶ゠㆏㑐ᢙ\ n ,A , m r , T , I 㧔ⶄ⚛ᢙ⴫␜㧕§ m ·a0 ' a0 ¨1  e ¸
© M¹

Z
r
a0 '

U

3

§ Z · U¨
¸e
© a0 ' ¹1\ 1,0,0S\ 2,0,01
4 2S\ 2,1,1

1

3

U

§ Z ·
2

2
e
U
¨
¸
© a0 ' ¹

3

8 S

U

§ Z ·
 iI
2
¨
¸ U e sin T e© a0 ' ¹\ 2,1,0

1
4 2S

U

§ Z ·
2
¨
¸ U e cos T
'
a
© 0 ¹

\ 2,1,1

1



3

3

8 S

U

§ Z ·
2
sin T e iI
e
U
¨
¸
'
a
© 0 ¹

3

\ 3,0,0

1
81 3S

U

§ Z ·
2
3


27
18
2
e
U
U
¨
¸
'
a
© 0 ¹

\ 3,1,1

1

81 S

U

§ Z ·
2
3

6
sin T eiIeUU¨¸
© a0 ' ¹\ 3,1,0\ 3,1,1\ 3,2,2

3

3

U

§ Z ·
2
3

6
cos T
e
U
U
¨
¸
'
a
© 0 ¹

2
81 S
1
81 S

3

U

§ Z ·
2
 iI
3
¨
¸ 6 U  U e sin T e© a0 ' ¹

1
162 S

3

§ Z · 2  U3
2
2 iI
¨
¸ U e sin T e© a0 ' ¹䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 31

3\ 3,2,1

181 S§ Z · 2  U3
 iI¨
¸ U e sin T cos T e© a0 ' ¹\ 3,2,0

1
81 6S§ Z · 2  U32¨
¸ U e 3cos T  1© a0 ' ¹\ 3,2,1

1
81 S§ Z · 2  U3
 iI¨
¸ U e sin T cos T e© a0 ' ¹\ 3,2,2

1
162 S

3

3

3

§ Z · 2  U3
2
2 iI
¨
¸ U e sin T e© a0 ' ¹‫ޤ⸼ޣ‬㧦᳓⚛㘃ૃේሶߩᵄേ㑐ᢙࠍⶄ⚛ᢙ⴫␜ߢ⴫ࠊߔߦᒰࠅ㧘Section 5.5 ߩ⚳ࠅߦᒝ⺞ߒߡ߅޿ߚ phase factor ߇㕖Ᏹߦ㊀ⷐߢ޽ࠆ㧚Section 5.5 ߢᒝ⺞ߒߚࠃ߁ߦᧄ
lecture note ߢߪ㧘ℂ⺰ߩ৻⽾ᕈࠍᄢ੐ߦߒߡ Condon-Shortley ᵹߩ phase ࠍណ↪ߒߡ޿ࠆ㧚ᓥߞߡ㧘Condon-Shortley ᵹߩ phase ࠍណ↪ߒߡ޿ߥ޿ᄙߊߩᢎ⑼ᦠߣߪᄸᢙߩ A 㧘
ᱜߩ m ߩ wavefunction ߦߟ޿ߡ╓ภ߇෻ኻߦߥߞߡ޿ࠆߎߣߦචಽᵈᗧߔߴ߈ߢ޽
ࠆ㧚଀߃߫㧘
Eyring, Walter, Kimball ߩ"Quantum Chemistry"ߢߪ page 88 ߩ 6.34 ᑼߦ\ 2,1,1 㧘\ 2,1,0 㧘߅ࠃ߮\ 2,1,1 ߩᒻ߇ਈ߃ࠄࠇߡ޿ࠆ߇㧘ߎߩ⚿ᨐߣ⴫ 9.7.1 ߣࠍᲧߴࠆߣ㧘\ 2,1,1
ߩ╓ภ߇෻ኻߦߥߞߡ޿ࠆߎߣ߇ಽ߆ࠆߢ޽ࠈ߁㧚Section 5.5 ߢ߽ㅀߴߚࠃ߁ߦ㧘ߎߩߎߣ߇㧘ᄢ߈ߥᷙੂࠍ᜗޿ߡ޿ࠆߩߢ㧘⚦ᔃߩᵈᗧ߇ᔅⷐߢ޽ࠆ㧚⴫ 9.7.1 ߦ␜ߒߚ orbital ߪ㧘ߘߩⷺᐲㇱಽ YA ,m T , I ߇ⶄ⚛ᢙߣߥߞߡ޿ࠆ㧚ߎࠇࠄ

orbital ߦߟ޿ߡߩ eigenvalue ߪᑼ(9.5.21)ߦ␜ߒߚࠃ߁ߦ n ߛߌߩ㑐ᢙߢ޽ࠆ㧚ߔߥࠊߜ㧘ห৻ߩ n ࠍ߽ߜ㧘 A 㧘 m ߇⇣ߥࠆ n 2 ୘ߩ orbital ߪߔߴߡ❗ㅌߒߡ޿ࠆ㧚❗ㅌߒߡ޿ࠆ㒢ࠅߘߩ㑆ߢߤߩࠃ߁ߥ✢ᒻ⚿วࠍߣߞߡ߽޿޿ࠊߌߢ޽ࠆ㧚wavefunction ߇ⶄ⚛ᢙߢ⴫␜ߐࠇߡ޿ߡ߽ਇㇺวߥὐߪ૗߽ߥ޿߇㧘ታⓨ㑆਄ߢ orbital ߩ࿑ࠍឬ޿ߚࠅ
ߔࠆ႐วߩߎߣࠍ⠨߃ࠆߣ wavefunction ߇ታᢙߦߥࠆࠃ߁ߦ✢ᒻ⚿วࠍߣߞߡ߅ߊߣ
૗߆ߣଢ೑ߢߪ޽ࠈ߁㧚଀߃߫ A 1 ߩ p ゠㆏ߦߟ޿ߡߪᰴߩࠃ߁ߥ 3 ߟߩ real ߩ orbital ࠍቯ⟵ߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 32

px1
Y1,1  Y1, 1
2
3
sin T cos I
4S(9.7.2a)

3 x
4S r
py

1
i Y1,1  Y1, 1
2
3
sin T sin I
4S(9.7.2b)

3 y
4S r
pz

Y1,0
3
cos T
4S

(9.7.2c)

3 z
4S r
‫ޤ⸼ޣ‬
㧦
ߔߋ਄ߩ㧔⸼㧕ߢ߽ㅀߴߚࠃ߁ߦ㧘ᧄ lecture note ߢߪ Condon-Shortley ߩ phase factorࠍ↪޿ߡ޿ࠆߚ߼ߦ㧘real ߥ wavefunction ߪᑼ(9.7.2a)㧘ᑼ(9.7.2b)ߩࠃ߁ߦ⴫ࠊߐࠇࠆ㧚⇣ߥߞߚ phase factor ࠍ↪޿ߡ޿ࠆ Eyring, Walter, Kimball ߩᢎ⑼ᦠߩ page 89, 6.35 ᑼߢߪ㧘 px , p y ߇⇣ߥߞߚᑼߢ⴫ࠊߐࠇߡ޿ࠆ߇㧘ᧄ lecture note ⥄૕ߪߔߴߡ consistentߦߥߞߡ޿ߡ㧘㑆㆑޿ߢ߽ࡒࠬࡊ࡝ࡦ࠻ߢ߽ߥ޿ߎߣࠍ㊀ߨߡᒝ⺞ߒߡ߅ߊ㧚A

2 ߩ d ゠㆏ߦߟ޿ߡߪ㧘ᰴߦ␜ߔࠃ߁ߥ 5 ߟߩ real ߥ orbital ࠍቯ⟵ߔࠆ㧚
d z2

Y2,0
5
3cos 2 T  1
16S
5 3z 2  r 2
r2
16S

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 33(9.7.3a)1
Y2,2  Y2,2
2

d x2  y 2

15
sin 2 T cos 2I
16S(9.7.3b)15 x 2  y 2
16S r 2d xy

1
i Y2,2  Y2, 22



15
sin 2 T sin 2I
16S(9.7.3c)

15 xy
16S r 2d yz

1
i Y2,1  Y2,1
2



15
sin T cos T sin I
4S(9.7.3d)

15 yz
4S r 2d zx

1
Y2,1  Y2, 1
2
15
sin T cos T cos I
4S(9.7.3e)

15 zx
4S r 2ߎࠇࠄߩ real ߥ㑐ᢙߩ߁ߜ㧘․ߦ d z 2 ߩ㑐ᢙᒻߦᵈᗧߔߴ߈ߢ޽ࠈ߁㧚ߎߩ㑐ᢙߪ਄ߢ␜ߒߚࠃ߁ߦ

3z 2  r 2
ߩᒻࠍߒߡ޿ࠆ㧚ઁߩ߽ߩߣหߓࠃ߁ߦ⴫ߔߥࠄ㧘 d3z 2  r 2 ߣ
r2ߢ߽␜ߖ߫޿޿ߩ߆߽ߒࠇߥ޿߇㧘notation ߩ◲⇛ൻߩߚ߼ߦ㧘ㅢᏱ d z 2 ߣ޿߁⸥ภࠍ
↪޿ߡ޿ࠆ㧚
એ਄ㅀߴߡ᧪ߚࠃ߁ߦ real ߥᵄേ㑐ᢙߪ㧘⴫ 9.7.1 ߦ␜ߒߚᵄേ㑐ᢙߩනߥࠆ✢ᒻ⚿วߢ޽ࠆ߇㧘ᔨߩߚ߼⴫ 9.7.2 ߦ߹ߣ߼ߡ߅ߊ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 34

⴫ 9.7.2 ᳓⚛㘃ૃේሶߩේሶ゠㆏㑐ᢙ㧔ታᢙ⴫␜㧕
Z
§ m ·
U
r
a0 ' a0 ¨1  e ¸
a0 '
© M¹

\ 1s

\ 2s

\2p

\2p

x

y

\2p

z

\3p

x

y

\3p

\ 3d

z

z2

S

1
4 2S

3

U

§ Z ·
2
¨
¸ 2 U e
'
a
© 0 ¹

3

1
4 2S

U

§ Z ·
2
¨
¸ U e sin T cos I
'
a
© 0 ¹

1
4 2S

U

§ Z ·
2
sin T sin I
e
U
¨
¸
'
a
© 0 ¹

3

3

U

§ Z ·
2
cos T
e
U
¨
¸
'
a
© 0 ¹

1
4 2S

1
81 3S

\ 3s

\3p

3

§ Z · U
¨
¸e
a
'
© 0 ¹

1

§ Z ·  U2¨
¸ xe
© a0 ' ¹1
4 2S§ Z ·  U2¨
¸ ye
© a0 ' ¹

1
4 2S

5

5

§ Z ·  U2¨
¸ ze
© a0 ' ¹

3

U

§ Z ·
2
3


27
18
2
e
U
U
¨
¸
© a0 ' ¹

3

2
81 2S

U

§ Z ·
2
3

6
sin T cos I
e
U
U
¨
¸
'
a
© 0 ¹

2
81 2S

U

§ Z ·
2
3
¨
¸ 6 U  U e sin T sin I
'
a
© 0 ¹

3

3

2
81 2S

U

§ Z ·
2
3
¨
¸ 6 U  U e cos T
'
a
© 0 ¹

1
81 6S

§ Z · 2  U32¨
¸ U e 3cos T  1
'
a
© 0 ¹

3

5

1
4 2S

5

2
81 2S

U

§ Z ·
3

6
x
e
U
¨
¸
'
a
© 0 ¹

2
81 2S

U

§ Z ·
3
¨
¸ y 6U e
'
a
© 0 ¹

5

5

2
81 2S

U

§ Z ·
3
¨
¸ z 6U e
'
a
© 0 ¹

1
81 6S

U

§ Z ·
2
2
3

3
z
r
e
¨
¸
'
a
© 0 ¹

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 35

7

\ 3d

2
81 2S

x2  y 2

\ 3d

\ 3d

\ 3d

yz

zx

xy

3

§ Z · 2  U32¨
¸ U e sin T cos 2I
'
a
© 0 ¹

2
81 2S

3

2
81 2S

§ Z · 2  U32¨
¸ U e sin T sin 2I
'
a
© 0 ¹

1
81 2S

§ Z · 2  U3¨
¸ U e sin T cos T sin I© a0 ' ¹1
81 2S§ Z · 2  U3¨
¸ U e sin T cos T cos I
'
a
© 0 ¹

3

3

§ Z ·
¨
¸
© a0 ' ¹

2
81 2S

7



U

x2  y 2 e 3

7

U

§ Z ·
3
xye
¨
¸
'
a
© 0 ¹

7

2
81 2S

U

§ Z ·
3
yze
¨
¸
© a0 ' ¹

2
81 2S

U

§ Z ·
3
zxe
¨
¸
'
a
© 0 ¹

7

‫ڏ‬Problem 9.7.1
᳓⚛ේሶߩၮᐩ⁁ᘒߦ߅޿ߡ
㧔a㧕 ේሶᩭߩ૏⟎߆ࠄ Bohr radius ࠃࠅ߽ㄭ޿ᚲߦ㔚ሶ߇ሽ࿷ߔࠆ⏕₸ࠍ᳞߼ࠃ㧚㧔b㧕 หߓߊ㧘Bohr radius ࠃࠅ߽㆙޿ᚲߦሽ࿷ߔࠆ⏕₸ࠍ᳞߼ࠃ㧚㧔ᔨߩߚ߼㧘㧔a㧕㧘㧔b㧕ࠍߘࠇߙࠇ᳞߼㧘ߘߩ๺߇ 1 ߦߥࠆߎߣࠍ⏕߆߼ࠃ㧕䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 36‫ ڏ‬Problem 9.7.2
޽ࠆ Hydrogen-like ion ߦߟ޿ߡߩ 1s  2 p ㆫ⒖ߩታ᷹ࠬࡍࠢ࠻࡞ߩᵄ㐳 On ߪᰴߩ⴫ߩㅢࠅߢ޽ߞߚ‫ޕ‬n

On / nm

2
3
4
5
6
7
8
9
10

30.3782
25.6547
24.322223.7264
23.4340
23.2540
23.1472
23.0691
23.0208એਅߩ໧޿ߦ╵߃ࠃ‫ޕ‬
㧔1㧕ࠛࡀ࡞ࠡ࡯ߩታ᷹୯ 1/ On ߣ Rydberg ߩᑼ
1

On

1 ·
§
RZ 2 ¨1  2 ¸© n ¹

ߦၮߠߊℂ⺰ᑼߣߩᏅߩੑਸ਼ߩ๺ࠍᦨዊߦߔࠆᦨዊੑਸ਼ᴺࠍ↪޿ߡ RZ 2 ࠍ᳿ቯߖࠃ‫ޕ‬
㧔2㧕

ฦ n ߦߟ޿ߡ‫ᦨޔ‬ዊੑਸ਼ᴺߦࠃࠆᦼᓙ୯㧔ߔߥࠊߜ㧔1㧕ߢ᳞߼ߚ RZ 2 ࠍ↪޿ߚ Rydberg ߩᑼߩ୯㧕ࠍᲧセߖࠃ‫ޕ‬

㧔3㧕ࠗࠝࡦߪ૗߆㧔 Z ߪ޿ߊߟ߆㧕

㧔4㧕
㧔5㧕

ߎߩࠗࠝࡦߦߟ޿ߡ‫ޔ‬ห૏૕ߦࠃࠆ R ߩߜ߇޿ߪߤߩ૏ߦߥࠆ߆‫ޕ‬
㧔2㧕ߣ㧔4㧕ߩ⚿ᨐ߆ࠄ‫ߩߎޔ‬ታ㛎ߢห૏૕ߩ඙೎߇಴᧪ࠆ߆ߤ߁߆ࠍี๧
ߖࠃ‫ޕ‬‫ޣ‬ᵈᗧ‫ޤ‬㧦৻⥸⊛ߦ㑐ᢙ㔚ථౝ⬿ߩࡊࡠࠣ࡜ࡓࠍ૶߁ߎߣߪ⦟޿ߎߣߢ޽ࠆߣᕁ߁‫ޔߒߛߚޕ‬ߎߩ໧㗴ߩ㧔1㧕ߩᦨዊੑਸ਼ᴺߦ㑐ߒߡߪ‫ޔ‬ౝ⬿ࡊࡠࠣ࡜ࡓࠍ↪޿ࠆߎߣߥߊ㧔ߒ߫ߒ߫૗ࠍ߿ߞߡ޿ࠆߩ߆ਇ᣿ߥߎߣ߇ᄙ޿߆ࠄ㧕⥄ಽߢℂ⺰ࠍዷ㐿ߔࠆߎߣ‫ޕ‬䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 37‫ڏ‬Problem 9.7.3
਄ߩ Problem 9.7.2 ߦ߅޿ߡ㧘ᵄ㐳ߦߟ޿ߡߩᦨዊੑਸ਼ᴺࠍ⹜ߺࠃ㧚਄ߢⴕߞߚᵄ
㐳ߩㅒᢙߦߟ޿ߡߩᦨዊੑਸ਼ᴺߣߪᄙዋ⇣ߥࠆ⚿ᨐ߇ᓧࠄࠇࠆߎߣߦᵈᗧߒࠃ߁㧚

9.8 ᳓⚛ේሶߩේሶ゠㆏㑐ᢙߩ࿑␜㧚㧔1㧕േᓘᵄേ㑐ᢙߩ⴫␜᳓⚛ේሶ߅ࠃ߮᳓⚛㘃ૃේሶߩ wavefunction ࠍ೨▵߹ߢߦ᳞߼ߡ᧪ߚࠊߌߢ޽ࠆ
߇㧘ᢙᑼߩ⴫⃻ߛߌߢߪᔅߕߒ߽ߪߞ߈ࠅℂ⸃಴᧪ߥ޿႐ว߇޽ࠆ㧚੹߹ߢ㧘▫ߩਛߩ☸ሶ㧘࿁ォㆇേ㧘⺞๺ᝄേሶ╬‫ߩߡోޘ‬႐วߦⴕߞߡ᧪ߚࠃ߁ߦ wavefunction \ ߅ࠃ߮ߘߩ⏕₸ಽᏓ㑐ᢙ \ ࠍᐳᮡߦኻߒߡࡊࡠ࠶࠻ߒߚ޿㧚․ߦ㧘ේሶߩᵄേ㑐ᢙࠍ
2ℂ⸃ߔࠆߎߣߪ㧘ൻቇߩ⃻⽎ࠍℂ⸃ߔࠆߩߦ㕖Ᏹߦ㊀ⷐߢ޽ࠆߎߣ߆ࠄ߽㧘ߥࠎߣ߆ߒߡ࿑ߢ⴫⃻ߒߚ޿߽ߩߢ޽ࠆ㧚࿑ߢ⴫ߔ႐ว㧘૗ࠍ࿑ߢ⴫ߔ߆ߣ޿߁ߣᄢ߈ߊಽߌߡᰴߩ 2 ㅢࠅ߇޽ࠆ㧚㧔a㧕ᵄേ㑐ᢙ\ ߘߩ߽ߩߩࡊࡠ࠶࠻
㧔b㧕ᵄേ㑐ᢙߩੑਸ਼㧘ߔߥࠊߜ㧘⏕₸ኒᐲ㑐ᢙ\ *\\ ߩࡊࡠ࠶࠻
2ߚߛߒ㧘ߎࠇࠄߩࡊࡠ࠶࠻ߪታ㓙໧㗴ߣߒߡߪߥ߆ߥ߆߻ߕ߆ߒ޿㧚ߣ޿߁ߩߪᰴߩࠃ߁ߥℂ↱߇޽ࠆ߆ࠄߢ޽ࠆ㧚\ ߪ৻⥸⊛ߦߪ 3 ᰴరߩᐳᮡ r , T , I㧔߹ߚߪ x, y, z 㧕ߩ㑐ᢙߢ޽ࠅ㧘\ ߹ߚߪ \ ࠍߎࠇࠄߩᐳᮡߣߒߡࡊࡠ࠶࠻ߒࠃ߁ߣߔࠆߣ 4 ᰴరߩ
2ᐳᮡ߇ᔅⷐߢ޽ࠆ㧚⚕ߩ਄ߦ↹ߊ㒢ࠅ㧘2 ᰴర㧘߹ߚߪᎿᄦࠍߒߡ߽ 3 ᰴరߩ࿑ߦ㒢ࠄࠇࠆ㧚ߘࠇߢߤ߁ߔࠆ߆ߣ޿߁ߣᄢ߈ߊಽߌߡ 2 ߟߩᣇᴺ߇޽ࠆ㧚1 ߟߪ r , T , I㧔߹ߚߪ x, y, z 㧕ߩ 3 ߟߩᐳᮡߩ߁ߜ 1 ߟ෶ߪ 2 ߟࠍ޽ࠆ୯ߦ࿕ቯߒߡߒ߹޿㧘ᱷࠅߩ 1
ߟ෶ߪ 2 ߟߩᐳᮡߦߟ޿ߡ \ ߹ߚߪ \ ࠍ࿑␜ߔࠆߣ޿߁ᣇᴺߢ޽ࠆ㧚߽߁ 1 ߟߪ㧘
2ᄌᢙಽ㔌ߩ⠨߃ߦࠃࠅ㧘ోᵄേ㑐ᢙߩ߁ߜ޽ࠆᐳᮡߩ㑐ᢙߩߺߦߟ޿ߡࡊࡠ࠶࠻ߔࠆ
ᣇᴺߢ޽ࠆ㧚޿ߕࠇߩᣇᴺ߽ᰳὐ߇޽ࠅ㧘૗ࠍࡊࡠ࠶࠻ߒߡ޿ࠆߩ߆ߦߟ޿ߡᱜ⏕ߦᛠីߒߡߥ޿ߣߣࠎߢ߽ߥ޿⺋⸃ࠍ↢ߕࠆෂ㒾ᕈ߇޽ࠆ㧚߹ߚ㧘ᓧࠄࠇߚ࿑߆ࠄ㧘

wavefunction ߩᧄᒰߩᆫࠍℂ⸃ߔࠆߎߣߪߥ߆ߥ߆㔍ߒ޿㧚ߒ߆ߒ㧘ਃᰴరⓨ㑆ߦ૑ࠎߢ޿ࠆᚒ‫ߪߡߞߣߦޘ‬ᱷᔨߥ߇ࠄߎࠇߒ߆಴᧪ߥ޿㧚ߚߛ㧘ߘࠇߛ߆ࠄߎߘ㧘޿ࠈ
޿ࠈߥ⒳㘃ߩ࿑␜ᣇᴺ߇⠨߃ࠄࠇߡ᧪ߚߣ޿߁ߎߣ߽੐ታߢ޽ࠈ߁㧚એਅ㧘ߘࠇࠄߩ߁ߜ޿ߊߟ߆ߦߟ޿ߡౕ૕⊛ߦ࿑ࠍ␜ߒߡ޿߈ߚ޿㧚ߎߩ▵ߢߪ߹ߕᚻᆎ߼ߦ㧘േᓘᵄേ㑐ᢙߩ r ߦߟ޿ߡߩࡊࡠ࠶࠻ࠍ⹜ߺࠃ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 38㧔a㧕േᓘᵄേ㑐ᢙ⥄り Rn ,A r ߩ r ߦߟ޿ߡߩࡊࡠ࠶࠻

േᓘᵄേ㑐ᢙ Rn ,A r ࠍ r ߦߟ޿ߡࡊࡠ࠶࠻ߔࠆᣇᴺߢ㧘࿑ 9.8.1 ߦ 1s ߆ࠄ 3d ߹ߢ 6⒳㘃ߩേᓘᵄേ㑐ᢙߦߟ޿ߡ␜ߒߡ޽ࠆ㧚㕖Ᏹߦࡐࡇࡘ࡜࡯ߥ࿑ߢߚ޿ߡ޿ߩᢎ⑼ᦠ
ߦタߞߡ޿ࠆ߇㧘ߎߩ࿑ߩᗧ๧ߔࠆߎߣߪᔅߕߒ߽ߪߞ߈ࠅߒߥ޿㧚ᵄേ㑐ᢙ\ ߪ\ n ,A ,m r , T , I

Rn ,A r 4A ,m T ) m I

Rn ,A r YA ,m T , I(9.8.1)ߢ␜ߐࠇࠆߣ޿߁ߩߦ㧘ߎߩ߁ߜߩ Rn ,A r ߛߌࠍขࠅ಴ߒߡࡊࡠ࠶࠻ߒߡ޿ࠆߩߢ޽
ࠆ㧚ᓥߞߡ㧘Rn ,A rߩࡊࡠ࠶࠻ߪ㧘YA , m T , I

1 ߦߥࠆࠃ߁ߥ᧦ઙਅߢߩ\ n ,A ,m r , T , Iߩࡊࡠ࠶࠻ߣ޿߁ߎߣߦߥࠆ㧚ߒ߆ߒ㧘YA , m T , I

1 ߦߥࠆߎߣߪ޽ࠅᓧߥ޿ߎߣߪ

᣿ࠄ߆ߢ޽ࠆ㧚ߒ޿ߡ⸒߃߫㧘T , I ߩ 2 ߟߩᐳᮡࠍ࿕ቯߒߡߒ߹ߞߡ㧘\ n ,A ,m r , T , Iࠍࡊࡠ࠶࠻ߔࠆ㧚ߘߒߡߘߩᤨߩ\ n ,A , m r , T , I ߩ୯ߪ㧘࿑ 9.8.1 ߩ❑ゲߦ␜ߐࠇߚ୯ߦ㧘࿕ቯߒߚ T 㧘 I ߦ߅ߌࠆ YA ,m T , I ࠍដߌߚ߽ߩߦߥࠆ㧚ߣ޿߁ߎߣߦߥࠆ㧚ᘠࠇߥ޿߁ߜߪߥ߆ߥ߆ಽ߆ࠅߦߊ޿࿑ߢ޽ࠆߣߪᕁ߁߇㧘ቯᕈ⊛ߦ޿ߊߟ߆ߩⷐὐࠍ߅ߐ߃
ߡ߅ߊߣࠃ߆ࠈ߁㧚߹ߕ㧘 s 㑐ᢙߢߪ R 0 z 0 㧘 s 㑐ᢙએᄖ㧔ߔߥࠊߜ p 㑐ᢙ߿ d 㑐ᢙ㧕ߢߪ R 00ߣ

޿߁ Section 9.3 ߢቯᕈ⊛ߦี๧ߒߚߎߣ߇ߜ߾ࠎߣࠣ࡜ࡈߦ⴫ߐࠇߡ޿ࠆ㧚߹ߚ㧘r0 એᄖߩ r ߢ㧘R 0

0 ߦߥࠆࠃ߁ߥ r ߩᐳᮡ߇޿ߊߟ޽ࠆ߆ࠍᢙ߃ߡߺࠃ߁㧚1s ゠

㆏ߢ 0 ୘㧘 2 s ゠㆏ߢ 1 ୘㧘 2 p ゠㆏ߢ 0 ୘㧘 3s ゠㆏ߢ 2 ୘㧘 3 p ゠㆏ߢ 1 ୘㧘 3d ゠㆏ߢ 0 ୘㧚ߎ߁޿߁ߎߣߢ㧘 R 0

0 ߦߥࠆ r ߩᐳᮡߪ n  A  1 ୘޽ࠆߣ޿߁ߎߣ߇ಽ

߆ࠆ㧚 n ߇ᄢ߈ߊߥࠆߦߟࠇߡ r ߩᄢ߈޿ᚲ߹ߢ wavefunction ߇ᐢ߇ߞߡ޿ࠆߣ޿߁
․ᓽ߽ℂ⸃ߒߡ߅ߊߣ޿޿ߛࠈ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 39

1s

2s

2

0.2

0.1
1
0

0

-0.1
5

10

5

10

3s

2p
0.1

0.2

0.05
0.1
0
-0.05

0
5

5

10

3p

10

3d

0.1
0.04
0.05
0.02

0

-0.05
5

࿑ 9.8.1

10

0
5

Rn ,A r ߩ㑐ᢙ୯ߩ r ߦߟ޿ߡߩࡊࡠ࠶࠻㧚❑ゲߪ a0 Rn ,A r 㧘ᮮゲߪ3䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 40

r
a0

10

2

2

㧔b㧕േᓘᵄേ㑐ᢙߩ⏕₸ኒᐲ㑐ᢙ Rn ,A r 㧔߹ߚߪ r 2 Rn ,A r 㧕ߩ r ߦߟ޿ߡߩࡊࡠ࠶
࠻
2

Rn ,A r ࠍ r ߦߟ޿ߡ plot ߒߚ߽ߩࠍ n 1, 2, 3 ߩߔߴߡߩ state ߦߟ޿ߡ࿑ 9.8.2 ߦ␜ߔ㧚ߎߩ࿑ߪ㧘ⷺᐲㇱಽ T ߅ࠃ߮ I ࠍ࿕ቯߒߚᤨߩ㧘޽ࠆ㒢ࠄࠇߚᣇะ਄ߢߩሽ࿷⏕₸ኒᐲߩ⋧ኻ⊛ᄌൻࠍ␜ߒߚ߽ߩߣℂ⸃ߐࠇࠆ㧚❑ゲߦߪ⋡⋓߇ઃ޿ߡ޿ࠆ߇㧘⌀ߩኒᐲߪ㧘ߎߩ୯ߦ YA ,m T , I

2ߩ୯ࠍដߌߚ߽ߩࠍߣࠆߣ޿߁ࠃ߁ߦℂ⸃ߔߴ߈ߢ޽ࠆ㧚ߎߩࠃ߁ߥ೙㒢ࠍ߽ߞߡ࿑ࠍ⷗ߥߌࠇ߫ߥࠄߥ޿߇㧘޿ߊߟ߆ߩቯᕈ⊛ߥᕈ⾰ߪ᣿⍎ߦ⃻ࠇߡ޿ࠆ㧚଀߃߫ s ゠㆏ߪේሶᩭߩ૏⟎ߢᄢ߈ߥ㔚ሶኒᐲࠍᜬߟ߇㧘p ゠㆏ߣ d゠㆏ߢߪේሶᩭߩ૏⟎ߢߩ㔚ሶߩሽ࿷⏕₸ߪ࠯ࡠߢ޽ࠆ㧚 s ゠㆏ߩߎߩࠃ߁ߥ․ᓽߪ
޿ࠈ޿ࠈߥᚲߢ㊀ⷐߥ㑐ࠊࠅࠍ߽ߞߡ޿ࠆ㧚଀߃߫㧘ṁᶧਛߩ࡜ࠫࠞ࡞ߩ㔚ሶࠬࡇࡦ౒㡆㧔electron spin resonance, ESR㧕ߩࠬࡍࠢ࠻࡞ߦ⷗಴ߐࠇࠆ╬ᣇ⊛⿥ᓸ⚦᭴ㅧ
㧔hyperfine structure㧕ߪ㧘ේሶᩭߩ૏⟎ߦ߅ߌࠆ㔚ሶߩሽ࿷⏕₸ߦᲧ଀ߔࠆߣ޿߁ߎ
ߣ߆ࠄ s ゠㆏ߩߺ߇ߘࠇߦነਈߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 41

2s

1s
0.6

4
3

0.4

2
0.2

1

0

0
5

5

10

2p

10

3s

0.03

0.15

0.02

0.1

0.01

0.05

0
5

0

10

5

3p

10

3d

0.008

0.002

0.006
0.004

0.001

0.002
0
5

࿑ 9.8.2

0

10

5

2

Rn ,A r ߩ㑐ᢙ୯ߩ r ߦߟ޿ߡߩࡊࡠ࠶࠻㧚2❑ゲߪ a0 Rn,A r 㧘ᮮゲߪ3䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 42

r
a0

10

਄ㅀߒߚࠃ߁ߦ Rn ,A r ߪ㧘 T ߅ࠃ߮ I ࠍ޽ࠆ୯ߦ࿕ቯߒߚᤨߩ⏕₸ኒᐲ㑐ᢙ22\ r ,T , Iߩ⋧ኻ୯ࠍ␜ߒߡ޿ࠆ߇㧘႐วߦࠃߞߡߪ㧘ߎߩ⏕₸ኒᐲ㑐ᢙࠍ T ߅ࠃ߮ Iߩోⓨ㑆ߦߟ޿ߡⓍಽߒ㧘ේሶᩭ߆ࠄߩ〒㔌߇ r ߢ޽ࠆ⃿㕙਄ߩ⏕₸ಽᏓ㑐ᢙࠍ⍮ࠅߚ޿႐ว߇޽ࠆ㧚ߎߩࠃ߁ߥ㑐ᢙࠍ‫ޟ‬േᓘಽᏓ㑐ᢙ‫߇߁⸒ߣޠ‬㧘ߘߩ㑐ᢙߩᒻࠍ᳞߼
ߡߺࠃ߁㧚
S

2S

2

³ dT ³ dI \ n,A,m r ,T ,I dW
0

0

S

2

³ 4A ,m T0

r 2 Rn ,A r

2S

sin T dT ³ ) m I

2

dI r 2 Rn ,A r

2

0

2

ߎߎߢ 2 ⴕ⋡ߩ 2 ߟߩⓍಽ߇޿ߕࠇ߽ 1 ߦߥࠆߎߣߪ㧘 4A ,m T߅ࠃ߮ ) m I ߩ

normalization condition ࠃࠅ᣿ࠄ߆ߢ޽ࠈ߁㧚ߎߩേᓘಽᏓ㑐ᢙࠍ n d 3 ߩ 6 ߟߩ゠㆏ߦߟ޿ߡ࿑ 9.8.3 ߦ࿑␜ߔࠆ㧚࿑ 9.8.3 ߦ␜ߒߚേᓘಽᏓ㑐ᢙߩ࿑ߪ㕖Ᏹߦ popular ߥ࿑ߢ޽ࠅ㧘s ゠㆏ߩߺߥࠄߕ㧘 p ゠㆏߿ d ゠㆏ߦߟ޿ߡ߽ᄙߊߩᢎ⑼ᦠߦ࿑߇಴ߡ޿ࠆߩߢ㧘࿑ 9.8.3 ߦ߽ n d 3 ߩߔߴߡߩ orbitalߦߟ޿ߡ࿑␜ߒߡߺߚ㧚ߒ߆ߒ㧘േᓘಽᏓ㑐ᢙߪ㧘⃿ኻ⒓ߢ޽ࠆ s ゠㆏ߦߟ޿ߡߪᗧ
๧߇ߪߞ߈ࠅߒߡ޿ࠆ߇㧘ᄢ߈ߥ⇣ᣇᕈࠍᜬߟ p ゠㆏߿ d ゠㆏ߦߟ޿ߡߤߩࠃ߁ߥᗧ
๧߇޽ࠆߩ߆ߪߪߥߪߛ⇼໧ߢ޽ࠆ㧚T ߅ࠃ߮ I ߦᄢ߈ߊଐሽߔࠆ⏕₸ಽᏓ㑐ᢙࠍ㧘T
߅ࠃ߮ I ߦߟ޿ߡⓍಽߒߡߒ߹ߞߡ r ߛߌߩ㑐ᢙߣߒߡ⴫ߒߚ߽ߩߦߤߩࠃ߁ߥᗧ๧
ઃߌࠍߒߚࠄࠃ޿ߩߢ޽ࠈ߁߆㧫

1 ߟߩ଀ࠍ⠨߃ߡߺࠃ߁㧚ᄥᐔᵗߩ⌀ਛߦ޽ࠆዊߐߥፉߦ߅޿ߡ㧘ඨᓘ 1 km㧘2 km㧘...
ߩ౞๟਄ߦ㟗ࠍ⷗಴ߔ⏕₸ߣ޿߁ߎߣࠍ⹤㗴ߦߔࠆߎߣߪᗧ๧ߩ޽ࠆߎߣߣᕁ߁㧚ߒ߆ߒ㧘᧻ፉ߆ࠄඨᓘ 1 km㧘2 km㧘㧚㧚㧚ߩ౞๟਄ߦ㟗ࠍ⷗಴ߔ⏕₸ߣ޿߁ߎߣߦߥࠆߣ㧘߶ߣࠎߤᗧ๧ߩߥ޿ળ⹤ߦߥߞߡߒ߹߁ߣᕁ߁㧚ߤ߁⠨߃ߚߞߡ㧘㒽࿾ߩᣇߦߪ㟗ߪ޿ߥ޿ࠊߌߢ޽ࠆ㧚㒽࿾ߩᣇߦ 1 km ㅴࠎߛ࿾ὐߢߩ⏕₸ߪቢోߦ࠯ࡠ㧚ᶏߩᣇߦㅴ
ࠎߛ࿾ὐߢߪ޽ࠆ⏕₸ࠍᜬߟ㧚ߘࠇࠄࠍᒻᑼ⊛ߦߔߴߡⓍಽߒߡߒ߹ߞߡඨᓘ 1 kmߩ౞๟਄ߢ㟗ࠍ⷗಴ߔ⏕₸ࠍ᳞߼ࠆߎߣߪਇน⢻ߢߪߥ޿㧚ߒ߆ߒ㧘ߘࠇࠃࠅ߽㧘߻ߒࠈ㧘ᣇะࠍᜰቯߒߡ㧘଀߃߫᧻ፉ߆ࠄ᧲ධ᧲ߩᣇะ߳ 1 km ㅴࠎߛ࿾ὐߢߩ⏕₸㧘ߣ޿߁ᣇ߇ߕߞߣᗧ๧ߩ޽ࠆ⹤㗴ߢ޽ࠈ߁㧚޽߹ࠅ޿޿଀ߢߪߥ߆ߞߚ߆߽ߒࠇߥ޿
22߇㧘೨⠪ߩ⹤㗴߇ r 2 Rn ,A r ߦ⋧ᒰߒ㧘ᓟ⠪ߩ⹤㗴߇ Rn ,A r ߦ⋧ᒰߔࠆࠊߌߢ޽ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 43

2s

1s

0.2

0.6

0.4
0.1
0.2
0

0
5

5

10

2p

3s

0.2

0.1

0.1

0.05

0
5

10

0

10

5

3p

10

3d

0.1

0.15

0.1
0.05
0.05

0

0
5

10

5

2

࿑ 9.8.3 േᓘಽᏓ㑐ᢙ r 2 Rn ,A r ߩ r ߦߟ޿ߡߩࡊࡠ࠶࠻㧚2❑ゲߪ a0 r 2 Rn ,A r 㧘ᮮゲߪ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 44

r
㧚
a0

10

ᦨᓟߦ㧘ᗧ๧ߩߪߞ߈ࠅߒߡ޿ࠆ⃿ኻ⒓ߩ 1s ゠㆏ߩേᓘಽᏓ㑐ᢙߦߟ޿ߡ㧘߽߁ዋߒี๧ߒߡߺࠃ߁㧚 r ߣ r  dr ߩ㑆ߦ㔚ሶࠍ⷗಴ߔ⏕₸ߪ㧘
2

R1s r r 2 dr

§ 2r ·
4 2
r exp ¨  ¸
3a0© a0 ¹ߢਈ߃ࠄࠇࠆ㧚ᓥߞߡ㧘ߎߩ curve ߩᭂᄢߪ§ 2r ·
d 2
r exp ¨  ¸ 0
dr© a0 ¹ࠃࠅr

a0ߦ޽ࠆߎߣ߇ಽ߆ࠆ㧚ߔߥࠊߜ㧘 r ߇ Bohr radius

a0

=2
4SH 0
me e 2ߦ╬ߒ޿ᚲߢ 1s ゠㆏ߪ߽ߞߣ߽ሽ࿷⏕₸߇㜞޿㧚Bohr ߩ઒⺑ߢߪ㧘 n 1 ߩ⁁ᘒߢߪ㔚ሶߪඨᓘ a0 ߢ౞ㆇേࠍߒߡ޿ࠆߣ⠨߃ߚ߇㧘㊂ሶജቇߦࠃࠆߣ㧘a0 ߩᚲ߇߽ߞߣ߽
ሽ࿷⏕₸߇㜞޿ߎߣߦߥࠆ㧚ߎߎߢ߽㧘㊂ሶജቇߩ⚿ᨐߣ Bohr model ߣߩኻᔕ߇⷗ࠄ
ࠇࠆ߇㧘ߎߩߎߣ߆ࠄ Bohr model ࠍㆊᄢ⹏ଔߒߡߪ޿ߌߥ޿ߎߣߪߔߢߦㅀߴߚㅢࠅ
ߢ޽ࠆ㧚

9.9 ᳓⚛ේሶߩේሶ゠㆏㑐ᢙߩ࿑␜㧚㧔2㧕⃿㑐ᢙ YA ,m T , I ߦ㑐ߔࠆᭂᐳᮡࡊࡠ࠶࠻

\ n ,A ,m r ,T , I ߩ߁ߜ㧘േᓘᵄേ㑐ᢙ Rn,A r ߦߟ޿ߡߪ⠨߃ߕ㧘ᱷࠅߩ YA ,m T , I ߩㇱಽࠍࡊࡠ࠶࠻ߔࠆᣇᴺߢ޽ࠆ㧚Rn ,A rߦߟ޿ߡߩࡊࡠ࠶࠻߆ࠄ\ n ,A ,m r , T , I ߩᒻࠍᛠីߔࠆߩ߇㔍ߒ߆ߞߚߎߣߣห᭽㧘YA ,m T , Iߩࡊࡠ࠶࠻߆ࠄ\ n ,A ,m r , T , I ߩᒻࠍᛠី

ߔࠆߎߣߪߥ߆ߥ߆߻ߕ߆ߒ޿㧚ߒ߆ߒ㧘㕖Ᏹߦࡐࡇࡘ࡜࡯ߥ⴫␜ᴺߢ޽ࠆߩߢએਅ
ߦ⚫੺ߔࠆ㧚

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 45ᭂᐳᮡ⴫␜ߢߪ㧘T 㧘I ߢ㒢ቯߐࠇࠆᣇะߦ㧘ේὐ߆ࠄߩ〒㔌߇ YA , m T , I ߩᚲߩὐߢ⴫ߔߎߣߦߥࠆ㧚ਃᰴర⴫␜ߢ޽ࠆ߇㧘ߎߎߢߪ㧘߹ߕ㧘ಽ߆ࠅ߿ߔ޿ࠃ߁ߦੑᰴ
రᐳᮡߢ⠨߃ࠆߎߣߦߔࠆ㧚ੑᰴరᐳᮡߢ⴫ߔߦߪ㧘 T ߹ߚߪ I ޿ߕࠇ߆ 1 ߟࠍ࿕ቯߒߡ㧘ߔߥࠊߜਃᰴరߩᭂᐳᮡⓨ㑆ߩਛߩ޽ࠆᢿ㕙ࠍ⠨߃ࠆߎߣߦߥࠆ㧚ߤ߁޿߁ᢿ
㕙ࠍߣߞߡ߽ࠃ޿߇㧘଀߃߫ I

0 ߩ xz 㕙ߣ߆ IS
2

ߩ yz 㕙ࠍߣࠆߩ߇⠨߃߿ߔ޿㧚ߎߩࡊࡠ࠶࠻ߦߟ޿ߡ߽㧘೨▵ߣห᭽㧘ᵄേ㑐ᢙ YA ,m T , I ߘߩ߽ߩߦߟ޿ߡߩࡊࡠ
࠶࠻ߣߘߩੑਸ਼ YA ,m T , I

2ߩࡊࡠ࠶࠻ߣ 2 ㅢࠅߩᣇᴺ߇޽ࠆ㧚એਅ㧘ߘࠇߙࠇߦߟ޿

ߡㅀߴࠆ㧚㧔a㧕ᵄേ㑐ᢙ YA ,m T , I ߘߩ߽ߩߦߟ޿ߡߩࡊࡠ࠶࠻
߹ߕ s ゠㆏߆ࠄ⠨߃ࠃ߁㧚㧔Exercise 5.5.1 ෳ㧕1s Y0,04Sߢ޽ࠆ㧚 T , I ߦή㑐ଥߢ޽ࠆ߆ࠄ㧘ਃᰴర⴫␜ߢߪඨᓘ
␜ߢߪ㧘ߤߩᢿ㕙ߢ߽ඨᓘ1ߩ⃿ߦߥࠆ㧚ੑᰴర⴫
4S

1ߩ౞ߣ޿߁ߎߣߦߥࠆ㧚࿑ 9.9.1 ߦ৻଀ࠍ␜ߔ㧚ᭂᐳ
4Sᮡ⴫␜ߢߪ㧘ේὐ߆ࠄߩ〒㔌ߢ߽ߞߡ⃿㑐ᢙߩ⛘ኻ୯ࠍ⴫ߔߩߢ㧘╓ภߦߟ޿ߡߪ⴫
ߒࠃ߁߇ߥ޿㧚ߘࠇߢ૗╬߆ߩᎿᄦࠍߔࠆ㧚଀߃߫㧔i㧕㧗㧘㧙ߩ╓ภࠍᦠ߈౉ࠇࠆ㧘
㧔ii㧕㧗ߩᣇߪታ✢ߢ㧘㧙ߩᣇߪὐ✢ߢ⴫ߔ㧚
㧔iii㧕㧗ߩᣇߦߪᢳ✢ࠍߟߌߚࠅ㧘ᓇࠍߟߌߚࠅߒ㧘㧙ߩᣇߪ⊕࿾ࠍᱷߒߡ߅ߊ㧘╬‫ߩޘ‬ᣇᴺ߇޽ࠆ㧚࿑ 9.9.1 એ㒠ߢߪ㧔i㧕ߩᣇᴺࠍ↪޿ߡ޿ࠆ㧚࿑ 9.9.1 s ゠㆏ߩੑᰴరᭂᐳᮡࡊࡠ࠶࠻㧚㨤㨦ᐔ㕙ౝ㧚ඨᓘ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 46

1ߩ౞ߣߥࠆ㧚
4S

ᰴߦ pz ゠㆏ࠍ⠨߃ࠃ߁㧚ߎࠇߪ
3
cos T
4S
ߩࠃ߁ߦ I ߦή㑐ଥߥᑼߢਈ߃ࠄࠇࠆ[(9.7.2c)ᑼ]ߢ޽ࠆ߆ࠄ㧘 z ゲࠍ฽߻ోߡߩᐔ㕙pzߢหߓᒻߣߥࠆ㧚࿑ 9.9.2 ߦ␜ߔࠃ߁ߦ㧘⋥ᓘ

3
ߩ౞ 2 ߟ߆ࠄߥࠆߎߣߪ⋥ߜߦಽ
4S߆ࠆߢ޽ࠈ߁㧚࿑ 9.9.2pz ゠㆏ߩੑᰴరᭂᐳᮡࡊࡠ࠶࠻㧚ඨᓘ

ห᭽ߥᣇᴺߢ㧘 px ゠㆏[(9.7.2a)ᑼ]px

3
sin T cos I
4S

߅ࠃ߮ p y ゠㆏[(9.7.2b)ᑼ]3
sin T sin I
4S
ࠍ࿑ 9.9.3㧙9.9.4 ߦ␜ߔ㧚py䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 473
ߩ౞ 2 ߟ߆ࠄߥࠆ㧚4S

࿑ 9.9. 3

px ゠㆏ߩੑᰴరᭂᐳᮡ⴫␜㧚ඨᓘ

3
ߩ౞ 2 ߟ߆ࠄߥࠆ㧚16S

࿑ 9.9.4

p y ゠㆏ߩੑᰴరᭂᐳᮡ⴫␜㧚ඨᓘ

3
ߩ౞ 2 ߟ߆ࠄߥࠆ㧚
16S

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 48

ห᭽ߦߒߡ㧘 A

2 ߩ 5 ߟߩ d ゠㆏[(9.7.3)ᑼ]ࠍ࿑ 9.9.5 ߦ␜ߔ㧚d x2  y 2 㧔 xy ᐔ㕙㧕d z 2 㧔 xz ᐔ㕙㧕d xy 㧔 xy ᐔ㕙㧕࿑ 9.9.5d yz 㧔 yz ᐔ㕙㧕d zx 㧔 xz ᐔ㕙㧕d ゠㆏ߩᭂᐳᮡ⴫␜ 㧙 ᜰቯߐࠇߚᐔ㕙਄ߩ୯䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 49એ਄㧘ੑᰴరᭂᐳᮡࡊࡠ࠶࠻ࠍ޿ߊߟ߆ߩ଀ߦߟ޿ߡ␜ߒߡ᧪ߚ߇㧘ߎߎ߹ߢ␜ߒ
ߡ᧪ߚ࿑ߪ㧘ᭂᐳᮡ࿑ߩ޽ࠆᐔ㕙਄ߦ㒢ߞߚ࿑ߢ޽ࠆ㧚ߘࠇߢߪಽ߆ࠅߦߊ޿ߣ޿߁
ߎߣߢ㧘ߘࠇࠄࠍ޿߆ߦ߽ 3 ᰴరߩ࿑ߩࠃ߁ߦ⷗ߖࠃ߁ߣ޿߁ᗧ࿑ߩ࿑߇ߒ߫ߒ߫ߺ
ࠄࠇࠆ㧚એਅߦ 2㧘3 ߩ଀ࠍ޽ߍࠃ߁㧚ⷺᐲㇱಽߩᵄേ㑐ᢙࠍ㧘਄ㅀߩࠃ߁ߦ㧘޽ࠆ 1 ߟߩᐔ㕙ౝߢߩ୯ߢߥߊ㧘3ᰴర⊛ߦ⴫ߔߩߪ㧘ታߪ㧘㐳޿ߎߣኈᤃߥߎߣߢߪߥ߆ߞߚ㧚਄ㅀߒߚ㧘޽ࠆᐔ㕙ౝߢ࿑ࠍ޿ߊߟ߆ឬ޿ߡ㧘ߘࠇ߆ࠄ㧘┙૕⊛ߥ࿑ࠍᮨᑼ⊛ߦ⴫ߒߡ޿ߚࠊߌߢ㧘ߘߩࠃ߁ߥ࿑ࠍឬߊߦߪ㧘ߜࠂߞߣߒߚ⛗ᔃ߇ᔅ↪ߢ޽ߞߚ㧚ߒ߆ߒ㧘⃻࿷ߪ㧘ࠦࡦࡇࡘ࡯
࠲ࠣ࡜ࡈࠖ࠶ࠢߩᚻᴺ߇⊒ዷߒ㧘ⷺᐲㇱಽߩ࿑ࠍឬߊߎߣߪ߆ߥࠅኈᤃߦ಴᧪ࠆࠃ߁
ߦߥߞߚ㧚એਅ㧘 s ゠㆏㧘 p ゠㆏㧘 d ゠㆏ߦߟ޿ߡ㧘ౕ૕⊛ߦࡊࡠ࠶࠻ߒߡߺࠃ߁㧚߹ߕ㧘 s ゠㆏ࠍ⠨߃ࠃ߁㧚 s ゠㆏ߪ㧘1
s Y0,04Sߣ⴫ߐࠇࠆ߆ࠄ㧘3 ᰴర⊛ߦ⠨߃ߡ߽㧘ේὐࠍਛᔃߣߒߚඨᓘ

࿑ 9.9.61ߩ⃿ߢ޽ࠆ㧚
4Ss ゠㆏ߩਃᰴరᭂᐳᮡࡊࡠ࠶࠻ࠍਃᰴర⊛ߦ⴫ߒߚ࿑㧚ඨᓘ

1
ߩ⃿㧚
4S

ᰴߦ p ゠㆏ࠍ⠨߃ࠃ߁㧚 px 㧘 p y 㧘 pz ゠㆏ߪߘࠇߙࠇ x ゲ㧘 y ゲ㧘 z ゲ߹ࠊࠅߩ࿁
ォߦኻߒߡᒻࠍᄌ߃ߥ޿㧚ߔߥࠊߜਃᰴరߢߪ 2 ߟߩ⃿ࠍߟߥߍߚᒻߣߥࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 50

࿑ 9.9.7

p ゠㆏ߩਃᰴరᭂᐳᮡࡊࡠ࠶࠻ࠍਃᰴర⊛ߦ⴫ߒߚ࿑㧚1
ߩ⃿ࠍ 2 ߟ㧘ߘࠇߙࠇ x ᣇะ㧘 y ᣇ16Sะ㧘 z ᣇะߦߟߥߍߚᒻߣߥߞߡ޿ࠆ㧚ฦゲߩᱜᣇะߩ⃿߇ᱜ㧘⽶ᣇะߩ⃿߇⽶px ゠㆏㧘 p y ゠㆏㧘 pz ゠㆏޿ߕࠇ߽ඨᓘ

ߩ╓ภ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 51

d ゠㆏ߦߟ޿ߡߪ㧘એਅߩ࿑ߩࠃ߁ߦߥࠆ㧚d x2  y 2

d z2

d xy
࿑ 9.9.8

d yz

d zx

d ゠㆏ߩⷺᐲㇱಽߩᵄേ㑐ᢙߩਃᰴరᭂᐳᮡࡊࡠ࠶࠻

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 52[Let’s play with Mathematica] ⷺᐲㇱಽߩᵄേ㑐ᢙߩᡆਃᰴరࡊࡠ࠶࠻
࿑ 9.9.6 ߆ࠄ࿑ 9.9.8 ߹ߢߩ࿑ߪ㧘Mathematica ࠍ૶ߞߚ߽ߩߢ޽ࠆ㧚Mathematica ߦߪSphericalPlot3D ߣ޿߁㑐ᢙ߇↪ᗧߐࠇߡ޿ࠆ㧚଀߃߫㧘࿑ 9.9.6 ߩ s ゠㆏ߦߟ޿ߡߪ㧘ᰴߩࠃ߁ߥࡊࡠࠣ࡜ࡓࠍ↪޿ߚ㧚s=SphericalPlot3D[Abs[SphericalHarmonicY[0,0, theta,phi]],[1/(2 Sqrt[Pi])],{theta,0, Pi},{phi, 0, 2 Pi},
AxesLabel -> {"x","y","z"},Mesh -> 50,
PlotLabel -> "s orbital",
LabelStyle ->Directive[Bold, FontFamily -> "Helvetica"],Ticks -> None]
࿑ 9.9.7 ߩ p ゠㆏ߪ㧘⃿㑐ᢙߘߩ߽ߩߪⶄ⚛ᢙߢ޽ࠆ႐ว߇ᄙ޿ߩߢ㧘ታᢙᒻᑼߦ⋥ߒߚ߽ߩࠍࡊࡠ࠶࠻ߔࠆ㧚਄ߣห᭽ߦ㧘SphericalHarmonicY ߣ޿߁㑐ᢙࠍ↪޿ߡቯ⟵ߒߡ߽޿޿߇㧘ᑼ㧔9.7.2a㧕߆ࠄᑼ㧔9.7.2c㧕ࠍ↪޿ߡ㧘ᄌᢙ T ߣ I ࠍ↪޿ߡቯ⟵ߒߡ߽޿޿㧚଀߃߫㧘 px ߦߟ޿ߡߪᰴߩࠃ߁ߦߥࠆ㧚

SphericalPlot3D[Abs[(Sqrt[3/Pi]/2 ) Sin[theta] Cos[phi]],{theta,0, Pi},{phi, 0, 2 Pi},
AxesLabel -> {"x","y","z"},Mesh -> 50,
Ticks -> None,
PlotLabel -> "px orbital",
LabelStyle -> Directive[Bold, FontFamily->"Helvetica"]]䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 53

SphericalPlot3D ߣ޿߁㑐ᢙߢߪ㧘㑐ᢙߩ୯߇⽶ߩ႐วߦߪ㧘ᚒ‫⷗ߦޘ‬ᘠࠇߚᭂᐳᮡࡊ
ࡠ࠶࠻ߣߪ⇣ߥࠆࡊࡠ࠶࠻ࠍਈ߃ߡ㕙ୟߢ޽ࠆߩߢ㧘਄ߩࡊࡠࠣ࡜ࡓߢߪ㑐ᢙ Absࠍ↪޿ߡ⛘ኻ୯ࠍߣߞߡ޿ࠆ㧚㧔b㧕ᵄേ㑐ᢙߩੑਸ਼ YA ,m T, I

2ߦߟ޿ߡߩࡊࡠ࠶࠻

ᧄ▵㧔2㧕ߩ㧔a㧕ߢ㧘 YA ,m T , I㧔ߊࠊߒߊߪߘࠇࠍታᢙߦߥࠆࠃ߁ߦ✢ᒻ⚿วߒߚ߽ߩ㧕ߩᭂᐳᮡࡊࡠ࠶࠻ࠍ⚫੺ߒ㧘ߘߩᒻ߆ࠄ wavefunction ో૕ߩᒻࠍᛠីߔࠆߎ
ߣߪ⥋㔍ߩᬺߢ޽ࠆ㧘ߣ޿߁ߎߣࠍㅀߴߚ߇㧘 YA ,m T , Iߩੑਸ਼ߦߟ޿ߡ߽ోߊหߓ

ߎߣ߇⸒߃ࠆ㧚㕖Ᏹߦಽ߆ࠅߦߊ޿࿑ߢ޽ࠆ߇㧘 YA ,m T , I⥄૕ߩࡊࡠ࠶࠻ߣߩᲧセ߽ᄢಾߥߩߢએਅߦ⚫੺ߔࠆ㧚ᭂᐳᮡࡊࡠ࠶࠻ߢ޽ࠆ߆ࠄ㧘ේὐ߆ࠄߩ〒㔌߇

YA ,m T , I

2ߢ޽ࠆὐࠍ⚿ࠎߢ޿ߊࠊߌߢ޽ࠆ㧚߹ߚ㧘ੑᰴరᐔ㕙ౝߦ࿑ࠍឬ߈ߚߌࠇ

߫㧘޽ࠆᐔ㕙ౝߩ YA ,m T , I ߩ୯ߦߟ޿ߡ⠨߃ࠆߎߣߪ㧔a㧕ߣหߓߢ޽ࠆ㧚߹ߕ s ゠㆏ࠍ⠨߃ࠆ㧚s2

Y0,0 2

1
4Sߢ޽ࠆ߆ࠄ㧘නߦඨᓘ1ߩ౞ߣ޿߁ߎߣߦߥࠆ㧚ᒰὼߩߎߣߥ߇ࠄ╓ภߪ޿ߚࠆᚲ
4S

ᱜߢ޽ࠆ㧚⷗߆ߌߪ㧘 s ⥄૕ߩࡊࡠ࠶࠻ߣ૗ߩ㆑޿߽ߥ޿㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 54

࿑ 9.9.9s ゠㆏ߩੑਸ਼ߩੑᰴరᭂᐳᮡࡊࡠ࠶࠻㧚ඨᓘ

1ߩ౞ߣߥࠆ㧚
4S

3 ߟߩ real ߥ p 㑐ᢙ px 㧘 p y 㧘 pz ߩੑਸ਼ߦߟ޿ߡ࿑ 9.9.10 ߦ␜ߔ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 55

࿑ 9.9.10 3 ߟߩ real ߥ p 㑐ᢙ㧦 px , p y , pz ߩੑਸ਼ߩᭂᐳᮡࡊࡠ࠶࠻㧚࿑ 9.9.10 ߢߪ㧘ߘࠇߙࠇߩ p ゠㆏ߦߟ޿ߡ㧘2 ߟߩ⇣ߥߞߚᐔ㕙ࠍขࠅ಴ߒߡ␜ߒ
ߡ޿ࠆ㧚ߎߩ࿑ߪ px , p y , pz ⥄૕ߩ࿑㧘㧔࿑ 9.9.2㧙࿑ 9.8.5㧕ߣᲧߴߡߜࠂߞߣ⷗ࠆߣ
หߓࠃ߁ߦ⷗߃ࠆ߆߽ߒࠇߥ޿߇㧘ᄢ߈ߊ⇣ߥࠆߎߣߦᵈᗧߔߴ߈ߢ޽ࠆ㧚px , p y , pz⥄૕ߩᤨߪ޿ߕࠇ߽ 2 ߟߩ౞߆ࠄ᭴ᚑߐࠇߡ޿ߚ㧚ߒ߆ߒߘࠇߦ෻ߒ㧘੹࿁ߩੑਸ਼ߩ
ࡊࡠ࠶࠻ߢߪᱜ⏕ߥ౞ߢߪߥߊ㧘⚦㐳ߊ㧘⤘ࠄ߹ߖߚ㘑⦁ߺߚ޿ߥᒻߦߥߞߡ޿ࠆߎ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 56ߣ߇⇣ߥࠆ㧚߹ߚ㧘╓ภߩߎߣߦ߽ᵈᗧߒߡ߶ߒ޿㧚 px , p y , pz ⥄૕ߢߪ㧘㧗߹ߚߪ㧙ߩ╓ภ߇ߟߊ㧚ੑਸ਼ߩ႐วߦߪᒰὼ╓ภߪ޿ߚࠆߣߎࠈ㧗ߢ޽ࠆ㧚ߎߩࠃ߁ߦ 2 ⒳㘃ߩ࿑ߪᄢ߈ߊ⇣ߥࠆߩߢ޽ࠆ߇㧘਎ߦ಴ߡ޿ࠆᢎ⑼ᦠߢߎߩ 2 ߟ߇ᷙੂߒߡ޿ࠆ࿑
߇ߒ߫ߒ߫⷗߆ߌࠄࠇࠆ㧚଀߃߫㧘⃿ߢߥߊߡ⚦㐳޿㘑⦁ߺߚ޿ߥᒻࠍߒߡ޿ߡ㧘ߒ߆߽㧘㧗ߣ߆㧙ߩ╓ภ߇ߟߌࠄࠇߡ޿ࠆ࿑߇޽ࠆ㧚⻉ำߪᷙੂߔࠆߎߣߥߊ߈ߜࠎߣ
ℂ⸃ߒߡ߶ߒ޿㧚࿑ 9.9.10 ߆ࠄ᣿ࠄ߆ߥࠃ߁ߦ㧘 px , p y , pz ߪߘࠇߙࠇ x ゲ㧘 y ゲ㧘 z ゲ߹ࠊࠅߩ࿁
222ォߦኻߒߡኻ⒓⊛ߢ޽ࠆ㧚࿑ 9.9.10 ߪ޽ࠆᐔ㕙ౝߩࡊࡠ࠶࠻ߢ޽ࠆ߇㧘ߎࠇࠍਃᰴర
⊛ߦ⴫ߔߣ࿑ 9.9.11 ߩࠃ߁ߦߥࠆ㧚

Py 2

px 2
࿑ 9.9.11

px , p y , pz ߩਃᰴరᭂᐳᮡࡊࡠ࠶࠻
222䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 57

pz 2

ห᭽ߦ㧘5 ୘ߩታᢙ⴫␜ߩ d ゠㆏ߩੑਸ਼ࠍ࿑ 9.9.12 ߦ␜ߔ㧚࿑ 9.9.5 ߦ␜ߒߚ d ゠㆏⥄૕ߩ࿑ߣᲧߴߡߺࠇ߫ߘߩᏅߪߪߞ߈ࠅߔࠆߢ޽ࠈ߁㧚ᒰὼߩߎߣߥ߇ࠄ╓ภߪߔߴߡᱜߢ޽ࠆ㧚࿑ 9.9.5 ߢߪ d z 2 ߇ࠃࠅᄢ߈ߊ⷗߃ࠆ߇㧘ߎߩᄢ߈ߐߩᏅߪ㧘࿑ 9.9.12
ߩੑਸ਼ߩ࿑ߢߪᦝߦ㗼⪺ߦ⷗߃ࠆ㧚d x2  y 2 2 㧔 xy ᐔ㕙㧕d z 2 2 㧔 xz ᐔ㕙㧕d xy 2 㧔 xy ᐔ㕙㧕࿑ 9.9.12

d yz 2 㧔 yz ᐔ㕙㧕d zx 2 㧔 xz ᐔ㕙㧕d ゠㆏ߩੑਸ਼ߩᭂᐳᮡ⴫␜ 㧙 ᜰቯߐࠇߚᐔ㕙਄ߩ୯㧚ᵄേ㑐ᢙࠍੑਸ਼ߒߡ޿ࠆ߆ࠄ╓ภߪᒰὼ޿ߚࠆߣߎࠈᱜ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 58ਃᰴర⊛ߦࡊࡠ࠶࠻ߒߚ߽ߩࠍએਅߦ␜ߔ㧚d x2  y 2 2

d z2 2d xy 2d yz 2
࿑ 9.9.13d ゠㆏ߩੑਸ਼ߩਃᰴరᭂᐳᮡࡊࡠ࠶࠻

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 59

d zx 2

9.10 ᳓⚛ේሶߩේሶ゠㆏㑐ᢙߩ࿑␜㧚㧔3㧕ᐔ㕙਄ߩᵄേ㑐ᢙߩ୯ߩᡆ 3 ᰴర⊛⴫␜㧔a㧕ᵄേ㑐ᢙߘߩ߽ߩߩࡊࡠ࠶࠻

Section 9.8 ߣ Section 9.9 ߢߪోᵄേ㑐ᢙ\ n ,A ,m r , T , I ࠍ⠨߃ࠆઍࠊࠅߦ㧘ߘߩᚑಽߢ޽ࠆ Rn ,A r ߅ࠃ߮ YA ,m T , I ޿ߕࠇ߆ߦߟ޿ߡߩߺ⠨߃ߡ᧪ߚ㧚ᧄ▵ߢߪ޿ࠃ޿ࠃ㧘ోᵄേ㑐ᢙ\ n ,A , m r , T , I ߦߟ޿ߡߩࡊࡠ࠶࠻ࠍ⹜ߺࠃ߁㧚3 ߟߩᐳᮡߦߟ޿ߡߩࡊࡠ
࠶࠻ߢߪ 4 ᰴరߩࠣ࡜ࡈ߇ᔅⷐߥߩߢߘߩ߹߹ߢߩࡊࡠ࠶࠻ߪ⥋㔍ߩᬺߢ޽ࠆ㧚ߘߎߢߤ߁ߔࠆ߆ߣ޿߁ߣ㧘3 ߟߩᐳᮡߩ߁ߜߩ 1 ߟࠍ࿕ቯߒߡ㧘1 ߟߩᐔ㕙ౝߦ㒢ߞߡ㑐ᢙߩ୯ࠍࡊࡠ࠶࠻ߔࠆߎߣࠍ⠨߃ࠆ㧚ߣ޿ߞߡ߽㧘2 ᰴరߩ⚕ߩ਄ߦ࿑␜ߔࠆߦߪ
Ꮏᄦ߇ᔅⷐߢ޽ࠆ㧚⠨߃ߡ޿ࠆᐔ㕙ࠍ⚕㕙ౝߦߣࠅ㧘wavefunction ߩ୯ࠍ⚕㕙ߦု⋥ߥゲߦߣࠆߎߣߦߥࠆ㧚ߎߩ႐ว㧘㜞ߐߪ㧘࿾࿑ߩࠃ߁ߦ╬㜞✢ࠍឬߊᣇᴺ߆㧘߹ߚߪ㧘ጊߩࠬࠤ࠶࠴࿑ߩࠃ߁ߦ 3 ᰴర⊛ߥ࿑ߢ⴫ߔᣇᴺ߽޽ࠆ㧚ᧄ▵ߢߪ߹ߕᓟ⠪ߩᣇ
ᴺࠍߣࠅ㧘೨⠪ߦߟ޿ߡߪ ᰴߩ▵ߢ⠨߃ࠃ߁㧚2 ᰴర⊛⴫␜ߣߪ޿ߞߡ߽㧘ߘ߽ߘ߽

2 ᰴరᐔ㕙ౝߩ㑐ᢙ୯ࠍ 3 ᰴర⊛ߦߦឬߊࠊߌߢ㧘‫ ⇼ޟ‬3 ᰴర⊛⴫␜‫ࠆࠇࠊ⸒ߣޠ‬㧚߹ߕ s ゠㆏㑐ᢙࠍ⠨߃ࠆ㧚ේὐࠍ฽߻ߤࠎߥᐔ㕙ߢ߽㑐ᢙ୯ߪหߓߦߥࠆߪߕߢ޽
ࠆ㧚࿑ 9.10.1㧙࿑ 9.10.3 ߦ\ 1s 㧘\ 2s 㧘߅ࠃ߮\ 3s ߦߟ޿ߡߩߎߩࠃ߁ߥ࿑ࠍ␜ߔ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 60

1s

࿑ 9.10.1 \ 1s ߩේὐࠍ฽߻㕙ౝߢߩ㑐ᢙ୯ߩᡆ 3 ᰴర⊛ࡊࡠ࠶࠻ᐔ㕙ߪߘࠇߙࠇߩㄝ߇ේὐ߆ࠄ r25a0 ߩ㐳ߐࠍ߽ߜ㧘ࡔ࠶ࠪࡘߩ 1 ߟ߇ 1 ㄝ a0 ߩᱜᣇᒻߦ⋧ᒰߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 61

2s࿑ 9.10.2 \ 2s ߩේὐࠍ฽߻㕙ౝߢߩ㑐ᢙ୯ߩᡆ 3 ᰴర⊛ࡊࡠ࠶࠻㧚ᐔ㕙ߪߘࠇߙࠇߩㄝ߇ේὐ߆ࠄ r25a0 ߩ㐳ߐࠍ߽ߜ㧘ࡔ࠶ࠪࡘߩ 1 ߟ߇ 1 ㄝ a0 ߩᱜᣇᒻߦ⋧ᒰߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 62

3s

࿑ 9.10.3 \ 3s ߩේὐࠍ฽߻㕙ౝߢߩ㑐ᢙ୯ߩᡆ 3 ᰴర⊛ࡊࡠ࠶࠻㧚ᐔ㕙ߪߘࠇߙࠇߩㄝ߇ේὐ߆ࠄ r25a0 ߩ㐳ߐࠍ߽ߜ㧘ࡔ࠶ࠪࡘߩ 1 ߟ߇ 1 ㄝ a0 ߩᱜᣇᒻߦ⋧ᒰߔࠆ㧚\ 2 p ߅ࠃ߮\ 3 p ߩ xy ᐔ㕙ౝߢߩ㑐ᢙ୯ߩᡆਃᰴర⊛ࡊࡠ࠶࠻ࠍ࿑ 9.10.4㧙࿑ 9.10.5
x

x

ߦ␜ߔ㧚

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 63

2px

࿑ 9.10.4 \ 2 px ߩ xy ᐔ㕙߹ߚߪ xz ᐔ㕙ౝߢߩ㑐ᢙ୯ߩᡆ 3 ᰴర⊛ࡊࡠ࠶࠻㧚ᐔ㕙ߪߘࠇߙࠇߩㄝ߇ේὐ߆ࠄ r25a0 ߩ㐳ߐࠍ߽ߜ㧘ࡔ࠶ࠪࡘߩ 1 ߟ߇ 1 ㄝ a0 ߩᱜᣇᒻߦ⋧ᒰߔࠆ㧚ߥ߅㧘\ 2 p y ߅ࠃ߮\ 2 pz ߦߟ޿ߡ߽⋧ᒰߔࠆᐔ㕙ࠍㆬߴ߫ోߊห
ߓ࿑ߣߥࠆ㧚

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 64

3px

࿑ 9.10.5 \ 3 px ߩ xy ᐔ㕙߹ߚߪ xz ᐔ㕙ౝߢߩ㑐ᢙ୯ߩᡆ 3 ᰴర⊛ࡊࡠ࠶࠻㧚ᐔ㕙ߪߘࠇߙࠇߩㄝ߇ේὐ߆ࠄ r25a0 ߩ㐳ߐࠍ߽ߜ㧘ࡔ࠶ࠪࡘߩ 1 ߟ߇ 1 ㄝ a0 ߩᱜᣇᒻߦ⋧ᒰߔࠆ㧚ߥ߅㧘\ 3 p y ߅ࠃ߮\ 3 pz ߦߟ޿ߡ߽⋧ᒰߔࠆᐔ㕙ࠍㆬߴ߫ోߊห
ߓ࿑ߣߥࠆ㧚

3d ゠㆏ߦߟ޿ߡߪᒻߩ⇣ߥࠆ 2 ߟߩ゠㆏ߦߟ޿ߡ࿑ 9.10.6㧙࿑ 9.10.7 ߦ␜ߔ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 65

3dz2

࿑ 9.10.6

3d z 2 ゠㆏ߩ xz ᐔ㕙ౝߢߩ㑐ᢙ୯ߩᡆ 3 ᰴర⊛ࡊࡠ࠶࠻㧚ᐔ㕙ߪߘࠇߙࠇߩㄝ߇ේὐ߆ࠄ r30a0 ߩ㐳ߐࠍ߽ߜ㧘ࡔ࠶ࠪࡘߩ 1 ߟ߇ 1
ㄝ a0 ߩᱜᣇᒻߦ⋧ᒰߔࠆ㧚⚕㕙ߩ߅߅ࠃߘᏀ߆ࠄฝ߳ะߊゲ߇ x ゲߢ㧘ᚻ೨߆ࠄᅏ߳ะߊゲ߇ z ゲߢ޽ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 66

3dxy࿑ 9.10.7

3d xy ゠㆏ߩ xy ᐔ㕙ౝߢߩ㑐ᢙ୯ߩᡆ 3 ᰴర⊛ࡊࡠ࠶࠻㧚ᐔ㕙ߪߘࠇߙࠇߩㄝ߇ේὐ߆ࠄ r30a0 ߩ㐳ߐࠍ߽ߜ㧘ࡔ࠶ࠪࡘߩ 1 ߟ߇ 1
ㄝ a0 ߩᱜᣇᒻߦ⋧ᒰߔࠆ㧚⚕㕙ߩ߅߅ࠃߘᏀ߆ࠄฝ߳ะߊゲ߇ x ゲߢ㧘ᚻ೨߆ࠄᅏ߳ะߊゲ߇ y ゲߢ޽ࠆ㧚 3d x2  y 2 ゠㆏ߪߎߩ࿑ࠍ z ゲ࿁ࠅߦ 45 ᐲ࿁ォߒߚ߽ߩߢ޽ࠅ㧘߹ߚ㧘 3d yz ߅ࠃ߮ 3d zx ゠㆏ߪߎߩ࿑ࠍߘࠇߙࠇ y ゲ
߅ࠃ߮ x ゲ࿁ࠅߦ 90 ᐲ࿁ォߒߚ߽ߩߣߥࠆ㧚[Let’s play with Mathematica]

ᡆਃᰴరࡊࡠ࠶࠻

ᧄ▵ߢ␜ߔᡆਃᰴరࡊࡠ࠶࠻ߪߔߴߡ Matematica ߢឬ޿ߚ߽ߩߢ޽ࠆ㧚⥄ಽߢࡊࡠ࠶
࠻ߩࡊࡠࠣ࡜ࡓࠍ૞ࠆߣߥࠆߣ߆ߥࠅᄢᄌߢ޽ࠆ߇㧘޽ࠅ߇ߚ޿ߎߣߦ Mathematica
ߦߪ Plot3D ߣ޿߁㑐ᢙ߇ቯ⟵ߐࠇߡ߅ࠅ㕖Ᏹߦ◲නߦ 3 ᰴరࡊࡠ࠶࠻߇ᓧࠄࠇࠆ㧚଀ߣߒߡ㧘࿑ 9.10.1 ߩ 1s ゠㆏ߪᰴߩࠃ߁ߦߒߡᓧࠄࠇߚ߽ߩߢ޽ࠆ㧚߹ߕ㑐ᢙ\ n,A ,m ࠍቯ⟵ߒߡ߅ߊ㧚R[n_,L_,r_]:=Sqrt[4 (n-L-1)! n^(-4) (n+L)!^(-1)]* (2r/n)^L Exp[-r/n]䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 67

LaguerreL[n-L-1,2L+1,2r/n];
Y[L_,m_]:=SphericalHarmonicY[L,m,theta,phi];
psi[n_,L_,m_]:=R[n,L,r] Y[L,m];
൩⺰㧘 1s ゠㆏ߣ߆ 2s ゠㆏ߣ߆ࠍౕ૕⊛ߥ㑐ᢙᒻࠍਈ߃ࠆߎߣߦߔࠆߣᑼߪ߽ߞߣ◲නߦߥࠆ߇㧘਄ߩࠃ߁ߦቯ⟵ߒߡߒ߹߁ߣ㧘ߔߴߡߩ㊂ሶᢙߦߟ޿ߡ౒ㅢߦ૶߃ࠆߩ
ߢ߻ߒࠈଢ೑ߢ޽ࠆ㧚࿑ 9.10.1 ߩࠃ߁ߦ xy 㕙ౝߢߩࡊࡠ࠶࠻ߢ޽ࠇ߫ᰴߩࠃ߁ߦᄌᢙ r ࠍቯ⟵ߔࠆ㧚

r:=Sqrt[x^2+y^2];
࿑ 9.10.1 ߩ 3 ᰴరࡊࡠ࠶࠻ߪએਅߩࠃ߁ߦⴕߞߚ߇㧘2 ⴕ⋡એ㒠ߩ⒳‫ࡦ࡚ࠪࡊࠝߩޘ‬ߪᧄ⾰⊛ߥ߽ߩߢߪߥ޿㧚Plot3D[psi[1,0,0],{x,-25,25},{y,-25,25}㧘
PlotRange->All,
Boxed->False,
Axes->False,
PlotPoints->50,
ViewPoint->{1.300,-2.400,0.05},
AspectRatio->Automatic,
BoxRatios->{1,1,1},
PlotLabel->"1s"]㧔b㧕ᵄേ㑐ᢙߩੑਸ਼ߦ㑐ߔࠆᡆਃᰴరࡊࡠ࠶࠻
࿑ 9.10.8㧙࿑ 9.9.14 ߩߘࠇߙࠇߦ⋧ᒰߔࠆᵄേ㑐ᢙߩੑਸ਼ߩࡊࡠ࠶࠻ࠍએਅߦ␜
ߔ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 68

1s

࿑ 9.10.8 \ 1s *\ 1s ߩේὐࠍ฽߻㕙ౝߢߩ㑐ᢙ୯ߩᡆਃᰴర⊛ࡊࡠ࠶࠻㧚ᐔ㕙ߪߘࠇߙࠇߩㄝ߇ේὐ߆ࠄ r25a0 ߩ㐳ߐࠍ߽ߜ㧘ࡔ࠶ࠪࡘߩ 1 ߟ߇ 1
ㄝ a0 ߩᱜᣇᒻߦ⋧ᒰߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 69

2s࿑ 9.10.9 \ 2s *\ 2s ߩේὐࠍ฽߻㕙ౝߢߩ㑐ᢙ୯ߩᡆਃᰴర⊛ࡊࡠ࠶࠻㧚ᐔ㕙ߪߘࠇߙࠇߩㄝ߇ේὐ߆ࠄ r25a0 ߩ㐳ߐࠍ߽ߜ㧘ࡔ࠶ࠪࡘߩ 1 ߟ߇ 1
ㄝ a0 ߩᱜᣇᒻߦ⋧ᒰߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 70

3s

࿑ 9.10.10 \ 3s *\ 3s ߩේὐࠍ฽߻㕙ౝߢߩ㑐ᢙ୯ߩᡆਃᰴర⊛ࡊࡠ࠶࠻㧚ᐔ㕙ߪߘࠇߙࠇߩㄝ߇ේὐ߆ࠄ r25a0 ߩ㐳ߐࠍ߽ߜ㧘ࡔ࠶ࠪࡘߩ 1 ߟ߇ 1
ㄝ a0 ߩᱜᣇᒻߦ⋧ᒰߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 71

2px

࿑ 9.10.11 \ 2 px *\ 2 px ߩ xy ᐔ㕙߹ߚߪ xz ᐔ㕙ౝߢߩ㑐ᢙ୯ߩᡆਃᰴర⊛ࡊࡠ࠶࠻㧚ᐔ㕙ߪߘࠇߙࠇߩㄝ߇ේὐ߆ࠄ r25a0 ߩ㐳ߐࠍ߽ߜ㧘ࡔ࠶ࠪࡘߩ 1 ߟ߇ 1 ㄝ a0 ߩᱜᣇᒻߦ⋧ᒰߔࠆ㧚ߥ߅㧘\ 2 px *\ 2 px ߅ࠃ߮\ 2 pz *\ 2 pz ߦߟ޿ߡ߽⋧ᒰߔࠆᐔ㕙ࠍㆬߴ߫ోߊหߓ࿑ߣߥࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 72

3px

࿑ 9.10.12 \ 3 px *\ 3 px ߩ xy ᐔ㕙߹ߚߪ xz ᐔ㕙ౝߢߩ㑐ᢙ୯ߩᡆਃᰴర⊛ࡊࡠ࠶࠻㧚ᐔ㕙ߪߘࠇߙࠇߩㄝ߇ේὐ߆ࠄ r25a0 ߩ㐳ߐࠍ߽ߜ㧘ࡔ࠶ࠪࡘߩ 1 ߟ߇ 1 ㄝ a0 ߩᱜᣇᒻߦ⋧ᒰߔࠆ㧚ߥ߅㧘\ 3 p y *\ 3 p y ߅ࠃ߮\ 3 pz *\ 3 pz ߦߟ޿ߡ߽⋧ᒰߔࠆᐔ㕙ࠍㆬߴ߫ోߊหߓ࿑ߣߥࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 73

3dz2

࿑ 9.10.13

3d z 2 ゠㆏ߩੑਸ਼ߩ xz ᐔ㕙ౝߢߩ㑐ᢙ୯ߩᡆਃᰴర⊛ࡊࡠ࠶࠻㧚ᐔ㕙ߪߘࠇߙࠇߩㄝ߇ේὐ߆ࠄ r30a0 ߩ㐳ߐࠍ߽ߜ㧘ࡔ࠶ࠪࡘߩ 1 ߟ߇ 1
ㄝ a0 ߩᱜᣇᒻߦ⋧ᒰߔࠆ㧚⚕㕙ߩ߅߅ࠃߘᏀ߆ࠄฝ߳ะߊゲ߇ x ゲߢ㧘ᚻ೨߆ࠄᅏ߳ะߊゲ߇ z ゲߢ޽ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 74

3dxy࿑ 9.10.14

3d xy ゠㆏ߩੑਸ਼ߩ xy ᐔ㕙ౝߢߩ㑐ᢙ୯ߩᡆਃᰴర⊛ࡊࡠ࠶࠻㧚ᐔ㕙ߪߘࠇߙࠇߩㄝ߇ේὐ߆ࠄ r30a0 ߩ㐳ߐࠍ߽ߜ㧘ࡔ࠶ࠪࡘߩ 1 ߟ߇ 1 ㄝ a0 ߩᱜᣇᒻߦ⋧ᒰߔࠆ㧚⚕㕙ߩ߅߅ࠃߘᏀ߆ࠄฝ߳ะߊゲ߇ x ゲߢ㧘ᚻ೨߆ࠄᅏ߳ะ
ߊゲ߇ y ゲߢ޽ࠆ㧚3d x2  y 2 ゠㆏ߪߎߩ࿑ࠍ z ゲ࿁ࠅߦ 45 ᐲ࿁ォߒߚ߽ߩߢ޽ࠅ㧘߹ߚ㧘 3d yz ߅ࠃ߮ 3d zx ゠㆏ߪߎߩ࿑ࠍߘࠇߙࠇ y ゲ߅ࠃ߮ x ゲ࿁ࠅߦ 90 ᐲ࿁ォߒߚ߽ߩߣߥࠆ㧚

9.11 ᳓⚛ේሶߩේሶ゠㆏㑐ᢙߩ⴫␜㧚㧔4㧕ᐔ㕙਄ߩᵄേ㑐ᢙߩ୯ߩ╬㜞✢࿑Section 9.10 ߢ⚫੺ߒߚ⇼ਃᰴర⊛ߥ࿑ߩઍࠅߦ㧘޽ࠆᐔ㕙ౝߢߩ㑐ᢙ୯ࠍ╬㜞✢ߢ␜ߒߚ࿑߽ᢥ₂ߦߪ಴ߡ޿ࠆ㧚଀߃߫㧘 Streitwieser, Jr, P. H. Owens "Orbital andElectron Density Diagrams"ߦߪߘߩࠃ߁ߥ╬㜞✢࿑߇޿ߊߟ߆ߩ゠㆏ߦߟ޿ߡタߖࠄࠇߡ޿ࠆ㧚ߒ߆ߒ㧘ߘࠇࠄߩ࿑ߪ㧘೨▵ߩਃᰴర⊛ߥ࿑ࠍ޽ࠆ㕙ߢベಾࠅߦߒߚߛߌ
ߩ࿑ߢ޽ࠆ㧚࿾࿑ࠍ⷗ߡጊߩᒻ߇ಽ߆ࠆࠃ߁ߦ㧘ߜࠂߞߣ㚔ࠇࠇ߫㧘╬㜞✢࿑ࠍߺࠇ߫ Section 9.10 ߢ␜ߒߚ⇼ਃᰴర⊛ߥ࿑߇⋥ߜߦᕁ޿ᶋ߆߱ߎߣߢߪࠈ߁߇㧘᥉ㅢߩ
ੱߦߪ㧘Section 9.10 ߩᣇ߇ߪࠆ߆ߦಽ߆ࠅ߿ߔ޿㧚ߘ߁޿߁ᗧ๧ߢ㧘ࠊߑࠊߑಽ߆ࠅ
ߦߊ޿࿑ࠍ⚫੺ߔࠆߎߣߪߒߥ޿㧚⥝๧ߩ޽ࠆᣇߪ㧘਄⸥ Streitwieser, Jr, P. H. Owensߩᧄࠍ⷗ߡ޿ߚߛ߈ߚ޿㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 75

9.12 ᳓⚛ේሶߩේሶ゠㆏㑐ᢙߩ⴫␜㧚㧔5㧕ᵄേ㑐ᢙߘߩ߽ߩߩ୯ߩ╬୯ᦛ㕙⴫␜੹߹ߢ㧘3 ᰴరߩᐳᮡߩ㑐ᢙߢ޽ࠆᵄേ㑐ᢙࠍ⚕ߩ਄ߦឬߊߚ߼ߦ 1 ߟߩᐳᮡࠍ࿕
ቯߒߡߒ߹ߞߡ޽ࠆᐔ㕙਄ߩᵄേ㑐ᢙ߹ߚߪߘߩੑਸ਼ߩࡊࡠ࠶࠻ࠍ⹜ߺߡ᧪ߚ㧚ߤ߁
ߒߡ 1 ߟߩᐳᮡࠍ࿕ቯߔࠆᔅⷐ߇޽ࠆ߆ߣ޿߁ߣ㧘ߔߢߦ⺑᣿ߒߚࠃ߁ߦ㧘3 ߟߩᐳᮡ r 㧘T 㧘I ߩ㑐ᢙߢ޽ࠆᵄേ㑐ᢙࠍ⴫ߔߎߣߪ 3 ᰴరⓨ㑆ߢߪ಴᧪ߥ޿߆ࠄߢ޽ࠆ㧚ߒ߆ߒ㧘ᵄേ㑐ᢙߩ୯߇৻ቯߢ޽ࠆ╬୯ዪ㕙ߥࠄ߫㧘3 ᰴరⓨ㑆ߢࡊࡠ࠶࠻಴᧪ࠆߪ
ߕߢ޽ࠆ㧚એਅߦ㧘ᵄേ㑐ᢙ߇޽ࠆ୯ߦ╬ߒ޿ὐࠍ⚿ࠎߢ૞ࠄࠇࠆ╬୯ᦛ㕙ࠍ 3 ᰴర⊛ߦឬ޿ߚ߽ߩࠍ⚫੺ߔࠆ㧚

2 px

2 py

2 pz

࿑ 9.12.1

2 px 㧘 2 py 㧘 2 pz ߩ╬୯ᦛ㕙⴫␜.
ᵄേ㑐ᢙߩ୯߇ atomic unit ߢ 0.03 ߢ޽ࠆᦛ㕙㧚 x ゲ㧘 y ゲ㧘 z ゲߘࠇߙࠇߦߟ޿
ߡ㧘 3 atomic unit ߆ࠄ 3 atomic unit ߹ߢߩ▸࿐㧚ߘࠇߙࠇߩゲߩᱜߩᣇะߩᦛ
㕙߇ᱜ㧘⽶ߩᣇะߩᦛ㕙߇⽶ߢ޽ࠆ㧚ߎߩ࿑ߪߜࠂߞߣ⷗ࠆߣ㧘ࠊߌ߇ಽ߆ࠄߥ޿࿑ߛߣ޿ߞߡ⚫੺ߒߚ࿑ 9.9.7 ߣૃߡ޿ࠆࠃ߁ߦᕁ߁߆߽ߒࠇߥ޿㧚ߒ߆ߒ㧘࿑ 9.9.7 ߪᱜ⏕ߥ⃿ࠍߟߥߍߚ߽ߩߢ޽ࠆ߇㧘ߎߩ࿑ߢߪ㧘߽ߞߣ߭ߒ߾ߍߚ㧘౞⋚ᛩߍߩ౞⋚ߺߚ޿ߥᒻߣߥߞߡ޿ࠆ㧚⃿ࠍ 2 ߟߟߥߍߚ࿑ߦ⷗ᘠࠇߡ޿ߚ⻉ำߪ㧘ᄸ⇣ߦᗵߓࠆ߆߽ߒࠇߥ޿߇㧘ታߪ㧘ߎߩ࿑߇ࠃ
ࠅ⃻ታߦㄭ޿ߩߢ޽ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 76หߓࠃ߁ߦߒߡឬ߆ࠇߚ 5 ߟߩ 3d ゠㆏ࠍ࿑ 9.11.2 ߦ␜ߔ㧚3d x2  y 2

3d z 2

3d xy

3d yz

3d zx

࿑ 9.12.2 5 ୘ߩ 3d ゠㆏ߩ╬୯ᦛ㕙⴫␜㧚ᵄേ㑐ᢙߩ୯߇ atomic unit ߢ 0.01 ߢ޽ࠆᦛ㕙㧚 x ゲ㧘 y ゲ㧘 z ゲߘࠇߙࠇߦߟ޿
ߡ㧘 18 atomic unit ߆ࠄ 18 atomic unit ߹ߢߩ▸࿐㧚એ਄㧘Section 9.8 ߆ࠄ Section 9.12 ߹ߢ޿ࠈ޿ࠈߥ࿑ࠍ␜ߒߡ᧪ߚ㧚ߤߩ࿑߇ᦨ߽
ಽ߆ࠅ߿ߔ޿߆ߪ৻᭎ߦߪ⸒߃ߥ޿ߢ޽ࠈ߁㧚ᘠࠇ߽޽ࠆߒ㧘wavefunction ߩߤ߁޿߁ᚲߦ⌕⋡ߒߡ޿ࠆ߆ߣ޿߁ߎߣߦࠃߞߡ߽⇣ߥߞߡ᧪ࠆߢ޽ࠈ߁㧚ߚߛᗵߓߢ޿߃
߫㧘wavefunction ⥄૕ߩ࿑␜߽㧘wavefunction ߩੑਸ਼ߩ࿑␜߽㧘࿑ 9.12.1㧘࿑ 9.12.2 ߩ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 77ࠃ߁ߥ࿑߇߆ߥࠅಽ߆ࠅ߿ߔ޿ࠃ߁ߦᕁ߁㧚ࠦࡦࡇࡘ࡯࠲࡯ࠣ࡜ࡈࠖ࠶ࠢߩᚻᴺߩ⊒㆐ߒߚ⃻࿷㧘ߎ߁޿߁࿑߇ߎࠇ߆ࠄߩਥᵹߦߥߞߡ޿ߊߩߢߪߥ߆ࠈ߁߆㧚[Let’s play with Mathematica] ᵄേ㑐ᢙߘߩ߽ߩߩ╬୯ᦛ㕙
࿑ 9.11.1 ߅ࠃ߮࿑ 9.11.2 ߦ␜ߒߚࠃ߁ߥ╬୯ᦛ㕙ߩ࿑ࠍឬߊߩߪ㧘ᓥ᧪㕖Ᏹߦ㕙ୟߢ޽ߞߚ㧚ߒ߆ߒ㧘ᐘ޿ߦ߽㧘Mathematica ߦߪ ContourPlot3D ߣ޿߁޽ࠅ߇ߚ޿㑐ᢙ߇ਈ߃ࠄࠇߡ޿ࠆ㧚଀߃߫㧘࿑ 9.11.1 ߩ 2 pz 㑐ᢙߪ㧘ᰴߩࠃ߁ߥࡊࡠࠣ࡜ࡓࠍ↪޿ߡឬ޿ߚ߽ߩߢ޽ࠆ㧚ContourPlot3D[
Abs[Sqrt[1/(32 Pi)] z Exp[-Sqrt[x^2+y^2+z^2]]] = = 0.03, {x, -3, 3}, {y,-3, 3}, {z, -3, 3},
Mesh -> 30,
AxesLabel {Style[x, Large, Bold, Black],Style[y, Large, Bold, Black],Style[z, Large, Bold,
Black]},PlotRange -> All,
Ticks -> None,
PlotLabel -> "2pz orbital",
LabelStyle -> Directive[Bold, FontFamily "Helvetica"]]䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 78

9.13 ᳓⚛ේሶߩේሶ゠㆏㑐ᢙߩ⴫␜㧚
㧔6㧕ᵄേ㑐ᢙߘߩ߽ߩߩ୯ߩੑਸ਼ߩ╬୯ᦛ㕙⴫␜೨▵ߣห᭽㧘ᵄേ㑐ᢙߘߩ߽ߩߩࡊࡠ࠶࠻ߢ㧘ߎߩ▵ߢߪ㧘ᵄേ㑐ᢙߩੑਸ਼ߩ╬୯ᦛ㕙⴫␜ࠍ␜ߔ㧚ੑਸ਼ߢ޽ࠆ߆ࠄ㧘⸒߁߹ߢ߽ߥߊ㧘╓ภߪߔߴߡߩ႐วߦᱜߢ޽ࠆ㧚

2 px 2

2 py 2
࿑ 9.13.1

2 pz 22 px 㧘 2 py 㧘 2 pz ߩੑਸ਼ߩ╬୯ᦛ㕙⴫␜㧚ᵄേ㑐ᢙߩੑਸ਼ߩ୯߇ atomic unit ߢ 0.001 ߢ޽ࠆᦛ㕙㧚x ゲ㧘 y ゲ㧘 z ゲߘࠇߙࠇߦߟ޿ߡ㧘 3 atomic unit ߆ࠄ 3 atomic unit ߹ߢߩ▸࿐㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 79

3d x2  y 2 2

3d z 2 2

3d xy 23d yz 23d zx 2

࿑ 9.13.2 5 ୘ߩ 3d ゠㆏ߩੑਸ਼ߩ╬୯ᦛ㕙⴫␜㧚ᵄേ㑐ᢙߩੑਸ਼ߩ୯߇ atomic unit ߢ 0.0001 ߢ޽ࠆᦛ㕙㧚 x ゲ㧘 y ゲ㧘 z ゲߘࠇߙࠇߦߟ޿ߡ㧘 18 atomic unit ߆ࠄ 18 atomic unit ߹ߢߩ▸࿐㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 9: Page 80

CHAPTER 10
៨േℂ⺰ߎࠇ߆ࠄੑਃߩ┨ߦࠊߚߞߡ㧘㊂ሶജቇߢ↪޿ࠄࠇࠆㄭૃᴺߦߟ޿ߡቇ⠌ߔࠆ㧚ߎߩࠃ߁ߥㄭૃᴺߦߟ޿ߡቇ⠌ߒߥߌࠇ߫ߥࠄߥ޿ߣ޿߁ߎߣߪ㧘৻㕙ߦ߅޿ߡߪ㧘ᖤߒ޿⃻ታߢ޽ࠆ㧚ߖߞ߆ߊ Schrödinger equation ߣ޿߁વኅߩቲಷࠍᜬߜߥ߇ࠄ㧘ߘࠇࠍ↪޿ߡ෩ኒߥ⸃ࠍᓧࠆߎߣߪ߶ߣࠎߤਇน⢻ߥߩߢ޽ࠆ㧚⃻ߦ㧘෩ኒߥ⸃ࠍ᳞߼ࠆ
ߎߣ߇಴᧪ࠆࠃ߁ߥ♽ߪ㧘Chapter 6 ߆ࠄ Chapter 9 ߹ߢߢ߶ߣࠎߤ಴ዧߊߒߡߒ߹ߞ
ߚࠃ߁ߥ߽ߩߢ޽ࠆ㧚⃻ታߦሽ࿷ߔࠆ♽ߢߪ㧘߿߻ࠍᓧߕ㧘ߥࠎࠄ߆ߩㄭૃࠍߒߡ෩
ኒߥ⸃ߦㄭ޿ߢ޽ࠈ߁⸃ࠍ᳞߼ࠆߎߣߢᚒᘟࠍߒߥߌࠇ߫ߥࠄߥ޿㧚ᢙቇߩ਎⇇ߩࠃ߁ߦ㧘߈ߜࠎߣߒߚℂ⺰૕♽ߦߪߥࠄߥ޿߇㧘⃻ታ໧㗴ߣߒߡ㧘ߎࠇ߇ᚒ‫⌀߇ޘ‬ታߦㄭઃߊ໑৻ߩᣇᴺߢ޽ࠇ߫㧘ߘࠇࠍቇ⠌ߒߥ޿ࠊߌߦߪ޿߆ߥ޿㧚ㄭૃᴺߣߒߡ․ߦ
㊀ⷐߢ޽ࠆߩߪߎߩ┨ߢቇ⠌ߔࠆ៨േℂ⺰ߣ㧘ᰴߩ┨ߢቇ⠌ߔࠆᄌಽᴺߢ޽ࠆ㧚㊂ሶ
ൻቇߩ਎⇇ߢ↪޿ࠄࠇࠆߔߴߡߩㄭૃߪߎߩ޿ߕࠇ߆㧘෶ߪߘߩਔᣇߦᏫ⌕ߐࠇࠆߣ޿޿ᓧࠆߢ޽ࠈ߁㧚ߘ߁޿߁ᗧ๧ߢ㧘ߎߩ 2 ߟߩㄭૃᴺࠍ߇ߞߜࠅቇ⠌ߔࠆߎߣ߇ᄢ
ಾߢ޽ࠆ㧚਄ㅀߒߚࠃ߁ߦ៨േℂ⺰㧔perturbation theory㧕ߪ㧘ᦨ߽ઍ⴫⊛ߥㄭૃᴺߩ 1 ߟߢ޽ࠅ㧘㕖Ᏹߦߒ߫ߒ߫૶ࠊࠇࠆ㧚ߎߩℂ⺰߇ㆡ↪ߐࠇࠆ᧦ઙߪ㧘ᚒ‫޿ߡߒߣ߁ߎ⸃߇ޘ‬ࠆ♽ߩ Hamiltonian ߇㧘ߔߢߦ eigenvalue ߣ eigenfunction ߇ಽ߆ߞߡ޿ࠆ♽ߩ
Hamiltonian ߩዊߐߥ⵬ᱜߢਈ߃ࠄࠇࠆߣ޿߁ߎߣߢ޽ࠆ㧚଀߃߫㧘Section 6.2 ߢߪ
0 d x d L ߩ㗔ၞߢ potential ߇࠯ࡠߢ޽ࠆࠃ߁ߥ৻ᰴరߩ▫ߩਛߩ☸ሶࠍᛒߞߚ߇㧘ߎߩ㗔ၞߢ㧘potential ߇࠯ࡠߦㄭ޿ߥ߇ࠄ߽ቢోߦ࠯ࡠߢߪߥߊ㧘޽ࠆ㑐ᢙߢ⴫ߐࠇࠆ
ࠃ߁ߥ♽ߦㆡ↪಴᧪ࠆ㧚ߎߩ႐ว㧘ߔߢߦ᳞߹ߞߡ޿ࠆ৻ᰴరߩ▫ߩਛߩ☸ሶߩ⸃ࠍ↪޿ߡ㧘ᣂߒ޿♽ߩ eigenvalue ߣ eigenfunction ࠍ⴫ߔߎߣࠍ⹜ߺࠆࠊߌߢ޽ࠆ㧚߽ߞߣ߽㧘਄ߢ㧘‫ޟ‬ዊߐߥ⵬ᱜ‫߇ߚ޿↪ࠍ⪲⸒߁޿ߣޠ‬㧘ߎߩࠃ߁ߥ potential ߩઃട㗄߇‫ޟ‬ዊߐߥ‫⵬ޠ‬ᱜߢ޽ࠆ߆ߤ߁߆ࠍߤ߁޿߁┙႐߆ࠄ್ᢿߒߚࠄ޿޿ߩ߆ߪ໧㗴ߢ޽ࠆ㧚ߒ߆ߒ㧘ߎߩ޽ߚࠅߩߎߣߦߟ޿ߡߪ㧘ᆎ߼ߩ߁ߜߪ޽޿߹޿ߦߒߡ߅޿ߡ㧘ߛࠎߛࠎߣീᒝߒߡ޿ߊߎߣߦߒࠃ߁㧚10.1 㕖❗ㅌ♽ߩ៨േ⺰⠨߃ߡ޿ࠆ♽ߩ Hamiltonian H ࠍ
H H0  H '䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 1(10.1.1)

ߩࠃ߁ߦ࠯ࡠᰴߩ Hamiltonian H 0 ߣ៨േߩ Hamiltonian H ' ߩ 2 ߟߦಽߌࠆ㧚ਈ߃ࠄࠇߚ Hamiltonian ࠍ 2 ߟߦಽߌࠆ߿ࠅᣇߪ޿ࠈ޿ࠈ޽ࠆߩߢ㧘ߤ߁޿߁ⷰὐ߆ࠄ H 0 ࠍㆬ߱߆ߣ޿߁ߎߣࠍ⠨߃ߥ޿ߣ޿ߌߥ޿㧚ᄢಾߥߎߣߪ 2 ߟ޽ࠆ㧚1 ߟߪ H 0 ߩ࿕᦭୯ߣ
࿕᦭㑐ᢙ߇ߔߢߦಽ߆ߞߡ޿ࠆ㧔߹ߚߪ᳞߼ᓧࠆ㧕ߣ޿߁ߎߣߢ޽ࠅ㧘߽߁ 1 ߟߪ H '߇ H 0 ߦኻߒߡዊߐߥ⵬ᱜ㗄ߢ޽ࠆߣ޿߁ߎߣߢ޽ࠆ㧚ੑਃߩ଀ࠍ޽ߍࠃ߁㧚Section 6.2 ߢ৻ᰴరߩ੗ᚭဳ potential ߦߟ޿ߡቇ⠌ߒߚ߇㧘ߘߩ႐ว㧘 0 d x d L ߩ▸࿐ߢ potential V x

V x

H sinSx
L0 ߢ޽ߞߚ㧚ߎࠇ߇ቢోߦ 0 ߢߥߊ㧘ߢ⴫ߐࠇࠆ႐วߦߟ޿ߡ⠨߃ࠆᔅⷐ߇޽ߞߚߣߔࠆ㧚࿑ 10.1.1 ៨േߩ߆߆ߞߚ੗ᚭဳࡐ࠹ࡦࠪࡖ࡞

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 2ߎߩ႐ว㧘Schrödinger equation ߪᒰὼ§ =2 d 2Sx·
 H sin¨¸\
2
L ¹© 2m dxE\(10.1.2)

ߢ޽ࠆ߇㧘ߎࠇࠍ⸃ߊߎߣߪ㧘ߪߚߒߡน⢻ߢ޽ࠆ߆ߤ߁߆ߜࠂߞߣ◲නߦߪ್ᢿ಴
᧪ߥ޿ߦߒߡ߽㧘ߘ߁◲නߢߥ޿ߎߣߪߪߞ߈ࠅߒߡ޿ࠆ㧚ߒ߆ߒ㧘potential ߩ㗄ࠍ฽߹ߥ޿♽ߦߟ޿ߡߪߔߢߦ Section 6.2 ߢ⸃޿ߡ޿ߡ㧘eigenvalue ߣ eigenfunction ߪ᳞߹ߞߡ޿ࠆࠊߌߢ޽ࠆ㧚ߘࠇߢߎߩ႐ว㧘=2 d 22m dx 2Sx
H ' H sin
L

H0



(10.1.3)ߩࠃ߁ߦ߅ߊߩ߇ࠃ߆ࠈ߁㧚߽߁ 1 ߟ଀ࠍ޽ߍࠃ߁㧚Section 7.1 ߢ Hooke ߩᴺೣߦᓥ߁߫ߨߦߟ޿ߡ Schrödinger
equation ࠍ⸃޿ߚ㧚߽ߒ㧘potential ߇ᑼ(7.1.1)ߩઍࠅߦ

1 2 1 3kx  gx2!
3!

V x(10.1.4)ߩࠃ߁ߦ x3 ߦᲧ଀ߔࠆ㗄߇޽ߞߚߣߔࠆ㧚ዋߥߊߣ߽ߎߩࠃ߁ߥ potential ߇ࠃࠅ⃻ታߦㄭ޿ߎߣߪ Section 7.7 ߦ␜ߒߚ㧚㧔ᑼ(7.7.5)ࠍෳᾖ㧚㧕ߎߩ႐วߦߪ
=2 d 2 1 2
 kx
2m dx 2 2

H0



H'

1 3
gx
6

(10.1.5)

ߣߒߡ㧘Section 7.1 ߢ᳞߼ߚ harmonic oscillator ߩ⸃ࠍ߽ߣߦߒߡ੹⠨߃ߡ޿ࠆ♽ߩ⸃ࠍ⴫ߔߎߣࠍ⠨߃ࠆࠊߌߢ޽ࠆ㧚਄ߩ⼏⺰ߢ H ' ߇ H 0 ߩዊߐߥ⵬ᱜ㗄ߢ޽ࠆᔅⷐ߇޽ࠆߣㅀߴߚ߇㧘ߎߩ್ᢿߪᔅߕ
ߒ߽߿ߐߒߊߥ޿߆߽ߒࠇߥ޿㧚଀߃߫ᑼ(10.1.3)ߩ H 0 ߣ H ' ߣߩᄢ߈ߐߩ㑐ଥߪߘ߁᣿ࠄ߆ߢߪ޽ࠆ߹޿㧚ᑼ(10.1.5)ߩ H 0 ߣ H ' ߩ㑐ଥߦߟ޿ߡ߽ߘ߁ߢ޽ࠈ߁㧚ᦝߦᑼ1(10.1.5)ߩ႐วߦߪ㧘ߎߩᄢዊ㑐ଥߪ᣿ࠄ߆ߦ x ߩ୯ߦଐሽߔࠆ㧚x ߩዊߐ޿ᚲߢ gx3
61
ߩ㗄߇ዊߐߊ㧘ߘࠇࠍ H ' ߦߔࠆߩ߇ࠃ߆ߞߚߣߒߡ߽㧘x ߇ᄢ߈ߊߥߌࠇ߫ㅒߦ kx 22ߩᣇ߇ዊߐߊߥࠅ㧘߻ߒࠈߘߩ㗄ࠍ H ' ߦߒߥߌࠇ߫޿ߌߥ޿߆߽ߒࠇߥ޿㧚ߎߩࠃ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 3

߁ߦ㧘Hamiltonian ߩߤߩㇱಽࠍ H 0 ߦߒߡᱷࠅࠍ H ' ߣߔࠆ߆ߪ㧘ታߩߣߎࠈ㧘⚿ᨐࠍ⷗ߥ߇ࠄ್ᢿߒߥߌࠇ߫ߥࠄߥ޿ㇱಽ߽޽ࠅ㧘ߜࠂߞߣා੺ߥ໧㗴ߢ޽ࠆ㧚ߒ߆ߒ㧘ߘ߁޿߁ߎߣߪ㧘߅޿߅޿᣿ࠄ߆ߦߥߞߡ޿ߊߢ޽ࠈ߁㧚೨⟎߈ߪߎߩ૏ߦߒߡ㧘ᧄ⺰ߦ౉ࠆߎߣߦߔࠆ㧚ߐߡ㧘ᑼ(10.1.1)ߢߪ៨േߩ Hamiltonian ࠍ H ' ߣ⟎޿ߚ߇㧘એᓟ㧘ߎࠇࠍ㧘޽ࠆଥᢙ O ߣ Hamiltonian H ߩⓍߢ⴫ߔߎߣߦߔࠆ㧚ߎ߁ߒߡ߅ߊߣᑼߩዉ಴ㆊ⒟ߢ㧘 O ߩ౹ዷ㐿ࠍߔࠆߎߣ߇಴᧪ࠆߩߢଢ೑ߢ޽ࠆ߆ࠄߢ޽ࠆ㧚ߔߥࠊߜ㧘ᑼ(10.1.1)ߩઍࠅߦ
1

H

H0  O H

1(10.1.6)ߣ߅ߊ㧚 H 0 ߩ eigenfunction ߅ࠃ߮ eigenvalue ߪ

H 0\ n 0

En 0\ n 0

(10.1.7)ߢቯ⟵ߐࠇࠆ㧚 E ߿\ ߩฝ⢋ߦ␜ߒߚ 0 ߣ޿߁ superscript ߪ‫ࡠ࠯ޟ‬ᰴ‫߁޿ߣޠ‬ᗧ๧ߢ

H 0 ߩ࿕᦭୯㧘࿕᦭㑐ᢙߢ޽ࠆߎߣࠍ␜ߒߡ޿ࠆ㧚ᚒ‫ߪޘ‬㧘
H0  O H 1 \ n

En\ n(10.1.8)ࠍ⸃߈ߚ޿ࠊߌߢ޽ࠆ㧚ߎߩᑼߢ O ࠍ࠯ࡠߣ߅ߌ߫ᑼ(10.1.7)ߦߥࠆ㧚ᓥߞߡ㧘 O ߇ዊߐ޿႐วߦߪᑼ(10.1.8)ߩ⸃ߪᑼ(10.1.7)ߦㄭ޿ߢ޽ࠈ߁ߣ⠨߃ࠄࠇࠆ㧚ߔߥࠊߜ㧘៨േ O H 1 ߩലᨐߪ㧘㕖៨േߩ energy En ߿ wavefunction \ n 0 ࠍ߶ࠎߩࠊߕ߆ߛߌᄌ߃ࠆ
0ߛߌߢ޽ࠈ߁ߣ⠨߃ࠄࠇࠆ㧚ߘߎߢ㧘ߎߩ㧘߶ࠎߩࠊߕ߆ߥᄌൻಽ߇ߤ߁޿߁ᒻߢ⴫
ߐࠇࠆ߆ࠍ᳞߼ࠃ߁ߣ޿߁ࠊߌߢ޽ࠆ㧚ߐߡ㧘 H 0 ߩ eigenfunction ߪ❗ㅌ߇޽ࠆ႐วߣߥ޿႐วߣ߇޽ࠈ߁߇㧘એᓟ❗ㅌ߇޽ࠆ߆ߥ޿߆ߢ೎‫ߦޘ‬⠨߃ࠆߎߣߦߔࠆ㧚❗ㅌ߇޽ࠆ႐วߦߪ Section 10.2 ߢᛒ߁ߎߣߦߒ㧘ᧄ▵ߢߪએᓟ❗ㅌ߇ߥ޿႐วࠍ⠨߃ࠆ㧚‫ޤ⸼ޣ‬㧦ߎߩ‫❗ޟ‬ㅌ߇޽ࠆ‫❗ޟޠ‬ㅌ߇ߥ޿‫ߪߣߎ߁޿ߣޠ‬㧘޽߹ࠅߪߞ߈ࠅߒߥ޿߆߽ߒࠇ
ߥ޿㧚ߎߎߢ޿߁ᗧ๧ߪ㧘ᵈ⋡ߒߡ޿ࠆ⁁ᘒ\ n ߩ࠯ࡠᰴ⁁ᘒ\ n 0 ߇❗ㅌߒߡ޿ࠆ߆޿
ߥ޿߆ߣ޿߁ᗧ๧ߢ޽ࠆ㧚\ n ߦᵈ⋡ߔࠆ႐ว㧘\ n 0 ߇❗ㅌߒߡ޿ߥߌࠇ߫㧘\ n 0 એᄖ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 4߇❗ㅌߒߡ޿ߡ߽‫❗ޟ‬ㅌ߇ߥ޿‫ߡߒߣ♽ޠ‬ขࠅᛒ߁ߩߢ޽ࠆ㧚ߘߩߎߣߪ㧘એਅߩᑼࠍᵈᗧߒߡ⷗ࠇ߫ಽ߆ࠆߣᕁ߁߇㧘ᔨߩߚ߼ᒝ⺞ߒߡ߅ߊ㧚\ n ߿ En ߪ O ߩ㑐ᢙߢ޽ࠅ㧘߹ߚ㧘 O ߇ዊߐ޿߶ߤ⵬ᱜߩലᨐߪዊߐ޿ߣ޿߁ߎߣ
߆ࠄ㧘ߎࠇࠄࠍ O ߩ౹⚖ᢙߢዷ㐿ߔࠆ㧚ߔߥࠊߜ\ n \ n  O\ n 1  O 2\ n 2  ...

(10.1.9)

En 0  O En  O 2 En

(10.1.10)

0

1

En

 ...

2

\ n 1 , \ n 2 ,... ߅ࠃ߮ En 1 , En 2 , ... ߪ O ߦή㑐ଥߢ㧘ߎࠇ߆ࠄ᳿߼ࠃ߁ߣߔࠆ߽ߩߢ޽
ࠆ㧚ᑼ(10.1.9)߅ࠃ߮ᑼ(10.1.10)ࠍᑼ(10.1.8)ߦઍ౉ߔࠆ㧚

^

`

^

`

H 0\ n 0  O H 1 \ n 0  H 0\ n 1  O 2 H 1 \ n 1  H 0\ n 2  ...

^En 0\ n 0  O En 1 \ n 0  En 0\ n 1^`

 O 2 En \ n 0  En \ n  En 0\ n2

1

1

(10.1.11)2`  ...

ߎߩᑼ߇ O ߩ୯ߦࠃࠄߕᚑࠅ┙ߚߥߌࠇ߫ߥࠄߥ޿ߩߢ㧘 O ߩฦ౹ਸ਼ߩ㗄߇Ꮐฝߘࠇ
ߙࠇ╬ߒߊߥߌࠇ߫޿ߌߥ޿㧚ߔߥࠊߜ㧘H 0\ n 0

En 0\ n 0

(10.1.12)

H 0  En 0 \ n

1En \ n 0  H \ n 0

H 0  En 0 \ n

2

En \ n 0  En \ n  H \ n

1

1

2

1

(10.1.13)
1

1

1(10.1.14)

ᑼ(10.1.12)ߪᑼ(10.1.7)ߘߩ߽ߩߢ޽ࠅߔߢߦ⸃ߌߡ޿ࠆ㧚ᑼ(10.1.13)ࠍ⸃ߊߎߣ߇಴
᧪ࠇ߫㧘1 ᰴߩ⵬ᱜ㗄\ n 1 ߣ En ߇᳞߹ࠆ㧚ห᭽ߦᑼ(10.1.14)߇⸃ߌࠇ߫ 2 ᰴߩ⵬ᱜ㗄1\ n 2 ߣ En 2 ߇᳞߹ࠆ㧚એਅห᭽ߦߒߡ 3 ᰴߩ⵬ᱜ㗄㧘4 ᰴߩ⵬ᱜ㗄㧘㧚㧚㧚ߣ᳞߼ࠆߎ
ߣߪේℂ⊛ߦน⢻ߢ޽ࠆ߇㧘⃻ታ໧㗴ߣߒߡߪ㧘޽߹ࠅ㜞ᰴߩ⵬ᱜ㗄߹ߢ᳞߼ࠆߎߣ
ߪᗧ๧߇ߥ޿㧚㧔߻ߒࠈ㧘ߘߎ߹ߢ޿߆ߥ޿߁ߜߦ෼᧤ߔࠆࠃ߁ߦ H 0 ߣ H ' ࠍㆬ߱ߴ߈ߢ޽ࠆ㧚㧕

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 5ᑼ(10.1.13)ࠍ⸃ߊߦ޽ߚߞߡ㧘\ n1ࠍߔߢߦಽ߆ߞߡ޿ࠆ࠯ࡠᰴߩ wavefunction^\ ` ߩ⚵ߢዷ㐿ߔࠆ㧚ߔߥࠊߜ
0

n

¦A\

\n 1

0

m

(10.1.15)mm

ߎߎߢ A1 , A2 , ... ߪዷ㐿ଥᢙߢߘࠇࠍߎࠇ߆ࠄ᳿߼ߡ޿ߊ㧚ᑼ(10.1.15)ߩ basis function^\ ` ߪ orthonormal set ࠍ૞ߞߡ޿ࠆߣ⠨߃ߡࠃ޿㧚
0

n

‫ޤ⸼ޣ‬
㧦

H 0 ߇ Hermitian ߢ޽ࠇ߫ߘߩ eigenfunction ߪ߅੕޿ߦ⋥੤ߒߡ޿ࠆߎߣߪߔߢߦቇ
⠌ߒߚߣ߅ࠅߢ޽ࠆ㧚H 0 ߇ Hermitian ߢߥ޿႐วߪා੺ߢ޽ࠆ߇㧘⃻ታ໧㗴ߣߒߡᚒ‫ޘ‬߇ Hermitian ߢߥ޿ H 0 ࠍᛒ߁ߎߣߪ߹ߕ޽ࠆ߹޿ߣ޿߁ߎߣߢ޽߹ࠅᷓߊᖠ߹ߥ޿ߎ
ߣߦߒࠃ߁㧚ᑼ(10.1.15)ߩዷ㐿ߢᄢಾߥߎߣߪ㧘ᚒ‫ߪޘ‬ቢో♽ࠍ૞ࠆోߡߩ

wavefunction ࠍ↪޿ߡዷ㐿ߒߥߌࠇ߫޿ߌߥ޿ߣ޿߁ߎߣߢ޽ࠆ㧚ᄙߊߩ႐ว H 0 ߩeigenfunction ߪή㒢୘޽ࠆߩߢ㧘ᑼ(10.1.15)ߪή㒢୘ߩ㗄ߩ๺ߣߥࠆ㧚 H 0 ߇᳓⚛ේሶߦߟ޿ߡߩ Hamiltonian ߢ޽ࠇ߫㧘energy ߇⽶ߩ bound state ߆ࠄ energy ߇ᱜߩ continuum

state ߹ߢోߡࠍ฽߼ߚ wavefunction ߩ⚵ߢ߽ߞߡዷ㐿ߔࠆࠊߌߢ޽ࠆ㧚⃻ታ໧㗴ߣߒߡߘ߁޿߁߰߁ߦή㒢୘ߩ㗄ߩ๺ࠍߣࠆߎߣ߇น⢻߆ߤ߁߆ߩี๧ߪᓟ࿁ߒߦߒߡ
ߣߦ߆ߊ⛯ߌࠃ߁㧚ᑼ(10.1.15)ࠍᑼ(10.1.13)ߦઍ౉ߔࠆߣ

¦A H E
m

0

0

n

\ m0

En 1  H 1 \ n 0(10.1.16)mߣߥࠆ߇㧘ߎߎߢᑼ(10.1.12)ࠍ↪޿ࠆߣ

¦A E E
0

m

m

n

0

\ m0

En  H
1

1

\ n0

m

ߣߥࠆ㧚Ꮐ߆ࠄ\ n 0 * ࠍ߆ߌߡⓍಽߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 6(10.1.17)

¦A E E0m

m

En 1  ³\ n 0* H 1 \ n 0 dW

G nm

0

n

m

(10.1.18)

Ꮐㄝߪ࠯ࡠߢ޽ࠆ߆ࠄ㧘1 ᰴߩ៨േ energy ߪ
En

³\

1

0*
n

{ H nnH \ n 0 dW
1

(10.1.19)

1ߣߥࠆ㧚㧔2 ߟ⋡ߩ╬ᑼߦߟ޿ߡߪᑼ(10.1.21)ࠍෳᾖ㧚㧕
ᰴߦ wavefunction ߦኻߔࠆ 1 ᰴߩ⵬ᱜ㗄\ 1 ࠍ᳞߼ࠃ߁㧚ᑼ(10.1.17)ߦᏀ߆ࠄ

\ k0 *

k z n ࠍ߆ߌߡⓍಽߔࠆߣ㧘¦A E E
0

m

m

1

³\

G km

0

n

En 1 G kn H kn 1

(10.1.20)

m

ߎߎߢ

H kn

0*
k

H \ n 0 dW
1

(10.1.21)

ߩࠃ߁ߥ notation ߩ◲⇛ൻࠍⴕߞߚ㧚ߎࠇࠃࠅ

Ak Ek  En00

 H kn1(10.1.22)

ߔߥࠊߜ
1

Ak

H kn
0
0
En  Ekkzn(10.1.23)߆ߊߒߡᑼ(10.1.15)ߩዷ㐿ߩଥᢙߩ߁ߜ\ n 0 ߩଥᢙ An ࠍ㒰޿ߡߔߴߡᑼ(10.1.23)ߢਈ
߃ࠄࠇࠆߎߣ߇ಽ߆ߞߚ㧚ߢߪ An ߪߤ߁ߒߚࠄ᳞߹ࠆߢ޽ࠈ߁߆㧫ߘࠇߦߪᑼ

(10.1.9)ߩ\ n ߇ⷙᩰൻߐࠇߡ޿ߥߌࠇ߫ߥࠄߥ޿ߣ޿߁᧦ઙࠍ↪޿ࠆ㧚ᑼ(10.1.15)ࠍᑼ
(10.1.9)ߦઍ౉ߔࠆߣ

\ n \ n 0  O ¦ Am\ m 0  O An\ n 0  O 2 ...  ...
mznߣߥࠆ㧚ߘߎߢ

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 7(10.1.24)

0³\ *\ dW ³\ *\ dW O ¦ A ³\ *\ dW  2O A ³\ *\ dW
0

n

n

n

n

0

0

m

n

0

0

m

n

n

n

mz n

(10.1.25)

 O 2 ...
...
ߢ޽ࠆ߇ ^\ n 0 ` ߇ orthonormal set ߢ޽ࠆߎߣ߆ࠄ

³\ \ dW 1  2O A  O ...  ...
*

2

n

n

(10.1.26)nߣߥࠆ㧚਄ߩᑼߩฝㄝߪ޿߆ߥࠆ O ߩ୯ߢ߽ 1 ߢߥߌࠇ߫޿ߌߥ޿㧚ߔߥࠊߜAn 0(10.1.27)
ߢߥߌࠇ߫ߥࠄߥ޿ߣ⚿⺰ߐࠇࠆ㧚߆ߊߒߡ㧘energy ߅ࠃ߮ wavefunction ߩਔᣇߦߟ޿ߡ㧘 O ߩ 1 ᰴߩ㗄߹ߢߩ⵬ᱜ㗄߇᳞߹ߞߚ㧚ᔨߩߚ߼㧘߹ߣ߼ߡ߅ߊ㧚En

En 0  O H nn 1  O 2 ...

(10.1.28)\n

H mn 1\n  O¦ 0\ 0  O 2 ...
0 m
E
E

mzn
n
m

(10.1.29)0ᰴߦ㧘ᑼ(10.1.14)ߦࠃࠅ㧘2 ᰴߩ៨േ㗄 En ߅ࠃ߮\ n ࠍ᳞߼ࠃ߁㧚ᣇᴺߪ਄ߣో22ߊหߓߢ޽ࠆ㧚߹ߕ\ n 2 ࠍ ^\ n 0 ` ߩ⚵ߢዷ㐿ߔࠆ㧚

¦B \

\n 2

m

0

(10.1.30)mm

ᑼ(10.1.30)߅ࠃ߮ᑼ(10.1.15)ࠍᑼ(10.1.14)ߦઍ౉ߒ㧘߆ߟᑼ(10.1.12)ࠍ↪޿ࠆߣ¦B E E
0

m

m

0

n

\ m0

En 2 \ n 0  ¦ Am En 1  H 1 \ m 0(10.1.31)mznmߣߥࠆ㧚ߎߎߢฝㄝ╙ 2 㗄⋡ߩ๺ߢ mn ࠍ㒰޿ߚߩߪᑼ(10.1.27)ߦࠃࠆ㧚Ꮐ߆ࠄ\ n 0 *ࠍ߆ߌߡⓍಽߔࠆ㧚¦B E E
0

m

m

m

n

0

G nm

En 2  ¦ Am En 1 G nm  H nm 1mznᏀㄝߪᒰὼ࠯ࡠߢ޽ࠆ㧚ᓥߞߡ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 8(10.1.32)

En

¦A H

2

m

1

(10.1.33)

nmmzn

ߣߥࠆ㧚ߎߎߢᑼ(10.1.23)ࠍ↪޿ࠆߣ
1

En

1

H nm H mn¦
00m z n En  Em

2

(10.1.34)

ߣ޿߁⚿ᨐ߇ᓧࠄࠇࠆ㧚߆ߊߒߡ㧘2 ᰴߩ៨േ energy ߇᳞߹ߞߚ㧚

wavefunction ߦߟ޿ߡߩ 2 ᰴߩ៨േ㗄\ n 2 ߽◲නߦ᳞߹ࠆ㧚ᑼ(10.1.31)ߦᏀ߆
ࠄ\ k 0k z n ࠍ߆ߌߡⓍಽߔࠆ㧚¦B E E
0m

m¦A E G H

G km

0

n

1

m

n

km

1

(10.1.35)

kzn(10.1.36)

km

mz n

m

ߔߥࠊߜ
Bk Ek  En
00

Ak En  ¦ Am H km
1

1

mz n

ߣߥࠆ߇㧘ߎߎߢᑼ(10.1.19)߅ࠃ߮ᑼ(10.1.23)ࠍ↪޿ࠆ㧚
1

Bk Ek  En
0

0

1

1

1

H nn H kn
H H ¦ km0 mn000
En  Ekm z n En  Em

(10.1.37)

⚿ዪ
1

Bk

1

1

1

H km H mn
H H kn
 nn
¦
2
0
0
0
0
En  Ekm z n En  Em
En 0  Ek 0ߣ޿߁⚿ᨐ߇ᓧࠄࠇࠆ㧚ᦝߦᑼ(10.1.30)ߩዷ㐿ߦ߅޿ߡ
1
2
Bn  ¦ Am
2 mzn

kzn

(10.1.38)(10.1.39)

ߢ޽ࠆߎߣߪ㧘\ n ߩⷙᩰൻ᧦ઙࠃࠅᑼ(10.1.27)ࠍ᳞߼ߚࠃ߁ߦߒߡ◲නߦዉ߆ࠇࠆ㧚એ਄ߢ㧘energy ߅ࠃ߮ wavefunction ౒ߦ O ߦ㑐ߒߡ 2 ᰴ߹ߢ⵬ᱜߔࠆߎߣ߇಴᧪ߚ㧚ᔨߩߚ߼એਅߦ߹ߣ߼ߡ߅ߊ㧚En

En  O H nn
01

H nm 1 H mn 1
O ¦
 O 3 ...  ...
0
0

E
E
mzn
n
m
2

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 9(10.1.40)

H mn
\ 0
0 m
E
E

mzn
n
m

\ n \ n0  O ¦

0

ª­
º
1
1
1
1 ½
H km H mn
H nn H kn ° 0 »« °®
¾\ k »
000
0 2« ° m¦
En 0  Em 0z n En  Ek

E
E
°¿
n
k
»
O 2 ¦ « ¯
11»
k zn ««  1 H kn H nk 2 \ n 0»« 2 E 0 E 0
»
n
k
¬
¼

(10.1.41)

 O 3 ...  ...

৻ᣤߎߩࠃ߁ߥᑼ߇᳞߹ߞߡߒ߹߃߫㧘ᓟߪ㧘O ߣ޿߁ parameter ߪࠊߕࠄࠊߒ޿㧚߻ߒࠈ㧘߽ߣߦᚯߒߡ㧘 H ' O H 1 ߦߒߡ㧘⠨߃ߚᣇ߇ಽ߆ࠅ߿ߔ߆ࠈ߁㧚ᒻᑼ⊛ߦߪ㧘ᑼ(10.1.40)߅ࠃ߮ᑼ(10.1.41)ߢ O ࠍ 1 ߦߒߡ㧘H

1

ߦ㑐ߔࠆⓍಽ H mn ߥߤࠍ H ' ߦ
1㑐ߔࠆⓍಽ H 'mn ߳⟎߈឵߃ࠆߛߌߢ޽ࠆ㧚ᑼ(10.1.40)߅ࠃ߮ᑼ(10.1.41)ߪ㧘ⶄ㔀ߦߺ߃ࠆ߆߽ߒࠇߥ޿㧚ߒ߆ߒ㧘energy ߦ㑐ߒߡߪ 2 ᰴ߹ߢ㧘wavefunction ߦߟ޿ߡߪ 1ᰴ߹ߢߪᑼࠍⷡ߃ߡ߅޿ߚᣇ߇ࠃ޿㧚ߘࠇߛߌ㗫❥ߦ૶ࠊࠇࠆ㊀ⷐߥᑼߢ޽ࠆ㧚➅ࠅ㄰ߒߦߥࠆ߇㧘ᔨߩߚ߼㧘ᦨ⚳⊛ߦᓧࠄࠇߚᑼࠍ㧘㧔bracket notation ࠍ↪޿ࠆ႐วࠍ߽฽߼ߡ㧕એਅߦ߹ߣ߼ߡ߅ߎ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 10៨േ⺰ߩ߹ߣ߼㧦࠯ࡠᰴߩ Hamiltonian ࠍ H 0 㧘៨േߩ Hamiltonian ࠍ H ' ߣߔࠆ㧚H0  H 'H(10.1.42)࠯ࡠᰴߩ࿕᦭୯߅ࠃ߮࿕᦭㑐ᢙߪ㧦

H 0\ n 0

En 0\ n 0

(10.1.43)

߹ߚ㧘bracket notation ߢߪ0

H 0 n0En n0(10.1.43a)energy ߪ 2 ᰴ߹ߢߩ⵬ᱜߢ㧦
H' H'
0
En En  H 'nn  ¦ 0nm mn0  ...
m z n En  Em

(10.1.44)߹ߚ㧘bracket notation ߢߪ

En  n0 H ' n0  ¦0En

n0 H ' m0 m0 H ' n0
En  Em
0

nzm

0

 ...

(10.1.44a)

ߎߎߢ

H 'nm

n0 H ' m0³\ * H '\ dW
0

0

n

m

ᵄേ㑐ᢙߪ 1 ᰴ߹ߢߩ⵬ᱜߢ㧦H'\ n \ n 0  ¦ 0 mn 0\ m 0  ...
m z n En  Em

(10.1.45)(10.1.46)߹ߚ㧘bracket notation ߢߪnn0  ¦ m0mz n

m0 H ' n0
En  Em
0

0

 ...

(10.1.46a)‫⵬ޣ‬ㆮ‫ޤ‬㧦
perturbation ߩ⚿ᨐ߇㧘ᱜߒ޿⸃ߣߤ߁޿߁㑐ଥߦߥߞߡ޿ࠆ߆ࠍࠃࠅ᣿ࠄ߆ߦߔ
ࠆߚ߼㧘2 Ḱ૏♽ߦߟ޿ߡਔ⠪ࠍᲧセߒߡߺࠃ߁㧚ߎߎߢ 2 Ḱ૏♽ߣ޿ߞߚߩߪ㧘 H 0
߅ࠃ߮ H

H 0  H ' ߩ eigenfunction ߇ߘࠇߙࠇ 2 ߟߒ߆ߥ޿ߣ޿߁႐วߢ޽ࠆ㧚ᓟ߆䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 111ࠄቇ⠌ߔࠆ spin ߩ♽߇ߘ߁޿߁႐วߢ޽ࠆ㧚2਄ߢቇ⠌ߒߚࠃ߁ߦోࡂࡒ࡞࠻࠾ࠕࡦߪ
H H0  H '(10.1.47)

H 0  H ' ߣߦಽߌࠄࠇࠆ㧚 H 0 ߩ orthonormal ߥ eigenfunction ߪᰴߩࠃ߁

ߩࠃ߁ߦ Hߦቯ⟵ߐࠇࠆ㧚H 0\ 10

E10\ 10

H 0\ 2 0

E2 0\ 2 0

(10.1.48)

੹㧘ߤ߁ߢ߽޿޿ߎߣߢ޽ࠆ߇㧘ᓟ߆ࠄߩଢ೑ߩߚ߼㧘 E1 ! E2 ߩࠃ߁ߦ 1 ߅ࠃ߮ 200ߩ⇟ภઃߌࠍߣࠆߎߣߦߔࠆ㧚 H ߩ eigenfunction \ 1 ߅ࠃ߮\ 2 ࠍ\ 10 ߣ\ 2 0 ߩ linear

combination ߣߒߡ⴫ߘ߁㧚ߔߥࠊߜ\ 1 c11\ 10  c12\ 20\2(10.1.49)

c21\ 10  c22\ 2 0\ 1 ߅ࠃ߮\ 2 ߦߟ޿ߡߩ eigenvalue ߇ߘࠇߙࠇ E1 ߅ࠃ߮ E2 ߢ޽ࠆߣߔࠆ㧚ߔߥࠊߜ
H\ 1

E1\ 1

H\ 2

E2\ 2

(10.1.50)

ᓥߞߡc11 H\ 10  c12 H\ 2 0

E1 c11\ 10  c12\ 20c21 H\ 10  c22 H\ 2 0

E2 c21\ 10  c22\ 2 0(10.1.51)

ᑼ(10.1.51)ߩ 1 ⴕ⋡ߩᑼߦᏀ߆ࠄ\ 10 * ߹ߚߪ\ 2 0 * ࠍ߆ߌߡⓍಽߔࠆ㧚

c11 H11  E1  c12 H120

c11 H 21  c12 H 22  E1

0(10.1.52a㧕ห᭽ߦᑼ(10.1.51)ߩ 2 ⴕ⋡ߩᑼߦᏀ߆ࠄ\ 10 * ߹ߚߪ\ 2 0 * ࠍ߆ߌߡⓍಽߔࠆ㧚

c21 H11  E2  c22 H120

c21 H 21  c22 H 22  E20䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 12(10.1.52b)

ߎߎߢ
H ij

³\ * H\ dW³\ * H  H ' \ dW
0

0

i

j

0

0

i

j

0

G ij E j 0  H 'ij
H 'ij³\ * H '\ dW00i(10.1.53)

j

ߔߥࠊߜ
H11

E1  H '11

H 22

E2  H '22

H12

H 21  H '1200(10.1.54)

ᑼ(10.1.52a)߅ࠃ߮ᑼ(10.1.52b)ߩ non-trivial solution ߪ޿ߕࠇ߽
H11  EH 21

H12
H 22  E(10.1.55)

0ߣ޿߁ secular equation ߩ⸃ࠃࠅᓧࠄࠇࠆ㧚߹ߚߪ

H11  H 22 H11  H 22

E
2
2

H12

H 21

H11  H 22 H11  H 22

E
2
2

0

(10.1.56)

ߎߎߢ energy ߩၮḰࠍ H11 ߣ H 22 ߣߩᐔဋ୯ߦࠪࡈ࠻ߒߡ

E'E

H11  H 222

(10.1.57)

ߣߒࠃ߁㧚ߘ߁ߔࠆߣ secular determinant ߪ

H11  H 22
E'
2

H12

H 21

H  H 22
 11
E'
2

0

(10.1.58)

ߎߩ secular determinant ࠍ⸃޿ߡ㧘eigenvalue ߣ eigenfunction ࠍ᳞߼ࠆߎߣߪ੹߹ߢ૗
࿁߽ⴕߞߚߩߢ⋥ߜߦታⴕߔࠆߎߣ߇಴᧪ࠆߛࠈ߁㧚⻉ำ⥄ࠄታⴕߒߡߺߡ߶ߒ޿㧚ߎߎߢߪߜࠂߞߣ߫߆ࠅ elegant ߥᣇᴺࠍવ᝼ߒࠃ߁㧚એਅ H ߪ real ߢ Hermitianߢ޽ࠆߣ઒ቯߔࠆ㧚ᑼ(10.1.58)ߢ

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 13

2§ H11  H 22 ·2¨¸  H122©
¹
H11  H 22
W cos 2T
2
H12 W sin 2T

W(10.1.59)ߣ߅ߊ㧚ߘ߁ߔࠆߣ secular determinant ߪ
W cos 2T  E 'W sin 2T0
W sin 2T
W cos 2T  E '(10.1.60)

ߣߥࠆ㧚⸃ߪ⋥ߜߦ᳞߹ࠆߢ޽ࠈ߁㧚ߔߥࠊߜ

E '1

W

\1

cos T \ 10  sin T \ 2 0

E '2

W

\2

 sin T \ 10  cos T \ 2 0(10.1.61)

ߎߎߢtan 2TH12
H11  H 222

(10.1.62)

߹ߚߪ㧘ᑼ(10.1.57)ࠍ⠨ᘦߒߡ

E1
E2

H11  H 22
W2
H11  H 22
W
2

\1

cos T \ 10  sin T \ 2 0\2

 sin T \  cos T \ 2(10.1.63)
01

0

ߎࠇ߇ 2 Ḱ૏♽ߦ߅ߌࠆᱜߒ޿ eigenvalue ߣ eigenfunction ߩᑼߢ޽ࠆ㧚એਅ㧘޿ࠈ޿ࠈߥㄭૃߩ߽ߣߢߎߩᑼߣ៨േ⺰ߣߩ⚿ᨐࠍᲧセߒߡߺࠃ߁㧚(1) ߹ߕ㧘perturbation ߇㕖Ᏹߦዊߐߊ㧘
H12 0(10.1.64)

ߣㄭૃ಴᧪ࠆ႐วࠍ⠨߃ࠃ߁㧚ᑼ(10.1.59)ࠃࠅ

W

H11  H 222ߢ޽ࠆ߆ࠄᑼ(10.1.63)ߪ

E1

H11

E1  H '11

E2

H 22

E2  H '22

0(10.1.65)0ߣߥࠆ㧚߹ߚ㧘ᑼ(10.1.59)ࠃࠅ 2T

0 ߢ޽ࠆ߆ࠄ㧘wavefunction ߪ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 14

\ 1 \ 10(10.1.66)\ 2 \ 20ߎࠇࠄߩ⚿ᨐࠍ㧘៨േ⺰ߣᲧߴߡߺࠆ㧚ᑼ(10.1.65)ߩ energy ߪ㧘៨േࠍ 1 ᰴ߹ߢߣߞ
ߚᑼߢ޽ࠅ㧘߹ߚᑼ(10.1.66)ߩ wavefunction ߪ࠯ࡠᰴߩ wavefunction ߘߩ߹߹㧘ߣ޿߁ߎߣߦߥࠆ㧚㧔2㧕 ᰴߦ߽߁ዋߒㄭૃࠍ㜞߼ߡ㧘 H ߦߟ޿ߡߩ㕖ኻⷺ㗄㧔off diagonal term㧕 H12 ߇

diagonal term ߩᏅߩඨಽߦᲧߴߡචಽήⷞ಴᧪ࠆ߶ߤዊߐ޿႐วࠍ⠨߃ߡߺࠃ߁㧚ߔߥࠊߜ
H12 

H11  H 222

(10.1.67)

ߎߩ႐ว2§ H11  H 22 ·2¨¸  H122©
¹

W

H11  H 22
4 H12
1
2
2
H11  H 22
2

ߎߎߢᑼ(10.1.67)ࠍ⠨ᘦߒߡ࡞࡯࠻ߩਛࠍዷ㐿ߒߡ╙ 2 㗄ߢᛂߜಾࠆߣ
2º
H11  H 22 ª
2 H12
«1 
»
2
2
H11  H 22 ¼»¬«

W

H11  H 22
H12
2
H11  H 222ᓥߞߡ㧘ᑼ(10.1.63)ࠃࠅ
2

E1

H11 

H12
H11  H 22
2

E1  H '11 
0

H '12
0
0
E1  E2  H '11  H '22
2

E2(10.1.68)

H12
H 22 
H 22  H112

E2  H '22 
0

H '12
0
0
E2  E1  H '22  H '11ߎߎߢ㧘߽߁ 1 ߟㄭૃߣߒߡ㧘 E1  E2 ߦኻߒߡ H '11  H '22 ߇ዊߐߊߡήⷞ಴᧪ࠆߣ
00䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 15ߔࠆߣ㧘

E1  H '11 
0

E1

H '12
0
E  E201(10.1.69)

H'
E2 E2  H '22  0 12 0E2  E10ߣߥࠆ㧚৻ᣇ㧘wavefunction ߦߟ޿ߡߪ㧘ᑼ(10.1.67)ࠍ⠨ᘦߒߡ2 H12
tan 2T | 2T
H11  H 22ߎࠇࠃࠅ
cos T | 1

H12
H11  H 22

sin T | Tᓥߞߡᑼ(10.1.61)ߪ

\ 1 \ 10 

H12
\ 20
H11  H 22

(10.1.70)

H12
\2 
\ 10
H 22  H11\20ߣߥࠆ㧚਄ߢⴕߞߚࠃ߁ߦ E1  E2 ߦኻߒߡ H '11  H '22 ߇ήⷞ಴᧪ࠆߣߔࠆߣ
0\ 10 \ 10 
\2

0

0

H '12
\ 20
E  E2 00
1H'\ 2  0 12 0 \ 10E2  E1(10.1.71)

0ᑼ(10.1.69)߅ࠃ߮ᑼ(10.1.71)ࠍߺࠆߣ㧘ߎࠇࠄߩᑼߪ㧘energy ߦߟ޿ߡߪ៨േߩ 2 ᰴ߹ߢ㧘wavefunction ߦߟ޿ߡߪ៨േߩ 1 ᰴ߹ߢ⠨ᘦߒߚᑼߣߥߞߡ޿ࠆ㧚એ਄㧘2 ߟߩ႐วߦߟ޿ߡ㧘ᱜߒ޿⸃ߣ៨േ⺰ߦࠃࠆㄭૃ⸃ࠍᲧセߒߡߺߚ㧚޿ߕࠇߩ႐ว߽㧘energy ߩᣇ߇ wavefunction ࠃࠅ߽៨േ߇ 1 ᰴಽߛߌ㜞ᰴߣߥߞߡ޿ࠆߎ
ߣߦᵈ⋡ߒߚ޿㧚৻⥸ߦ៨േ⺰ߢ energy ߣ wavefunction ࠍ᳞߼ࠆ႐ว㧘wavefunction
ࠍ n ᰴߩ៨േ߹ߢߣࠇ߫ energy ߩᣇߪ n  1 ᰴ߹ߢขࠆߣߟߓߟ߹߇ว߁㧘ߣ޿߁ߎߣ߇ዉ߆ࠇࠆ㧚਄ߦ޽ߍߚ 2 ߟߩ଀ߪ߹ߐߒߊߎߩߎߣࠍ␜ߒߡ޿ࠆ㧚‫ڏ‬Exercise 10.1.1
ᄢ߈ߐ L ߩ 1 ᰴరߩ particle-in-a-box ߦ߅޿ߡ㧘potential ߇ᰴߩࠃ߁ߥ step functionߢ⴫ߐࠇࠆ႐วࠍ⠨߃ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 16

La
2

V x

0

0d x

V x

H

La
La
dxd
2
2

V x

0

La
xdL
2

࿑ 10.1.2aLߣߒߡ n 1 ߅ࠃ߮ n
10ࡐ࠹ࡦࠪࡖ࡞

2 ߩ⁁ᘒߩߘࠇߙࠇߦߟ޿ߡ㧘৻ᰴߩ៨േ energy ࠍ᳞߼

ࠃ㧚
‫ޤ⸃ޣ‬
㧦

1 ᰴߩ៨േ energy ³\ n 0 H '\ n 0 dx ࠍ᳞߼ࠇ߫ࠃ޿㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 17

wavefunction ߪ(6.3.12)ߦ␜ߐࠇࠆࠃ߁ߦ
2
nS x
sin
L
L
ߢ޽ࠆ㧚ᓥߞߡ

\ n0

La

En

1

2 2
nS x
H dx
sin 2
³
L La
L
2

nS a ·
n 1
§a
 1
sin
¸
nS
L ¹
©L

H¨
a
L

0.1 ࠍઍ౉ߒߡ
E1 1

0.1984H

1

0.0065H

E2
ᓥߞߡ

E1
E2

h2
 0.1984H
8mL2
h2
 0.0065H
8mL2‫ڏ‬Exercise 10.1.2
ᄢ߈ߐ L ߩ৻ᰴరߩ particle-in-a-box ߦ߅޿ߡ㧘potential ߇V x

H sinSx
Lߢ⴫ߐࠇࠆࠃ߁ߥ៨േ߇߆߆ߞߚ႐วࠍ⠨߃ࠆ(࿑ 10.1.1)㧚 n 1 ߩ⁁ᘒߦߟ޿ߡ 1 ᰴ
ߩ៨േ energy㧘߅ࠃ߮ 2 ᰴߩ៨േ energy ࠍ᳞߼ࠃ㧚ߚߛߒ㧘ᓟ⠪ߩ⸘▚ߦ޽ߚߞߡߪ
n 9 ߹ߢߩ๺ࠍߣࠇ߫ࠃ޿ߎߣߦߔࠆ㧚‫ޤ⸃ޣ‬㧦
n 1 ߩ࠯ࡠᰴߩ wavefunction ߪ

2
Sx
sin
L
L
ᓥߞߡ㧘1 ᰴߩ៨േ energy ߪ\ 10䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 18

2
Sx Sx
 H ³ sin 2
sin
dx
L 0
L
L
L

E1

1

8H
3S
2 ᰴߩ៨േ energy ߪ㧘


E1

f

H '1n2

¦E E

2

n 2

0
1

0

n

ߢ޽ࠆ߇㧘ಽᲣߩ energy Ꮕߪh2
1  n28mL2৻ᣇ㧘ಽሶߦߟ޿ߡߪ
E10  En 0

2H
Sx
nS x
sin 2
sin
dx
³
L 0
L
L
L



H '1n

ߢ޽ࠆ߇ n ߇஧ᢙߩߣ߈ߦߪ࠯ࡠߦߥࠆߎߣ߇⋥ߜߦಽ߆ࠆ㧚৻ᣇ㧘 n ߇ᄸᢙߩߣ߈
ߦߪ



H '1n

2H § 1n ·¨  2S © n n  4 ¸¹

2H'ߣߥࠆ㧚 0 1n 0 ࠍౕ૕⊛ߦ n 3, 5, 7, 9 ߦߟ޿ߡ⸘▚ߔࠆ㧚
E1  En

2

n

H '1n

H

§ H '1n ·¨ H ¸©
¹
n2  1

30.16983.60 u103

5

0.02432.45 u105

7

0.00811.36 u106

9

0.0037

1.69 u107ᧄ᧪ߥࠄ n 11, 13, ... ߣ nf ߹ߢߩ㗄ࠍ⸘▚ߒߥߌࠇ߫ߥࠄߥ޿߇㧘 n 9 ߹ߢขࠆ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 19ߣ޿߁ㄭૃࠍขࠆߎߣ߇໧㗴ߦ␜ߐࠇߡ޿ࠆߩߢ n 9 ߢᛂߜಾߞߚࠊߌߢ޽ࠆ㧚ߎࠇࠄߩ๺ࠍߣߞߡ⚿ዪ

E1 2

3.63 u103 H 2
h 2 / 8mL2

ߣߥࠆ㧚0 ᰴ㧘1 ᰴ㧘2 ᰴࠍ߹ߣ߼ࠆߣᰴߩࠃ߁ߦߥࠆ㧚
E1

h2
8H
103 H 2
3.63uh 2 / 8mL2
8mL2 3S8h2 §·1  J  3.63 u 103 J 2 ¸
2 ¨
8mL © 3S
¹

ߚߛߒ

J

H
2

h / 8mL2៨േ⺰ࠍౕ૕⊛ߦ૶߁ߦ޽ߚߞߡ޿ߊߟ߆ߩᛛⴚ⊛ߥ໧㗴߇޽ࠆ㧚1 ߟߪ㧘੹߹ߢ޽޿߹޿ߦㆊߏߒߡ᧪ߚ໧㗴ߢ㧘៨േߩ series ߪߪߚߒߡ෼᧤ߔࠆ߆ߣ޿߁ߎߣߢ޽
ࠆ㧚ߎࠇߪታߪ H ' ߩㆬ߮ᣇߦ㑐ଥߔࠆ㧚਄ߩ Exercise ࠍߺࠆߣ㧘energy ߪ

H
h2

ߦ

8mL2

ߟ޿ߡߩ౹ዷ㐿ߩᒻߣߥߞߡ޿ࠆ㧚ߎߩ୯߇ዊߐߌࠇ߫ዊߐ޿߶ߤ෼᧤ߪ޿޿ࠊߌߢ
޽ࠆ㧚ߒ߆ߒ㧘ᑼ(10.1.3)ߦ␜ߒߚ H 0 ߣ H ' ࠍ⷗ߡ㧘⋥ߜߦ

H
h2

ߣ޿߁㊂ࠍᕁ޿

8mL2

ߟߊ߆ߤ߁߆㧘ߥ߆ߥ߆߻ߕ߆ߒ޿ߣߎࠈߢ޽ࠈ߁㧚߹޽㧘ߣߦ߆ߊ߿ߞߡߺࠆߩ߇
޿޿ߩߢߪߥ߆ࠈ߁߆㧚ߘߒߡ㧘߽ߒ߽ 1 ᰴ㧘2 ᰴ㧘㧚㧚㧚ߣ⸘▚ߒߡ޽߹ࠅߦ߽෼᧤߇
ᖡߌࠇ߫㧘߽ߣߦᚯߞߡ⇣ߥߞߚ H 0 ߣ H ' ߩขࠅᣇࠍ⠨߃ߡߺࠇ߫޿޿ߢ޽ࠈ߁㧚

1 ᰴߩ៨േ energy ࠍ᳞߼ࠆߎߣߪ޿ߟ߽㔍ߥߊ಴᧪ࠆ㧚ߎࠇߪ㧘operator H ' ߦ㑐ߔࠆ average ߢ޽ࠆ߆ࠄ૗ߩ໧㗴߽ߥ޿㧚ߒ߆ߒ㧘2 ᰴએ㒠ߩ៨േ energy㧘߹ߚߪ 1ᰴએ㒠ߩ wavefunction ࠍ᳞߼ࠆ႐วߦߪ㧘޿ߟ߽ή㒢୘ߩ๺ࠍߤ߁߿ߞߡ⸘▚ߔࠆ߆㧘ߣ޿߁ߎߣ߇໧㗴ߣߥࠆ㧚energy Ꮕ߇޿ߟ߽ಽᲣߦ⃻ࠇࠆߩߢ㧘ߒ߫ߒ߫‫ޟ‬energy Ꮕ߇޽ࠆ⒟ᐲࠃࠅᄢ߈޿⁁ᘒ߆ࠄߩ៨േߪ⠨߃ߥߊߡ޿޿‫߁޿ߣޠ‬ㄭૃ߇ߣࠄࠇࠆ㧚ߒ߆ߒ㧘ಽᲣߩ energy Ꮕ߇ᄢ߈ߊߥߞߚᤨ㧘ಽሶߩⓍಽ߇ߘࠇએ਄ߦᄢ߈ߊߥࠄߥ޿ߣ
޿߁଻㓚ߪᄙߊߩ႐วᓧࠄࠇߥ޿㧚Exercise 10.1.2 ߩ႐วߪ㧘߻ߒࠈ଀ᄖ⊛ߦ߁߹ߊ޿ߞߡ޿ࠆߣ޿߃ࠃ߁㧚ߎߩ႐วߦߪ㧘n ߇ᄢ߈ߊߥࠆߣ㧘ಽᲣߩ energy Ꮕߪ 1  n 2 ߢ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 20ᕆỗߦᄢ߈ߊߥߞߡ޿ߊߒ㧘ಽሶߩ H '1n ߽ n ߣ౒ߦዊߐߊߥߞߡ޿ࠆ㧚ౕ૕⊛ߥᢙ୯ࠍߺߡ߽ n 3, 5, 7, 9 ߦߥࠆߦߟࠇߡᕆỗߦዊߐߊߥߞߡ޿ࠆ㧚ߎࠎߥߦ߁߹ߊ޿ߊ
႐วߪ߼ߞߚߦߥ޿㧚ߒ߆ߒ㧘ߎߩ㧘଀ᄖ⊛ߦ߁߹ߊ޿ߞߚ႐วߢ߽㧘 n ࠍή㒢ᄢ߹ߢ⿷ߒวࠊߖߚࠄߤ߁ߥࠆߢ޽ࠈ߁߆㧫ዊߐߥዊߐߥᢙࠍή㒢୘㓸߼ߚࠄߤ߁ߥࠆ
ߢ޽ࠈ߁߆㧫ߎߩ޽ߚࠅߩߎߣߪߥ߆ߥ߆߻ߕ߆ߒߊ㧘߶ߣࠎߤಽ߆ࠄߥ޿㧘ߣ޿߁ߩ߇ታᖱߢ޽ࠆ㧚ᄙߊߩ႐วߪ㧘޽߹ࠅߜ߾ࠎߣߪี๧ߒߥ޿ߢ૶ߞߡ޿ࠆߺߚ޿ߢ
޽ࠆ㧚ή㒢୘ߩ๺ࠍߤߎ߆ߢߜࠂࠎಾࠆߎߣߦਇḩࠍᜬߟੱߪߒ߫ߒ߫ 2 ᰴߩ៨േ energyࠍ‫' ޟ‬E ㄭૃ‫ߊ⸃ߡ޿↪ࠍޠ‬㧚closure approximation ߣ޿߁⸒⪲߽૶ࠊࠇࠆ㧚2 ᰴߩ៨േ energy
H 'nm H 'mn
2(10.1.72)
En¦00m z n E n  Emࠍ᳞߼ࠆߦᒰࠅ㧘઒ߦ㧘ಽᲣߩ energy Ꮕ߇ m ߦή㑐ଥߢ 'E ߣ޿߁୯ߦ⟎ߌࠆߣߒࠃ
߁㧚઒ߦ㧘ߘ߁޿߁ߎߣ߇น⢻ߢ޽ࠆߣߔࠆߣ㧘
1
2
En¦ H 'nm H 'mn'E m z n(10.1.73)
1 ­2½®¦ H 'nm H 'mn  H 'nn ¾'E ¯ m¿ߣߥࠆ߇㧘ฝㄝߩ╙ 1 㗄⋡ߪ¦H ' H '
nm

mn

H'

2

(10.1.74)

nn

m

2

ߩࠃ߁ߦ H ' ߣ޿߁ operator ߦ㑐ߔࠆᦼᓙ୯ߢ⴫ߐࠇࠆ㧚(Problem 10.1.1)
ᓥߞߡ

En

2

1
'E

^ H'

2

nn

 H 'nn2`(10.1.75)2ߣߥࠆ㧚ߎߩ⚿ᨐߪ㧘ή㒢㗄ߩ๺߇ߥߊߥߞߡ޿ߡ㧘 H ' ߅ࠃ߮ H ' ߦ㑐ߔࠆᦼᓙ୯
ߩߺ߆ࠄ 2 ᰴߩ៨േ energy ࠍ᳞߼ࠆߎߣ߇಴᧪ࠆߣ޿߁ὐߢ㧘ାᄺ⠪߽ᄙ޿㧚ߒ߆ߒ㧘
ߎߩᣇᴺߩℂ⺰⊛ᰳὐߪ㧘૗ߣ޿ߞߡ߽ En  Em ߣ޿߁㧘⃻ߦ m ߦଐሽߔࠆ energy00Ꮕࠍ m ߦή㑐ଥߦ߅ߊ㧘ߣ޿߁઒ቯߦ޽ࠆ㧚ቢో♽ࠍߥߔోߡߩ⁁ᘒ߇หߓ energyࠍᜬߟߥࠎߡ޿߁ߎߣߪ޽ࠅ߃ߥ޿ߎߣߪ⸒߁߹ߢ߽ߥ޿㧚(10.1.73)ߪ߻ߒࠈ㧘ߎߩᑼߢቯ⟵ߐࠇࠆࠃ߁ߥ 'E ߣ޿߁‫ޟ‬ᐔဋߩ energy Ꮕ‫ࠆ޽߇ޠ‬㧘ߣ⠨߃ߡߏ߹߆ߔߩߢ
޽ࠆ߇㧘ߘߎߦℂ⺰⊛ߥᩮ᜚߇޽ࠆࠊߌߢߪߥ޿㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 21‫ڏ‬Exercise 10.1.3
Exercise 10.1.2 ߩ♽ߦߟ޿ߡ 2 ᰴߩ៨േ energy ࠍ closure approximation ߢ᳞߼ࠃ㧚

Exercise 10.1.2 ߩ⚿ᨐ߇ᱜߒ޿ߣߔࠆߣ 'E ߪߤߩ૏ߩᄢ߈ߐߦߥࠆ߆㧚‫ޤ⸃ޣ‬㧦
Exercise 10.1.2 ߢ᳞߼ߚࠃ߁ߦ



H 'nn
H'

2
nn

8H
3S
2 2 L 4 Sx
dx
H sin
L ³0
L
2 2 S 4H ³ sin xdx

S0(10.1.76)

3S
H
S
8
2
3H
4
2

2

ᓥߞߡᑼ(10.1.75)ࠃࠅ

H 2 °­ 3 § 8 · °½
® ¨ ¸ ¾'E °¯ 4 © 3S ¹ °¿
2

E1

2

0.0295

ߎߩ୯ࠍ 3.63 u'EH(10.1.77)2'E

103 H 2ߣ╬ߒ޿ߣ߅ߊߣ
h2
8mL2

8.13

h2
8mL2

ߣߥࠆ㧚ߎࠇߪ n 3 ߩ⁁ᘒߣߩ energy Ꮕ 1  9(10.1.78)h2ߦㄭ޿㧚ߒ߆ߒ㧘ߎߩߎߣߦ૗
8mL2╬߆ߩ‛ℂ⊛ᗧ๧߇޽ࠆࠊߌߢߪߥ޿㧚એ਄㧘㐳‫ߣޘ‬ㅀߴߡ᧪ߚࠃ߁ߦ㧘៨േℂ⺰ߩᑼߩዷ㐿ߪ㕖Ᏹߦ෩ኒߦⴕߞߚ㧚ߘߎߢᓧࠄࠇߚᑼߪᱜߒ޿㧚ߒ߆ߒ㧘ߘࠇࠍ೑↪ߔࠆᲑߦߥࠆߣ㧘޿ߊߟ߽ߩ⃻ታ⊛ߥ࿎㔍ߦㆣㆄߔࠆ㧚៨േߩᰴᢙࠍ 1 ᰴߣ߆ 2 ᰴߣ߆ߢ඙ಾࠆߎߣ߇߹ߕㄭૃߢ޽ࠆߩߦ㧘ߘߩㄭૃౝߢߐ߃෩ኒߥ⸃ࠍᓧࠆߎߣߪᄙߊߩ႐วߦ಴᧪ߥ޿ߩߢ޽ࠆ㧚⚿ዪ㧘㊂ሶജቇߦ㑐ㅪߒߚℂ⺰ߣ޿߁߽ߩߪ㧘ᅤ૗ߦㄭૃࠍዉ౉ߔࠆ߆㧘ߩᎿᄦ 1 ߟߦ߆߆ߞߡ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 22޿ࠆ㧚෩ኒߥߎߣࠍᦸ߻ੱߦߣߞߡߪ⠴߃㔍޿߶ߤዮㄗ⊛ߥߎߣߦᕁ߃ࠆ㧚ߎߩ┨ߩ౨㗡ߢ㧘‫ߥ߁ࠃߩߎޟ‬ㄭૃᴺߦߟ޿ߡቇ⠌ߒߥߌࠇ߫ߥࠄߥ޿ߣ޿߁ߎߣߪᖤߒ޿⃻ታߢ޽ࠆ‫ߣޠ‬ㅀߴߚߩߪ਄ㅀߒߚࠃ߁ߥߎߣ߇㗡ߦ޽ߞߚ߆ࠄߢ޽ࠆ㧚‫ڏ‬Exercise 10.1.4
߫ߨߦ߅޿ߡ㧘potential ߇V x

1 2 1 3 1 4kx  gx  hx
2
6
24

(10.1.79)

ߢ⴫ߐࠇࠆߣߔࠆ㧚㕖⺞๺ᕈ㧘ߔߥࠊߜ potential ߩ╙ 2 㗄⋡એ㒠ࠍ៨േߣߒߡ 1 ᰴߩ៨േ energy ࠍ᳞߼ࠃ㧚
‫ޤ⸃ޣ‬
㧦

H'

1 3 1 4gx  hx
624
1
1
g[ 3 
h[ 4
2
3
E
24
6 E

1 ᰴߩ៨േ energy ߪEn

1³\ [ H \' [ d[nnߢ޽ࠆ߇㧘ኻ⒓ᕈࠃࠅ [ 3 ߩ៨േ㗄ߪነਈߒߥ޿ߎߣ߇ಽ߆ࠆ㧚ᓥߞߡh1\ n [ [ 4\ n [ d [En
24E 2 ³ߢਈ߃ࠄࠇࠆߎߣߦߥࠆ㧚(7.5.5),(7.5.6)ࠃࠅ
1[b  b
2
ᓥߞߡ

En 1

4
h
\ [ b  b \ n [ d[
2 ³ n
96E

ߣߥࠆ㧚ᑼ(7.5.30)߅ࠃ߮ᑼ(7.5.31)ࠃࠅ

b  b \ n2b  b \ n

n\ n 1  n  1\ n 1
n b  b  \ n 1  n  1 b  b  \ n 1
n n  1 \ n  2  n\ n  n  1 \ n 䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 23

n  1 n  2 \ n2

3b  b \ nn n  1 b  b  \ n  2  2n  1 b  b  \ n n  1 n  2 b  b \ n2n n  1 n  2 \ n 3  n n  1 \ n 1  2n  1

n\ n 1  2n  12

4b  b \ nn  1 n  2 \ n 1 n  1\ n 1

n  1 n  2 n  3 \ n 32

n n  1 n  2 b  b  \ n 3
3 n3 b  b  \ n 1  3


n 1

3

b  b  \ n 1

n  1 n  2 n  3 b  b  \ n 3

3 n3 n\ n  3

n 1

3

n  1\ n  terms other than \ n

2
3 ª n 2  n  1 º\ n  terms other than \ n
¬
¼

ᓥߞߡ

³\ b  b 4n\ n d[

2
3 ªn2  n  1 º
¬
¼

߆ߊߒߡ
En 1

h
2
3 ªn2  n  1 º
2
¬
¼
96 E
h ª 2
2
n  n 1 º
2 ¬
¼
32 E

(10.1.80)‫ڏ‬Problem 10.1.1
ᑼ(10.1.74)ࠍ⸽᣿ߖࠃ㧚

Hint㧦ዷ㐿ቯℂࠍ↪޿ࠆ㧚‫ڏ‬Problem 10.1.2

Exercise 10.1.4 ߩᑧ㐳ߣߒߡ㧘៨േ
H'

1 3
gx
6

ߦߟ޿ߡߩ 2 ᰴߩ៨േ energy ࠍ᳞߼ࠃ㧚ߎߩ႐ว㧘ή㒢㗄ߩ๺ߩ߁ߜ㧘߶ࠎߩᢙ㗄ߩߺ߇࠯ࡠߢߥߊ㧘2 ᰴ៨േߩᨒౝߢߪᱜ⏕ߥ⚿ᨐ߇᳞߹ࠆ⃟ߒ޿଀ߢ޽ࠆ㧚‫ڏ‬Problem 10.1.3
Section 6.5 ߢᱜ⏕ߦ⸃ߊߎߣࠍ⹜ߺߚᲑᏅߩ޽ࠆ੗ᚭဳࡐ࠹ࡦࠪࡖ࡞㧔࿑ 6.5.1㧕ߩ࿕᦭୯ࠍ៨േ⺰ߢ⸃߈㧘ᱜ⏕ߥ⸃ߣᲧセߖࠃ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 24‫ڏ‬Problem 10.1.4㊂ሶᢙ n ߩ harmonic oscillator ߩ energy ߪ

hQ

En

n 1
2

(10.1.81)

ߢਈ߃ࠄࠇࠆߎߣߪߔߢߦቇ⠌ߒߚ㧚ߎߩᑼࠍᵄᢙන૏ߢ⴫ߖ߫

Q n 1Enhc(10.1.82)

c 2ߣߥࠆ߇㧘ಽሶಽశቇߩಽ㊁ߢߪ㧘Herzberg ߩ⪺ᦠߦᓥߞߡߎࠇࠍ

Ze

G n

n 1
2

(10.1.83)ߩࠃ߁ߦ⴫ߔߎߣ߇ᄙ޿㧚਄ᑼ߇ᚑࠅ┙ߟ㒢ࠅᒰὼ㧘G n  G n 1

Ze

(10.1.84)

ߩࠃ߁ߦ㧘㊂ሶᢙ߇ 1 ߟ⇣ߥࠆ⁁ᘒߩ energy Ꮕߪ৻ቯߦߥࠆߎߣ߇ᦼᓙߐࠇࠆ㧚ߒ߆ߒ㧘⃻ታߩಽሶߢߪߘ߁ߪ޿߆ߥ޿㧚HCl ߦߟ޿ߡߩታ᷹࠺࡯࠲ࠍ⴫ߦ␜ߔ߇㧘ߎߩ⴫߆ࠄಽ߆ࠆࠃ߁ߦ n ߇ᄢ߈ߊߥࠆ
ߦߟࠇߡ energy Ꮕߪዋߒߕߟዊߐߊߥߞߡ޿ߊ௑ะߦ޽ࠆ㧚n

G n G 0cm 1

G n  G n 1
cm 1

1

2885.92885.925668.027821.

38347.02679.04109231
.

25761
.

5

13396.52473.4ߎߩߎߣߪ㧘potential ߩ 3 ᰴߩ㗄ߦଐሽߔࠆ anharmonicity ߩߚ߼ߣ⠨߃ࠄࠇߡ޿ࠆ
㧔Exercise 10.1.4 ߢⴕߞߚࠃ߁ߦ㧘4 ᰴߩ㗄ߩ៨േ energy ߪᱜߢ޽ࠆ߆ࠄ energy 㑆㓒ߪ n ߇ᄢ߈ߊߥࠆߦߟࠇߡᄢ߈ߊߥࠆߪߕߢ޽ࠆ㧚ߎߩߎߣࠍߚߒ߆߼ࠃ㧕㧚਄ߩ⴫ߩࠃ߁ߥታ㛎⚿ᨐࠍ⴫ߔ⚻㛎⊛ߥᑼߣߒߡᰴߩࠃ߁ߥᑼ߇ਈ߃ࠄࠇߡ޿ࠆ㧚

G n

n 1
§ n 1 ·Ze
 Ze xe ¨
¸
2
© 2 ¹

2

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 25(10.1.85)

ߎߎߢ Ze xe ߪ㧘Ze ߦ xe ࠍដߌߚ߽ߩߣ⠨߃ߡ߽ࠃ޿߇㧘ㅢᏱߪ㧘ߘ߁ߪ⠨߃ߕ㧘Ze xeߣ޿߁ 2 ߟߩᢥሼߢ 1 ߟߩᄌᢙࠍ⴫ߔߣ⷗ߥߒߡ޿ࠆ㧚એਅߩ໧㗴ߦ╵߃ࠃ㧚

(1)਄⴫ߩታ㛎࠺࡯࠲ࠍ߽ߞߣ߽ࠃߊ⴫ߔࠃ߁ߦ Ze ߅ࠃ߮ Ze xe ࠍ᳞߼ࠃ㧚ߘߩᓟ㧘ߘߩᑼ߆ࠄߩ⸘▚୯ߣታ㛎୯㧔਄ߩ⴫㧕ߣࠍ⴫ߩᒻߢᲧセߖࠃ㧚

(2)
(3)

(4)

Problem 10.1.2 ࠍෳ⠨ߦߒߡ㧘ᑼ(10.1.85)߇ 2 ᰴߩ៨േ⺰ߦ޽ࠆㄭૃࠍዉ౉ߔ
ࠆߎߣߦࠃߞߡᓧࠄࠇࠆߎߣࠍ␜ߖ㧚ߎߩ႐ว㧘ߤ߁޿߁ㄭૃࠍዉ౉ߔࠆᔅⷐ߇޽ࠆ߆ࠍ᣿ࠄ߆ߦߒ㧘߆ߟ㧘ߘߩㄭૃߩᅷᒰᕈࠍี๧ߖࠃ㧚ߎߩ႐ว Ze xe ߪߤ߁޿߁ᑼߢ⴫ߐࠇࠆ߆㧚ߥ߅㧘ߎߩ⚿ᨐߪ଀߃߫ Herzberg ߩ᦭ฬߥᢎ⑼ᦠ㧔G. Herzberg, "Spectra of Diatomic

Molecules"㧕ߩ p.93 ߦ಴ߡ޿ࠆ㧚⸥ㅀߪ޿ߚߞߡ◲නߢ㧘‫ޟ‬៨േ⺰ࠍ↪޿ࠆߣ⚿ᨐߪߎ߁ߥࠆ‫߁޿ߣޠ‬૏ߒ߆ᦠ޿ߡߥ޿㧚㊂ሶൻቇߩ౉㐷⊛ߥᢎ⑼ᦠߣߪ㆑ߞߡ⎇ⓥ⠪߇૶߁ᢎ⑼ᦠߢߪ㧘ߘߩ⒟ᐲߩ⸥ㅀߒ߆ߥ޿ߩ߇৻⥸⊛ߢ޽ࠅ㧘ᓥߞߡၮ␆߇ߒߞ߆ࠅ
ߒߡߥ޿ߣߥ߆ߥ߆ᢎ⑼ᦠ߽ℂ⸃಴᧪ߥ޿ߎߣߦߥࠆ㧚‫ڏ‬Problem 10.1.5

Problem 10.1.4 ߦ߅޿ߡߪ㧘ᑼ(10.1.85)ߩ⚻㛎⊛ߥᑼࠍ potential ߩ 3 ᰴߩ㗄ߩ 2 ᰴ៨േ energy ߢ⸃㉼ߒߚ㧚ታߪ potential ߩ 4 ᰴߩ㗄ߩ 1 ᰴ៨േ energy㧔Exercise 10.1.4㧕
ߣ potential ߩ 3 ᰴߩ㗄ߩ 2 ᰴ៨േ energy㧔Problem 10.1.4㧕ࠍวࠊߖߚ㗄ߦߟ޿ߡ߽ᑼ
(10.1.85)ߩ⚻㛎ᑼߪℂ⺰⊛ߦ⸃㉼ߐࠇࠆ㧚ߎߩ႐วߪ Ze xe ߪߤ߁޿߁ᑼߢ⴫ߐࠇࠆ߆㧚‫ڏ‬Problem 10.1.6
Exercise 10.1.4 ߩ⸃ᴺ߅ࠃ߮⸃߇⸵ߐࠇࠆ n ߩ▸࿐ࠍ␜ߖ㧚㧔 n

0, 1, 2, ... ߩߔߴߡ

ߩ㊂ሶᢙߢᚑࠅ┙ߟ߆ࠍี๧ߖࠃ㧕10.2 ❗ㅌߒߡ޿ࠆ♽ߦߟ޿ߡߩ៨േ⺰೨ߩ▵ߢߪ㧘࠯ࡠᰴߩ Hamiltonian H 0 ߩ eigenfunction ߦ❗ㅌ߇ߥ޿ߣ઒ቯߒߡ᧪
ߚ㧚ߥߗߘ߁઒ቯߒߥߌࠇ߫ߥࠄߥ߆ߞߚ߆㧘޽߹ࠅߪߞ߈ࠅߣㅀߴߡߪ޿ߥ߆ߞߚ
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 26ߣᕁ߁㧚ᦝߦߎߩ⴫⃻ߪᄙዋ޽޿߹޿ߢ߽޽ߞߚ㧚❗ㅌߒߡ޿ࠆߣߤ߁޿߁ߣߎࠈߢ߹ߕ޿߆ߪᦨ⚳⚿ᨐߩᑼ(10.1.40),ᑼ(10.1.41)ࠍ⷗
ࠇ߫᣿ࠄ߆ߢ޽ࠆ㧚઒ࠅߦ\ n 0 ߇❗ㅌߒߡ޿ࠆ⁁ᘒߩ 1 ߟߢ޽ࠆߣߔࠆ㧚ߎߩ႐ว㧘ᑼ(10.1.40)߅ࠃ߮ᑼ(10.1.41)ߦ⴫ߐࠇࠆ๺ߩਛߢಽᲣ߇࠯ࡠߩ㗄߇ 1 ߟએ਄޽ࠆߎߣ
ߦߥࠆ㧚ᓥߞߡ㧘ߘߩ㗄ߦߟ޿ߡ߽ߒ߽ಽሶ߇࠯ࡠߢߥ޿㒢ࠅ⊒ᢔߒߡߒ߹߁㧚ߎߩࠃ߁ߦ㧘៨േℂ⺰ߩ⚿ᨐ߇⊒ᢔߒߡߒ߹߁ߣ޿߁ߎߣߪ㧘࠯ࡠᰴߩ wavefunction ߩขࠅᣇ߇ㆡᒰߢߥ߆ߞߚߣ޿߁ߎߣࠍᗧ๧ߒߡ޿ࠆ㧚৻⥸ߦߪ㧔߽ߒ߽❗ㅌ߇ߥߌࠇ
߫㧕㧘࠯ࡠᰴߩ wavefunction ߣ޿߁߽ߩߪ H 0 ࠍਈ߃ࠇ߫৻⟵⊛ߦ᳿߹ߞߡߒ߹߁㧚ߒ߆ߒ㧘❗ㅌ♽ߦߟ޿ߡߪ੐ᖱ߇⇣ߥࠆ㧚੹߹ߢ૗࿁߽ቇ⠌ߒߡ᧪ߚࠃ߁ߦ㧘❗ㅌߒߡ޿ࠆ wavefunction ߩㆬ߮ᣇߦߪή㒢୘ߩน⢻ᕈ߇޽ࠆࠊߌߢ㧘޽ࠆㆬ߮ᣇࠍߔࠇ߫៨
േ㗄߇⊒ᢔߒߡߒ߹߁ߣ޿߁޽ࠊࠇߥ⚿ᨐߦߥߞߡߒ߹߁߇㧘ㆡᒰߥㆬ߮ᣇࠍߔࠆ
ߣ㧘ߘߩࠃ߁ߥ⊒ᢔࠍ㒐ߋߎߣ߇಴᧪ࠆߩߢ޽ࠆ㧚એᓟ㧘ߘߩߎߣࠍౕ૕⊛ߦี๧ߒࠃ߁㧚ߥ߅㧘 H 0 ߩ eigenfunction ߩਛߦ㧘ߚߣ߃❗ㅌߒߚ߽ߩ߇޿ߊߟ߆޽ߞߚߣߒߡ߽㧘
੹ᑼ(10.1.41)ߢ᳞߼ࠃ߁ߣߒߡ޿ࠆ\ n ߇ߘߩ❗ㅌߒߚ wavefunction ߩ 1 ߟߢߥ޿㒢ࠅ㧘ᑼ(10.1.40)߿ᑼ(10.1.41)ࠍ↪޿ࠆߎߣߪ૗╬Ꮕߒᡰ߃ߥ޿ࠊߌߢ޽ࠆ㧚੹߹ߢ㧘‫❗ޟ‬ㅌ߇ߥ޿႐ว‫߇ߚߞ߆ߥ᧪ߡߒ߆ߒ⃻⴫ߥ޿߹޿޽ߣޠ‬㧘ᱜ⏕ߦߪߎ߁޿߁ߎߣࠍ⸒
ߞߡ޿ࠆߎߣߦߥࠆ㧚੹㧘❗ㅌߒߡ޿ࠆ࠯ࡠᰴߩ wavefunction ߪ\ 10 , \ 2 0 , ..., \ m 0 ߢ޽ࠆߣߒߡ৻⥸ᕈߪᄬࠊࠇߥ޿㧚ߔߥࠊߜH 0\ j 0

E j 0\ j 0

j 1, 2, ..., m(10.2.1)

0(10.2.2)

ߦ߅޿ߡ㧘
0

E1

E2

0

... Emᦝߦ㧘ߎࠇࠄ m ୘ߩ wavefunction ߪ੕޿ߦ⋥੤ߒߡ޿ࠆߣ⠨߃ߡࠃ޿㧚㧔޿ߟߢ߽⋥
੤ߔࠆࠃ߁ߦߔࠆߎߣ߇น⢻ߢ޽ࠆ㧚㧕ߐߡ㧘ߎࠇࠄ m ୘ߩ wavefunction ߩㆡᒰߥ✢ᒻ⚿ว߆ࠄ㧘੕޿ߦ⁛┙ߥ m ୘ߩ wavefunction

Ii 0

m

¦c \
ij

0
j

i 1, 2, ..., m(10.2.3)

j 1

ࠍ૞ࠆߎߣ߇಴᧪ࠆ㧚ߎ߁ߒߡ૞ࠄࠇߚᣂߒ޿ wavefunction ߪ߿ߪࠅ❗ㅌߒߡ޿ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 27ߎߩࠃ߁ߥ wavefunction ߩ⚵ߩߣࠅᣇߪή㒢ㅢࠅߩน⢻ᕈ߇޽ࠆ߇㧘ߘߩਛߢ H ' ߦ㑐ߔࠆⓍಽ ³ Ii 0 * H 'I j 0 dW ߇ߔߴߡ࠯ࡠߦߥࠆࠃ߁ߥ⚵ࠍㆬ߱㧚ߤ߁߿ߞߡߎߩࠃ߁ߥ

wavefunction ߩ⚵ࠍ᳞߼ࠆ߆ߪᓟ߆ࠄ⠨߃ࠆߎߣߣߒߡ㧘ߣߦ߆ߊ㧘 H 0 ߩeigenfunction ࠍ I10 , I2 0 , ..., Im 0 , \ m 10 , \ m  2 0 , ... ߩࠃ߁ߦߣࠆ㧚ߔߥࠊߜᦨೋߩ m ୘ࠍ❗ㅌߒߡ޿ࠆ߽ߩߦㆬ߱ࠊߌߢ޽ࠆ㧚ߎࠇ߆ࠄⴕ߁ᔅⷐߩ޽ࠆߎߣߪ㧘I10 ߆ࠄ Im 0 ߹ߢߩ wavefunction ߅ࠃ߮ energy ߦߟ޿ߡߩ៨േࠍ᳞߼ࠆߎߣߢ޽ࠆ㧚\ m 10 એ㒠ߦߟ޿ߡߪ❗ㅌߪߥ޿ߩߢ೨▵ߩᑼࠍߘߩ߹߹૶޿ᓧࠆࠊߌߢ޽ࠆ߆ࠄᡷ߼ߡㅀߴࠆᔅ
ⷐߪߥ޿㧚೨▵ߣห᭽ߦ㧘H0  O H 1 \ i

Ei\ ii 1, 2, ..., m(10.2.4)ࠍ⸃ߊࠊߌߢ޽ࠆ߇㧘ߎߎߢ\ i ߅ࠃ߮ Ei ࠍᑼ(10.1.9)߅ࠃ߮ᑼ(10.1.10)ߩࠃ߁ߦ㧘O ߩ౹ዷ㐿ࠍⴕ߁㧚\ i Ii 0  O\ i 1  O 2\ i 2  ...

(10.2.5)

Ei 0  O Ei  O 2 Ei

(10.2.6)Ei

12ߎࠇࠄࠍᑼ(10.2.4)ߦઍ౉ߒ㧘ᑼ(10.1.12)߅ࠃ߮ᑼ(10.1.13)ࠍ಴ߒߚߩߣหߓࠃ߁ߦߒ
ߡᰴߩ⚿ᨐࠍᓧࠆ㧚

H 0Ii 0E10Ii 0H 0  E10 \ i

1(10.2.7)
Ei Ii 0  H Ii 01

1

(10.2.8)

ᑼ(10.2.7)ߪᑼ(10.2.2)ࠃࠅߔߢߦᚑࠅ┙ߞߡ޿ࠆ㧚ᑼ(10.2.8)ࠃࠅ Ii 1 ߅ࠃ߮ Ei 1 ࠍ᳞߼ࠆࠊߌߢ޽ࠆ߇㧘ߎߎߢ߽ᑼ(10.1.15)ߣหߓࠃ߁ߦ㧘 H 0 ߩ eigenstate

I10 , I20 , ..., Im 0 , \ m 10 , \ m  2 0 , ... ߢዷ㐿ߔࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 28

m¦ A I  ¦ A\

\i 1

0

j

j

j

0

(10.2.9)jj t m 1

j 1

ߎߩᑼߢ㧘ฝㄝߩ╙ 1 㗄߇❗ㅌߒߡ޿ࠆ wavefunction ߩ⚵ߦߟ޿ߡߩ๺㧘╙ 2 㗄߇ߘߩઁߩ wavefunction ߦߟ޿ߡߩ๺ߢ޽ࠆ㧚j i ߩ㗄ߪᒰὼ╙ 1 㗄ߩ๺ߩਛߦ฽߹ࠇࠆ㧚ߎߩᑼࠍᑼ(10.2.8)ߦઍ౉ߔࠆ㧚m¦A E E I  ¦ A E E \
0
1

j

0
1

0

0

i

j

0
1

j

0
j

j t m 1

j 1

m

¦c E
ij

H

1

i

\j

1

(10.2.10)0

j 1

ߎߎߢᑼ(10.2.3)ࠍ↪޿ߚ㧚Ꮐㄝߩ╙ 1 㗄⋡ߪᒰὼ࠯ࡠߢ޽ࠆ㧚ᱷࠅߩฦ㗄ߦ㧘Ꮐ߆ࠄ❗ㅌߒߡ޿ࠆ wavefunction\ 10 , \ 2 0 , ..., \ m 0 ߩ߁ߜߩ 1 ߟ\ k 0 ࠍ߆ߌߡⓍಽߔࠆ㧚\ k 0
ߣ\ j 0 ߪߘ߽ߘ߽⋥੤ߔࠆࠃ߁ߦㆬࠎߢ޽ࠆࠊߌߢ㧘ᑼ(10.2.10)ߩᏀㄝ╙ 2 㗄⋡ߪ࠯ࡠߣߥࠆ㧚ᓥߞߡ
m

¦c H

 Ei

1

1

k 1, 2, ..., m

0

ij

kj

1

³\ * H \ dW

(10.2.11)

j 1

ߚߛߒ

H kj

1

0

0

k

(10.2.12)jߪ㧘❗ㅌߒߡ޿ࠆ⚵ౝߩ 2 ߟߩ wavefunction \ k 0 ߣ\ j 0 ߩ⚵ߦߟ޿ߡߩⓍಽߢ޽ࠆ㧚ᑼ(10.2.11)ࠍౕ૕⊛ߦᦠ߈ਅࠈߔߣᰴߩࠃ߁ߥㅪ┙ᣇ⒟ᑼߣߥࠆ㧚

H11  Ei11

ci1  H12 ci 2  ...  H1m cim
1

H 21 ci1  H 22  Ei
1

1

1

1

ci 2  ...  H 2 m cim
1

0
0

...

(10.2.13)...
...
H m1 ci1  H m 2ci 2  ...  H mm  Ei
1

1

1

cim

0

ߎߩㅪ┙ᣇ⒟ᑼߩ non-trivial solution ߪ㧔ߔߢߦ Section 4.4 ߢቇ⠌ߒߚࠃ߁ߦ㧕

H11  Ei
1

H 21
:

1

H m1

1

1

H12

1

H 22  Ei
1

H m2

1

1

...

H1m

1

...

H 2m

1

... H mm  Ei101䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 29(10.2.14)

ߣ޿߁ secular equation ߩ⸃ࠃࠅᓧࠄࠇࠆ㧚ߔߥࠊߜ㧘ߎߩ secular equation ࠍ⸃޿ߡᓧ
ࠄࠇࠆ m ୘ߩ Ei ߇ 1 ᰴߩ៨േ energy ߣߥࠆࠊߌߢ޽ࠆ㧚 m ୘ߩ Ei ߇᳞߹ߞߚࠄ㧘11ߘߩ߁ߜߩ 1 ߟ 1 ߟߩ Ei ࠍᑼ(10.2.13)ߩㅪ┙ᣇ⒟ᑼߦઍ౉ߒ㧘߆ߟ1ci1 * ci1  ci 2 * ci 2  ...  cim * cim

1(10.2.15)

ߣ޿߁ normalization condition ࠍട߃ߡ㧘ߔߴߡߩዷ㐿ଥᢙ ci1 , ci 2 , ..., cim ࠍ᳞߼ࠆߎߣ
߇಴᧪ࠆ㧚ߎߩࠃ߁ߦߒߡ㧘ᑼ(10.2.3)ߩߔߴߡߩ Ii 0 ߇᳞߹ߞߚࠊߌߢ޽ࠆ㧚ߎ߁ߒߡ Ii 0 ߇᳞߹ࠇ߫㧘wavefunction ߦኻߔࠆ 1 ᰴߩ⵬ᱜ㗄ߪ⋥ߜߦ᳞߹ࠆ㧚ᑼ

(10.2.10)㧘ߔߥࠊߜ¦ A E E \
0

j

0
1

j

Ei 1  H 1 Ii 00j(10.2.16)

j t m 1

ߦᏀ߆ࠄ\ j 0 * 㧔ߚߛߒ j t m  1 㧕ࠍ߆ߌߡⓍಽߔࠆ㧚

 ³\ j 0 * H Ii 0 dW

Aj E j 0  E101(10.2.17)

߹ߚߪ

³\ * H I dW
1

0

j

Aj

0

i

j t m 1

E10  E j 0(10.2.18)⚿ዪ㧘❗ㅌߒߚ࡟ࡌ࡞ߩ wavefunction ߳ߩ៨േߪ㧘ߎߩᑼ߅ࠃ߮ᑼ(10.2.5)߅ࠃ߮ᑼ

(10.2.9)ࠃࠅ

\ * H I dW
I O ¦ ³\  O ...  ...
1

0

\i

0

j

0

i

E10  E j 0i

j t m 102

jߣߥࠆ㧚ߚߛߒ㧘ߎߎߢ㧘ᑼ(10.2.9)ߩ╙ 1 㗄ߩ๺ߦ⃻ࠇࠆߔߴߡߩ Aj(10.2.19)
jdm ߪ

normalization condition ࠃࠅ࠯ࡠߦߥࠆߎߣࠍ↪޿ߡ޿ࠆ㧚
2 ᰴߩ៨േ energy ߪᰴߩࠃ߁ߦߥࠆ㧚^³\ * H I dW `^³ I * H \ dW `
0

E

2

O

2

¦

j t m 1

j

1

0

i

0

i

1

0

j

E10  E j 0䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 30(10.2.20)

‫ڏ‬Exercise 10.2.1

Section 8.1 ߦ߅޿ߡ㧘౞๟਄ࠍ⥄↱ㆇേ㧔ߔߥࠊߜ potential ࠍฃߌߥ޿ㆇേ㧕ߒߡ޿ࠆ☸ሶߩ໧㗴ࠍᛒߞߚ߇㧘ߎߎߢߪ㧘ߎߩ☸ሶ߇ potential
H ' H sin 2 Iࠍฃߌߡ޿ࠆ႐วࠍ៨േ⺰ߢ⠨߃ࠆ㧚m 1 ߩ 2 ߟߩ⁁ᘒߦߟ޿ߡ㧘㧔a㧕energy ߅ࠃ߮

wavefunction ߦߟ޿ߡߩ 1 ᰴߩ⵬ᱜ߅ࠃ߮㧔b㧕energy ߦߟ޿ߡߩ 2 ᰴߩ⵬ᱜࠍ᳞߼ࠃ㧚‫ޤ⸃ޣ‬㧦perturbation H ' ߪ
H ' H sin 2 I
H

e 2iI  e 2iI  24❗ㅌߒߡ޿ࠆ 2 ߟߩ wavefunction ߩขࠅᣇߦߪ޿ࠈ޿ࠈ޽ࠅ㧘଀߃߫(8.1.16)ߢ߽޿޿

ߒ㧘(8.1.17)ߢ߽޿޿ߒ㧘ߎࠇࠄߩߤࠎߥ✢ᒻ⚿วߢ߽޿޿ࠊߌߢ޽ࠆ㧚ߎߎߢߪ(8.1.16)㧘ߔߥࠊߜ
eimI
\ m
2S
 imI
e
\ m
2S
ࠍၮᐩ㑐ᢙߣߒߡㆬ߱ߎߣߦߔࠆ㧚(a) ߹ߕ㧘 m 1 ߩ 2 ߟߩ⁁ᘒߦ㑐ߔࠆ secular determinant ᑼ(10.2.14)ߦᔅⷐߥⓍಽ㧔ⴕ೉ⷐ⚛㧕ߪᰴߩࠃ߁ߦߥࠆ㧚2S

H '1 ,1

1 H
e 2iI  e 2iI  2 dI
2S 4 ³0

H '1 ,1

1 H
e 2iI  e 2iI  2 dI³
2S 4 0H '1 ,1

1 He 2iI e 2iI  e 2iI  2 dI³
2S 4 0H '1 ,1

1 H
e 2iI e 2iI  e 2iI  2 dI
2S 4 ³0

2S

H
2

H
2

2S

2S




H
4

H
4

ᓥߞߡ㧘⸃ߊߴ߈ secular determinant ߪ

H
2

E



H

H



H
4

0

E
4
2
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 31ߎࠇࠍ⸃޿ߡ㧘1 ᰴߩ៨േ energy ߣߒߡᰴߩ⚿ᨐࠍᓧࠆ㧚

E
E ߇
1

3H H
,
4 4

1

3H
H
ߩ wavefunction ࠍ I3 㧘 ߩ wavefunction ࠍ I 1 ߣߔࠆߣ㧘ᑼ(10.2.13)ߩㅪ┙4444

ᣇ⒟ᑼߦ E ࠍઍ౉ߔࠆߎߣߦࠃࠅ
\ 1 \ 1
1

I3

I1

4

2
\ 1 \ 12ࠍᓧࠆ㧚ߎࠇ߇❗ㅌߒߚ level ߦߟ޿ߡㆬ߱ߴ߈࠯ࡠᰴߩ wavefunction ߢ޽ࠆ㧚
wavefunction ߦߟ޿ߡߩ 1 ᰴߩ⵬ᱜࠍⴕ߁ߦߪᦝߦᰴߩⴕ೉ⷐ⚛߇ᔅⷐߢ޽ࠆ㧚
42S

H 'm  , 3

4

³\

m

* H ' I 3 dI
4

0

2S

1 1 H
e  imI e 2iI  e 2iI  2 eiI  e iI dI³
2 2S 4 0
1 1 H
2SG m,3
2 2S 4


H
4 2

2S

H 'm , 3

4

³\

m

G m,3
* H ' I 3 dI
4

0

2S

1 1 H
eiI e 2iI  e 2iI  2 eiI  e  iI dI³
2 2S 4 0

H
4 2

G m ,32S

H 'm  , 1

4

³\

m



H
4 2

2S

H 'm  , 1

4* H ' I 1 dI
4

0

³\

m

G m ,3* H ' I 1 dI
4

0



H

G m ,34 2ᓥߞߡ㧘ᑼ(10.2.19)ߦ⃻ࠇࠆή㒢୘ߩ๺ߪ㧘 j

3 ߣ j䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 32

3  ߩ 2 㗄ߩߺߩ๺ߣߥࠆ㧚

߆ߊߒߡ

H
\3

4

\ 3

4

\ 3

I3  4 0 2 0  4 0 2 04
E1  E3 E1  E3
H

\1

H

\ 3

H

\ 3

I1  4 0 2 0  4 0 2 04
E1  E3 E1  E3

ߣߥࠆ㧚ߚߛߒ

E1  E30=2 2 2
1 3
2I

0

ߢ޽ࠆߩߢ

\3
\1

4

4

2I H
\ 3  \ 3 
4= 2 32 2
2I H
I1  2\ 3  \ 34
= 32 2I3 (b) 2 ᰴߩ៨േ energy ߪ
2

E3 2
4

§ H · § H ·¨¸ ¨¸©4 2¹ ©4 2¹
E10  E30


E1 2
4

2

H 2 2I

128 = 2H 2 2I
128 = 2

⚿ዪ 2 ᰴ߹ߢߩ⵬ᱜࠍ౉ࠇߡ㧘energy ߪᰴߩࠃ߁ߦߥࠆ㧚

E3
E1

4

4

= 2 3H H 2 2 I 
2 I 4 128 = 2=2 H H 2 2 I 
2 I 4 128 = 2‫ڏ‬Exercise 10.2.2⏛႐ B ߇ሽ࿷ߔࠆᤨ㧘㔚ሶߩฃߌࠆ⋧੕૞↪ߪ

H ' PB B

L=ߢ⴫ߐࠇࠆ㧔ߚߛߒ㧘ߎߎߢߪ chapter 14 એ㒠ߢቇ⠌ߔࠆ㔚ሶࠬࡇࡦ߿ᩭࠬࡇࡦߩߎ
ߣߪ⠨߃ߥ޿㧕㧚ߎߎߢ㧘 L ߪ゠㆏ⷺㆇേ㊂ߩṶ▚ሶߢ޽ࠅ㧘 PB ߪ Bohr magneton ߣ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 33

๭߫ࠇ

PB

e
2me

ߣቯ⟵ߐࠇࠆ㧚᳓⚛ේሶߩ n 1 ߅ࠃ߮ n 2 ߩ⁁ᘒߩ energy ߇ߎߩ⏛႐ߦࠃࠅߤߩࠃ߁ߦᓇ㗀ฃߌࠆ߆ࠍ৻ᰴߩ៨േ⺰ߢี๧ߖࠃ㧚ߚߛߒ㧘⏛႐ߪ z ゲᣇะߦ߆߆ߞߡ޿ࠆߣ⠨߃ߡࠃ
޿㧚
‫ޤ⸃ޣ‬
㧦
⏛႐߇ z ゲ਄ߦ߆߆ߞߡ޿ࠆߩߢ៨േࡂࡒ࡞࠻࠾ࠕࡦߪH'

PB Bz Lz=ߣߥࠆ㧚߹ߕ n 1 ߩ⁁ᘒ㧘ߔߥࠊߜ 1s ゠㆏ࠍ⠨߃ࠃ߁㧚\ 1s

R1s r Y0,0 T , Iߢ޽ࠆ߆ࠄ㧘ߎߩ⁁ᘒߦ㑐ߔࠆ৻ᰴߩ៨േ energy ߪ'E1s³\ 1s H '\ 1s dW³ R r r dr ³ Y T , I H 'sin T dI dT
2

1s

2

0,0

ߢ޽ࠆ߇㧘ߎߎߢ㧔ᑼ(5.3.3)ࠍෳᾖߔࠆ߹ߢ߽ߥߊ㧕
LzY0,0 T, I

0ߢ޽ࠆߩߢ਄ߩⓍಽߪ࠯ࡠߢ޽ࠆߎߣߪ᣿ࠄ߆ߢ޽ࠆ㧚ߔߥࠊߜ'E1s 0ᰴߦ n

2 ߩ⁁ᘒࠍ⠨߃ࠆ㧚 n

2 ߩ⁁ᘒߪ 2s ߅ࠃ߮ 3 ߟߩ 2 p 㧘ㇺว 4 ߟߩ⁁ᘒ߇❗

ㅌߒߡ޿ࠆ㧚ᓥߞߡ❗ㅌ♽߳ߩ៨േ⺰ࠍㆡ↪ߒߥߌࠇ߫ߥࠄߥ޿㧚3 ߟߩ 2 p ゠㆏ l ߪ㧘ⶄ⚛ᢙ⴫␜ࠍ↪޿ߡ߽ታᢙ⴫␜ߩ㑐ᢙࠍ↪޿ߡ߽ࠃ޿߇㧘ߎߎߢߪ೨⠪ࠍߣࠆߎߣߦ
ߔࠆ㧚ߔߥࠊߜ㧘ၮᐩ㑐ᢙߣߒߡ\ 2s

R2 s r Y0,0 T , I\ 2 p1R2 p r Y1,1 T , I\ 2 p0R2 p r Y1,0 T , I\ 2 p1R2 p r Y1,1 T , Iࠍ↪޿ࠃ߁㧚

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 34

LzY1,1=Y1,1

LzY1,00

LzY1, 1

 =Y1,1ࠃࠅ⋥ߜߦ

H '\ 2 p1

P B Bz\ 2 p1

H '\ 2 p00

H '\ 2 p1

 P B Bz\ 2 p1߹ߚ㧘ᒰὼߩߎߣߥ߇ࠄ
H '\ 2s0ߣߥࠆ㧚ᓥߞߡᑼ(10.2.14)ߦ⋧ᒰߔࠆ secular determinant ߪᰴߩࠃ߁ߦߥࠆ㧚
E

0
PB Bz  E
0
0

0
0
0

0
0
0
0
E
0
0  PB Bz  E

0ߎߩ secular equation ߩ⸃ߣߒߡ
E 0

E

0

E

PB Bz
 PB Bz

E߇ᓧࠄࠇࠆ㧚⏛႐ߦࠃࠆ energy level ߩಽⵚࠍ Zeeman ಽⵚߣ⸒߁㧚‫ޤ⸼ޣ‬㧦਄ߩ⸘▚⚿ᨐࠍߺࠆߣ㧘ߚ߹ߚ߹ off diagonal term ߇ߔߴߡ࠯ࡠߢ޽ࠆߩߢ㧘㑆㆑ߞߡ㕖❗ㅌ♽ߩ៨േ⺰ࠍㆡ↪ߒߚߣߒߡ߽ᱜߒ޿⸃߇ᓧࠄࠇߡߒ߹߁㧚ߒ߆ߒߎߩ႐
ว㧘ߚ߹ߚ߹ off diagonal term ߇࠯ࡠߦߥߞߚߣ޿߁ߛߌߢ޽ࠅ㧘޽ߊ߹ߢ 4 ᰴߩⴕ೉ߩኻⷺൻ߇ᔅⷐߢ޽ࠆ㧚ߥ߅㧘ࠬࡇࡦߦࠃࠆ Zeeman ലᨐࠍ⠨߃ߥ߆ߞߚߩߪᄢ߈
ߥᚻ⪭ߜߢ޽ࠆ㧚Chapter 14 ߢᡷ߼ߡߎߩ໧㗴ࠍขࠅᛒ߁ߎߣߦߔࠆ㧚‫ڏ‬Problem 10.2.1
Exercise 10.2.2 ߩ໧㗴ࠍ㧘2 p ߩၮᐩ㑐ᢙߣߒߡታᢙ⴫␜ߩ wavefunction ࠍ↪޿ࠆߎߣߦࠃߞߡ⸃޿ߡߺࠃ㧚⚿ᨐߪ Exercise 10.2.2 ߣหߓߦߥࠆߪߕߢ޽ࠆ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 35‫ڏ‬Problem 10.2.2᳓⚛ේሶߦ E ߩᄢ߈ߐߩ㔚႐ࠍ߆ߌߚߣ߈㧘(a) n 1 ߩၮᐩ⁁ᘒ㧘߅ࠃ߮(b) n 2 ߩബ⿠⁁ᘒߩ energy ߪߤߩࠃ߁ߦᄌൻߔࠆ߆ࠍ㧘৻ᰴߩ៨േ⺰ࠍ↪޿ߡ⠨ኤߖࠃ㧚៨േߩࡂࡒ࡞࠻࠾ࠕࡦ H ' ߪ㧘㔚႐߇ z ゲߦ߆ߌࠄࠇߚ႐ว
H ' eEz eEr cos Tߢਈ߃ࠄࠇࠆ㧚ߥ߅㧘േᓘᵄേ㑐ᢙ Rnl ࠍ฽߻Ⓧಽߪf³ R rR r dr
2nl

n 'l '

I nl  n ' l '0ߩࠃ߁ߥ⸥ภߢ⴫ߒ㧘Ⓧಽ୯ߩ⸘▚ߪⴕࠊߥߊߡࠃ޿㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 10: Page 36

CHAPTER 11
ᄌಽේℂᄌಽේℂ㧔variation theory㧕ߪ㧘៨േℂ⺰㧔perturbation theory㧕ߣਗࠎߢ㧘 Schrödinger
equation ࠍㄭૃ⊛ߦ⸃ߊ߽ߞߣ߽ၮᧄ⊛ߥᣇᴺߢ޽ࠆ㧚ߎߩ 2 ߟߩᣇᴺߐ߃ℂ⸃ߒߡߒ߹߃߫㧘㊂ሶൻቇߩᔕ↪㕙ߢߪ߹ߕ࿎ࠆߎߣߪߥ޿㧚ߘࠇ߶ߤᄢ੐ߥℂ⺰ߢ޽ࠆ߆ࠄߘߩᧄ⾰ࠍࠃߊℂ⸃ߒߡ߅޿ߡ߽ࠄ޿ߚ޿㧚11.1 ᄌಽේℂ
Schrödinger equationߩᱜߒ޿⸃߇ಽ߆ࠄߥ޿ᤨ㧘1ߟߩㄭૃ⸃ߣߺߥߖࠆ߆߽ߒࠇߥ
޿ߣ޿߁޽ࠆ㑐ᢙࠍ߹ߕ⠨߃㧘ߘࠇࠍ⹜ⴕ㑐ᢙ㧔trial function㧕ߣߔࠆ㧚⹜ⴕ㑐ᢙࠍߎ
ߎߢߪ\ trial ߣ⴫ߔߎߣߦߔࠆ㧚ߐߡ㧘ᰴᑼߢ⴫ߐࠇࠆ H ࠍRayleigh ratioߣቯ⟵ߔࠆ㧚

H³\ * H\ dW³\ *\ dW
trial

trial

trial

(11.1.1)

trial

ߎߎߢ H ߪHamiltonianߢ޽ࠆ㧚ߎߩ H ߪනߦ2ߟߩⓍಽߩᲧߢ޽ࠅ㧘 Schrödinger
equationߩ⸃ߣߪߥߞߡ޿ߥ޿ߎߣߪ൩⺰ߢ޽ࠆ߇㧘߽ߒ߽\ trial ߇ߚ߹ߚ߹Schrödinger
equationߩ࿕᦭㑐ᢙߣ৻⥌ߒߡ޿ࠇ߫Rayleigh ratioߪSchrödinger equationߩ࿕᦭୯ߣ޿

߁ߎߣߦߥࠆ㧚ߘ߁޿߁ߎߣߢ㧘߽ߒ઒ߦᱜߒ޿࿕᦭୯߇ಽ߆ߞߡ޿ࠆࠃ߁ߥ႐วߦ
ߪ㧘ߘߩᱜߒ޿࿕᦭୯ߣRayleigh ratioߣࠍᲧセߔࠆߎߣߦࠃߞߡ㧘\ trial ߇ᱜߒ޿࿕᦭
㑐ᢙߦߤߩ૏ㄭ޿߆㧘ߣ޿߁ߎߣߩ߅߅߹߆ߥ⋡቟ߦߥࠆߛࠈ߁ߣߩᗐ௝ߪߟߊߢ޽
ࠈ߁㧚ߎߩ⋡቟ߩߟߌᣇࠍએਅ߽߁ዋߒℂ⺰⊛ߦᛒ߁㧚߽ߜࠈࠎ㧘⃻ታߩ႐วߦߪᱜߒ޿࿕᦭㑐ᢙ߽ᱜߒ޿࿕᦭୯߽ಽ߆ߞߡ޿ߥ޿ࠊߌߢ
޽ࠆ㧚㧔߽ߒಽ߆ߞߡ޿ࠇ߫㧘૗߽੹ᦝㄭૃᴺߢ᳞߼ࠆᔅⷐߪߥ޿㧚㧕ߒ߆ߒ㧘ᱜߒ޿
eigenfunctionߣeigenvalue߇ሽ࿷ߔࠆߎߣߪ⏕߆ߢ޽ࠆߩߢ㧘ߎߩeigenvalueࠍenergyߩ㗅ߦ㧘
E0 d E1 d E2 d ....

(11.1.2)

ߩࠃ߁ߦ⇟ภઃߌࠍߔࠆߎߣߦߔࠆ㧚ߤ߁ߢ߽޿޿ߎߣߢߪ޽ࠆ߇㧘ߎߎߢߪߎߩ⇟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 11: Page 1ภࠍ࠯ࡠ߆ࠄߟߌߡ޿ࠆ㧚ߔߥࠊߜ㧘E0 ߪၮᐩ⁁ᘒ㧔ground state㧕ߦߟ޿ߡߩeigenvalueߢ޽ࠆ㧚ߘߒߡ㧘 E0 , E1 , ... ߩeigenvalueࠍᜬߟeigenfunctionࠍߘࠇߙࠇ\ 0 , \ 1 , ... ߣߒࠃ߁㧚ߔߥࠊߜ
H\ i Ei\ ii0, 1, 2, ...

(11.1.3)㧔ᧄ▵ᦨᓟߩ[coffee break]ࠍෳᾖ㧚㧕ߎߎߢ\ i ߪ߅੕޿ߦ⋥੤ߒߡ޿ࠆߣߔࠆ㧚㧔H ߇Hermitianߢ޽ࠆ㒢ࠅ㧘ߘߩeigenfunctionߪ߅੕޿ߦ⋥੤ߒߡ޿ࠆ߆㧘߹ߚߪ⋥੤ߔࠆࠃ߁ߦߣࠆߎߣ߇น⢻ߢ޽ࠆߎߣߪߔߢߦSection 4.2ߢቇ⠌ߒߚ㧚㧕ߐߡ㧘ᧄ┨ߢቇ⠌ߔࠆvariation theoryߣߪ㧘ᰴᑼߢ␜ߐࠇࠆᑼߩߎߣߢ޽ࠆ㧚ᄌಽේℂ㧦

H t E0for any trial function(11.1.4)ߔߥࠊߜ㧘ߤࠎߥtrial functionߦߟ޿ߡ߽㧘ߘߩRayleigh ratio H ߪground stateߦߟ޿ߡߩᱜߒ޿eigenvalueࠃࠅߪዊߐߊߥࠄߥ޿㧘ߣ޿߁ߎߣߢ޽ࠆ㧚╬ภߪ㧘trial function
߇ᱜߒ޿eigenfunctionߦ╬ߒ޿႐วߦᚑࠅ┙ߟ㧚ߎߩᑼߩ⸽᣿ߪએਅߩࠃ߁ߦߥߐࠇ
ࠆ㧚
trial function \ trial ࠍ H ߩeigenfunction ^\ n ` ߢዷ㐿ߔࠆ㧚¦c \\ trialn(11.1.5)nnߐߡ㧘ᰴߩࠃ߁ߥintegralࠍ⠨߃㧘ߘߩ୯ࠍ⹏ଔߒߡߺࠃ߁㧚

³\

trial

* H  E0 \ trial dW㨇ᑼ(11.1.5)ࠍ↪޿ߡ㨉¦¦ c * c ³\ * H  E \ dW㨇ᑼ(11.1.3)ࠍ↪޿ߡ㨉¦¦ c * c㨇 ^\ n ` ߩ⋥੤ᕈࠃࠅ㨉

n

n

n

n

m

n

0

m

m

m

¦c

n

2

m

Em  E0 ³\ n *\ m dW

En  E0㨇ᑼ(11.1.2)ࠃࠅ㨉nt0

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 11: Page 2(11.1.6)

ߔߥࠊߜ

³\

trial

* H\ trial dW t E0 ³\ trial *\ trial dW(11.1.7)߇ᓧࠄࠇࠆ㧚ฝㄝߩⓍಽߪᔅߕᱜߢ޽ࠆ߆ࠄ㧘H³\ * H\ dW t E³\ *\ dW
trial

trial

0

trial

(11.1.8)trialߔߥࠊߜ㧘variation theory߇⸽᣿ߐࠇߚ㧚ߐߡ㧘ᱜߒ޿eigenfunction߇᳞߹ࠄߕߦ޽ࠆㄭૃ⸃߇ਈ߃ࠄࠇߚᤨ㧘ߘߩㄭૃ⸃߇
ᱜߒ޿eigenfunctionߦߤߩ૏ㄭ޿߆㧘ߔߥࠊߜ㧘ㄭૃ⸃ߩ⏕߆ࠄߒߐࠍ␜ߔዤᐲࠍߤ
߁ขࠆ߆ߪߥ߆ߥ߆߻ߕ߆ߒ޿໧㗴ߢ޽ࠆߣ⸒߃ࠃ߁㧚ߒ߆ߒ㧘ㅢᏱߪ㧘variation
theoryߦၮߠ߈㧘Rayleigh ratio߇⌀ߩeigenvalueߦㄭߌࠇ߫ㄭ޿߶ߤtrial functionߪ⌀ߩ
eigenfunctionߦㄭ޿ߣߺߥߔ㧚ߔߥࠊߜ㧘Rayleigh ratio H ߇ዊߐߌࠇ߫ዊߐ޿߶ߤ⌀ߩeigenfunction߳ߩࠃ޿ㄭૃ⸃ߢ޽ࠆߣ⠨߃ࠆ㧚
variation theoryࠍㆡ↪ߒߡㄭૃ⸃ࠍ᳞߼ࠆ৻⥸⊛ߥᣇᴺߪ㧘߹ߕtrial functionࠍ޿ߊߟ߆ߩparameterࠍ฽߻ᒻߢ⴫ߒߡ߅߈㧘ߘߩparameterࠍᄌൻߐߖߡRayleigh ratioߩ߽ߞߣ߽ዊߐߥὐࠍ⷗಴ߔߎߣࠍⴕ߁㧚ߔߥࠊߜ㧘ߘࠇࠄparameterࠍ p1 , p2 , ... ߣߔࠆߣwHwH
0,0, ... ࠍ⸃޿ߡ H ߩᭂዊὐࠍ⷗ߟߌࠆߎߣߦߥࠆ㧚ߎ߁ߒߡᓧࠄࠇߚ
wp1
wp2
wavefunction߇㧘᦭㒢ߥᢙߩparameterߩ▸࿐ౝߢbestߥwavefunctionߣߺߥߐࠇࠆߣ޿

߁ࠊߌߢ޽ࠆ߇㧘parameterߩㆬቯ߇޿޿߆ߤ߁߆ߩี๧߇೨߽ߞߡචಽߦߥߐࠇߡ޿
ߥ޿ߣ㧘ߣࠎߢ߽ߥ޿⸃߇ᓧࠄࠇࠆߎߣ߽޽ࠅᓧࠆߩߢᵈᗧ߇ᔅⷐߢ޽ࠆ㧚‫ڏ‬Exercise 11.1.1
harmonic oscillatorߦ㑐ߔࠆHamiltonian=2 d 2 1 2
 kx
2m dx 2 2ߦߟ޿ߡߩㄭૃ⸃ࠍ
H\ trial x

N exp O x 2(11.1.9)(11.1.10)

ߣߒ㧘 O ࠍᄌಽparameterߣߒߡᄌಽߢ᳞߼ࠃ㧚‫ޤ⸃ޣ‬㧦
߹ߕᑼ(11.1.10)ࠍⷙᩰൻߔࠆࠃ߁ߦ N ࠍ᳿߼ߡ߅ߌ߫㧘Rayleigh ratioߪᑼ(11.1.1)ᑼ
ߩಽሶߛߌߢ⸘▚಴᧪ࠆߩߢଢ೑ߢ޽ࠈ߁㧚ᑼ(7.4.18)ࠍ↪޿ߡ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 11: Page 3

f

1
N2

³ exp 2O x dx
2

f

S
2O

(11.1.11)ߣߥࠆ㧚ᓥߞߡtrial functionߣߒߡߪ\ trial

2O

4S

exp O x 2(11.1.12)

ᓥߞߡRayleigh ratio H ߪᰴߩࠃ߁ߦߥࠆ㧚
f

H

³\

trial

* H\ trial dx

f

2O

f

2O

f

=2 d 2 1 2 ·
2 §
x
exp

O

 kx ¸ exp O x 2 dx
¨
2
³
S f
© 2m dx 2
¹

º
1
=2
2 ª
exp

O

4O 2 x 2  2O  kx 2 » exp O x 2 dx
x
«
³
S f
2
¬ 2m
¼

ߎߎߢᑼ(7.4.18)߅ࠃ߮ᑼ(7.4.21)ࠍ↪޿㧘
k
=2
O
2m
8O
ࠍᓧࠆ㧚ߎࠇߩᭂዊ୯ࠍ᳞߼ࠆ㧚

H

wH
wO

k
=2
 2
2m 8O

(11.1.13)

0

ࠃࠅ

1 mk
2 =
ࠍᓧࠆ㧚ᓥߞߡRayleigh ratioߪ

O(11.1.14)= k hQ(11.1.15)
2 m
2ߚߛߒߎߎߢᑼ(7.1.4)ࠍ↪޿ߡ޿ࠆ㧚trial functionߪᑼ(11.1.14)ࠍᑼ(11.1.12)ߦઍ౉ߔࠆ

H

ߎߣߦࠃࠅ

\ trial4§ 1 mk x 2 ·mkexp ¨¨ ¸¸=S© 2 = ¹
§ [2 ·
E
exp ¨  ¸
S
© 2 ¹

(11.1.16)ࠍᓧࠆ㧚ߎߎߢ E ߦ㑐ߔࠆ(7.2.2)ᑼ߅ࠃ߮ [ ߦ㑐ߔࠆ(7.2.4)ᑼࠍ↪޿ߡ޿ࠆ㧚ᑼ(11.1.16)ࠍߺࠆߣ㧘ߎࠇߪChapter 7ߢ᳞߼ߚground stateߩwavefunction㧔(7.4.12)
䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 11: Page 4

ߢ n 1 ߣ߅޿ߚ߽ߩ㧕ߘߩ߽ߩߣߥߞߡ޿ࠆ㧚ߎࠇߪ㧘ߚ߹ߚ߹㧘ᦨೋߦㆬࠎߛtrial
functionᑼ(11.1.10)߇㑐ᢙᒻߣߒߡ⦟ߔ߉ߚߚ߼ߦ㧘ㄭૃ⸃ߩߟ߽ࠅ߇⌀ߩ⸃ߦ৻⥌ߒߚߦߔ߉ߥ޿㧚৻⥸ߦߪߎࠎߥߦ߁߹ߊߪ޿߆ߥ޿ߩ߇᥉ㅢߢ޽ࠆ㧚ߒ߆ߒ㧘߽߁1ߟߛߌ߁߹ߊ޿ߊ଀ࠍᰴߦ␜ߘ߁㧚‫ڏ‬Exercise 11.1.2᳓⚛ේሶߩၮᐩ⁁ᘒߩwavefunctionࠍvariation theoryߢ᳞߼ࠃ㧚
trial functionߣߒߡߪ\ trialN exp O r

(11.1.17)ࠍ↪޿ࠃ㧚ߎߎߢ N ߪnormalization factor㧚‫ޣ‬ᵈ‫ޤ‬㧦᳓⚛ේሶߩwavefunctionߪߔߢߦᱜ⏕ߥ⸃ࠍChapter 9ߢ᳞߼ߡ߅ࠅ㧘ground stateߦߟ޿ߡߪ㧘਄ߩࠃ߁ߦ r ߩᜰᢙ㑐ᢙߩᒻߦߥࠆߎߣ߇ಽ߆ߞߡ޿ࠆ㧚㧔⴫ 9.7.1㧕ᓥߞߡ㧘variation theoryࠍ↪޿ߡ㧘ߪߚߒߡᄌಽparameter O ߇ᱜߒߊ᳞߹ࠆ߆ߤ߁߆߇⥝๧ߩ޽ࠆߣߎࠈߢ޽ࠆ㧚‫ޤ⸃ޣ‬㧦
߹ߕtrial functionࠍnormalizeߒߡ߅ߎ߁㧚

1
N2

2S

S

f

0

0

0

(11.1.18)

O3
exp O r
S(11.1.19)

2³ dI ³ sin T dT ³ exp 2O r r dr

S
O3
ᓥߞߡ

\ trialHamiltonianߪᑼ(9.1.5)ߢਈ߃ࠄࠇߡ޿ࠆ㧚H

=2 § 1 w 2 w
e2
w
w
w2 ·
1
1
r



sin
T
¨
¸
wT r 2 sin 2 T wI 2 ¹ 4SH 0 r
2 P © r 2 wr wr r 2 sin T wT
(11.1.20)

ߎࠇࠄߩᑼࠍ↪޿ߡRayleigh ratioࠍ⸘▚ߔࠆࠊߌߢ޽ࠆ߇㧘଀߃߫

³\

trial

³\

trial

1
* \ trial dWrO*  2\ trial dWO 2䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 11: Page 5ߥߤࠍ↪޿㧘㧔ߎ߁ߥࠆߎߣࠍ⏕߆߼ࠃ㧍㧕ᦨ⚳⊛ߦᰴᑼࠍᓧࠆ㧚

H

=2 2
e2
O 
O
2P
4SH 0

ߎࠇߩᭂ୯ߪ

wH
wO

0 ࠃࠅ

e4 P
2
2
4SH 0 2=
1

H

(11.1.21)(11.1.22)

ߎߎߢBohr radius a0 ߦreduced massߩ⵬ᱜࠍߒߚ a0 ' 㨇(9.3.24)ᑼ㨉
a0 ' 4SH 0

=2
P e2

ࠍ↪޿ࠆߣ

O

1
a0 '

(11.1.23)ߣ޿߁⚿ᨐ߇ᓧࠄࠇࠆ㧚ᓥߞߡ\ trial§ r ·exp ¨ ¸
a0 '3 S© a0 ' ¹
1(11.1.24)

ߣߥࠆ㧚ߎߩ⚿ᨐߪ㧘⴫9.7.1ߦਈ߃ࠄࠇߡ޿ࠆ\ 1s ߩwavefunctionߘߩ߽ߩߢ޽ࠆ㧚‫ڏ‬Exercise 11.1.3
ᄢ߈ߐ L ߩ੗ᚭဳࡐ࠹ࡦࠪࡖ࡞ߩground stateߩeigenfunction߅ࠃ߮eigenvalueߪ

\ x

2
Sx
sin
L
L

(11.1.25)

h2(11.1.26)
8mL2ߣߥࠆߎߣࠍߔߢߦSection 6.2ߢቇ⠌ߒߚ㨇ᑼ(6.2.12)߅ࠃ߮ᑼ(6.2.7)㨉㧚ߎߎߢߪߘߩ
E

⸃ࠍ⍮ࠄߥ޿ߣߒߡ㧘޿ߊߟ߆ߩtrial functionࠍ⠨߃variation theoryߦࠃࠅߘߩᅷᒰᕈ
ࠍี๧ߒߡߺࠃ߁㧚
Schrödinger equationࠍ⸃߆ߥߊߣ߽㧘Ⴚ⇇᧦ઙࠃࠅ㧘wavefunctionߪ x
x

0 ߅ࠃ߮

L ߦ߅޿ߡ࠯ࡠߦߥࠄߥߌࠇ߫ߥࠄߥ޿ߎߣ㧘߹ߚ㧘ground stateߢ޽ࠆ߆ࠄ xL2ߦߟ޿ߡኻ⒓ߢߥߌࠇ߫ߥࠄߥ޿ߎߣߪ੹߹ߢߩቇ⠌߆ࠄᗐ௝߇ߟߊߢ޽ࠈ߁㧚ߘߎߢ㧘ߎߩ2ߟߩⷐ⺧ࠍḩߚߔwavefunctionߣߒߡᰴߩ2ߟߩtrial functionࠍ⠨߃ࠃ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 11: Page 6

\1 x

x Lx

\2 x

x2 L  x(11.1.27)2(11.1.28)

ᄌಽᴺࠍ↪޿ߡ㧘ߎߩ\ 1 x , \ 2 x ߩ޿ߕࠇ߇ࠃࠅࠃ޿wavefunctionߢ޽ࠆ߆ࠍี๧
ߖࠃ㧚‫ޤ⸃ޣ‬㧦
߹ߕ\ 1 x ߦߟ޿ߡnormalizeߒߡ߅ߎ߁㧚normalization constantࠍ N1 ߣߒߡ
L

1
2
N1

³ x L  x dx
2

2

0

L5
30

ᓥߞߡnormalizeߐࠇߚ\ 1 x ߪ
30
x Lx
L5

\1 x

(11.1.29)

ห᭽ߦ\ 2 x ߦߟ޿ߡ߽normalizeߒߡ߅ߎ߁㧚
L

1
2
N2

4
³ x L  x dx
4

0

L9
630

ᓥߞߡ
630 2
2
x Lx
9
L

\2 x

(11.1.30)\ 1 x ߦߟ޿ߡߩRayleigh ratioࠍ᳞߼ߡߺࠆ㧚§=2 · d 2¸ 2 \ 1 x dx
© 2m ¹ dx

LH1³\ 1 x ¨ 
0

5=
mL2
2

ห᭽ߦ\ 2 x ߦߟ޿ߡߩRayleigh ratioࠍ᳞߼ߡߺࠃ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 11: Page 7(11.1.31)

L

H2

§

=2 · d 2¸ 2 \ 2 x dx
© 2m ¹ dx³\ 2 x ¨ 
0

6= 2
mL2

(11.1.32)\ 1 x ߅ࠃ߮\ 2 x ߦߟ޿ߡߩRayleigh ratioࠍᲧセߔࠆߣ\ 1 x ߦߟ޿ߡߩᣇ߇ዊߐ޿㧚ߘߎߢ㧘\ 1 x

ߣ\ 2 x ߩ2ߟߛߌߢᲧߴࠆߥࠄ߫\ 1 x ߩᣇ߇ࠊߕ߆ߥ߇ࠄ⦟޿

wavefunctionߣ޿߁ߎߣߪ಴᧪ࠆ㧚ߚߛߒ㧘ᱜߒ޿eigenvalueࠍ⍮ࠄߥߌࠇ߫㧘ߎߩࠃ߁ߦ⚿⺰ߒߡ቟ᔃߒߡ޿ࠄࠇࠆߛࠈ߁߇㧘ߎߎߢ᳞߼ߚ2ߟߩRayleigh ratioࠍᱜߒ޿eigenvalueߣᲧߴߡߺࠆߣ㧘 H1 ߽ H 2 ߽ߤߜࠄ߽߆ߥࠅᄢ߈޿ߎߣ߇ಽ߆ࠆ㧚\ 1 x ߹ߚߪ\ 2 x ߩtrial functionߣᱜߒ޿eigenfunction\ exact x ߣߩᏅ⇣ࠍߪߞ߈ࠅߔࠆߚ߼ߦ㧘࿑11.1.1ߦ\ exact x 㧘\ trial x 㧘߅ࠃ߮\ exact x \ trial x ࠍplotߒߡ޿ࠆ㧚ߎߩ࿑߆ࠄ߽\ 1 x ߩᣇ߇߿߿⦟޿wavefunctionߢߪ޽ࠆ߇㧘ߤߜࠄ߽⌀ߩ
wavefunction߆ࠄߪ߆ߥࠅߕࠇߡ޿ࠆߎߣ߇ಽ߆ࠆߢ޽ࠈ߁㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 11: Page 8

2.0

1.5

1.0

0.5

0.0

-0.5
0.0

0.2

0.4

0.6

0.8

1.0

Z.

㧔a㧕\ exact 㧔ታ✢㧕㧘\ 1 㧔⎕✢㧕㧘\ exact \ 1 㧔ὐ✢㧕

2.0

1.5

1.0

0.5

0.0

-0.5
0.0

0.2

0.4

0.6

0.8

1.0

Z.

㧔b㧕\ exact 㧔ታ✢㧕㧘\ 2 㧔⎕✢㧕㧘\ exact \ 2 㧔ὐ✢㧕

࿑ 11.1.1⌀ߩeigenfunction \ exact ߣtrial function \ trial
㧔\ 1 ߹ߚߪ\ 2 㧕ߣߩᲧセ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 11: Page 9‫ڏ‬Problem 11.1.1
hydrogen-like wavefunctionߩground stateߩwavefunctionࠍ㧘trial functionߣߒߡ\N exp D r 2

ߣ޿߁Gaussian functionࠍㆬ߮㧘variation theoryߢ᳞߼ࠃ㧚[Coffee Break] wavefunctionߩ⇟ภઃߌߎߎߢߪeigenvalue߿eigenfunctionߩ⇟ภઃߌࠍ㧘ᑼ(11.1.2)ߩࠃ߁ߦenergyߩૐ޿ᣇ߆ࠄ㜞޿ᣇ߳0㧘1㧘2㧘㧚㧚㧚ߣ㗅⇟ߦઃߌߡ޿ࠆ㧚ߎߩᣇᑼߪ㧘ࠃߊߣࠄࠇࠆpopularߥᣇᴺߢߪ޽ࠆ߇㧘ᧄ⻠⟵ߣߒߡߪᆎ߼ߡߩᣇᴺߢ޽ࠆ㧚ߔߥࠊߜ㧘੹߹ߢߪ㧘
eigenvalue߿eigenfunctionߪ૗ࠄ߆ߩ㊂ሶᢙߢ⴫ߐࠇߡ޿ߚ㧚଀߃߫Chapter 6ߢቇ⠌ߒߚ੗ᚭဳpotentialߩ႐ว㨇ᑼ(6.2.7)߅ࠃ߮ᑼ(6.2.12)㨉E ߿\ ߪ n ߣ޿߁㊂ሶᢙߢቯ⟵ߐࠇ㧘 n ߪ1㧘2㧘㧚㧚㧚ߣ޿߁ᱜߩᢛᢙߦ㒢ࠄࠇߡ޿ߚ㧚Chapter 7ߩ⺞๺ᝄേሶߢ߽㨇ᑼ
(7.3.15)߅ࠃ߮ᑼ(7.4.10)߹ߚߪᑼ(7.4.12)㨉E ߿\ ߪ㊂ሶᢙ n ߢⷙቯߐࠇ㧘ߎߩ႐ว n ߪ
0㧘1㧘2㧘
㧚㧚㧚ߣ㧘࠯ࡠ߆ࠄߪߓ߹ࠆᱜߩᢛᢙߢ޽ߞߚ㧚߹ߚ㧘Section 8.1ߢቇ⠌ߒߚ౞๟਄ߩ⥄↱☸ሶߢߪ㧘㊂ሶᢙ m ߪ0㧘r1㧘r2㧘㧚㧚㧚ߣ࠯ࡠࠍ฽ߺ㧘߆ߟᱜ㧘⽶ߩᢛᢙߢ޽ߞߚ㧚
㨇ᑼ(8.2.8)߅ࠃ߮ᑼ(8.2.7)㨉⃿⴫㕙਄ߩ⥄↱㊂ሶߢߪ㊂ሶᢙ j ߪ0㧘1㧘2㧘㧚㧚㧚ߣ޿߁࠯ࡠࠍ฽߻ᱜߩᢛᢙ㧚ᦝߦChapter 9ߢቇ⠌ߒߚ᳓⚛ේሶߩ႐ว㨇ᑼ(9.3.22)߅ࠃ
߮⴫9.7.1㨉ߦߪ㊂ሶᢙ n ߪ1߆ࠄᆎ߹ࠆᱜߩᢛᢙߢ޽ߞߚ㧚ߎߩࠃ߁ߦ㧘޿ߕࠇߩ႐
ว߽㧘 E ߿\ ߪ޽ࠆ᳿߹ߞߚᣇᴺߦࠃࠅቯ⟵ߐࠇߚ㊂ሶᢙߢⷙቯߐࠇߡ޿ߚ㧚ߘࠇࠍฦ⥄ߩᅢߺߢᄌ߃ࠆߎߣߪ಴᧪ߥ޿㧚଀߃߫㧘޽ࠆੱ߇‫ޟ‬harmonic oscillatorߩ E ߿\ ߇
0߆ࠄߪߓ߹ࠆߩߪߤ߁߽޿߿ߛ㧚᳓⚛ේሶߥߤߣหߓࠃ߁ߦ1߆ࠄᆎ߹ࠆࠃ߁ߦߒߚ

޿‫ߡߞ޿ߣޠ‬㧘 En ߿\ n ࠍ⴫ߔᑼߢᓥ᧪ n ߣ߅޿ߡ޿ߚ߽ߩࠍ n  1 ߦ⟎߈឵߃ࠆߎߣ
ߪℂ⺰⊛ߦߪਇน⢻ߢߪߥ޿㧚ߒ߆ߒ㧘㐳޿㑆ߩ⠌ᘠߣߒߡ㧘harmonic oscillatorߩ㊂ሶᢙߪ࠯ࡠ߆ࠄᆎ߹ࠆߎߣߦߥߞߡ޿ࠆࠊߌߢ㧘ߘࠇࠍ୘ੱߩ⿰๧ߢߘ߁◲නߦ⠌ᘠߣ⇣ߥࠆᣇᴺࠍߣߞߡ޿޿߽ߩߣߪᕁ߃ߥ޿㧚ߘࠎߥߎߣࠍߐࠇߚࠄ㧘޿ߚߕࠄߦᷙ
ੂ߇ᄢ߈ߊߥࠆ߫߆ࠅߢᄙߊߩੱ㆐߇ㅅᖺࠍⵍࠆߢ޽ࠈ߁㧚ߎߩߎߣߪ㧘buildingߩ㓏ߩᢙ߃ᣇߩߎߣࠍ⠨߃ߡߺࠇ߫ಽ߆ࠆߢ޽ࠈ߁㧚ᣣᧄߩ1㓏ߪ㧘ࠕࡔ࡝ࠞߢߪfirst floor㧘ࠗࠡ࡝ࠬ㧔߅ࠃ࡛߮࡯ࡠ࠶ࡄߩᄙߊߩ࿖㧕ߢߪgroundfloorߣ⸒߁㧚ᣣᧄߩ2㓏ߪࠕࡔ࡝ࠞߢߪsecond floor㧘ࠗࠡ࡝ࠬ㧔߅ࠃ࡛߮࡯ࡠ࠶ࡄߩ
ᄙߊߩ࿖㧕ߢߪfirst floorߣ⸒߁㧚ᣣᧄ߿ࠕࡔ࡝ࠞߩᣇᑼߣࠗࠡ࡝ࠬߩᣇᑼߣ߇⇣ߥࠆ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 11: Page 10ߎߣߪ޽߹ࠅ޿޿ߎߣߢߪߥ޿߆߽⍮ࠇߥ޿߇㧘ੱ‫ߩߘߪޘ‬㆑޿ߩߎߣࠍ⦟ߊ⍮ߞߡ޿ࠆ߆ࠄߚ޿ߒߚᷙੂߪߥ޿㧚ߒ߆ߒ㧘ᣣᧄߢ㧘޽ࠆbuildingߩᜬਥ߇㧘⥄ಽߪᣣ☨
ߩᣇᴺࠃࠅ߽ࠗࠡ࡝ࠬᵹ߇ᅢ߈ߛ߆ࠄߣ޿߁ℂ↱ߢ㧘1㓏ߩߎߣࠍၮᐩ㓏ߣ߆࠯ࡠ㓏ߣ޿޿㧘2㓏ߩߎߣࠍ1㓏ߣ޿߁㘑ߦฬઃߌ㧘ㇱደ⇟ภ߿ࠛ࡟ࡌ࡯࠲࡯ߩ⴫␜ࠍߘ߁ߒ
ߡߒ߹ߞߚࠄ㧘ᄙಽᴺᓞ਄ߪ૗ߩ໧㗴߽ߥ޿ߩߛࠈ߁߇㧘߿ߪࠅᷙੂߪㆱߌࠄࠇߥ޿
ߢ޽ࠈ߁㧚㊂ሶᢙߩ⠌ᘠࠍ◲නߦᄌ߃ࠄࠇߥ޿ߩ߽ૃߚࠃ߁ߥߎߣߢ޽ࠆ㧚ߎߩࠃ߁ߥ㧘㐳޿ᱧผߦၮߠ޿ߚ⠌ᘠ߇޽ࠆᣇᴺߣߪ⇣ߥࠅ㧘ᑼ(11.1.2)ߩ⇟ภઃߌߦߟ޿ߡߪ㧘ߘࠇ߶ߤᷓ޿ᗧ๧߇޽ࠆࠊߌߢߪߥ޿㧚⇟ภઃߌࠍࠛࡀ࡞ࠡ࡯ߩ㗅ߦ0㧘1㧘2㧘
㧚㧚㧚ߣߒߚߩߪߎߩlecture noteߩଢቱ਄ߘ߁ߒߚ߽ߩߢ޽ࠅ㧘⺕߆߇1߆ࠄᆎ߹ࠆᢛᢙߢ⇟ภࠍઃߌߚߣߒߡ߽ᄢߒߚᷙੂߪ⿠ࠄߥ޿㧚߽ߞߣ߽ၮᐩ⁁ᘒࠍ0㧘ബ⿠⁁ᘒࠍ1㧘2㧘㧚㧚㧚ߣ⇟ภઃߌࠍⴕ߁⠌ᘠ㧘ߔߥࠊߜࠗࠡ࡝ࠬᵹߩ㓏ߩ๭߮ᣇߦᓥߞ
ߡ⁁ᘒߩ⇟ภઃߌࠍߔࠆᣇᑼߪ㧘⃻࿷㧘ᣣ☨ࠍ฽߼㧘਎⇇ਛߢ߶߷ቯ⌕ߒߟߟ޽ࠆࠃ
߁ߦᕁ߃ࠆ㧚

11.2 ✢ᒻᄌಽේℂ㧔Rayleigh-Ritzߩᣇᴺ㧕trial function \ trial ࠍ㧘޽ࠆ㒢ࠄࠇߚᢙߩwavefunction㧔basis functionߣ⸒ࠊࠇࠆ㧕ߩ⚵㧘\ 1 , \ 2 , ..., \ n 㧔basis setߣ⸒ࠊࠇࠆ㧕ߩlinear combinationߣߒߡ⴫ߔߎߣࠍ⠨߃ࠆ㧚ߔߥࠊߜ

\ trialn¦ c\i(11.2.1)ii 1ߎߎߢbasis function \ 1 , \ 2 , ..., \ n ߪ㧘޽ࠆቯ߹ߞߚ㑐ᢙᒻࠍߒߡ޿ߡ㧘ߎߩਛߦߪ
ᄌಽparameterߪ฽߹ࠇߡ޿ߥ޿㧚߹ߚ㧘ߘࠇࠄߪ৻⥸ߦߪorthonormalߢߪߥ޿㧚ߎߩtrial functionߦߟ޿ߡ ci i 1, 2, ..., n ࠍᄌಽparameterߣߔࠆvariation theoryࠍ
linear variation theoryߣ޿߁㧚ߔߥࠊߜ㧘Rayleigh ratio H ߇ᦨዊߦߥࠆࠃ߁ߥ ci ߩ⚵ࠍㆬ߱ߎߣߦߔࠆ㧚਄ㅀߒߚࠃ߁ߦ\ i ߪਇᄌߢ޽ࠆ㧚H³\ * H\ dW³\ *\ dW
trial

trial

trial

trial

䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 11: Page 11(11.2.2)

ߦᑼ(11.2.1)ࠍઍ౉ߔࠆߣᰴᑼߩࠃ߁ߦߥࠆ㧚¦¦ c * c H¦¦ c * c S

H

i

i

j

ij

i

j

ij

j

i

(11.2.3)

j

ߎߎߢ

H ij

³\ * H\ dW(11.2.4)

Sij³\ *\ dW

(11.2.5)

i

i

j

j

ߣ߅޿ߚ㧚ᑼ(11.2.3)ࠃࠅ

H ¦¦ ci * c j Sijij¦¦ c * c Hiij(11.2.6)ijjਔㄝࠍ ck * ߢᓸಽߔࠆ㧚H ¦ c j Skj jwH¦¦ ci * c j Sij
wck * i j¦c Hj

(11.2.7)kjjᄌಽේℂࠃࠅ
wH
0
wck *(11.2.8)ߣߥࠆࠃ߁ߦ c1 , c2 , ... ࠍㆬ߱ߎߣߦߥࠆ㧚(11.2.8)ࠍᑼ(11.2.7)ߦઍ౉ߒ㧘¦c H HS
j

kj

kj

0

(11.2.9)jߣ޿߁ㅪ┙ᣇ⒟ᑼ߇ᓧࠄࠇࠆ㧚ᔨߩߚ߼㧘ᑼ(11.2.9)ࠍౕ૕⊛ߦᦠ߈ਅࠈߔߣ㧘ᰴߩࠃ߁ߦߥࠆ㧚
H11  H S11 c1  H12  H S12 c2  .....  H1n  H S1n cn

0

H 21  H S 21 c1  H 22  H S 22 c2  .....  H 2 n  H S 2 n cn

0(11.2.10)

.....
.....
H n1  H S n1 c1  H n 2  H S n 2 c2  .....  H nn  H S nn cn

0ߎߩࠃ߁ߥㅪ┙ᣇ⒟ᑼ߆ࠄ H ࠍ᳞߼ࠆߦߪ㧘ߔߢߦSection 4.4߿Section10.2ߢቇ⠌ߒߚࠃ߁ߦ㧘એਅߦ␜ߔsecular equationߩ⸃ࠍ᳞߼ࠇ߫ࠃ޿㧚ߔߥࠊߜ䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 11: Page 12

H11  H S11
H 21  H S 21.....

H12  H S12
H 22  H S 22..... H1n  H S1n..... H 2 n  H S 2 n

H n1  H S n1

H n 2  H Sn 2 ..... H nn  H S nn

(11.2.11)0ߎࠇߪ H ߦ㑐ߒߡ n ᰴᣇ⒟ᑼߣߥࠆ㧚 n ߇3એ਄ߩ႐วߩᩮߪ㧘࿃ᢙಽ⸃߇಴᧪ߥߌࠇ
߫◲නߦߪ᳞߹ࠄߥ޿߆߽ߒࠇߥ޿߇㧘ᢙ୯⸘▚╬ߢߣߦ߆ߊᩮࠍ᳞߼ࠆߎߣߪน⢻ߢ޽ࠆ㧚ᦝߦ H ߇Hermitianߢ޽ࠆ㒢ࠅ㧘 n ୘ߩ⸃ߪߔߴߡታᢙߣߥࠆࠊߌߢ޽ࠆ㧚ߎߩ n ୘ߩታᢙ⸃ߩ߁ߜ㧘ᦨዊߩ H ߇㧘߽ߞߣ߽ࠃ޿ㄭૃ⸃ߣߥࠆ㧚H ߇᳞߹ߞߚࠄ㧘ߘߩ H ࠍᑼ(11.2.10)ߦઍ౉ߒߡଥᢙ c1 , c2 , ..., cn ࠍ᳞߼ࠆߎߣߦߥ
ࠆ㧚 n ୘ߩᣇ⒟ᑼ߇޽ࠅ㧘ᧂ⍮ᢙ߇ n ୘ߢ޽ࠆ߆ࠄ⸃ߪᓧࠄࠇߘ߁ߦᕁ߃ࠆ߇㧘ታߪᑼ(11.2.10)ߩ n ୘ߩᣇ⒟ᑼߪ੕޿ߦ⁛┙ߢߪߥߊ㧘⁛┙ߥߩߪ n  1 ୘ߒ߆ߥ޿㧚ߘߎ
ߢ߽߁1ߟᣇ⒟ᑼ߇ᔅⷐߢ㧘ㅢᏱߪnormalization conditionࠍ↪޿ࠆ㧚ᑼ(11.2.1)߇
normalizeߐࠇࠆߣ޿߁᧦ઙ㧘ߔߥࠊߜ³\*\ trial dW

1(11.2.12)¦¦ c * c S

1(11.2.13)

trial

ࠃࠅ
i

i

j

ij

j

߇ᔅⷐ᧦ઙߣߥࠆ㧚਄ㅀߒߚࠃ߁ߦ㧘basis functionߪ৻⥸ߦߪ߅੕޿ߦorthonormalߢߪߥ޿ߩߢ㧘 Sij

Gij ߩࠃ߁ߦߪߥࠄߥ޿ߎߣߦᵈᗧߒߚ޿㧚߽ߜࠈࠎ㧘basis function⥄૕ࠍnormalizeߒߡ߅ߌ߫ Sij ߩኻⷺ㗄ߦߟ޿ߡߪ Sii

1 ߣ߅޿ߡ޿޿ߎߣߦߥࠆ㧚‫ڏ‬Problem 11.2.1
Exercise 11.1.3ߢ㧘੗ᚭဳࡐ࠹ࡦࠪࡖ࡞ߩground stateߩ⸃ߣߒߡ㧘2ߟߩtrial functionߦߟ޿ߡี๧ߒߚ㧚ߔߥࠊߜ㧘normalizeߐࠇߚtrial functionᑼ(11.1.29)߅ࠃ߮ᑼ(11.1.30)

\1 x

30
x Lx
L5

630 2
2
x Lx
9
L
ߩ2ߟࠍ⠨߃ߚࠊߌߢ޽ࠆ߇㧘ߘߩ޿ߕࠇߦߟ޿ߡߩRayleigh ratio߽ᱜߒ޿\2 xeigenfunctionࠃࠅߪ߆ߥࠅᄢ߈޿㧘ߔߥࠊߜ2ߟߣ߽ߘࠇ߶ߤ޿޿ㄭૃߢߪߥ޿ߎߣ߇

ಽ߆ߞߚ㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 11: Page 13ᧄProblemߢߪ㧘\ 1 x

ߣ\ 2 x ߩlinear combinationߣߒߡtrial functionࠍ⴫ߒ㧘߽ߞߣ߽ࠃ޿ㄭૃ⸃ࠍ᳞߼ߡߺࠃ㧚ᦝߦ㧘࿑11.1.1ߩࠃ߁ߥ࿑ࠍឬ޿ߡ㧘Exercise 11.1.3ߩ
⚿ᨐߣᲧセߖࠃ㧚‫ڏ‬Problem 11.2.2
ᐩ߇ᢳ߼ߦߥߞߡ޿ࠆ৻ᰴరߩ੗ᚭဳpotentialߩਛߩ☸ሶࠍ⠨߃ࠆ㧚ߔߥࠊߜ㧘potentialߪ

V x

V
x
L

0d xdLߎߩ♽ߩ࿕᦭㑐ᢙߦኻߔࠆ⹜ⴕ㑐ᢙࠍ

\1 x

2
Sx
sin
L
L

2
2S x
sin
L
L
ߩ2ߟߩ✢ᒻ⚿วߣߒߡ㧘ၮᐩ⁁ᘒߩࠛࡀ࡞ࠡ࡯ࠍ✢ᒻᄌಽᴺߢ᳞߼ࠃ㧚⸃ࠍ⸃ᨆ⊛ߦ᳞߼ߚᓟ㧘ౕ૕⊛ߦ V ࠍ㧘ᐩߩᐔࠄߥ੗ᚭဳpotentialߩၮᐩ⁁ᘒߩenergyߩ10ಽߩ1
ߦߣࠅenergyࠍ᳞߼ࠃ㧚ߥ߅㧘⸘▚ࠍ◲නߦߔࠆߚ߼ߦ L 1 ߣߒߡࠃ޿㧚\2 x[Coffee Break] basisߣbaseᑼ(11.2.1)ߩࠃ߁ߦ㧘޽ࠆwavefunctionࠍ㧘૗୘߆ߩwavefunctionߩlinear combinationߣߒߡ⴫ߔᤨ㧘ᓟ⠪ߩwavefunctionࠍၮᐩ㑐ᢙ㧔basis function㧕߹ߚߪනߦၮᐩ㧔basis㧕ߣ޿߁⸒⪲ߢ⴫ߔ㧚ߎߩbasisߩߎߣࠍ㑆㆑ߞߡbaseߣ⸒߁ੱ߇޿ࠆ߇㧘ߘ߁޿߁㑆㆑޿ߪߺߞߣ߽ߥ޿ߩߢᵈᗧࠍߒߡ߶ߒ޿㧚⺆ቇߩ┙႐߆ࠄ⷗ࠆߣ㧘base߽basis߽ߤߜࠄ߽૗╬߆ߩ‫ޟ‬ၮ␆‫߁޿ߣޠ‬ᗧ๧ߦ↪޿ࠄࠇࠆ㧚ߢߪ㧘ߤ߁㆑߁ߩ߆ߣ޿߁ߣ㧘baseߪᒻߩ޽ࠆ߽ߩߦኻߒߡ㧘basisߪ᛽⽎⊛ߥ߽ߩߦ↪޿ࠄࠇࠆ㧚଀߃߫‫ߩߎޟ‬ᑪ‛ߩၮ␆ߪߒߞ߆ࠅ૞ࠄࠇߡ޿ࠆ‫ߩᤨ߁޿ߣޠ‬‫ޟ‬ၮ␆‫"ߪޠ‬base"ߢ޽ࠅ㧘‫ޟ‬㊂ሶജቇߩၮ␆ߪ٤٤ᐕߦ૞ࠄࠇߚ‫ޟߩᤨ߁޿ߣޠ‬ၮ␆‫ޠ‬
ߪ"basis"ߢ޽ࠆ㧚ᔨߩߚ߼㧘Webster's New Dictionary of Synonyms߆ࠄߩ৻ㇱࠍᛮ☴ߒߡએਅߦ␜ߔ㧚
Base may be applied to the lowest part or bottom of something without strong
implication of purpose as a support or prop (the base of a tree) (base of amonument) but more often it implies specific reference to a broad bottom or to a䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 11: Page 14

substructure on which a thing rests or seems to rest for support or by which it is
kept upright or stable (the base of a pyramid) (the base of a lamp) (the base of atriangle).
Basis like base may be used in reference to something that underlies and supports
or to something that serves as a starting point, but the term is rarely applied to aphysical or material thing; thus, one may speak of the base (but not the basis) of amonument, or of the basis (not the base) for a certain belief.

ߥ߅㧘baseߣ޿߁⸒⪲ߪ㧘ൻቇ⠪ߦߣߞߡߪ‫ޟ‬Ⴎၮ‫߁޿ߣޠ‬ᗧ๧ߢ߽ߞߣ߽ࡐࡇࡘ
࡜࡯ߦ૶ࠊࠇࠆߎߣߪᓮᛚ⍮ߩߣ߅ࠅߢ޽ࠆ㧚߹ߚ㧘਄ߩWebsterߩㄉᦠߦ߽಴ߡ᧪ࠆ
ਃⷺᒻߩ‫ޟ‬ᐩㄝ‫߁޿ߣޠ‬ᗧ๧ߢbaseߣ޿߁⸒⪲߇⑼ቇߩ⺰ᢥߦ૶ࠊࠇࠆߎߣ߽޽ࠆߢ
޽ࠈ߁㧚ߒ߆ߒ㧘ߎࠇએᄖߦߪbaseߣ޿߁⸒⪲ߪ߶ߣࠎߤ಴ߡ᧪ߥ޿ߩߢߪߥ߆ࠈ߁
߆㧚ߎࠇߛߌ඙೎߇ߪߞ߈ࠅߒߡ޿ࠆߩߦ㧘basisߩߎߣࠍbaseߣ㑆㆑ߞߡ⸒߁ߩߪߤ߁
ߒߡߢ޽ࠈ߁߆㧫߭ࠂߞߣߔࠆߣ㧘ⶄᢙᒻߦߟ޿ߡߩή⍮߇ߘߩේ࿃߆߽ߒࠇߥ޿㧚basisߩⶄᢙᒻߪbasesߢ޽ࠆ㧚basesߣ޿߁ⶄᢙᒻࠍߺߡ㧘ߘߩනᢙᒻࠍbaseߣᕁߞߡߒ

߹ߞߚߩ߆߽ߒࠇߥ޿߇㧘ߥࠎߣ߽ᖤߒ޿⹤ߛ㧚ߎ߁޿߁㑆㆑޿ߪ㧘޽ߚ߆߽indicesߣ޿߁ⶄᢙᒻߩනᢙᒻߪindiceߢ޽ࠆߣᕁߞߚࠅ㧘analysesߣ޿߁ⶄᢙᒻߩනᢙᒻߪ
analyseߢ޽ࠆߣൊ㆑޿㧔ᱜߒ޿නᢙᒻߪ㧘⸒߁߹ߢ߽ߥߊ೨⠪ߪindex㧘ᓟ⠪ߪanalysis㧕ߔࠆߩߣૃߚࠃ߁ߥ⹤ߢߺߞߣ߽ߥ޿ߩߢߊࠇߋࠇ߽ᵈᗧߒߡ߶ߒ޿㧚䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟䌟
Lecture Note on Quantum Chemistry by T. Azumi
Chapter 11: Page 15