ࠛ࠺ࡘࠤ࡯࡚ࠪࡦ㧽࠙ࠚ࡞࠽࡯࡮ࡂࠗ࠯ࡦࡌ࡞ࠢ(1901~1976)ߩഞ❣
A meritorious deed of Werner Heisenberg (1901-1976)᧛਄ 㓁৻㧔᧲੩Ꮏᬺᄢቇ㧕
Yoichi MURAKAMI (Tokyo Institute of Technology)
e-mail: murakami.y.af@m.titech.ac.jp㧞೨ᦼ㊂ሶ⺰ߩ㑐ㅪᚑᨐࡏ࡯ࠕߩኻᔕේℂ᳓⚛ේሶߩ㔚ሶߦߟ޿ߡඨᓘ r ߩฎౖ౞ㆇേࠍ
⠨߃ࠆߣ㧘ߘߩోࠛࡀ࡞ࠡ࡯㧔ㆇേ㧗૏⟎㧕ߪe22rߢ޽ࠅ㧘゠㆏ㆇേߩ๟ᵄᢙߪ

(1)

E=−

e2
(2)
4π 2 mr 3ߣዉ߆ࠇ߹ߔ㧔ࠤࡊ࡜࡯ߩ╙ 3 ᴺೣ㧕㧚ᑼ(1)ߣᑼ(2)߆ࠄ r ࠍᶖ෰ߔࠆߣ౞ㆇേߩᝄേᢙߪ
ν2 =࠙ࠚ࡞࠽࡯࡮ࡂࠗ࠯ࡦࡌ࡞ࠢ㧔1901~1976㧕㧝ߪߓ߼ߦ
ࡊ࡜ࡦࠢߩᴺೣ[1]ߦࠃߞߡ᐀ࠍ㐿ߌߚ㊂ሶ⺰ߪ㧘1913 ᐕߦࡏ࡯ࠕߦࠃߞߡ⊒⴫ߐࠇߚේሶ⺰[2]ࠍᄾᯏߦᕆㅦߥዷ㐿ࠍ⷗ߖᆎ߼߹ߔ㧚೨ᦼ㊂ሶ⺰
ߣ⒓ߐࠇࠆߎߩㆊᷰ⊛ߥᤨᦼߦ߅޿ߡ㧘ੱ‫ߪޘ‬㊂ሶ⺰ߩฎౖ⺰ߦࠃࠆ㘃ផࠍഥߌࠆࡏ࡯ࠕߩኻᔕේ
ℂࠍ㆏ౕߣߒߡ᭽‫ߥޘ‬໧㗴ߦขࠅ⚵ࠎߢࠁ߈߹ߔ㧚ߔߥࠊߜ㧘ᒰᤨߪ‫ޟ‬ൊߣᛛᏁߢ߽ߞߡ୘‫ߩޘ‬໧㗴ࠍኻᔕේℂ⊛ߦಣℂߒ㧘޽ߣ߆ࠄߺߡᧄ⾰⊛ߦᱜߒ޿㑐ଥᑼࠍ⚿⺰ߔࠆ‫[ޠ‬3] ⁁ᴫߦ޽ߞߚߣ޿߃
߹ߔ㧚ߎߩࠃ߁ߥขࠅ⚵ߺߪ৻ቯߩᚑᨐࠍ᜼ߍߚ
߽ߩߩ㧘᣿⏕ߥ౏ℂ♽ࠍ઻ࠊߥ޿ߎߣߦࠃࠆ৻⥸ᕈߩᰳᅤ߇໧㗴ߣߒߡઃ߈߹ߣ޿߹ߒߚ㧚
1925 ᐕߩೋᄐ㧘ࠥ࠶࠴ࡦࠥࡦᄢቇߩ⑳⻠Ꮷߛߞߚࡂࠗ࠯ࡦࡌ࡞ࠢߪේሶౝ㔚ሶߩ゠㆏ߩ᭎ᔨࠍ໧
㗴߆ࠄឃߒ㧘ቯᏱ⁁ᘒ㑆ߩㆫ⒖ࠍਛᔃߦᝪ߃ߡኻ
ᔕේℂߩᔀᐩ⊛ߥ♖ኒൻࠍⴕ߁ߎߣߢ㧘⥄Ꮖ⍦⋫ߩߥ޿㚂የ৻⽾ߒߚᢙቇᒻᑼ㧘ߔߥࠊߜ㊂ሶജቇߩ᭴▽ߦᚑഞߒ߹ߔ㧚ߎࠇߦࠃࠅ೨ᦼ㊂ሶ⺰ߩᤨઍߪ⚳ࠊࠅࠍㄫ߃㧘ߘߩ⠉ᐕ⊒⴫ߐࠇߚ㧔․ᱶ⋧ኻᕈℂ⺰ࠍᜰዉේℂߣߒߚ࠼࡮ࡉࡠࠗߩℂ⺰[4]ߦ
ၮ޿ߚ㧕ࠪࡘ࡟࠺ࠖࡦࠟ࡯ߩᵄേℂ⺰[5]ߣߣ߽ߦ
㊂ሶജቇߩᤨઍ߇᐀ࠍ㐿ߌࠆߎߣߦߥࠅ߹ߔ㧚વᾲ 2011 ᐕ 7 ᦬- 36 -

ν =−

3

2
E2
π 2 me 4(3)ߣ⴫ߐࠇ߹ߔ㧚ฎౖ⺰ߢߪ๟ᦼㆇേࠍⴕ߁㔚⩄ߪᑼ(3)ߢ⴫ߐࠇࠆᝄേᢙߩశࠍ᡼಴ߔࠆߣ⠨߃ࠄࠇ㧘৻⥸ߦᬦ౞゠㆏ㆇേࠍ⠨߃ߚ႐วߦߪ㧘㔚⩄ߩㆇേߦࠃࠅᑼ(3)ߩၮᧄᝄേᢙνߦട߃ߡߘߩτ୚㜞⺞ᵄ㧔τ = 1, 2, ⋅⋅⋅㧕ν τ ≡ τν = −

3

2
E 2τ
π 2 me 4(4)߇૬ߖߡ᡼಴ߐࠇࠆߎߣߦߥࠅ߹ߔ㧚৻ᣇ㧘㊂ሶ⺰ߦ߅޿ߡߪࡃ࡞ࡑ࡯ߩ౏ᑼ

1 ν
1 ⎞
⎛ 1
= = R⎜ 2 − 2 ⎟
λ c
n ⎠
⎝m

(5)

߆ࠄ㧘᳓⚛ේሶߦ߅ߌࠆ n ⇟⋡ߩ㔚ሶ゠㆏߆ࠄ m⇟⋡ߩ゠㆏߳ߩㆫ⒖ߦ઻߁⊒శߩᝄേᢙν n→m ߪ

1 ⎞⎛ 1
ν n → m = Rc ⎜ 2 − 2 ⎟
n ⎠
⎝m

(6)

ߣᦠ߆ࠇ߹ߔ㧚ᑼ(6)ߣࡏ࡯ࠕߩᝄേᢙ᧦ઙ[2]ν n→m =

En − Emh(7)

J. HTSJ, Vol. 50, No. 212

ࠛ࠺ࡘࠤ࡯࡚ࠪࡦ㧽ᐳᮡࠍ q ߣߔࠆߣ㧘♽߇࿕᦭⁁ᘒߦ޽ࠆߣ߈ p(t)
ߣ q(t)߇ p-q ᐔ㕙਄ߦឬߊ࠻࡜ࠫࠚࠢ࠻࡝ߪ࿑ 1(a)ߩࠃ߁ߥᬦ౞ߦߥࠅ߹ߔ㧚⾰㊂ࠍ m㧘ࡃࡀቯᢙࠍκ ߣߔࠆߣ㧘ࠛࡀ࡞ࠡ࡯ߣᝄേᢙߪߘࠇߙࠇߣߩᲧセ߆ࠄ㧘᳓⚛ේሶߩࠛࡀ࡞ࠡ࡯Ḱ૏hRc
En = − 2n(8)߇ᓧࠄࠇ߹ߔ㧚ߎߎߢτ ≡ n - m㧔τ = 1, 2, ⋅⋅⋅㧕ࠍቯ⟵ߒ㧘n >> τ ߩ႐วࠍ⠨߃ࠆߣ㧘n → m (= n - τ)ߩㆫ⒖ߢ᡼಴ߐࠇࠆశߩᝄേᢙߪ

⎧ 2nτ − τ 2 ⎫ 2 Rc
ν n → n −τ = Rc ⎨
≅ 3 τ
2 2 ⎬
⎩ (n − τ ) n ⎭ n

E=

(9)

ν n → n −τ = −h 3 Rc

3
2

E τ.

1
2π

(12)

κ
m

ߣ⴫ߐࠇ߹ߔ㧚৻ᣇ㧘࿑ 1(a)ߩ࠻࡜ࠫࠚࠢ࠻࡝ߢ
࿐߹ࠇߚ㕙Ⓧ J ߪ(10)ᑼ(4)ߣᑼ(10)ߪ⋧ૃᒻߢ޽ࠅ㧘ߎࠇߪ㜞޿㊂ሶᢙ
㑆ߩㆫ⒖ߦ઻ߞߡ᡼಴ߐࠇࠆశߩᝄേᢙ߇ฎౖ⺰ߩᝄേᢙߦẋㄭߔࠆߎߣࠍ␜ໂߒ߹ߔ㧚ታ㓙㧘ᑼ(4)ߣᑼ(10)ߩฝㄝࠍ╬ภߢ߅޿ߡ⸃ߊߣᱜߒ޿࡝
ࡘ࡯࠼ࡌ࡝ቯᢙ R = 2π2me4/(h3c) = 1.09 × 107 [m-1]߇ᓧࠄࠇࠆߎߣ߆ࠄ㧘㊂ሶᢙ n ߇චಽᄢ߈޿႐ว
ߦߪνn→n-τ ≅ ντ ߇ᚑ┙ߔࠆߣ⠨߃ࠄࠇ߹ߔ㧚ߔߥࠊߜ㧘㊂ሶ⺰ߦ߅ߌࠆ n → n - τ ߩㆫ⒖ߦኻ
ߒߡߪㆇേ⁁ᘒ n ߦ޽ࠆ♽߇ฎౖ⺰ߦᓥߞߡ᡼಴
ߔࠆశߩၮᧄᝄേᢙνߩτ ୚㜞⺞ᚑಽ߇ኻᔕߔࠆ㧘৻⥸ߦ⸒޿឵߃ࠇ߫‫ޟ‬n → ∞ߩᭂ㒢ߢ㊂ሶ⺰ߪฎౖ⺰ߩ⚿ᨐߦẋㄭߔࠆ‫ߩࠕ࡯ࡏ߇ߩ߁޿ߣޠ‬ኻᔕ
ේℂߩⷐὐߢߔ㧚‛ℂ⊛ߦߪ㧘n ߇චಽᄢ߈޿ߣ
߈ߦߪࠛࡀ࡞ࠡ࡯ߩߣ߮ߣ߮ߩ㑆㓒߇ᓸዊߦߥࠅ
ࠛࡀ࡞ࠡ࡯ߩਇㅪ⛯ᕈ߇⋡┙ߚߥߊߥࠆߎߣ㧘߹ߚߘߩࠃ߁ߥ⁁ᴫߢߪ㔚ሶߪේሶᩭ߆ࠄචಽߦ㔌ࠇ⥄↱㔚ሶߦㄭ޿ㆇേߔࠆߣ⠨߃ࠄࠇࠆߎߣ߆ࠄ㧘n → ∞ߩᭂ㒢ߢߪ⁁ᴫ߇ฎౖ⺰ߢ⸥ㅀน⢻ߥ႐วߦẋㄭߔࠆߣ⸃㉼ߐࠇ߹ߔ㧚ࠛ࡯࡟ࡦࡈࠚࠬ࠻ߩᢿᾲ઒⺑࿕᦭ᝄേࠍⴕߞߡ޿ࠆജቇ♽ߦᄖㇱ߆ࠄචಽࠁ
ߞߊࠅߣߒߚᄌൻࠍട߃ࠆ႐ว㧔଀߃߫ᝄࠅሶߩ
♻ߩ㐳ߐ߿ᝄേᒏߩᒛജࠍᝄേ๟ᦼࠃࠅචಽ㐳޿ᤨ㑆߆ߌߡ✭߿߆ߦᄌൻߐߖࠆ႐ว㧕ࠍ⠨߃ࠆߣ㧘
ߎߩᄌൻߣߣ߽ߦᝄേᢙν߿♽ߩࠛࡀ࡞ࠡ࡯E ߪᄌൻߒ߹ߔ߇㧘E ߩᄌൻߩഀวߣνߩᄌൻߩഀวߪ
╬ߒ޿ߎߣ߆ࠄ㧘ᄌൻߩㆊ⒟ߢ E/νߪ৻ቯߩ୯ࠍ
ᜬߜ⛯ߌ߹ߔ㧚ߎߩࠃ߁ߥᄌൻߦ߅ߌࠆਇᄌ㊂ࠍ
ᢿᾲਇᄌ㊂ߣ๭߮߹ߔ㧚ౕ૕⊛ߦ㧘৻ᰴరߩᱜᒏᝄേሶߩㆇേ㊂ࠍ p㧘

વᾲ 2011 ᐕ 7 ᦬

(11)

ν =

ߣߥࠅ߹ߔ㧚ᑼ(8)ߣᑼ(9)߆ࠄ n ࠍᶖ෰ߔࠆߣ㧘2

1 2 κ 2p + q
2m2

- 37 -J = ∫ pdq = 2πE

m
κ

(13)

ߣ᳞߼ࠄࠇ߹ߔ߇㧘ߎࠇߪᑼ(12)ߦࠃߞߡ

E
=J
v

(14)

ߣᦠ߈឵߃ࠄࠇ㧘J ߇ᢿᾲਇᄌ㊂ߢ޽ࠆߎߣ߇ಽ
߆ࠅ߹ߔ㧚ട߃ࠄࠇߚᢿᾲ⊛ᄌൻߦࠃࠅ࠻࡜ࠫࠚ
ࠢ࠻࡝ߪ࿑ 1(b)ߩࠃ߁ߦᄌൻߒ߹ߔ߇㧘ߘߩㆊ⒟ߢ㕙Ⓧ J ߪਇᄌߢ޽ࠅ㧘♽ߪᄌൻߩㅜਛ߅ࠃ߮ᄌ
ൻᓟ߽࿕᦭ᝄേࠍⴕ߁ߎߣߦߥࠅ߹ߔ㧚ࠛ࡯࡟ࡦࡈࠚࠬ࠻㧔P. Ehrenfest, 1880 − 1933㧕ߪ㊂ሶ⺰ߦ߅ߌࠆᢿᾲਇᄌ㊂ߩ⎇ⓥࠍⴕ޿㧘ᰴߩ઒⺑ߩឭ᩺ߦࠃࠅ㧘㊂ሶജቇ߇⏕┙ߐࠇࠆ߹ߢߩㆊ
ᷰ⊛ᣇᴺ⺰ࠍឭଏߒ߹ߒߚ㧚‫ޟ‬ജቇ♽ߦᄖㇱ߆ࠄᄌൻࠍਈ߃ࠆߣ߈㧘ߎߩᄌൻ߇ή㒢ߦࠁߞߊࠅⴕ
ࠊࠇࠆߥࠄ߫㧘ߘߩ⚻ㆊߩㅜਛ㧘ജቇ♽ߩㆇേࠍㅢᏱߩജቇࠍ↪޿ߡ⺰ߕࠆߎߣ߇⸵ߐࠇࠆ㧚ߎߩߣ߈㧘ᄌൻߩᆎ߹ࠆ೨ߦ♽߇㊂ሶ⊛ߦ⸵ߐࠇߚ⁁ᘒߦ޽ࠇ߫㧘ߎߩᄌൻߩㅜਛ߅ࠃ߮ߘߩᓟ߽♽ߪ
߿ߪࠅ㊂ሶ⊛ߦ⸵ߐࠇߚ⁁ᘒߦ޽ࠆ㧚‫[ޠ‬6]
৻⥸ൻߐࠇߚ㊂ሶ᧦ઙࠛ࡯࡟ࡦࡈࠚࠬ࠻ߩᢿᾲ઒⺑ࠍၮ␆ߣߒߡ㧘
1910 ᐕઍߩਛ⋚ߦ߆ߌߡⶄᢙߩ‛ℂቇ⠪ߦࠃߞߡᰴߩࠃ߁ߥ৻⥸ൻߐࠇߚ㊂ሶ᧦ઙ

J = ∫ pdq =nh(15)߇ዉ߆ࠇ߹ߔ㧚ᑼ(15)ߦᑼ(14)ࠍㆡ↪ߔࠇ߫ࡊ࡜ࡦ
ࠢᝄേሶߩ E = nhν ࠍ⏕⹺ߢ߈߹ߔ㧚ᑼ(15)ߪ㧘࿑ 1(c)ߩࠃ߁ߦ㊂ሶ⊛ߦ⸵ߐࠇࠆ⁁ᘒߪ࠻࡜ࠫࠚ
ࠢ࠻࡝ߦࠃߞߡ࿐߹ࠇࠆ㕙Ⓧ߇ࡊ࡜ࡦࠢቯᢙߩᢛ
ᢙ୚ߦߥࠆ߽ߩߦ㒢ࠄࠇࠆ㧔㊂ሶᢙ߇ᢿᾲਇᄌ㊂

J. HTSJ, Vol. 50, No. 212

ࠛ࠺ࡘࠤ࡯࡚ࠪࡦ㧽᡼ߐࠇࠆߴ߈ߢ㧘⊒శߩᝄേᢙ߿ᒝᐲߣߒߡታ㓙ߦ᷹ⷰߐࠇࠆੑߟߩቯᏱ⁁ᘒ㑆ߩ‫ޟ‬ㆫ⒖‫ࠍߺߩޠ‬ᛒࠊߥߌࠇ߫ߥࠄߥ޿ߣ޿߁⠨߃ߦ⥋ࠅ߹ߔ[6,7]㧚
ߎߩࠃ߁ߥ⠨߃ߩਅ㧘ᜰዉේℂߣߔࠆࡏ࡯ࠕߩኻ
ᔕේℂࠍᔀᐩ⊛ߦ♖ኒൻߒߡࠁߊߎߣߢ㧘‛ℂ⊛ᗧ๧߇ਇ᣿⏕ߥ㔚ሶߩ゠㆏ㆇേߣ޿߁᭎ᔨߦ❈ࠄ
ࠇߚฎ޿㊂ሶ⺰߆ࠄߩ⣕ළࠍ⹜ߺߚߩߢߔ㧚ࡂࠗ࠯ࡦࡌ࡞ࠢߩᒰೋߩ⋡⊛ߪ᳓⚛ේሶߦ߅ߌ
ࠆࡃ࡞ࡑ࡯✢ߩ⊒శᒝᐲࠍ⸘▚ߔࠆߎߣߦ޽ࠅ߹
ߒߚ[8]㧚ฎౖ⊛ߥࡑࠢࠬ࠙ࠚ࡞ℂ⺰ߢߪ᡼኿ᒝᐲ
ߪᝄേ෺ᭂሶߩᝄ᏷ߩ⛘ኻ୯ߩੑਸ਼ߦᲧ଀ߒ߹ߔ㧚৻ᣇ㧘㊂ሶ⺰ߦ߅޿ߡߪㆫ⒖ߩ㗫ᐲ㧔න૏ᤨ㑆޽
ߚࠅߩㆫ⒖⏕₸㧕߇᡼኿ߩᒝᐲߦ޽ߚࠅ߹ߔ㧚ࡂࠗ࠯ࡦࡌ࡞ࠢߪኻᔕේℂߦᴪߞߡㅴ߻ߚ߼㧘߹ߕ㧔ߘࠇએ೨ߩኻᔕේℂ⊛ߥ⎇ⓥߢⴕࠊࠇߡ޿ߚࠃ
߁ߦ㧕‛ℂ㊂ࠍ゠㆏ㆇേߩឬ௝ߩਅߢࡈ࡯࡝ࠛ⚖ᢙߩᒻߦᦠ߈಴ߔߎߣ߆ࠄᆎ߼߹ߔ㧚਄ㅀߩ⊒శᒝᐲࠍ᳞߼ࠆ⋡⊛߆ࠄ㧘╙৻ߦ㊀ⷐߥ‛ℂ㊂ߪ㔚
ሶߩᝄേᝄ᏷ߦ㑐ㅪߔࠆ૏⟎ᐳᮡ x ߣߥࠅ߹ߔ㧚ౕ૕⊛ߦ㧘n ⇟⋡ߩቯᏱ⁁ᘒߦ޽ࠆ㔚ሶߩ૏⟎ xߪฎౖ๟ᦼㆇേߩ㜞⺞ᚑಽν(n,τ)ߦߟ޿ߡߩࡈ࡯
࡝ࠛ⚖ᢙߣߒߡᰴߩࠃ߁ߦᦠ߆ࠇ߹ߔ㧚࿑ 1 (a) ᱜᒏᝄേ♽ߩ࠻࡜ࠫࠚࠢ࠻࡝㧚(b) ᢿᾲᄌൻߦ߅ߌࠆផ⒖ߩ଀㧚(c) ㊂ሶ᧦ઙ㧚+∞x (t ) = ∑ X ( n, τ )e 2 πiν ( n ,τ ) t(19)

τ = −∞

ߦߥߞߡ޿ࠆ㧕ߣ޿߁ߎߣࠍ⴫ߒ߹ߔ㧚৻ᣇ㧘๟ᦼ♽ߩᝄേᢙνߪ E ߣ J ߦࠃߞߡν =dE ( J )dE ( J )⇔ ν τ = τν = τdJ
dJࡂࠗ࠯ࡦࡌ࡞ࠢߪߎߩࠃ߁ߥฎ޿⠨߃ߦ߅ߌࠆ㔚ሶߩ゠㆏ㆇേߩࡈ࡯࡝ࠛ⚖ᢙߩฦ㗄߇㊂ሶ⊛ߥฦ‫ޟߩޘ‬ㆫ⒖‫ߦޠ‬ኻᔕߒߡ޿ࠆߣ⠨߃߹ߔ㧚ߘߒߡᓐߪߎߩࠃ߁ߥฎ޿⠨߃ߦ߅ߌࠆ゠㆏ㆇേߩࡈ
࡯࡝ࠛᚑಽߦኻᔕߒߡ n → n - τ ߩㆫ⒖ࠍ⴫ߔ㊂ሶജቇ⊛ߥ㧔ㆫ⒖ᚑಽߣߢ߽๭߱ߴ߈㧕㊂(16)

ߣ⴫ߐࠇ[6]㧘ߎࠇߦᑼ(15)ࠍઍ౉ߔࠆߣ

ν τ J = nh ≡ ν ( n,τ ) =τ dE ( n )
h dn

(17)

X ( n; n − τ )e 2 πiν ( n ;n −τ ) t߇ᓧࠄࠇ߹ߔ㧚ฎౖ⊛ߥ୚㖸㑐ଥߦၮߠ޿ߚߎߩ
ᓸಽᑼߦኻߒߡߪ㧘㊂ሶ⺰ߢߪࡏ࡯ࠕߩᝄേᢙ᧦
ઙ㧔ᑼ(7)㧕ߦࠃࠅᏅಽᑼ

ν n→n −τ ≡ ν ( n; n − τ ) =E (n) − E (n − τ )h߇ሽ࿷ߒߡ޿ߡ㧘ߘߩ⛘ኻ୯ߩੑਸ਼߇ㆫ⒖ߩ⏕₸ࠍਈ߃ࠆߩߦၮᧄ⊛ߥᓎഀࠍᨐߚߔߩߦ㆑޿ߥ޿㧘ߣ޿߁⷗ᒰ߆ࠄ಴⊒ߒ߹ߔ[3,6]㧚ߔߥࠊߜฎ޿⠨߃ߢߩࡈ࡯࡝ࠛᚑಽ㧔ᑼ(19)ߩฦ㗄㧕ࠍㆫ⒖ᚑಽ㧔ᑼ(20)㧕ߦࠃߞߡ⟎߈឵߃ߡࠁߊߣ޿߁ᣇ㊎ࠍߣࠅ߹ߔ㧚એਅ㧘ኻᔕ㑐ଥࠍ⸥ภ‫ߡߞࠃߦޠ↔ޟ‬⚿߮㧘Ꮐ஥ߦ゠㆏ឬ௝ߦ┙⣉ߒߚฎ޿㊂ሶ⺰ߩᒻᑼ㧘ฝ஥ߦߘࠇߦኻᔕߔࠆᣂߒ޿ᒻᑼࠍ␜ߔߎߣ
ߦߒ߹ߔ㧚ߎࠇ߆ࠄ㧘ᑼ(19)ߣᑼ(20)ߩኻᔕ㑐ଥߪX(n,τ) ↔ X(n; n - τ)ߣ⴫ߐࠇ߹ߔ㧚(18)߇ኻᔕߔࠆߎߣߦߥࠅ߹ߔ㧚ߎߩࠃ߁ߥኻᔕ㑐ଥߪࡏ࡯ࠕࠄߦࠃߞߡ⏕┙ߐࠇߡࠁ߈߹ߒߚ㧚㧟㊂ሶജቇߩ⊒⷗ࡂࠗ࠯ࡦࡌ࡞ࠢߩᜰ㊎ࡂࠗ࠯ࡦࡌ࡞ࠢߪߘࠇ߹ߢߩ⚻㛎ߣ⠨ኤ߆ࠄ㧘ታ㛎⊛ߦ᷹ⷰߐࠇߥ޿ฎ޿㊂ሶ⺰ߦ߅ߌࠆේሶౝ
㔚ሶߩ゠㆏ㆇേߣ޿߁߽ߩߪℂ⺰ߩᨒ⚵ߺ߆ࠄㅊવᾲ 2011 ᐕ 7 ᦬(20)- 38 -J. HTSJ, Vol. 50, No. 212

ࠛ࠺ࡘࠤ࡯࡚ࠪࡦ㧽ᓸಽṶ▚ߦߟ޿ߡߩኻᔕ㑐ଥ
2.3 ▵ߦ␜ߒߚᑼ(17)ߣᑼ(18)ߩኻᔕ㑐ଥ㧔ᓸಽߣᏅಽߩኻᔕ㧕ߪ㧘ߎࠇࠍቯᏱ⁁ᘒߏߣߦ୯ࠍ߽
ߟ n ߩછᗧߩ㑐ᢙ F(n)ߦ᜛ᒛߒߡ

ߣ޿߁ኻᔕ㑐ଥ߇ᓧࠄࠇ߹ߔ㧚ᑼ(28)ߪࡂࠗ࠯ࡦ
ࡌ࡞ࠢ߇㊂ሶജቇࠍ⊒⷗ߔࠆ਄ߢ㊀ⷐߥᓎഀࠍᨐ
ߚߒߚࠞࠡߣߥࠅ߹ߒߚ[3,6]㧚

dF ( n)↔ F (n) − F (n − τ )dnⓍṶ▚ߦߟ޿ߡߩኻᔕ㑐ଥߎߩࠃ߁ߦ㔚ሶߩ゠㆏ㆇേߩឬ௝ߦ┙⣉ߒߚฎ
޿ℂ⺰ࠍᣂߒ޿ᒻᑼߦኻᔕ⊛ߦ⸃⺒ߒߡࠁߊ㓙㧘⺒ߺ⸃ߊߴ߈ฎ޿ℂ⺰ߦᄌᢙห჻ߩⓍ߇⃻ࠇࠆߎߣ߇޽ࠅ߹ߔ㧚ౖဳ⊛ߥ଀ߪ x2 ߢ޽ࠅ㧘ߎࠇࠍᣂߒ޿ᒻᑼߢߤߩࠃ߁ߦኻᔕߐߖࠆߴ߈߆߇໧㗴ߣߥࠅ߹ߔ㧚߹ߕ㧘゠㆏ㆇേߩឬ௝ߢࡈ࡯࡝ࠛ⚖ᢙߦࠃࠅ⴫
ߐࠇߚ x㧔ᑼ(19)㧕ߩੑਸ਼ࠍ⠨߃߹ߔ㧚ߎߩࠃ߁ߥ
x2 ߦ߅޿ߡ߽߿ߪࠅหߓᝄേᢙν(n,τ)ࠍ฽߻⚖ᢙߢ⴫ߐࠇࠆߎߣ߆ࠄ㧘τ ⇟⋡ߩࡈ࡯࡝ࠛᚑಽߪ

τ

(21)

ߣᦠ߆ࠇ߹ߔ㧚
ߘߩઁߩ㊀ⷐߥኻᔕ㑐ଥߣߒߡߪ㧘
ᑼ(17)ߣᑼ(18)ࠍᲧセߒߚ㓙ߣหߓ⼏⺰߆ࠄν (n,τ ) = −ν (n,−τ ) ↔ ν (n; n − τ ) = −ν (n − τ ; n) .
(22)㧔Ꮐ஥ߩ㑐ଥߪν(n,τ) = τν(n,1)߆ࠄ㧚㧕߹ߚ㧘x(t)߇ታᢙߢ޽ࠆߣ޿߁᧦ઙ߆ࠄࡈ࡯࡝ࠛଥᢙߩⶄ⚛
౒ᓎߦߟ޿ߡߩኻᔕ㑐ଥX (n,τ ) = X * (n,−τ ) ↔ X (n; n − τ ) = X * (n − τ ; n)+∞

∑ X (n,τ ′)e(23)= ∑ X ( n,τ ′) X ( n,τ − τ ′)e 2 πiν ( n ,τ ) t
τ ′= −∞

ߣᦠ߆ࠇ߹ߔ㧚ߎߎߢτ ′ߪනߥࠆ࠳ࡒ࡯ᄌᢙߢ޽
ࠅ㧘ਔㄝߩᜰᢙㇱߩ㑐ଥߪฎౖ⊛ߥ୚㖸ᝄേೣν ( n, τ ′) + ν ( n, τ − τ ′) = ν ( n, τ )

(24)

(25)

d2 f
{ f ( x + Δx ) − f ( x )} − { f ( x ) − f ( x − Δx )}
= lim
2
0
Δ
x
→
dx
( Δx ) 2(26)

ߦᥧ␜ࠍฃߌߡ㧘ᑼ(24)ߩᣂߒ޿㊂ሶ⺰ߦ߅ߌࠆᒻᑼߣߒߡߪ㧘ᑼ(26)ߢΔx ࠍτ (= Δn)ߣ⺒ߺ឵߃

τ 2 {E ( n + τ ) − E ( n )} − {E ( n ) − f ( n − τ )}
(27)
h
τ2
= ν ( n + τ ; n ) − ν ( n; n − τ )ν ( n; n − τ ′) + ν ( n − τ ′; n − τ ) = ν ( n; n − τ )વᾲ 2011 ᐕ 7 ᦬(31)߇ኻᔕߔߴ߈ߣ⠨߃߹ߔ㧔‫ޟ‬AψB ߩㆫ⒖ᝄേᢙߣ
BψC ߩㆫ⒖ᝄേᢙߩ๺ߪ AψC ߩㆫ⒖ᝄേᢙߦ╬ߒ޿‫ߣߎ߁޿ߣޠ‬㧕㧚ߎߩኻᔕ㑐ଥߩ᣿ኤ߇ࡂࠗ
࠯ࡦࡌ࡞ࠢࠍ⊒⷗ߦዉ޿ߚ㎛ߣߥࠅ߹ߒߚ㧚ߎࠇ߆ࠄ⋥ߜߦᣂߒ޿ᒻᑼߦ߅ߌࠆ x2 ߩᚑಽ㧔n → n -߇㘃ផߐࠇ㧔ฝㄝ߳ߩ⒖ⴕߪᑼ(7)ߦࠃࠆ㧕㧘ߐࠄߦߎߩ㑐ଥࠍછᗧ n ߣτ ߩ㑐ᢙ F(n,τ)ߦ᜛ᒛߒߡ

∂F ( n , τ )↔ F ( n + τ ; n ) − F ( n; n − τ )
∂n

(30)

ߦၮߠ޿ߡ޿߹ߔ㧔‫ޟ‬A ୚ᵄߣ B ୚ᵄߩᝄേᢙߩ
๺ߪ A+B ୚ᵄߩᝄേᢙ‫ߣߎ߁޿ߣޠ‬㧕㧚ߎߎߢߪ
2‫ޟ‬x ߩࡈ࡯࡝ࠛᚑಽߩᝄേᢙਛߦ x ߩࡈ࡯࡝ࠛᚑ
ಽߦ⃻ࠇߚᝄേᢙએᄖߩ߽ߩ߇৻ߟ߽⃻ࠇߥ޿‫ޠ‬ߣ޿߁ߎߣ߇ᧄ⾰⊛ߥὐߣߥߞߡ޿߹ߔ㧚 ৻ᣇ㧘ᣂߒ޿ᒻᑼߢߪᑼ(20)ߩㆫ⒖ᚑಽࠍ⠨߃ࠆߎߣߦߥࠅ߹ߔ߇㧘ߎߎߢ߽ห᭽ߦ‫ޟ‬x2 ߩṶ▚ࠍⴕߞߚ㓙ߦߘߎߦ x ߩᚑಽਛߦሽ࿷ߒߥ߆ߞߚ
ᝄേᢙ߇ᣂߚߦ⃻ࠇߡߪ޿ߌߥ޿‫ⷐ߁޿ߣޠ‬᳞ࠍ߅߈߹ߔ㧚ߎߩⷐ᳞ࠍḩߚߔⓍṶ▚ࠍ⠨᩺ߔࠆߦ޽ߚࠅ㧘ࡂࠗ࠯ࡦࡌ࡞ࠢߪᑼ(30)ߩ୚㖸ᝄേೣߦߪ㊂ሶ⺰ߦ߅ߌࠆᝄേᢙ⚿วೣ㧔࡝࠶࠷ߩ⚿วೣ㧘
ᑼ(7)ߦ↱᧪㧕

ߣ޿߁ 2 㓏ᓸಽߣߥࠆߩߢ㧘2 㓏ᓸಽߩቯ⟵ᑼ

τ

(29)

+∞

ࠍ⠨߃߹ߔ㧚ᑼ(24)ߦᑼ(17)ߩνࠍઍ౉ߔࠆߣ

τ 2 d 2 E (n)h dn 2X ( n,τ − τ ′)e 2 πiν ( n ,τ −τ ′) t

τ ′ = −∞

߇⋥ߜߦᓧࠄࠇ߹ߔ㧚
ᑼ(21)ߢߪቯᏱ⁁ᘒ n ߩߺߩ㑐ᢙࠍ⠨߃߹ߒߚ߇㧘ࠃࠅ㊀ⷐߥ㑐ᢙߣߒߡ㧘ㆫ⒖ߦ㑐ߔࠆ n ߣτ ߩਔᣇࠍ฽߻㑐ᢙ㧔଀߃߫ᑼ(19)ߩࡈ࡯࡝ࠛଥᢙ㧕߇޽ࠅ߹ߔ㧚ߎߎߢ㧘ฎ޿㊂ሶ⺰ߦ߅޿ߡᰴߩࠃ
߁ߥᓸಽ㗄߇⃻ࠇߚ႐ว∂ν ( n , τ )τ∂n2 πiν ( n ,τ ′ ) t(28)- 39 -J. HTSJ, Vol. 50, No. 212

ࠛ࠺ࡘࠤ࡯࡚ࠪࡦ㧽τߩㆫ⒖ᚑಽ㧕ߣߒߡ+∞+∞∑ X (n; n − τ ′) X (n − τ ′; n − τ )e

2 πiν ( n ; n −τ ) t

nh = −2πi ∑ P ( n,τ )Q * ( n,τ )τ

τ ′ = −∞

ߣᦠ߈឵߃ࠄࠇ㧘ߎࠇߪࡈ࡯࡝ࠛᚑಽߢ⴫ߐࠇߚ
㊂ሶ᧦ઙߣߥࠅ߹ߔ㧚⛯޿ߡᑼ(28)ߩࠞࠡࠍ↪޿ࠆ⋡⊛ߢᑼ(37)ߩਔㄝࠍ n ߢᓸಽߔࠆߣ㧘߇ዉ߆ࠇ߹ߔ㧚ߎߩ⠨߃ߪછᗧߩੑߟߩ㊂ Y㧘ZߩⓍߦ৻⥸ൻߐࠇ㧘Ⓧ YZ ߩㆫ⒖ᚑಽߪ
+∞∑ Y (n; n − τ ′) Z (n − τ ′; n − τ )e

2 πiν ( n ; n −τ ) t+∞

h = −2πi ∑ τ(33)

τ ′ = −∞

τ = −∞

{

}

(38)

+∞
h
= − ∑ P ( n + τ ; n )Q * ( n + τ ; n )2πiτ = −∞(39)+∞+ ∑ P ( n ; n − τ ) Q * ( n; n − τ )τ = −∞

ߣᦠ߈឵߃ࠄࠇ߹ߔ㧚ߎߎߢ੹ᐲߪᑼ(23)ߩฝ஥
ߩ㑐ଥࠍ↪޿㧘ฝㄝ╙ੑ㗄ߩ๺ߢτ → -τߩᄌᢙᄌ
឵ࠍⴕ߁ߣ
+∞
h
= ∑ P ( n; n + τ )Q ( n + τ ; n)2πi τ = −∞

(40)

+∞

− ∑ Q ( n; n + τ )P ( n + τ ; n )τ = −∞㧠ᣂℂ⺰ߩᕈ⾰ߣᅷᒰᕈ
ᣂℂ⺰ߦ߅ߌࠆ㊂ሶ᧦ઙࡂࠗ࠯ࡦࡌ࡞ࠢߩ⠨߃ߢߪ‛ℂ㊂ߪㆫ⒖ᚑಽߩ㓸วߣߒߡਈ߃ࠄࠇࠆߎߣ߆ࠄ㧘ᣂߒ޿ᒻᑼߦ὾
߈⋥ߔߦ޽ߚࠅ㧘ฎ޿ℂ⺰ߦ߅ߌࠆ‛ℂ㊂ߪࡈ࡯࡝ࠛ⚖ᢙߦዷ㐿ߒߡ߅ߊᔅⷐ߇޽ࠅ߹ߔ㧚ฎ޿㊂ሶ᧦ઙ㧔ᑼ(15)㧕ࠍ⺒ߺ⸃ߊߚ߼ߦ㧘p ߣ q ࠍࡈ࡯࡝ࠛ⚖ᢙ

p = ∑ P ( n ,τ ) e∂
P ( n , τ )Q * ( n , τ ) .
∂nߎࠇߦᑼ(28)ߩኻᔕ㑐ଥࠍㆡ↪ߔࠆߣ㧘ᑼ(38)ߪ

ߣߥࠅ߹ߔ㧚
ߎߩᣂߒ޿Ṷ▚ⷙೣߩ⪺ߒ޿․ᓽߪ㧘n → n - τߩㆫ⒖ࠍ⴫ߔⓍṶ▚ߢߪ㧘⋥ធ㑐ଥߔࠆᆎ⁁ᘒ㧔n㧕ߣ⚳⁁ᘒ㧔n - τ㧕એᄖߦ㧘ฎౖ⊛ߦߪή㑐ଥߢ޽ࠆߪߕߩߘߩઁోߡߩ⁁ᘒ㧔n - τ ′, τ ′:- ∞ ⋅⋅⋅ 0 ⋅⋅⋅ ∞㧕߇ਛ㑆⁁ᘒߣߒߡㆫ⒖ߦ㑐ଥߒߡߊࠆߣ޿߁ὐߦ޽ࠅ߹ߔ㧚ߘߒߡ㧘ߎߩࠃ߁ߥᕈ⾰ߦࠃߞߡᑼ(33)ߢߪ YZ ߣ ZY ߩⓍṶ▚߇৻⥸ߦ╬ߒߊߥࠄߥ޿ߎߣ߇ಽ߆ࠅ߹ߔ㧚ߎࠇߪ㊂ሶജቇߦ߅ߌࠆṶ▚ߩ㕖น឵ᕈࠍ⴫ߔ߽ߩߣߥߞߡ޿߹
ߔ㧚ߎߩࠃ߁ߥ৻ㅪߩṶ▚ⷙೣߩ⊒⷗ߣᅷᒰᕈߩ
⏕⹺㧔ᰴ▵㧕ࠍ⚻ߡ㧘ࡂࠗ࠯ࡦࡌ࡞ࠢߪᣂߒ޿ᢙ
ቇᒻᑼߩ⊒⷗ߦᚑഞߒߚߩߢߒߚ㧚

+∞

(37)

τ = −∞(32)ߣ⴫ߐࠇ߹ߔ㧚ߐࠄߦ㧘ㆇേ㊂ p ߇ mq& ߢ⴫ߐࠇࠆ
႐วߦߪ㧘ᑼ(35)ࠍᤨ㑆ᓸಽߒߚ߽ߩࠍᑼ(15)ߦઍ
౉ߒߡห᭽ߦ⸃߈㧘⸘▚ߩㅜਛߢ Q ∗Q = QQ ∗ = Q
߅ࠃ߮ᑼ(23)ߩฝ஥ߩ㑐ଥࠍ↪޿ߡ+∞{

h = 4 π 2 m ∑ Q (n + τ ; n) ν (n + τ ; n)2

τ = −∞

2− Q ( n ; n − τ ) ν ( n; n − τ )

}

(41)

2πiν ( n ,τ ) t(34)ߣ⴫ߐࠇ߹ߔ㧚ᑼ(40)߅ࠃ߮ᑼ(41)߇ᣂߒ޿ℂ⺰ߦ
߅ߌࠆ㊂ሶ᧦ઙߣߥࠅ߹ߔ㧚q = ∑ Q ( n,τ )e 2πiν ( n ,τ ) t(35)ᣂℂ⺰ߦ߅ߌࠆᝄേᝄ᏷
⛯޿ߡ 3 ▵ߢ⊒⷗ߐࠇߚṶ▚ⷙೣࠍ↪޿ߡฎ޿
ㆇേᣇ⒟ᑼࠍᦠ߈឵߃ࠆߎߣࠍ⠨߃߹ߔ㧚◲නߥ႐วߣߒߡ⺞๺ᝄേሶ㧔ࡊ࡜ࡦࠢߩᝄേሶ㧕ࠍ⠨߃ࠆߣ㧘ߘߩฎౖ⊛ㆇേᣇ⒟ᑼߪ

τ = −∞+∞

τ = −∞

ߢ⴫ߒ㧘ᑼ(15)ߦઍ౉ߒߡ๟࿁Ⓧಽࠍ⸃ߊߣ
+∞nh = 2πi ∑ P ( n,τ )Q ( n,−τ )τ = −∞

ν ( n,−τ )ν ( n,1)(36)

κ
x = &x& + ( 2 πν ) 2 x = 0 .
mฎ޿㊂ሶ⺰ߦ߅ߌࠆ &x& ߪ㧘ᑼ(19)߆ࠄ
&x& +߇ᓧࠄࠇ߹ߔ㧚ߎߎߢฝㄝߩಽᢙߦᑼ(22)ߩᏀ஥
ߩ㑐ଥߣ୚㖸㑐ଥν(n,τ) = τν(n,1)ࠍ↪޿㧘ߐࠄߦ
Q(n,-τ)ߦ߅޿ߡᑼ(23)ߩᏀ஥ߩ㑐ଥࠍㆡ↪ߔࠆߣ
ᑼ(36)ߪ

વᾲ 2011 ᐕ 7 ᦬

2&x& = −{2 πν ( n,τ )}2 X ( n,τ )e 2 πiν ( n ,τ ) t- 40 -(42)(43)

J. HTSJ, Vol. 50, No. 212

ࠛ࠺ࡘࠤ࡯࡚ࠪࡦ㧽ߣᦠ߆ࠇ㧘ᑼ(42)ߦᑼ(43)ߣᑼ(19)ࠍઍ౉ߒߡ

{}( 2 π ) 2 ν 2 − ν ( n,τ ) 2 X ( n,τ ) = 0

߇ᓧࠄࠇ߹ߔ㧚ᑼ(51)ߪਈ߃ࠄࠇߚ n ߦ߅޿ߡ xߩᝄ᏷ࠍቯ߼ࠆ㑐ଥᑼߢ޽ࠅ㧘㊂ሶ᧦ઙ㧔ᑼ(15)㧕
ࠍㆫ⒖ᝄ᏷ߦࠃߞߡኻᔕේℂ⊛ߦᦠ߈⋥ߒߚᒻߣ⸒߃߹ߔ㧚ታ㓙㧘ฎౖ⊛ߥᱜᒏᝄേߩᑼ q = Xe2πiνt+ X *e-2πiνt㧔X: ᝄ᏷㧕ߦᑼ(15)ࠍㆡ↪ߒߡዉ߆ࠇࠆ
㊂ሶ⊛ߦ⸵ߐࠇߚᝄ᏷ߩੑਸ਼㧔|X|2㧕߇㧘ᑼ(51)ߩ
ฝㄝߣ৻⥌ߔࠆߎߣ߇⏕߆߼ࠄࠇ߹ߔ㧚(44)߇ዉ߆ࠇ߹ߔ߇㧘ν = ν(n, 1)߅ࠃ߮ᑼ(22)ߩᏀ஥ߩ
㑐ଥߦࠃߞߡτ = ±1 ߩ႐วߩߺࡈ࡯࡝ࠛଥᢙ
X(n,τ)߇㕖࠯ࡠߦߥࠆ੐߇ࠊ߆ࠅ߹ߔ㧔‛ℂ⊛ߦߪනᝄേߦߪ㜞⺞ᚑಽ߇ሽ࿷ߒߥ޿ߎߣߦኻᔕ㧕㧚 ৻ᣇ㧘ᣂߒ޿ᒻᑼߦߟ޿ߡߎࠇࠍኻᔕ⊛ߦ⠨߃ࠆߣ㧘 &x& ߪᑼ(20)߆ࠄ

− {2 πν ( n; n − τ )} X ( n; n − τ )e 2 πiν ( n ;n −τ ) t2ᣂℂ⺰ߦ߅ߌࠆࠛࡀ࡞ࠡ࡯
ࡂࠗ࠯ࡦࡌ࡞ࠢߪ㧘ᓐ߇ዉ಴ߒߚᣂߒ޿ᒻᑼߦ
߅޿ߡቯᏱ⁁ᘒߩࠛࡀ࡞ࠡ࡯଻ሽೣ߇ᚑࠅ┙ߞߡ޿ࠆߎߣ⏕߆߼ࠆᔅⷐ߇޽ࠅ߹ߒߚ㧚⺞๺ᝄേሶߩࠛࡀ࡞ࠡ࡯E ߪฎౖ⊛⴫⃻ߢ

(45)ߣᓧࠄࠇ㧘ߎࠇࠍᑼ(42)ߦઍ౉ߒߡ

{}( 2 π ) 2 ν 2 − ν ( n; n − τ ) 2 X ( n; n − τ ) = 0

(46)

1
1
2
(53)
mx& 2 + m (2 πν ) x 2
22ߣᦠ߆ࠇ߹ߔ㧚߹ߕ㧘x& 2 ߩ n → n - τ ߦߟ޿ߡߩㆫ
⒖ᚑಽߦߟ޿ߡߪ㧘ᑼ(20)߆ࠄ᳞߼ߚ x& ࠍᑼ(32)E=߇ᓧࠄࠇ߹ߔ߇㧘ν = ν ( n; n − 1) = −ν ( n − 1; n ) = −ν ( n; n + 1)

(47)߇ᚑ┙ߔࠆߩߢ㧘ฎ޿⠨߃ߩ႐วߣห᭽㧘τ = ±1
ߩ႐วߦߩߺㆫ⒖ᝄ᏷ X(n; n - τ)߇㕖࠯ࡠߦߥࠆ
੐߇ࠊ߆ࠅ߹ߔ
㧔⺞๺ᝄേሶߦ߅ߌࠆㆫ⒖ㆬᛯೣ㧕㧚
X(n; n ± 1)ߩౕ૕⊛ߥ୯ߪએਅߩࠃ߁ߦ᳞߼ࠄࠇ
߹ߔ㧚߹ߕ㧘ᑼ(41)ߩ⚖ᢙߦ߅޿ߡτ = +1 ߣ-1 ߩ㗄ߩߺࠍࡇ࠶ࠢࠕ࠶ࡊߔࠆߣ

{2

h = 8 π 2 mν X ( n + 1; n ) − X ( n; n − 1)

2

}

(48)

⎧ +∞
− 2 π 2 m ⎨ ∑ν ( n; n − τ ′)ν ( n − τ ′; n − τ )(54)⎩τ ′= −∞⋅ X ( n; n − τ ′) X ( n − τ ′; n − τ )e 2 πiν ( n ;n −τ ) t}߇ᓧࠄࠇ߹ߔ㧚ߎࠇߦㆫ⒖ᝄ᏷ X(n; n - τ)߇τ = ±1
ߩ႐วߩߺ㕖࠯ࡠߣߥࠆ੐㧔4.2 ▵㧕㧘ᑼ(23)ߩฝ
஥ߩ㑐ଥ㧘߅ࠃ߮ᑼ(47)ࠍㆡ↪ߔࠆߣ㧘 x& 2 ߪ{߇ᓧࠄࠇ߹ߔ㧚ฝㄝߩ{ }᜝ᒐౝߪᑼ(28)ߩᓸಽߦ
㑐ߔࠆኻᔕ㑐ଥߦ߅޿ߡτ = 1 ߣߒߚ߽ߩߢ޽ࠆ
߆ࠄ㧘ߎߩㇱಽߦߟ޿ߡߩኻᔕ㑐ଥߪ2

4 π 2ν 2 X ( n; n − 1) + X ( n; n + 1)

2

} for τ = 0− 4 π 2ν 2 X ( n; n m 1) X ( n m 1; n m 2 )e 2 πiν ( n ;n m 2 ) t
for τ = ± 2 .

∂2
2
2
X ( n,1) ↔ X ( n + 1; n ) − X ( n; n − 1) (49)∂nߢ޽ࠅ㧘ߐࠄߦ㧘X(n, 1) ↔ X(n; n - 1)ߩኻᔕ㑐ଥ
㧔3.1 ▵㧕߇ᚑࠅ┙ߞߡ޿ࠆߩߢ㧘ᑼ(48)ߪ
h
∂
2
(50)
=
X ( n; n − 1)
8 π 2 m ν ∂nߣᦠ߈⋥ߐࠇ߹ߔ㧚ߎߩਔㄝࠍ n ߢⓍಽߔࠆߣ㧘
ᑼ(46)ߦ߅ߌࠆτ = +1 ߩ႐วߩㆫ⒖ᝄ᏷ߩੑਸ਼
1
2(51)
X ( n; n − 1) = 2nh
8 π mν
߇᳞߹ࠅ߹ߔ㧚৻ᣇ㧘ᑼ(51)Ꮐㄝߩ|X|2 ౝߢⶄ⚛౒ᓎࠍߣߞߡ࠮ࡒࠦࡠࡦߢ඙ಾࠄࠇߚⷐ⚛ߩ㗅⇟ࠍ౉ࠇᦧ߃㧘n → n + 1 ߩᄌᢙᄌ឵ࠍⴕ߁ߣ㧘ᑼ(46)ߦ߅ߌࠆτ = -1 ߩ႐วߩㆫ⒖ᝄ᏷ߩੑਸ਼
1
2(52)(n + 1)hX ( n; n + 1) = 2
8 π mν

વᾲ 2011 ᐕ 7 ᦬ߩੑਸ਼ߩṶ▚ⷙೣߦઍ౉ߒߡ

(55)
2

x ߦߟ޿ߡ߽ᑼ(32)ࠍ↪޿ߡห᭽ߥṶ▚ࠍⴕ޿㧘X ( n; n − 1) + X ( n; n + 1)

2

2

X ( n; n m 1) X ( n m 1; n m 2)e2 πi ν ( n ; n m 2 ) t

for τ = 0(56)

for τ = ±2
߇ዉ߆ࠇ߹ߔ㧚ᦨᓟߦᑼ(55)ߣᑼ(56)ࠍᑼ(53)ߦઍ
౉ߔࠆߣ㧘ࠛࡀ࡞ࠡ࡯E ࠍ⴫ߔㆫ⒖ᚑಽ

{2

4π 2ν 2 m X ( n; n − 1) + X ( n; n + 1)
0

2

} for τ = 0
for τ ≠ 0(57)

߇ᓧࠄࠇ߹ߔ㧚ߎߎߢ㊀ⷐߥߩߪ㧘ࠛࡀ࡞ࠡ࡯ߩ
ㆫ⒖ᚑಽߣߒߡߪτ = 0 ߩ߽ߩ㧘ߔߥࠊߜ n → n ߦኻᔕߒߡᤨ㑆 t ࠍ฽߹ߥ޿߽ߩߛߌ߇㕖࠯ࡠߩ୯- 41 -J. HTSJ, Vol. 50, No. 212

ࠛ࠺ࡘࠤ࡯࡚ࠪࡦ㧽ࠍᜬߜ㧘ઁߩ߽ߩߪోߡ࠯ࡠߦߥࠆߣ޿߁ὐߢߔ㧚ߎࠇߪቯᏱ⁁ᘒߦ߅ߌࠆࠛࡀ࡞ࠡ࡯଻ሽೣ߇ᚑࠅ
┙ߞߡ޿ࠆߎߣࠍ␜ߔ߽ߩߢ޽ࠅ㧘ߎࠇߦࠃࠅࡂ
ࠗ࠯ࡦࡌ࡞ࠢߪᢙቇ⊛ߦ⥄Ꮖή⍦⋫ߢ㚂የ৻⽾ߒߚ૕♽ߩ᭴▽ߦᚑഞߒߚߎߣࠍ⏕ାߒߚߩߢߒߚ㧚㧔‫ᦨޟ‬ೋߩ৻㗄ߢࠛࡀ࡞ࠡ࡯ೣ߇ᧄᒰߦ⏕⹺ߐࠇߚߣ߈ߦ㧘
⑳ߪ޽ࠆ⥝ᅗ⁁ᘒߦ߅ߜ޿ߞߡߒ߹޿㧘ߘࠇ߆ࠄవߩ⸘▚ߢߪ૗ᐲ߽૗ᐲ߽⸘▚ߩࡒࠬࠍ➅ࠅ㄰ߒߡߒ߹ߞߚ߶ߤߛߞߚ㧚⋅⋅⋅ ⑳ߪᔃᐩ߆ࠄ
㛳ᗭߒߚ㧚⑳ߪේሶ⃻⽎ߩ⴫㕙ࠍ⓭߈ᛮߌߡ㧘ߘߩ⢛ᓟߦᷓߊᮮߚࠊࠆ⁛․ߩౝㇱ⊛ߥ⟤ߒߐࠍ߽ߞߚ࿯บࠍߩߙ߈ߺߚࠃ߁ߥᗵߓ߇ߒߚ㧚‫[ޠ‬7]㧕
ߥ߅㧘ᑼ(57)ߦᑼ(51)ߣᑼ(52)ࠍઍ౉ߔࠆߣ㊂ሶജቇߦ߅ߌࠆ n ⇟⋡ߩ⁁ᘒߦ޽ࠆࡊ࡜ࡦࠢᝄേሶ
ߩࠛࡀ࡞ࠡ࡯

1⎞⎛
E n = ⎜ n + ⎟ hν
2⎠
⎝

(58)

(59)

n′

ߣ⴫ߐࠇࠆߎߣ߆ࠄ㘃ផߐࠇ߹ߔ㧔n′: 1, 2, ⋅⋅⋅㧕㧚ᑼ(59)ߦ߅ߌࠆ n′ߩᓎഀߪ㧘ᑼ (33)ߦ߅ߌࠆਛ㑆⁁ᘒ n - τ ′ߩߘࠇߦ㘃ૃߔࠆ߽ߩߣߥߞߡ޿߹ߔ㧚೨▵߹ߢߪኻᔕේℂࠍ↪޿ࠆᢥ⣂ߢฎౖ⊛ឬ௝ࠍᗧ⼂ߒߚ“n → n - τ”ߣ޿߁⴫⸥ࠍ↪޿ߡ߈߹ߒ
ߚ߇㧘੹ᓟߪ n - τ ≡ m ߣߒߡㆫ⒖ࠍ“n → m”ߣ⴫ߔߎߣߦߒ߹ߔ㧚ࡂࠗ࠯ࡦࡌ࡞ࠢߩ㊂ሶജቇߢߪ
ᐳᮡ x ߿ㆇേ㊂ p ߥߤߩ‛ℂ㊂ߪߘࠇߙࠇߩㆫ⒖ᚑಽߩ㓸วߣ⠨߃ࠄࠇ߹ߔ㧚੹㧘޽ࠆ‛ℂ㊂ A ࠍ⠨߃ࠆߣ㧘n → m ߩㆫ⒖ᚑಽ An,m ≡ an,mexp(2πiνn,mt)ߩ㓸วߪ㧘ฦᚑಽࠍ n ⴕ m ೉ߩⷐ⚛ߣߔࠆᄢ߈ߐ
ή㒢ᄢߩⴕ೉

⎡ a1,1 exp (2 πiν 1,1t ) K a1,m exp (2 πiν 1,m t ) ⎤⎢⎥ (60)A=⎢
M
O
M⎥⎢⎣ an ,1 exp (2 πiν n ,1t ) L an ,m exp (2 πiν n ,m t )⎥⎦ߦࠃߞߡ⴫ߔߎߣ߇ߢ߈߹ߔ㧚ߎߎߢฦ‫ⴕߩޘ‬೉ⷐ⚛ߪνn,m ߩㆫ⒖ᝄേᢙߢᝄേߒߡ߅ࠅ㧔n = m ߩߣ߈ߩߺᝄേᢙ 0㧕㧘ߘࠇࠄߪ࡝࠶࠷ߩ⚿วೣ㧔ᑼ(31)㧕ࠍḩߚߒ߹ߔ㧚A ߇᷹ⷰߦ߆߆ࠆࠃ߁ߥታ
࿷⊛‛ℂ㊂ࠍ⴫ߔⴕ೉ߥࠄ߫ߘߩኻⷺᚑಽ㧔An,n㧕ߪታᢙߢ޽ࠅ㧘ⴕ೉ⷐ⚛ An,m ߪᰴߩ㑐ଥ߇⋥ߜߦዉ߆ࠇ㧘ߎߎߦߪએ೨ߩታ㛎⚿ᨐ߆ࠄሽ࿷߇ᦼᓙߐࠇߡ޿ߚ࠯ࡠὐᝄേࠛࡀ࡞ࠡ࡯
㧔1/2hν㧕߇⚵ߺㄟ߹ࠇߡ޿ࠆߎߣ߇ࠊ߆ࠅ߹ߔ㧚ࡂࠗ࠯ࡦࡌ࡞ࠢߪߎߩ⊒⷗ࠍ‫ޟ‬ㆇേቇ⊛߆ߟജቇ⊛ߥ㑐ଥߩ㊂ሶ⺰⊛ౣ⸃㉼‫⴫߁޿ߣޠ‬㗴ߩ⺰ᢥ[9]ߦ߹ߣ߼㧔1925 ᐕ 7 ᦬ᛩⓂ㧕㧘ේⓂࠍࠥ࠶࠴ࡦ
ࠥࡦᄢቇߢࡂࠗ࠯ࡦࡌ࡞ࠢߩ਄มߢ޽ߞߚࡏ࡞ࡦ
㧔M. Born, 1882 − 1970, 1954 ᐕࡁ࡯ࡌ࡞‛ℂቇ
⾨㧕ߦ⸤ߒߡ࿖ᄖߩᣏ߳ߣ಴߆ߌ߹ߔ[3]㧚

An ,m = Am,n

∗

(61)

ࠍḩ⿷ߔࠆߎߣ㧘ߔߥࠊߜࠛ࡞ࡒ࡯࠻ⴕ೉ߢ޽ࠆ
ᔅⷐ߇޽ࠅ߹ߔ[10]㧚ᑼ(61)ߪࡂࠗ࠯ࡦࡌ࡞ࠢ߇↪
޿ߚⶄ⚛౒ᓎ㑐ଥ㧔ᑼ(23)ߩฝ஥ߩ㑐ଥ㧕ߩⴕ೉ߣ޿߃߹ߔ㧚
੤឵㑐ଥ߅ࠃ߮‛ℂ㊂ߩ⴫⃻ࡂࠗ࠯ࡦࡌ࡞ࠢ߇ዉ޿ߚᒻᑼߪࡏ࡞ࡦߣ࡛࡞࠳
ࡦߦࠃߞߡ⷗ㅢߒߩ⦟޿ᒻᑼߦ઀਄ߍࠄࠇߡࠁ߈
߹ߔ㧚ᑼ(40)ߩ㊂ሶ᧦ઙߪᑼ(59)ߩࠕ࠽ࡠࠫ࡯߆ࠄ
૏⟎߅ࠃ߮ㆇേ㊂ࠍ⴫ߔⴕ೉ X ߣ P ߦߟ޿ߡ㧡ⴕ೉ജቇ߳ߩ⒖ⴕ
ⴕ೉⴫⸥ߩዉ౉
ࡂࠗ࠯ࡦࡌ࡞ࠢߪⴕ೉ജቇߩഃᆎ⠪ߣ⠨߃ࠄࠇߡ޿߹ߔ߇㧘ᓐߪ㊂ሶജቇߩ⊒⷗ᤨὐߢߪⴕ೉ࠍ
⍮ࠅ߹ߖࠎߢߒߚ
㧔‫ⴕ߇ࠢ࡞ࡌࡦ࠯ࠗࡂ޿⧯ޟ‬೉ࠍ⍮ࠄߕߦⴕ೉ജቇࠍ⊒⷗ߒߚߎߣߪ㕙⊕޿㧚ᣂߒ޿ℂ⺰ࠍ⊒⷗ߔࠆߩߦ㧘ᔅߕߒ߽޽ࠄ߆ߓ߼ߘࠇ
ߦ߰ߐࠊߒ޿ᢙቇࠍ⍮ߞߡ޿ߥߊߣ߽ࠃ޿଀ߢ޽
ࠆ‫[ޠ‬3]㧕㧚ࡂࠗ࠯ࡦࡌ࡞ࠢ߇⊒⷗ߒߚᒻᑼࠍⴕ೉
⴫⸥ߦ઀਄ߍߚߩߪ㧘ߎߩℂ⺰߇ⴕ೉ߦࠃࠅ◲ଢߦ⴫⸥ߢ߈ࠆߎߣߦ᳇ߠ޿ߚࡏ࡞ࡦ߅ࠃ߮ᓐߣද
หߒߚ࡛࡞࠳ࡦ㧔P. Jordan, 1902 − 1980㧕ߦࠃࠆ߽ߩߢߔ㧚ࡂࠗ࠯ࡦࡌ࡞ࠢߩᒻᑼ߇ⴕ೉ߢ⸥ㅀߐࠇ
߁ࠆߎߣߪ㧘ⓍṶ▚㧔ᑼ(33)㧕ߦ߅ߌࠆ㕖น឵ᕈ
㧔YZ ≠ ZY㧕㧘޽ࠆ޿ߪᱜᣇⴕ೉ߩⓍ X = YZ ࠍ⠨߃ߚߣ߈ߦⴕ೉ X ߩ i ⴕ j ೉ߩⷐ⚛ xij ߇વᾲ 2011 ᐕ 7 ᦬

xij = ∑ yin ′ z n ′j

(PX )n ,n − (XP )n ,n = −ih,

h≡

h
2π

(62)

ߣ⴫ߐࠇ㧔ᷝ߃ሼߩ n, n ߪⓍߩኻⷺᚑಽࠍ⴫ߔ㧕㧘ߐࠄߦߘߩ㕖ኻⷺⷐ⚛߇ 0 ߣߥࠆߣ޿߁઒ቯࠍઃ
ߔߎߣߢ㧘ᑼ(40)ߦઍࠊࠆᣂߒ޿㊂ሶ᧦ઙߣߒߡ

PX − XP = −ihI(63)ߣ޿߁◲ẖߥᒻ߇ዉ߆ࠇ߹ߔ[6]㧔I ߪන૏ⴕ೉㧕㧚

- 42 -J. HTSJ, Vol. 50, No. 212

ࠛ࠺ࡘࠤ࡯࡚ࠪࡦ㧽
ᑼ(63)ߪᱜḰ੤឵㑐ଥߣ⒓ߐࠇ߹ߔ㧚X ߅ࠃ߮ P ߪએਅߩࠃ߁ߦዉ߆ࠇ߹ߔ㧚ᑼ(46)߆ࠄτ = ±1 ߩߣ߈ߩߺ X(n; n - τ)߇㕖࠯ࡠߦߥࠆߎ
ߣ㧘߅ࠃ߮ᑼ(51)ߣᑼ(52)߆ࠄ㧘⺞๺ᝄേሶߩ૏⟎ⴕ೉ X ߩⴕ೉ⷐ⚛ Xn,m ߪ

X n , n −1 =nhe 2 πiνt ,8 π 2 mνX n −1, n =nhe − 2 πiνt
8 π 2 mν∂A i
∂Ai
= (PA − AP ) ,= − (XA − AX ) (69)
∂X h
∂P
h
߇ᚑ┙ߔࠆ੐߇␜ߐࠇ߹ߔ㧔⸽᣿ߪᢥ₂ 6㧕㧚ᑼ(69)
ߦ߅޿ߡ A = H ߣߒߚ߽ߩࠍᑼ(68)ߦઍ౉ߔࠆߣ

dX i
dP i
= (HX − XH ) ,= (HP − PH )
dt h
dt h

(64)

߇ᓧࠄࠇ㧘ߎࠇߪ৻⥸ߩⴕ೉ A(X, P)ߦ᜛ᒛߒߡ

- ihߢ޽ࠅ㧘n ≠ ±1 એᄖߩⷐ⚛ߪ 0 ߣߥࠅ߹ߔ㧔╙ੑᑼߩᜰᢙㇱߩ⽶ภߪᑼ(22)ߩฝ஥ߩ㑐ଥߦࠃࠆ㧕㧚ߐࠄߦ p = mx& = 2 πi mν x ࠍ↪޿ࠆߣ㧘⺞๺ᝄേሶnhm ν 2 πiνt
e
,
2

(65)

nhm ν − 2 πiνt
Pn −1, n = −i
e
2

X=

⎡0
⎢
hm ν ⎢ 1
P=i
2 ⎢0
⎢
⎣⎢ M

1
0

0
2

2

0
3

− 1
0

0

2

− 2
0
3

L⎤
⎥
⎥
3⎥
⎥
O ⎥⎦
L ⎤
⎥
⎥
− 3⎥
⎥
O ⎦⎥

& = 0 ߇ᓧࠄࠇ㧘ߎࠇߪࡂࠗ࠯ࡦࡌ࡞ࠢ߇ዉ޿ߚ
Hࠛࡀ࡞ࠡ࡯଻ሽೣ㧔ᑼ(57)㧕ࠍ⴫⃻ߔࠆ߽ߩߣߥ
ࠅ߹ߔ㧚ࡏ࡞ࡦߣ࡛࡞࠳ࡦߪⴕ೉⴫⸥ߣᱜḰᒻᑼߦࠃࠆ
Ṷ▚ߩ⊒⷗㧔ᦨ߽㊀ⷐߥ߽ߩߪᑼ(63)ߩ⊒⷗㧕ࠍ⺰ᢥߦ߹ߣ߼㧘ࡂࠗ࠯ࡦࡌ࡞ࠢߩ⺰ᢥᛩⓂ߆ࠄ௖
߆ 2 ߆᦬ᓟߩ 1925 ᐕ 9 ᦬㧘‫ޟ‬㊂ሶജቇߦߟ޿ߡ‫ޠ‬ߣ޿߁⴫㗴ߢᛩⓂߒ߹ߔ[11]㧚ߐࠄߦ 1926 ᐕೋ㗡ߦߪࡏ࡞ࡦ࡮ࡂࠗ࠯ࡦࡌ࡞ࠢ࡮࡛࡞࠳ࡦߩ౒⪺ߦ
ࠃࠆ 59 ࡍ࡯ࠫߦ෸߱㓸ᄢᚑߣ߽޿߃ࠆ⺰ᢥ߇಴ߐࠇ㧔ᛩⓂߪ 1925 ᐕ 11 ᦬㧕[12]㧘ࡂࠗ࠯ࡦࡌ
࡞ࠢ߇⊒⷗ߒߚ㊂ሶജቇߪ߭ߣ߹ߕߩቢᚑࠍߺࠆ
ߎߣߦߥࠅ߹ߔ㧚

(66)

(67)

ߣ⴫ߐࠇ߹ߔ㧚ᑼ(66)ߣᑼ(67)ߪᱜḰ੤឵㑐ଥ㧔ᑼ(63)㧕ࠍḩ⿷ߔࠆ߽ߩߣߥߞߡ޿߹ߔ㧚㧢ࡂࠗ࠯ࡦࡌ࡞ࠢߩ↢ᶦ↢޿┙ߜ㧦ᄢቇ౉ቇ߹ߢ
ࡂࠗ࠯ࡦࡌ࡞ࠢߪ 1901 ᐕ࠼ࠗ࠷ߩࡧࡘ࡞࠷ࡉ
࡞ࠣߢ↢߹ࠇ㧘ࠡ࡝ࠪࡖ⺆ࠍኾ㐷ߣߔࠆῳⷫߩࡒ
ࡘࡦࡋࡦᄢቇᢎ᝼ዞછߦ઻޿ࡒࡘࡦࡋࡦߦ⒖ࠅ߹ߔ㧚චઍਛ㗃ߪ╙৻ᰴ਎⇇ᄢᚢ㧔1914 − 1918㧕ߦ
㊀ߥࠅ㧘ᄢᚢᧃᦼߦߪ㘫㙈߆ࠄㅏࠇࠆߚ߼㜞࿾ࡃࠗࠛ࡞ࡦߩㄘ႐ߢ૞↵ߣߒߡ௛߈㧘⚳ᚢᓟߪౝᚢ
⁁ᘒߩᷙੂߦ㒱ߞߚࡒࡘࡦࡋࡦߢ⍴ᦼ㑆᡽ᐭァߦදജߒߡァ㓌ߩ༡⥢ߦൕോߔࠆ↢ᵴࠍㅍࠅ߹ߔ㧚ࡂࠗ࠯ࡦࡌ࡞ࠢߪᒰᤨߩ㔓࿐᳇ࠍ‫ޟ‬ᐔ๺ߥᤨઍߩ
⧯⠪ߚߜࠍߣࠅ߹޿ߡ޿ߚኅᐸ߿ቇᩞߦࠃࠆ଻⼔ߪ㧘߽ߪ߿ߎߩᷙੂߩᤨઍߦߪ߶ߣࠎߤᄬࠊࠇߡ㧘ࡂࠗ࠯ࡦࡌ࡞ࠢߩㆇേᣇ⒟ᑼ
ㆇേᣇ⒟ᑼߦߟ޿ߡߪࡂࡒ࡞࠻ࡦߩᱜḰᣇ⒟ᑼ

∂Hd X ∂H d P
=−
=
,
∂X
∂P
dt
dt

(68)

߇㊂ሶജቇߢ߽ߘߩ߹߹ᒰߡߪ߹ࠆߣ޿߁઒ቯ߆
ࠄ಴⊒ߒ߹ߔ㧚H ߪฎౖജቇߩ႐วߣห᭽ࡂࡒ࡞
࠻࠾ࠕࡦߣ๭߮߹ߔ㧚ߘߒߡ㧘߽ߒᑼ(63)߇ᚑࠅ
┙ߞߡ޿ࠇ߫㧘X ߣ P ߩ㑐ᢙߢ޽ࠆછᗧߩⴕ೉
A(X, P)ߦߟ޿ߡ

વᾲ 2011 ᐕ 7 ᦬(71)ኻᔕࠍߥߔ߽ߩߢߔ㧚ᑼ(71)ߢ A = H ߣ߅ߊߣߣ᳞߼ࠄࠇ߹ߔ㧔ห᭽ߦ n ≠ ±1 એᄖߩⷐ⚛ߪ 0㧕㧚ᑼ(64)ߣᑼ(65)߆ࠄ㧘⺞๺ᝄേሶߩ t = 0 ߦ߅ߌࠆ૏⟎ⴕ೉ X ߅ࠃ߮ㆇേ㊂ⴕ೉ P ߪ

⎡0
⎢
h
⎢ 1
2
8π mν ⎢ 0
⎢
⎢⎣ M

dA
= HA − AH
dtߣᦠ߆ࠇ߹ߔ[3,6]㧚ᑼ(71)ߪ㊂ሶജቇߦ߅ߌࠆㆇ
േᣇ⒟ᑼߢ޽ࠅ㧘ࡂࠗ࠯ࡦࡌ࡞ࠢߩㆇേᣇ⒟ᑼߣ
๭߫ࠇࠆ߽ߩߢߔ㧚ߎߩᑼߪജቇ㊂ߦߟ޿ߡ 1 㓏ߩᤨ㑆ᓸಽߩᒻߢᦠ߆ࠇߡ߅ࠅ㧘⠉ᐕߦዉ߆ࠇߚ
ࠪࡘ࡟࠺ࠖࡦࠟ࡯ߩㆇേᣇ⒟ᑼ ihϕ& = Hϕ [5]ߣߩㆇേ㊂ⴕ೉ P ߩⴕ೉ⷐ⚛ Pn,m ߪ

Pn , n −1 = i(70)- 43 -J. HTSJ, Vol. 50, No. 212

ࠛ࠺ࡘࠤ࡯࡚ࠪࡦ㧽ߘߩઍࠊࠅߦ⧯⠪ߚߜ߇޽ࠆ⒟ᐲ⥄↱ߦ⠨߃ࠆ௑ะ߇↢߹ࠇߡ޿ߚ㧚ᓐࠄߪߘࠇߦࠃߞߡ㧘߹ߛචಽߥၮ␆߇ߢ߈ߡ޿ߥ޿႐วߢ߽㧘⥄ಽ⥄りߩ್ᢿࠍା↪ߔࠆࠃ߁ߦߥߞߡ޿ߚߩߢ޽ࠆ‫[ޠ‬7]ߣ⸥
ߒߡ޿߹ߔ㧚
ߎߩࠃ߁ߥᔅߕߒ߽ቇᩞߦᱜᏱߦㅢቇߢ߈ߥ޿
⁁ᴫߦ޽ߞߡᢙቇߦߪ․ߦ㑐ᔃࠍᛴ߈ീᒝࠍ㊀ߨߚߣવ߃ࠄࠇߡ޿߹ߔ㧚ߎߩᤨᦼߦߪࡊ࡜࠻ࡦߩ
‫ߣޠࠬࠝࠗࡑࠖ࠹ޟ‬಴ળ޿㧘‛⾰ߩ৻⇟ዊߐߥ᭴ᚑⷐ⚛ࠍ┙૕ᐞ૗ቇߩᱜᄙ㕙૕ߦࠃࠅ⺑᣿ߒߚ▎ᚲߦਇḩࠍᛴ߈߹ߔ㧚㧔‫ߦ߁ࠃߩࡦ࠻࡜ࡊޟ‬㕖Ᏹߦᛕ್⊛ߦ㧘߆ߟ㍈ߊ߽ߩࠍ⠨߃ࠆߎߣߩߢ߈ࠆື
ቇ⠪߇㧘ߤ߁ߒߡߘߩࠃ߁ߥ߻ߜ߾ߥᕁᑯߦ㒱ߞߡߒ߹ߞߚߩ߆ߣ㧘⑳ߪ߭ߤߊℂ⸃ߦ⧰ߒࠎߛ㧚⋅⋅⋅ ߅ߘࠄߊߎߩ⺒ᦠߦࠃࠆᦨᄢߩ෼ⓠߪ㧘߽ߒ߽‛⾰਎⇇ࠍℂ⸃ߒࠃ߁ߣ޿߁ߥࠄ㧘ߘߩ৻⇟ዊߐߥㇱಽߦߟ޿ߡ૗߆ࠍ⍮ࠄߥߊߡߪߥࠄߥ޿ߣ޿
߁⏕ାߢ޽ߞߚ㧚‫[ޠ‬7]㧕
ࡂࠗ࠯ࡦࡌ࡞ࠢߪ㜞ᩞතᬺ⹜㛎ࠍㅢㆊᓟ㧘ᢙቇ⠪ࡋ࡞ࡑࡦ࡮ࡢࠗ࡞߇⋧ኻᕈℂ⺰ߦߟ޿ߡ⸥ߒߚ
‫ޟ‬ⓨ㑆࡮ᤨ㑆࡮‛⾰‫ߦᧄ߁޿ߣޠ‬ᄞਛߦߥࠅ㧘ࡒࡘࡦࡋࡦᄢቇߦ߅޿ߡᢙቇࠍീᒝߔࠆߣ޿߁᳿ᔃ
ࠍ৻ጀᒝ޿߽ߩߦߒߡࠁ߈߹ߔ㧚⎇ⓥߩ㐿ᆎ㨪ࡏ࡯ࠕߣߩ಴ળ޿
ᄢቇ౉ቇߣหᤨߦࡂࠗ࠯ࡦࡌ࡞ࠢߪῳߩ⚫੺ߦࠃࠅᢙቇ⠪࡝ࡦ࠺ࡑࡦߩᢎ᝼ቶࠍ⸰ߨ㧘࠮ࡒ࠽࡯߳ߩෳട⸵นࠍ⺧޿߹ߔ߇ᢿࠄࠇߡߒ߹޿߹ߔ㧚⪭⢙ߒߚᓐߪౣᐲῳߣ⋧⺣ߒ㧘੹ᐲߪℂ⺰‛ℂቇಽ㊁ߩ⪺ฬߥᢎ᝼ߢ޽ߞߚ࠱ࡦࡑ࡯ࡈࠚ࡞࠻㧔A.
Sommerfeld, 1868 − 1951㧕ࠍ⸰ߨ߹ߔ㧚ᓐߪࡂࠗ࠯
ࡦࡌ࡞ࠢࠍᥦ߆ߊㄫ߃㧘⋧ኻᕈℂ⺰ߩ໧㗴㗔ၞߦᗖ߆ࠇࠆࡂࠗ࠯ࡦࡌ࡞ࠢߦഥ⸒ࠍਈ߃߹ߔ㧚‫৻ޟ‬⇟㔍ߒ޿ᚲ߆ࠄ߿ߞߚ߆ࠄߣ޿ߞߡ㧘߽ߞߣ߽߿
ߐߒ޿ߎߣ߇߭ߣࠅߢߦℂ⸃ߢ߈ࠆࠃ߁ߦߥࠆߣ
ᦼᓙߔࠆߎߣߪߢ߈߹ߖࠎࠃ㧚⋅⋅⋅ ߘߎ߳ߩ㆏ߪ޽
ߥߚ߇੹⠨߃ߡ޿ࠆࠃࠅ߽ߕߞߣ㆙޿ߩߢߔ㧚޽ߥߚߪવ⛔⊛ߥ‛ℂቇߩ㗔ၞߩਛߩ㧘߽ߞߣᚻ㗃ߥ㧘⚦߆޿઀੐߆ࠄᆎ߼ߥߊߡߪ޿ߌ߹ߖࠎ㧚‫[ޠ‬7]
ߎߩࠃ߁ߦߒߡ㧘1920 ᐕ⑺߆ࠄ࠱ࡦࡑ࡯ࡈࠚ࡞
࠻ߩ⎇ⓥቶߦ߅޿ߡℂ⺰‛ℂߩ⎇ⓥࠍ㐿ᆎߒ߹ߔ㧚⎇ⓥቶߩవヘߦߪࡄ࠙࡝㧔W. Pauli, 1900 − 1958,
1945 ᐕࡁ࡯ࡌ࡞‛ℂቇ⾨㧕[13]߇߅ࠅ㧘ᱦ߇ㄭߊ᳇቟޿⋧⺣⋧ᚻߛߞߚᓐ߆ࠄೝỗߣഥ⸒ࠍฃߌߚ

વᾲ 2011 ᐕ 7 ᦬- 44 -

ߎߣߪࡂࠗ࠯ࡦࡌ࡞ࠢߦߣߞߡᄢ߈ߥᐘㆇߢߒߚ㧚ࡄ࠙࡝ߪᒰೋ⋧ኻᕈℂ⺰ߦ⥝๧߇௑޿ߡ޿ߚࡂࠗ
࠯ࡦࡌ࡞ࠢߦኻߒ㧘ᧂ⸃᳿ߥ⻘߇ᄙߊᱷࠆේሶ⺰ߦะ߆߁ᣇ߇⾫᣿ߢ޽ࠆߎߣࠍ⺑߈߹ߔ㧚
1922 ᐕߩೋᄐ㧘ࡂࠗ࠯ࡦࡌ࡞ࠢߪࠥ࠶࠴ࡦࠥࡦ
ߢⴕࠊࠇࠆࡏ࡯ࠕߦࠃࠆㅪ⛯⻠⟵ߦ಴Ꮸߔࠆᯏળ
ࠍᓧ߹ߔ㧚ߘߒߡ޽ࠆ࿁ߩ⻠⟵ߩ⚳ࠊࠅ㗃ߦᕁ޿ಾߞߡᛕ್⊛ߥ⇼ᔨࠍㅀߴߚߣߎࠈ㧘ߘࠇ߇⊛⏕ߥ߽ߩߢ޽ߞߚߚ߼㧘ࡏ࡯ࠕߪࡂࠗ࠯ࡦࡌ࡞ࠢߣ
⸛⺰ࠍ⛯ߌࠆߚ߼⻠⟵ᓟߦᓐࠍᢔᱠߦ⺃޿߹ߔ㧚ߘߎߢࡂࠗ࠯ࡦࡌ࡞ࠢߪࡏ࡯ࠕߣේሶ⺰ߩ໧㗴ߦߟ޿ߡ⹤ߒว޿㧘޿ߕࠇࡏ࡯ࠕߩࠦࡍࡦࡂ࡯ࠥࡦ
ߩ⎇ⓥᚲࠍ⸰ࠇࠆࠃ߁⺃ࠊࠇ߹ߔ㧚㧔‫ߩߎޟ‬ᢔᱠߪ㧘
⑳ߩߘࠇએᓟߩቇ໧⊛↢㐳ߦߣߞߡᦨ߽ᒝ޿ᓇ㗀ࠍਈ߃ߚ߽ߩߢ޽ߞߚ㧚޽ࠆ޿ߪ㧘߻ߒࠈ⑳ߩቇ໧⊛↢㐳ߪߎߩᢔᱠࠍᯏળߦ㧘ࠃ߁߿ߊᆎ߹ߞߚ
ߩߛߣ޿ߞߚᣇ߇ߐࠄߦㆡಾ߆߽ߒࠇߥ޿㧚‫[ޠ‬7]㧕
ࡒࡘࡦࡋࡦ߳ߩᏫㅜ㧘࠱ࡦࡑ࡯ࡈࠚ࡞࠻ߪࡂࠗ
࠯ࡦࡌ࡞ࠢߦᵹ૕ജቇߩ⺖㗴ࠍਈ߃߹ߔ㧚ࡂࠗ࠯ࡦࡌ࡞ࠢߪ 1923 ᐕ㧘1 ቇᦼ㧔ඨᐕ㧕ߢੂᵹߦ߅ߌ
ࠆࡐࠕ࡙࠭ࠗᵹࠇߩ቟ቯᕈߦ㑐ߔࠆቇ૏⺰ᢥߩ઀੐ߣߘߩክᩏࠍᷣ߹ߖߡඳ჻ภࠍขᓧߒ߹ߔ[14]㧚
1923 ᐕߩᓟቇᦼࠍࠥ࠶࠴ࡦࠥࡦߢࡏ࡞ࡦߩഥᚻ
ߣߒߡ௛޿ߚᓟ㧘1924 ᐕᤐ㧘ࠦࡍࡦࡂ࡯ࠥࡦߩࡏ
࡯ࠕࠍ⸰ߨࠆߴߊ࠺ࡦࡑ࡯ࠢ߳ߣᣏ┙ߜ߹ߔ㧚
㊂ሶജቇߩ⊒⷗㨪ࡁ࡯ࡌ࡞⾨ߩฃ⾨
1924 ᐕ߆ࠄ 1925 ᐕ㧘ࡂࠗ࠯ࡦࡌ࡞ࠢߪࠦࡍࡦ
ࡂ࡯ࠥࡦߣࠥ࠶࠴ࡦࠥࡦࠍⴕ߈᧪ߒߥ߇ࠄ⎇ⓥߦขࠅ⚵ߺ߹ߔ㧚ߘߒߡ㧘ࠦࡍࡦࡂ࡯ࠥࡦߦ߅޿ߡ
ࠢ࡜ࡑ࡯ࠬ㧔H. A. Kramers, 1894 − 1952㧕ߣදหߒߡశᢔੂߩಽᢔ౏ᑼࠍዉ޿ߚ⚻㛎ࠍ߰߹߃ߡ㧘3▵ߦ⸥ㅀߒߚ㆏╭ߦᴪ޿㧘ኻᔕේℂࠍ♖ኒൻߒߡࠁߊ઀੐ߦขࠅដ߆ࠅ߹ߔ㧚1925 ᐕߩೋᄐ㧘ࡂࠗ࠯ࡦࡌ࡞ࠢߪ㊀޿ᨗ⨲ᾲ㧔⧎☳∝㧕ߦ߆߆ߞߚߎ
ߣ߆ࠄࡏ࡞ࡦߦ 2 ㅳ㑆ߩભᥜࠍ㗿޿಴ߡ㧘ർᶏߦ޽ࠆዊߐߥፉߦ≮㙃ߦ⸰ࠇ߹ߔ㧚ߘߩ㕒߆ߥኋደߢ໧㗴ߦขࠅ⚵ࠎߢ޿ߚ޽ࠆ᥅㧘ࡂࠗ࠯ࡦࡌ࡞ࠢ
ߪߟ޿ߦ㊂ሶജቇࠍ⊒⷗ߒ߹ߔ[7]㧚
1926 ᐕߩᤐ㧘ࡂࠗ࠯ࡦࡌ࡞ࠢߪࡌ࡞࡝ࡦᄢቇߩ
‛ℂቇ⺣⹤ળߢ㊂ሶജቇࠍ⊒⴫ߔࠆࠃ߁ⷐ⺧ߐࠇ㧘ߘߎߢࠕࠗࡦࠪࡘ࠲ࠗࡦߣ⸛⺰ߔࠆᯏળࠍᜬߜ߹
ߔ㧚ߒ߆ߒ㧘ේሶౝ㔚ሶߩ゠㆏ㆇേࠍߘࠇ߇᷹ⷰ
ߢ߈ߥ޿ߣ޿߁ℂ↱ߦࠃࠅᩮᧄ߆ࠄឃ㒰ߒߡߒ߹

J. HTSJ, Vol. 50, No. 212

ࠛ࠺ࡘࠤ࡯࡚ࠪࡦ㧽ߞߚࡂࠗ࠯ࡦࡌ࡞ࠢߩℂ⺰ߪࠕࠗࡦࠪࡘ࠲ࠗࡦߦ
ߪฃߌ౉ࠇ㔍޿߽ߩߢߒߚ㧚ߎߩὐߦ㑐ߔࠆືቇߩ⋧㆑ߪ㧘ߘߩᓟࠕࠗࡦࠪࡘ࠲ࠗࡦߣࠦࡍࡦࡂ࡯
ࠥࡦቇᵷߣߩ㑆ߦ㐳ߊᮮߚࠊࠆߎߣߦߥࠅ߹ߔ㧚ࡂࠗ࠯ࡦࡌ࡞ࠢߪ 1927 ᐕߦߪਇ⏕ቯᕈේℂࠍ⊒⴫ߒ㧘㊂ሶജቇߩ⊒ዷߦ㊀ⷐߥ⽸₂ࠍⴕ޿߹ߔ㧚
1927 ᐕߩᧃߦߪ࡜ࠗࡊ࠴ࡅᄢቇᢎ᝼ߦዞછߒ㧘
1932 ᐕߦߪ 31 ᱦߣ޿߁⧯ߐߢࡁ࡯ࡌ࡞‛ℂቇ⾨
ࠍฃ⾨ߒ߹ߔ㧚╙ੑᰴ਎⇇ᄢᚢᦼߣߘߩᓟ
1930 ᐕઍೋ㗡߆ࠄ࠼ࠗ࠷࿖ౝࠍⷒ޿ᆎ߼ߚ࠽
࠴ࠬบ㗡ߩᓇߪࡂࠗ࠯ࡦࡌ࡞ࠢߦᷓ޿ ᔨߣᄬᦸࠍਈ߃߹ߔ㧚࠽࠴ࠬߦࠃࠆᄢቇ߳ߩᐓᷤ߇ᒝ߹ࠆ
ਛ㧘ࡂࠗ࠯ࡦࡌ࡞ࠢߪ᛫⼏ࠍ␜ߔߚ߼ห௥ߚߜߣ
ㄉ⡯ߔࠆߎߣࠍ⠨߃߹ߔ߇㧘ߘࠇߦߟ޿ߡഥ⸒ࠍ
᳞߼ߚࡊ࡜ࡦࠢ߆ࠄ㧘ߘߩࠃ߁ߥㄉ⡯ࠍߒߡ߽ല
ᨐߪᦼᓙߢ߈ߕ⎕ዪߪ޿ߕࠇㆱߌࠄࠇߥ޿ߎߣ㧘߻ߒࠈ⎕ዪ߇⚳ࠊߞߚᓟߩౣ↢ࠍߤ߁ߔࠆ߆ߦ⋡ᮡࠍะߌࠆߴ߈ߢ޽ࠆߎߣࠍ⺑߆ࠇᕁ޿ߣߤ߹ࠅ
߹ߔ[7]㧚╙ੑᰴᄢᚢߪ᡽ᮭ߳ߩදജ⟵ോ߽฽߼ࡂ
ࠗ࠯ࡦࡌ࡞ࠢߦ⪺ߒ޿⧰ᖠࠍਈ߃ߚ߫߆ࠅߢߥߊ㧘ᧃᦼߦߪ⥄りߩኅ߽ⓨⷅߦࠃࠅ὾ᄬߒߡ޿߹ߔ㧚⚳ᚢᓟ㧘ࡂࠗ࠯ࡦࡌ࡞ࠢߪࡑ࠶ࠢࠬ࡮ࡊ࡜ࡦࠢ
⎇ⓥᚲߩᚲ㐳ߦዞછߒ㧘‛ℂቇߣේሶജ೑↪ߦ㑐ߔࠆ᭽‫⺖ߥޘ‬㗴ߦขࠅ⚵ߺߥ߇ࠄ㧘1970 ᐕߩㅌ⡯߹ߢቇⴚ⇇ߩౣ↢ߣ⊒ዷߦዧߒߡࠁ߈߹ߔ㧚ෳ⠨ᢥ₂
[1] ⧎᧛సᖗ㧘ࡑ࠶ࠢࠬ࡮ࡊ࡜ࡦࠢߩഞ❣㧘વᾲ㧘
48-205 (2009) 32㧚
[2] ᧛਄㓁৻㧘࠾࡯࡞ࠬ࡮ࡏ࡯ࠕߩഞ❣㧘વᾲ㧘
49-206 (2010) 25㧚
[3] ㊂ሶജቇߩ⊒ዷผ㧘㜞ᨋᱞᒾ⪺㧘ਛᄩ౏⺰␠
(1977)㧚
[4] ᧛਄㓁৻㧘࡞ࠗ࡮࠼࡮ࡉࡠࠗߩഞ❣㧘વᾲ㧘
49-208 (2010) 52㧚
[5] ⦼ේᱜᒾ㧘ࠛ࡞ࡧࠖࡦ࡮ࠪࡘ࡟࠺ࠖࡦࠟ࡯ߩ
ഞ❣㧘વᾲ㧘49-209 (2010) 72㧚

વᾲ 2011 ᐕ 7 ᦬- 45 -[6] ㊂ሶജቇ㧵㧘ᦺ᳗ᝄ৻㇢⪺㧘ߺߔߕᦠᚱ (1969)㧚
[7] ㇱಽߣో૕㧘ࡂࠗ࠯ࡦࡌ࡞ࠢ⪺㧘ጊፒ๺ᄦ⸶㧘ߺߔߕᦠᚱ (1999)㧚
[8] ߎߩ⸘▚ߪࡂࠗ࠯ࡦࡌ࡞ࠢߩዉ޿ߚᒻᑼ㧔߅ࠃ߮ⴕ೉ജቇ㧕ߢߪᢙቇ⊛ߦᭂ߼ߡⶄ㔀ߦߥࠅ㧘⚿ዪᓐ⥄りߦࠃߞߡߪ⸃߆ࠇ߹ߖࠎߢߒߚ㧚ߎߩ໧㗴ߪᛛᏁ⊛ᢙቇࠍ㚟૶ߒߚࡄ࠙࡝ߦࠃߞ
ߡⴕ೉ജቇࠍ↪޿ߡㇱಽ⊛ߦ⸃߆ࠇ㧘ߎࠇߦࠃߞߡࡂࠗ࠯ࡦࡌ࡞ࠢߩℂ⺰ߩᅷᒰᕈ߇⏕⹺ߐࠇࠆߎߣߦߥࠅ߹ߔ㧚[9] W. Heisenberg, Z. Physik, 33 (1925) 879.
[10] ৻ᣇ㧘᷹ⷰߦߪ߆߆ࠄߥ޿㕖ታ࿷⊛‛ℂ㊂㧘
଀߃߫↢ᚑ࡮ᶖṌࠍ⴫ߔⴕ೉ߥߤߪࠛ࡞ࡒ࡯࠻
ⴕ೉ߦߪߥࠅ߹ߖࠎ㧚[11] M. Born, P. Jordan, Z. Physik, 34 (1925) 858.
[12] M. Born, W. Heisenberg, P. Jordan, Z. Physik,
35 (1926) 557.
[13] ࡄ࠙࡝ߪ㜞ᩞතᬺߩ 2 ߆᦬ᓟߩ 18 ᱦߩߣ߈ߦ
৻⥸⋧ኻᕈℂ⺰ߦ㑐ߔࠆੱ↢ᦨೋߩ⺰ᢥࠍ಴ߒ㧘21 ᱦߩ⧯ߐߢ᳓⚛ಽሶࠗࠝࡦߩ㊂ሶ⺰ߦ㑐ߔࠆቇ૏⺰ᢥߦࠃࠅඳ჻ภࠍขᓧߒߡ޿
߹ߔ㧚
[14] ᢥ₂ 7 ߩ⸶⠪߇ㅀߴࠆࠃ߁ߦ㧘ㅢᏱ࠼ࠗ࠷ߢ
‛ℂߩቇ૏ࠍߣࠆߩߦᄢቇ౉ቇ߆ࠄ 8㨪9 ᐕ߆߆ࠆߣߎࠈ㧘౉ቇ߆ࠄ 3 ᐕ㧘ታ⾰ඨᐕߢቇ૏
ࠍขߞߚࡂࠗ࠯ࡦࡌ࡞ࠢߩᚽ⢻߇޿߆ߦਗᄖ
ࠇߚ߽ߩߢ޽ߞߚ߆߇ુ߃߹ߔ㧚ߎߩቇ૏⺰ᢥߪᢙቇ⊛ߥᛛᏁࠍ㚟૶ߒߡⶄ㔀ߥ㕖✢ᒻ໧㗴ࠍᛒߞߚ߽ߩߢ㧘ߘߩ⚿ᨐߪᱜߒ޿߽ߩߢ
޽ߞߚߦ߽߆߆ࠊࠄߕ㧘ቢోߦℂ⸃ߐࠇࠆ߹
ߢߦߪ߶߷ 30 ᐕࠍⷐߒߚߘ߁ߢߔ㧚ߎߩࠃ߁ߥቇ૏⎇ⓥߩ⺖㗴⸳ቯߪ㧘㊂ሶ⺰ߩࠃ߁ߥవ
┵⎇ⓥࠍⴕ߁㓙ߦ߽ฎౖ‛ℂቇߩၮ␆ߪᒻᚑ
ߐࠇߡ޿ࠆߴ߈ߣߒߚ࠱ࡦࡑ࡯ࡈࠚ࡞࠻ߩᣇ
㊎ߦࠃࠆ߽ߩߣᕁࠊࠇ߹ߔ㧚ߥ߅㧘࠱ࡦࡑ࡯ࡈࠚ࡞࠻⥄り߽ࡒࡘࡦࡋࡦߦ⒖ࠆએ೨ߪᵹ૕
ẢṖߩ⎇ⓥࠍⴕߞߡ߅ࠅ㧘ߔߴࠅゲฃߩẢṖ⁁ᘒߩ⹏ଔߦ↪޿ࠄࠇࠆ࠱ࡦࡑ࡯ࡈࠚ࡞࠻ᢙ
ߣ޿߁ήᰴరᢙࠍᱷߒߡ޿߹ߔ㧚

J. HTSJ, Vol. 50, No. 212