ฏ੒ 31 ೥౓େֶӃೖֶࢼ‫ݧ‬໰୊ɹ Iʢ3 ࣌ؒʣ஫ҙ(1) ໰୊ I–1ɼI–2ɼI–3 ͷղ౴͸ͦΕͧΕࢦఆ͞Εͨղ౴༻ࢴ 1 ຕʹ‫ه‬ೖͤΑɽ(2) ໰୊ I–1 ͷղ౴ʹ͸ཪ໘Λ༻͍ͯ΋Α͍ɽ(3) ໰୊ I–2 ͸ಠཱͨ͠ 2 ͭͷখ໰ I–2AɼI–2B ͔Βɼ໰୊ I–3 ͸ಠཱͨ͠ 4 ͭͷখ໰ I–3AɼI–3BɼI–3CɼI–3D ͔ΒͳΔɽͦΕͧΕͷখ໰ͷղ౴͸ղ౴༻ࢴͷࢦఆ͞Εͨ৔ॴ (ཪ໘Λ‫ؚ‬Ή) ʹ‫ه‬ೖͤΑɽ(4) ֤ղ౴༻ࢴ͸ԣ௕ʹ࢖༻ͯ͠ɼදଆͷࠨ্෦ʢઢΑΓ্ʣʹड‫ݧ‬൪߸ɼࢯ໊
Λ‫ه‬ೖͤΑɽղ౴༻ࢴͷଞͷ෦෼ʹड‫ݧ‬൪߸ɼࢯ໊Λॻ͍ͯ͸͍͚ͳ͍ɽղ
౴༻ࢴ্෦ͷઢΑΓ্ͷཝʹ͸දɼཪͱ΋ղ౴Λॻ͍ͯ͸͍͚ͳ͍ɽ
(5) ղ౴༻ࢴ͸ 3 ໰ʢ‫ ܭ‬3 ຕʣ͢΂ͯఏग़͢Δ͜ͱɽͳ͓ɼ໰୊࡭ࢠ͓ΑͼԼॻ͖༻ࢴ͸ճऩ͠ͳ͍ɽ
(6) ໰୊࡭ࢠ͸දࢴΛ‫ؚ‬Ίͯ 7 ϖʔδ·Ͱ͋Δɽ

1

I-1 (ి࣓‫( )ֶؾ‬100 ఺)ਅۭྖҬ͔Βɼ‫ࢹڊ‬తʹ͸ి‫ؾ‬తʹதੑͳϓϥζϚ͕ଘࡏ͢ΔྖҬ (ి཭૚) ΁఻೻͢Δి࣓೾Λߟ͑Δɽ2 ͭͷྖҬ͸ z = 0 ͷฏ໘ͰִͯΒΕɼz < 0 ͷྖҬ͸ਅۭɼz > 0 ͷྖҬ͸ۭؒతʹҰ༷ͳϓϥζϚ͕෼෍͓ͯ͠Γɼి཭ిࢠͷ਺ີ౓͸n Ͱ͋Δͱ͢Δɽ؆୯ͷͨΊɼϓϥζϚͷ೤ӡಈ͸ͳ͍΋ͷͱ͠ɼΠΦϯͷӡಈ͸ແࢹ͢Δɽͳ͓ɼిࢠͷมҐ͸ඍখͰ͋Δͱ͢Δɽి࣓೾͸ϚοΫε΢Σϧํఔࣜ

~ ·E
~ = 0
∇

(A)

~
~ ×E
~ = − ∂B
∇
∂t

(B)

~
~ ×B
~ = µ0 J~ + µ0 ε0 ∂ E∇∂t~~∇·B = 0

(C)
(D)

~ B
~ ͸ͦΕͧΕి৔ɼ࣓ଋີ౓ɼJ~ ͸ిྲྀີ౓ɼµ0 , ε0 ͸ͦΕʹै͏ɽ͜͜ͰɼE,ͧΕਅۭͷಁ࣓཰ɼਅۭͷ༠ి཰Ͱ͋Δɽ
ి཭૚தͷిࢠ͸ೖࣹ͢Δి࣓೾ͷి৔ʹΑΓՃ଎͞ΕΔɽ͜ͷͱ͖ɼిࢠͷిՙΛ eɼ࣭ྔΛ mɼ଎౓Λ ~v ͱ͢ΔͱɼҰͭͷిࢠͷӡಈ͸ɼӡಈํఔࣜ

m

d~v
~
= eE
dt

ʹै͏΋ͷͱ͢Δ (͜͜Ͱɼి࣓೾ͷ࣓৔ʹΑΔྗ͸ແࢹͰ͖Δͱͨ͠)ɽͳ͓ɼղ
౴ʹ͸ SI ୯Ґ‫ܥ‬Λ༻͍Αɽ~~ Λ༻͍ͯɼ
(1) ి཭૚தͰͷిྲྀີ౓ J~ ͷ࣌ؒඍ෼ ∂ J/∂t
͸ి৔ E
∂ J~
=
∂t
ͱॻ͚Δɽ ͋

͋~E

ʹೖΔద੾ͳࣜΛ‫ٻ‬ΊΑɽ~ Λফ‫͢ڈ‬Δ͜ͱͰɼి
(2) ࣜ (A)∼(D) ͷϚοΫε΢Σϧํఔ͔ࣜΒ࣓ଋີ౓ B~ ʹ͍ͭͯͷ೾ಈํఔࣜΛ‫ٻ‬ΊΔͱɼ
৔E
~ =
~ 2E
∇

͍

2~
~ + µ0 ε 0 ∂ E
E∂t2

ͱͳΔɽ͜ͷͱ͖ɼn, µ0 , ε0 , e, m ͷ͏ͪඞཁͳ΋ͷΛ༻͍ͯద੾ͳࣜΛ‫ٻ‬ΊΑɽ

2

͍

ʹೖΔ

(3) ి཭૚த (z > 0) Ͱͷฏ໘೾ (֯ৼಈ਺ ω ɼి཭૚தͷ೾਺ k 0 ) ͷ෼ࢄؔ܎͸ɼ√ޫ଎ c (= 1/ ε0 µ0 ) ͱϓϥζϚৼಈ਺ ωp Λ༻͍ͯɼω 2 = c2 k 02 + ωp2
ͱ༩͑ΒΕΔɽ͜ͷͱ͖ɼωp Λ n, µ0 , ε0 , e, m ͷ͏ͪඞཁͳ΋ͷΛ༻͍ͯදͤɽ~ i = Re[E0 ei(kz−ωt) ]~exҎԼͰ͸ɼਅۭத (z < 0) Λ +z ํ޲ʹਐΉೖࣹ೾ͷి৔Λ E
ͱ͢Δɽͨͩ͠ɼE0 ͸ఆ਺Ͱ͋Γɼ~ex ͸ x ํ޲ͷ୯ҐϕΫτϧͰ͋Δɽ(4) ω > ωp ͷͱ͖ɼి཭૚தͷి৔͸೾਺ k 0 ͷฏ໘೾Ͱ͋Δɽz = 0 Ͱͷ઀ଓ~ r, E~ t ͱ͢Δ) Λ‫ٻ‬ΊΑɽ৚͔݅Βɼ൓ࣹ೾ͱಁա೾ͷి৔ (ͦΕͧΕ E~(5) খ໰ (4)D Eʹ͓͍ͯɼి཭૚தͰͷಁա೾ͷϙΠϯςΟϯάϕΫτϧ S ͷ࣌ؒ~ ͷ޲͖ͱେ͖͞Λɼz = +0 ʹ͓͍ͯ‫ٻ‬ΊΑɽฏ‫ ۉ‬S(6) ω < ωp ͷͱ͖ɼz = +0 ͰͷϙΠϯςΟϯάϕΫτϧͷ࣌ؒฏ‫ۉ‬Λ‫ٻ‬ΊΑɽ

3

I-2A (ྗֶ) (50 ఺)࣭ྔ Ms ͷଠཅͷपΓΛ࣭ྔ Mp ( Ms ) ͷ࿭੕͕ެస͍ͯ͠Δɽέϓϥʔͷ๏ଇ͔Βɼສ༗Ҿྗͷ๏ଇ
Ms Mp
F = −G 2rΛಋ͍ͯΈΑ͏ɽଠཅ͸ෆಈͰ͋Δͱۙࣅ͠ɼެస໘಺ͰଠཅΛ‫͢ͱ఺ݪ‬Δ‫࠲ۃ‬ඪ (r, θ) Λ༻͍Δͱɼ࿭੕ʹ͸ͨΒ͘Ճ଎౓͸ɼҰൠʹͦͷӡಈঢ়ଶ͔Βɼ
 2


dθ
1d
d2 r
2 dθ
,
aθ =
r
ar = 2 − r
dt
dt
r dt
dt
ͱͳΔɽέϓϥʔͷୈ 2 ๏ଇʹΑΕ͹࿭੕ͷ໘ੵ଎౓ (࿭੕ͱଠཅΛ݁Ϳ௚ઢ͕୯1 dθҐ࣌ؒʹ૟͘໘ੵ = r2ʣ͸ҰఆͰ͋Δɽ͕ͨͬͯ͠ɼaθ = 0 Ͱ͋Γɼ࿭੕ʹ2 dt͸ͨΒ͘ྗͷ θ ੒෼͸θϩͰ͋Δɽ࿭੕ʹ͸ͨΒ͘ྗ͸ r ํ޲ͷΈͱߟ͑ΒΕΔɽ
έϓϥʔͷୈ 1 ๏ଇʹΑΕ͹ɼ͜ͷ࿭੕ͷެస‫ي‬ಓ͸ପԁͰ͋Δɽͦͷ໘ੵ͸ɼପ√ԁͷ௕൒‫ ܘ‬aɼ཭৺཰ ε Λ࢖ͬͯɼπa2 1 − ε2 ͱදͤΔɽ(1) ࿭੕ͷެసप‫ ظ‬T ͱପԁͷ໘ੵ͔Β໘ੵ଎౓Λ‫ٻ‬Ίɼදͤɽ(2) r ͷ࣌ؒʹؔ͢Δ 1 ֊ඍ෼ͱ 2 ֊ඍ෼͸ɼ
 
d 1
d2 r
dr
= A
=
,
dt
dθ r
dt2
ͱදͤΔɽ A ,

B

 
1
r

ʹೖΔద੾ͳࣜΛ T Λ‫ؚ‬Μͩ‫ٻͰܗ‬ΊΑɽ(3) ପԁͷ r ͱ θ ͷؔ܎ r =Λ༻͍ͯදͤɽ

B

d2
dθ2

dθ
Λ T Λ‫ؚ‬ΜͩࣜͰdtksp
a(1 − ε2 )Λ༻͍Δͱɼar = 2 ͱͳΔɽ ksp Λ a, T
1 + εcosθrҎ্Λ౿·͑ͯɼχϡʔτϯ͕ߦͬͨͰ͋Ζ͏ΞϓϩʔνΛߟ͑ͯΈΑ͏ɽ࿭੕͸
ksp MpଠཅʹΑͬͯ Fp = ar Mp =ͷେ͖͞ͷྗͰҾ͔Ε͍ͯΔɽέϓϥʔͷୈ 3r2
๏ଇʹΑΕ͹ɼksp ͸࿭੕ʹΑΒͣҰఆͰ͋ΔɽҰํɼଠཅͱ࿭੕ͷ໾ׂΛೖΕ͔͑ͯ΋ಉ༷ʹྗͷද͕ࣜ੒ΓཱͭͱԾఆ͢Δͱɼଠཅ͕࿭੕͔Βड͚Δྗ Fs ͸ɼ
kps MsC͜ͷͰ͋Γɼkps ͸ଠཅͷੑ࣭ʹΑΒͳ͍ͱਪ࿦͞ΕΔɽFs =r2Α͏ʹߟ͑Δͱɼksp = −GMs ɼkps = −GMp ͱͳΔ͜ͱΛࣔ͢͜ͱ͕Ͱ͖ɼສ༗
Ҿྗͷ๏ଇ͕ಋ͔ΕΔɽ

(4)Cʹ͋ͯ͸·Δద੾ͳઆ໌Λ؆ܿʹॻ͚ɽ

4

I-2B (ྔࢠྗֶ) (50 ఺)࣭ྔ mɼ֯ৼಈ਺ ω ͷҰ࣍‫ݩ‬ௐ࿨ৼಈࢠΛߟ͑Δɽ͜ͷ‫ܥ‬ͷΤωϧΪʔ‫ݻ‬༗ঢ়ଶ͸ɼަ‫[ ܎ؔ׵‬â, â† ] = 1 Λຬͨ͢ੜ੒ԋࢉࢠ â† ͱফ໓ԋࢉࢠ â ʹΑΓɼ1|ni = √ (â† )n |0i (n = 0, 1, 2, 3, · · · ) ͱද͢͜ͱ͕Ͱ͖Δɽ͜͜Ͱɼ|0i ͸‫ج‬ఈঢ়n!
ଶΛද͢ɽͳ͓ɼҎԼͰɼ~ ͸ϓϥϯΫఆ਺Λ 2π Ͱׂͬͨ΋ͷͰ͋Δɽ
r~(1) Ґஔԋࢉࢠ͕ x̂ =(â + â† ) ͱॻ͚Δ͜ͱΛ༻͍ͯɼhn0 | x̂ |niɼhn0 | x̂2 |ni
2mωΛΫϩωοΧʔͷσϧλΛ࢖ͬͯදͤɽ

(2) ௐ࿨ৼಈࢠ͕ҎԼͷ 4 ͭͷϙςϯγϟϧ V1 (x) = AxɼV2 (x) = Bx2 ɼV3 (x) =Cx3 ɼV4 (x) = Dx4 ͷͦΕͧΕʹΑΓઁಈΛड͚ͨͱ͢Δɽ͜ΕΒͷ͏ͪɼ|ni ͷ‫ݻ‬༗ΤωϧΪʔ En ʹରͯ̍࣍͠ͷઁಈʹΑΔมԽ͕ੜ͡Δ΋ͷ͸ͲΕ͔Λ౴͑Αɽ·ͨɼͦͷཧ༝ΛɼϋϛϧτχΞϯͷରশੑͷ‫͔఺؍‬Β؆୯ʹઆ໌ͤΑɽ͞Βʹɼ̍࣍ͷઁಈʹΑΔΤωϧΪʔมԽ͕ੜ͡Δ΋ͷʹ͍ͭͯɼͦͷେ͖͞Λ ~, m, ω, n ͓Αͼ A, B, C, D ͷ͏ͪඞཁͳ΋ͷΛ༻͍ͯදͤɽ
(3) ௐ࿨ৼಈࢠʹ V1 (x) + V2 (x) ͷઁಈ͕ՃΘͬͨͱ͖ͷΤωϧΪʔ‫ݻ‬༗஋ͷ‫ݫ‬ີղΛ‫ٻ‬ΊΑɽ

5

I-3A (ྗֶ) (25 ఺)Ұ༷ͳॏྗՃ଎౓ g ͕Ԗ௚Լ޲͖ʹ͸ͨΒ͍͍ͯΔ‫ڥ؀‬ԼͰɼॳ‫ ྔ࣭ظ‬M0 ͷϩέοτΛॳ଎ 0 ͰԖ௚্޲͖ʹଧ্ͪ͛Δɽ೩ྉ͸ɼϩέοτʹରͯ͠Ұఆͷ଎͞u Ͱ‫ʹํޙ‬෾ग़͞ΕΔɽϩέοτͷ࣭ྔ͸೩ྉͷ෾ग़ʹͱ΋ͳ͍‫ݮ‬গ͢Δɽଧ্ͪ͛࣌ࠁΛ t = 0 ͱ͠ɼ࣌ࠁ t Ͱͷϩέοτͷ࣭ྔ͕ M (t) ͱ༩͑ΒΕ͍ͯΔͱ͢ΔɽϩέοτͷӡಈํఔࣜΛಋग़͠ɼϩέοτͷ଎౓ v(t) Λ‫ٻ‬ΊΑɽͨͩ͠ɼԖ௚্
޲͖Λਖ਼ͱͤΑɽ

I-3B (‫ࠩޡ‬౷‫( )ܭ‬25 ఺)֬཰ม਺ k ͷ֬཰෼෍͕, Pλ (k) = e−λ λk /k! ɹ (ͨͩ͠ɼk = 0, 1, 2, ...) ʹै͏ͱ͖ɼ͜ΕΛύϥϝʔλ λ ͷϙΞιϯ෼෍ͱ‫͏ݴ‬ɽ(1) ύϥϝʔλ λ ͷϙΞιϯ෼෍ͷฏ‫ͱۉ‬෼ࢄΛ‫ٻ‬ΊΑɽ(2) ͋Δ෺ମ͔Β์ग़͞ΕΔཻࢠΛ‫ݕ‬ग़͢ΔଌఆΛߟ͑Δɽ‫ݕ‬ग़͞ΕΔཻࢠͷ਺͸ϙΞιϯ෼෍ʹै͏ͱ͢Δɽ‫ݕ‬ग़͞ΕΔཻࢠͷ਺ͷඪ४ภ͕ࠩฏ‫ۉ‬஋ͷ 1%ҎԼʹͳΔΑ͏ʹ͍ͨ͠ɽͲΕ͚ͩͷཻࢠ਺Λ‫ݕ‬ग़͢Δඞཁ͕͋Δ͔ɽ

6

I-3C (ྔࢠྗֶ) (25 ఺)ϋϛϧτχΞϯ͕

~2
H=−
2m



∂2
∂2
+
∂x21 ∂x22



+ λ|x1 − x2 |Ͱ༩͑ΒΕΔɼ̍࣍‫ʹ޲ํݩ‬ଋറ͞Εͨεϐϯ 1/2 ͷ̎ͭͷಉछཻࢠ͔ΒͳΔྔ
ࢠྗֶ‫ܥ‬Λߟ͑Δɽ͜͜Ͱɼx1 , x2 ∈ R ͸̎ͭͷཻࢠͷҐஔ࠲ඪɼm ͸ཻࢠͷ࣭
ྔɼ~ ͸ϓϥϯΫఆ਺Λ 2π Ͱׂͬͨ΋ͷɼλ ͸ਖ਼ͷఆ਺Ͱ͋Δɽॏ৺‫Ͱܥ‬ͷ‫ج‬ఈঢ়ଶͱୈҰྭ‫ى‬ঢ়ଶʹ͓͚Δશεϐϯͷେ͖͞ΛͦΕͧΕ౴͑ɼͦͷཧ༝Λ؆ܿʹड़΂Αɽ

I-3D (෺ཧ਺ֶ) (25 ఺)ϕΫτϧྔ ~r = (x, y, z) ͕ɼΨ΢ε෼෍ f (~r) = A exp(−β|~r|2 ) ʹै͍෼෍͍ͯ͠
Δͱ͢Δɽf (~r) ͕‫֨ن‬Խ৚݅Z
d3~rf (~r) = 1
Λຬͨ͢Α͏ʹ܎਺ A ΛܾఆͤΑɽ͜͜Ͱɼੵ෼ެࣜ√Z ∞π
2 −as2s e
ds = 3/2
4a0Λ༻͍ͯΑ͍ɽ·ͨɼ|~r| ͷೋ৐ฏ‫ ۉ‬h|~r|2 i Λ‫ٻ‬ΊΑɽ

7

ฏ੒ 31 ೥౓େֶӃೖֶࢼ‫ݧ‬໰୊ɹ IIʢ3 ࣌ؒʣ஫ҙ(1) ໰୊ II–1ɼII–2ɼII–3 ͷղ౴͸ͦΕͧΕࢦఆ͞Εͨղ౴༻ࢴ 1 ຕʹ‫ه‬ೖͤΑɽ(2) ໰୊ II–1ɼII–2 ͷղ౴ʹ͸ཪ໘Λ༻͍ͯ΋Α͍ɽ(3) ໰୊ II–3 ͸ಠཱͨ͠ 2 ͭͷখ໰ II–3AɼII–3B ͔ΒͳΔɽͦΕͧΕͷখ໰ͷղ౴͸ղ౴༻ࢴͷࢦఆ͞Εͨ৔ॴ (ཪ໘Λ‫ؚ‬Ή) ʹ‫ه‬ೖͤΑɽ(4) ֤ղ౴༻ࢴ͸ԣ௕ʹ࢖༻ͯ͠ɼදଆͷࠨ্෦ʢઢΑΓ্ʣʹड‫ݧ‬൪߸ɼࢯ໊
Λ‫ه‬ೖͤΑɽղ౴༻ࢴͷଞͷ෦෼ʹड‫ݧ‬൪߸ɼࢯ໊Λॻ͍ͯ͸͍͚ͳ͍ɽղ
౴༻ࢴ্෦ͷઢΑΓ্ͷཝʹ͸දɼཪͱ΋ղ౴Λॻ͍ͯ͸͍͚ͳ͍ɽ
(5) ղ౴༻ࢴ͸ 3 ໰ʢ‫ ܭ‬3 ຕʣ͢΂ͯఏग़͢Δ͜ͱɽͳ͓ɼ໰୊࡭ࢠ͓ΑͼԼॻ͖༻ࢴ͸ճऩ͠ͳ͍ɽ
(6) ໰୊࡭ࢠ͸දࢴΛ‫ؚ‬Ίͯ 7 ϖʔδ·Ͱ͋Δɽ

1

II-1 (౷‫( )ֶྗܭ‬100 ఺)
࣭ྔ m ͷೋͭͷ࣭఺͕ଋറ͞Ε͍ͯΔ໛‫ʹܕ‬Αͬͯ̎‫ࢠݪ‬෼ࢠΛߟ͑Δɽ͜ͷʮ෼ࢠʯ͕ମੵ V ͷശͷதʹ N ‫͋ݸ‬Δͱ͢Δɽ࣭఺ؒ‫཭ڑ‬ͷಛ௃తͳ௕͞ a ͸෼ࢠؒ‫཭ڑ‬ͷయ‫ܕ‬తͳ௕͞ (V /N )1/3 ΑΓ΋े෼খ͍͞ͱ͠ɼ෼ࢠؒͷ૬‫༻࡞ޓ‬͸ແࢹͰ
͖Δͱ͢ΔɽϘϧπϚϯఆ਺Λ kB ͱ͢Δɽ(1) ෼ࢠ಺ͷ‫ؒࢠݪ‬૬‫ͯ͠ͱ༻࡞ޓ‬ɼ࣭఺ؒ‫͕཭ڑ‬ৗʹҰఆ஋ a ʹอͨΕ͍ͯΔ໛‫ܕ‬Λߟ͑Δɽ‫ݹ‬య౷‫ֶྗܭ‬ͷൣғͰɼԹ౓ T ͷฏߧঢ়ଶʹ͓͚Δѹྗ
P (T, V, N ) ͓Αͼఆੵ೤༰ྔ CV (T, V, N ) Λॻ͚ɽ·ͨɼ͜ΕΒΛಋ͘‫ج‬ຊతͳߟ͑ํΛ؆ܿʹ‫ͤه‬ɽ‫ࢉܭ‬ͷৄࡉΛઆ໌͢Δඞཁ͸ͳ͍ɽ(2) ࣍ʹɼ෼ࢠ಺ͷ‫ؒࢠݪ‬૬‫͕༻࡞ޓ‬ɼࣗવ௕ aɼ͹Ͷఆ਺ k ͷ͹ͶͰ‫ه‬ड़͞Ε
Δ໛‫ܕ‬Λߟ͑Δɽ͢ͳΘͪɼ~r1 ʹ͋Δ࣭఺ͱ ~r2 ʹ͋Δ࣭఺͸
1k (|~r1 − ~r2 | − a)22ͱ͍͏૬‫༻࡞ޓ‬ΤωϧΪʔΛ࣋ͭɽka2  kB T ͷͱ͖ɼ‫ݹ‬య౷‫ֶྗܭ‬ͷൣғͰɼԹ౓ T ͷฏߧঢ়ଶʹ͓͚Δѹྗ P (T, V, N ) ͓Αͼఆੵ೤༰ྔ CV (T, V, N )Λ‫ٻ‬ΊΑɽ·ͨɼ͜ΕΒΛಋ͘ߟ͑ํΛ؆ܿʹ‫ͤه‬ɽ‫ࢉܭ‬ͷৄࡉΛઆ໌͢Δ
ඞཁ͸ͳ͍ɽ
খ໰ (2) ͷઃఆʹରͯ͠ྔࢠ౷‫͑ߟͰֶྗܭ‬Δɽͨͩ͠ɼ࣭఺͸ಉछཻࢠͰ͋Δͱ͠ɼϘʔε౷‫͏ैʹܭ‬΋ͷͱ͢ΔɽϓϥϯΫఆ਺Λ 2π Ͱׂͬͨ΋ͷΛ ~ ͱ͢Δɽ(3) ࣍‫ݩ‬ղੳΛ༻͍ͯɼҎԼͷ̏ͭͷԹ౓ T0 , T1 , T2 Λ‫ٻ‬ΊΑɽT0 ͸࣭఺ͷ࣭ྔm ͱ਺ີ౓ n ʹΑܾͬͯ·ΔԹ౓ɼT1 ͸࣭఺ͷ࣭ྔ m ͱࣗવ௕ a ʹΑͬͯ
ܾ·ΔԹ౓ɼT2 ͸࣭఺ͷ࣭ྔ m ͱ͹Ͷఆ਺ k ʹΑܾͬͯ·ΔԹ౓Ͱ͋Δɽͨ
ͩ͠ɼ~ ΍ kB Λ࢖ͬͯΑ͍ɽ(4) Թ౓ྖҬ T  T0 Ͱ‫؍‬ଌ͞ΕΔ‫ݦ‬ஶͳ‫ݱ‬৅ͷ໊শΛ‫ͤه‬ɽ(5) T0  T1  T2 ͱ͢Δɽ̏ͭͷԹ౓ྖҬ (i) T0  T  T1 , (ii) T1  T  T2 ,(iii) T  T2 ʹ͓͍ͯͦΕͧΕఆੵ೤༰ྔΛॻ͚ɽ·ͨɼ͜ΕΒΛಋ͘‫ج‬ຊతͳߟ͑ํΛ؆ܿʹ‫ͤه‬ɽ
5N kB(6) Ұ‫ؾ‬ѹৗԹ (300 K ఔ౓) ͷ̎‫ࢠݪ‬෼ࢠ‫ؾ‬ମͷఆੵ೤༰ྔ͸͓ΑͦͰ͋2pΔɽ͜ͷࣄ࣮ͱ੔߹͢ΔͨΊʹ͸ɼ͹Ͷͷ֯ৼಈ਺ ω = k/m ͸͋Δ஋ ωcΑΓ΋े෼େ͖͘ͳ͚Ε͹ͳΒͳ͍ɽͦͷ֯ৼಈ਺ͷ஋ ωc Λ‫ੵݟ‬΋Εɽͨͩ͠ɼ~ = 10−34 m2 kg s−1 , kB = 10−23 m2 kg s−2 K−1 ͱͯ͠Α͍ɽ

2ʢ͜ͷϖʔδ͸നࢴͰ͋Δʣ

3

II-2 (ྔࢠྗֶ) (100 ఺)ਤ̍ͷΑ͏ʹฏ໘্ʹ৐Δ൒‫ ܘ‬R ͷԁ C ͷ্ʹଋറ͞Ε࣭ͨྔ mɼిՙ e(e > 0 ͱ͢Δ) ͷՙిཻࢠͷӡಈΛྔࢠྗֶతʹߟ࡯͢Δɽ͜ͷฏ໘ʹਨ௚ͳํ޲ʹԁ C ͷத৺Λ؏͘Α͏ʹࡉ௕͍ιϨϊΠυ͕ઃஔ͞Ε͓ͯΓɼԁ C ͷ಺෦Λਤͷ্޲͖ʹ؏࣓͘ଋ Φ Λࣗ༝ʹίϯτϩʔϧͰ͖ΔΑ͏ʹͳ͍ͬͯΔɽͨͩ͠ɼ࣓৔͸ιϨϊΠυͷ಺෦ͷΈʹੜ͡ɼԁ C ্Ͱ͸ແࢹͰ͖Δ΋ͷͱ͢ΔɽҎԼͰ͸ɼ͜ͷՙిཻࢠͷఆৗঢ়ଶΛද͢೾ಈؔ਺Λ ψ(σ) Ͱද͠ɼप‫ڥظ‬ք৚݅ ψ(σ + 2πR) = ψ(σ)ʹै͏Ұ࣍‫ͯ͠ͱܥֶྗࢠྔݩ‬औΓѻ͏ɽ͜͜Ͱɼσ = Rθ ͸ C ্ͷ࠲ඪͰ͋Δɽ
ͳ͓ɼඞཁͰ͋Ε͹ϓϥϯΫఆ਺Λ 2π Ͱׂͬͨ΋ͷͱͯ͠ɼ~ Λ༻͍Αɽ
6
Φ

σ
θ
R

C

ਤ̍

(1) Φ = 0 ͷͱ͖ɼՙిཻࢠͷ೾ಈؔ਺ ψ(σ) ͕ै͏γϡϨσΟϯΨʔํఔࣜΛɼΤωϧΪʔ‫ݻ‬༗஋Λ E ͱͯ͠ॻ͖Լͤɽ·ͨɼऔΓಘΔ͢΂ͯͷΤωϧΪʔ‫ݻ‬༗஋ͱ‫ݻ‬༗ঢ়ଶͷ೾ಈؔ਺Λ‫ٻ‬ΊΑɽ(2) Φ 6= 0 ͷͱ͖ɼϕΫτϧϙςϯγϟϧͷ σ ํ޲੒෼ Aσ ͷ஋͸ɼC ্ͰҰఆ஋ΛऔΔΑ͏ʹબͿ͜ͱ͕Ͱ͖Δɽ͜ͷͱ͖ͷ Aσ Λ Φ ͱ R Λ༻͍ͯදͤɽ
(3) Aσ ͕Ұൠͷ஋ͷ৔߹ͷϋϛϧτχΞϯ͸ɼAσ = 0 ͷ৔߹ͷϋϛϧτχΞϯ
ʹ‫·ؚ‬ΕΔ σ ํ޲ͷӡಈྔ pσ Λ pσ − eAσ Ͱஔ͖‫ͳʹܗͨ͑׵‬Δɽ͜ͷ͜ͱΛ༻͍ͯɼAσ ͕ C ্ͰҰఆ஋Ͱ͋Δ৔߹ʹɼऔΓಘΔΤωϧΪʔ‫ݻ‬༗஋ΛAσ Λ༻͍ͯॻ͚ɽ
(4) ࣓ଋ Φ ͕͋Δ஋ͷ੔਺ഒͷͱ͖ɼΤωϧΪʔεϖΫτϧ͸ Φ = 0 ͷ৔߹ͱಉ͡ʹͳΔɽΦ > 0 ͷൣғͰɼΤωϧΪʔεϖΫτϧ͕ Φ = 0 ͱಉ͡ʹͳΔ࠷খͷ Φ ͷ஋Λ‫ٻ‬ΊΑɽҎԼɼ͜ͷ஋Λ Φ0 ͱ͓͘ɽ(5) ࣓ଋ Φ Λ Φ = 0 ͔ΒΏͬ͘Γͱ Φ = Φ0 ΁มԽͤ͞Δɽ͜ͷͱ͖ɼॎ࣠ΛΤωϧΪʔɼԣ࣠Λ Φ ͱͯ͠ɼΤωϧΪʔ‫ݻ‬༗஋ͷมԽΛਤࣔͤΑɽͨͩ͠ɼ
ΤωϧΪʔ४Ґ͸௿͍΋ͷ͔Βॱʹ̏൪໨·ͰΛඳ͚ɽ·ͨɼਤதͰ̎ͭҎ্ͷ४ҐͷΤωϧΪʔ͕ॖୀ͢Δ෦෼͕͋Δ৔߹ʹ͸ɼͦͷ෦෼Λ‫Ͱؙ‬ғΜͰࣔͤɽ(̐ͭ໨ͷΤωϧΪʔ४Ґ͕̏ͭ໨ͷΤωϧΪʔ४Ґͱॖୀ͢Δ఺
͕͋Δ৔߹ʹ͸ɼͦͷ఺΋‫Ͱؙ‬ғΉ͜ͱɽ)4࣍ʹɼ͜ͷ‫ʹܥ‬ϙςϯγϟϧ

V (σ) = λ cos



2σ
R



ʢλ ͸ਖ਼ͷఆ਺ʣ͕ՃΘͬͨ৔߹ʹɼΤωϧΪʔεϖΫτϧ͕ͲͷΑ͏ʹมԽ͢Δͷ͔ΛɼҎԼͷϚγϡʔؔ਺ͷੑ࣭Λ༻͍ͯߟ࡯͢Δɽν ࣍ͷϚγϡʔؔ਺ meν (ξ, q) ͸ɼq Λ‫ݻ‬ఆͯ͠ ξ ͷؔ਺ͱΈͳͨ͠ͱ͖ʹϚγϡʔͷඍ෼ํఔࣜ

d2
ϕ(ξ) + (a − 2q cos(2ξ))ϕ(ξ) = 0
dξ 2

(A)

ͷղͰ͋Γɼmeν (ξ + π, q) = eiνπ meν (ξ, q) ,meν (ξ, 0) = eiνξΛຬͨ͢ɽ͜͜Ͱɼν = ν(a, q) ͸࣮ఆ਺ a, q Ͱఆ·Δಛੑࢦ਺ͱ‫ݺ‬͹ΕΔఆ਺Ͱ͋Δɽ·ͨɼν ͕੔਺Ͱͳ͍৔߹ʹ͸ɼmeν (ξ, q) ͱ me−ν (ξ, q) ͕ඍ෼ํఔࣜ (A) ͷ̎ͭͷಠཱղΛ༩͑Δɽ(6) ͜ͷ‫ܥ‬ͷγϡϨσΟϯΨʔํఔࣜ͸ɼξ = σ/R, ψ(σ) = eikξ ϕ(ξ) ͱஔ͖ɼఆ਺ k Λద੾ʹબ΂͹ɼϚγϡʔͷඍ෼ํఔࣜ (A) ʹ‫ؼ‬ண͞ΕΔɽͦͷͱ͖ͷ
k, a, q Λ Φ, Φ0 , λ ͱΤωϧΪʔ‫ݻ‬༗஋ E ͷ͏ͪඞཁͳ΋ͷΛ༻͍ͯදͤɽ
(7) k, a, q ͕খ໰ (6) Ͱ‫ٻ‬Ίͨ΋ͷͰ͋Δͱ͖ɼψ(σ) = eikξ meν (ξ, q) ͸γϡϨσΟ
ϯΨʔํఔࣜͷղͰ͋Δɽ͞Βʹɼ͜ͷ ψ(σ) ͕प‫ڥظ‬ք৚݅Λຬͨ͢ͱ͖ɼν ͷऔΓಘΔ஋Λ‫ٻ‬ΊΑɽ(8) ਤ̎͸ɼ0 < q  1 ͱΈͳͤΔ͋Δ q ͷ஋Λ‫ݻ‬ఆͨ͠ͱ͖ͷ a ͱ ν ͷؔ܎Λදͨ͠άϥϑͰ͋Δɽಛʹɼν = 1.0 Λ༩͑Δ a ͸ɼa = 1.0 ͷ෇ۙʹ̎ͭ͋Γɼa ͕ͦͷؒͷ஋ΛऔΔͱ͖ʹ͸ ν ͸࣮਺ʹͳΒͳ͍ɽ͜ͷ͜ͱ͸ 0 < q  1ͷͱ͖ʹ͸ৗʹ੒Γཱͭɽ͜ͷάϥϑͷ‫͔ܗ‬Βɼλ ͕े෼খ͞ͳਖ਼ͷ஋ΛͱΔͱ͖ʹɼখ໰ (5) Ͱ‫ٻ‬Ίͨਤ͕ͲͷΑ͏ʹมΘΔͷ͔Λਪఆ͠ɼਤࣔͤΑɽͳ͓ɼখ໰ (5) ͱಉ༷ʹɼਤதͰ̎ͭҎ্ͷ४ҐͷΤωϧΪʔ͕ॖୀ͢Δ෦
෼͕͋Δ৔߹ʹ͸ɼͦͷ෦෼Λ‫Ͱؙ‬ғΜͰࣔͤɽ
ν

ਤ̎

5

a

II-3A (෺ཧ਺ֶ) (50 ఺)
1 ࣍‫[ ؒ۠ݩ‬0, L] Ͱఆٛ͞ΕͨԹ౓৔ T (x, t) ͕֦ࢄํఔࣜ
∂T
∂ 2T
=D 2
∂t
∂x

(B)

∂T (x, t)∂T (x, t)
=
=0
∂x
∂x
x=0
x=L

(C)

ʹै͏ͱ͠ɼ‫ڥ‬ք৚݅Λຬͨ͢ͱ͢Δɽ(1) ̍࣍‫[ ؒ۠ݩ‬0, L] ʹ͓͍ͯ༗քͰ࿈ଓͳ೚ҙͷؔ਺ T (x, t) ʹରͯ͠ɼద੾ͳ
࣮ؔ਺ͷ૊ φn (x) (n = 0, 1, 2, · · · ) ʹΑͬͯT (x, t) =∞Xan (t)φn (x)n=0ͱද͢͜ͱ͕Ͱ͖ΔɽT (x, t) ͕ࣜ (B) ͱ‫ڥ‬ք৚݅ (C) Λຬͨ͢Թ౓৔ͷͱ͖ɼ࣮ؔ਺ͷ૊ φn (x) (n = 0, 1, 2, · · · ) Λద੾ʹબͿ͜ͱʹΑΓɼan (t) ͷै͏ඍ෼ํఔ͕ࣜ am (m 6= n) Λ‫͍ͳ·ؚ‬Α͏ʹ͢Δ͜ͱ͕Ͱ͖ΔɽͦͷΑ͏ͳؔ਺ͷ૊ φn (x) ͷ۩ମྫΛॻ͚ɽɹ·ͨɼͦͷͱ͖ͷ an (t) ʹର͢Δඍ෼ํఔࣜΛղ͍ͯɼan (t) Λ an (0) Λ༻͍ͯදͤɽ

(2) Tn (t) ΛTn (t) =Z L

dx φn (x)T (x, t)0

Ͱఆٛ͢Δɽan (0) 6= 0 ͷ৔߹ɼTn (t)/an (0) Λ‫ٻ‬ΊΑɽͨͩ͠ɼ݁Ռ͸ੵ෼Λ‫͍ͳ·ؚ‬ද‫ͤͱݱ‬Αɽ(3) ࣌ࠁ t = 0 ʹ͓͚ΔԹ౓৔͕"T (x, 0) = A 1 −





3 #
1 2x
2x
−1 +
−1
L
3 L

RL
Ͱ༩͑ΒΕΔͱ͢ΔɽA ͸ఆ਺Ͱ͋Δɽ೚ҙͷ t ʹ͓͍ͯɼ 0 dx T (x, t) Λ‫ٻ‬ΊΑɽ

6

II-3B (࣮‫( )ݧ‬50 ఺)ՙిཻࢠͷӡಈྔΛଌఆ͢Δํ๏ʹ͍ͭͯߟ͑Δɽਤ̍ͷΑ͏ʹɼ0 ≤ y ≤ L ͷྖ
Ҭ D ʹɼ࣓ଋີ౓ͷେ͖͕͞ B Ͱ࣌ؒతʹҰఆ͔ۭͭؒతʹҰ༷ͳ࣓৔͕ɼxy ฏ໘ʹਨ௚ʹखલ͔ΒԞ΁ͷ޲͖ʹҹՃ͞Ε͍ͯΔͱ͢Δɽਖ਼ͷిՙ e Λཻ࣋ͭࢠ͕ y < 0 ͷྖҬ͔Β y = 0 ͷ໘ʹਨ௚ʹೖࣹ͠ɼྖҬ D Ͱ࣓৔ͷӨ‫ڹ‬Λड͚ͯӡ
ಈ͢Δ৔߹Λߟ͑Δɽ͜͜Ͱɼཻࢠ͸࣓৔Ҏ֎ͷӨ‫ڹ‬͸ड͚ͳ͍΋ͷͱ͢Δɽ·ͨɼ૬ର࿦తͳޮՌΛߟྀ͢Δඞཁ͸ͳ͍ɽਤ̍(1) ྖҬ D ಺Ͱೖཻࣹࢠͷ‫੻ي‬͸ԁͷҰ෦Ͱ͋Δԁ‫ހ‬Λඳ͘ɽԁͷ൒‫ܘ‬Λ ρ ͱͯ͠ɼཻࢠͷӡಈྔͷେ͖͞Λ ρ Λ༻͍ͯදͤɽ
(2) y = 0 ͓Αͼ y = L ʹ͓͍ͯɼཻࢠͷ x ํ޲ͷ௨աҐஔΛଌఆͨ͠ͱ͢Δɽ
y = 0, L Ͱͷଌఆ஋ΛͦΕͧΕ x1 , x2 ͱͯ͠ɼԁͷ൒‫ܘ‬Λ‫ٻ‬ΊΑɽͨͩ͠ӡಈྔͷେ͖͕͞े෼େ͖͍৔߹Λߟ͑ΔͨΊɼਤ̍ʹࣔ͢ภ޲֯͸े෼খ͍͞ͱͯ͠Α͍ɽ
(3) খ໰ (2) Ͱड़΂ͨҐஔͷଌఆ஋͔ΒಘΒΕΔཻࢠͷӡಈྔͷେ͖͞Λ p ͱ͢Δɽଌఆ஋ x1 , x2 ͷ֤ʑʹ͸ɼඪ४ภࠩ σx ͷಠཱͳଌఆ‫͕͋ࠩޡ‬Δͱ͢Δɽ‫఻ࠩޡ‬೻Λߟྀͯ͠ɼp ͷ૬ର‫ ࠩޡ‬σp /p Λ‫ٻ‬ΊΑɽͨͩ͠ɼx1 , x2 ͷଌఆ஋Ҏ֎ʹ‫ࠩޡ‬ͷ‫ݯى‬͸ͳ͘ɼಘΒΕͨ૬ର‫े͕ࠩޡ‬෼ʹখ͍͞৔߹Λߟ͍͑ͯΔ΋ͷͱ͢Δɽ7

ฏ੒ 31 ೥౓େֶӃೖֶࢼ‫ݧ‬໰୊ɹ IIIʢ3 ࣌ؒʣ஫ҙ(1) ໰୊ III–1ɼIII–2ɼIII–3 ͷղ౴͸ͦΕͧΕࢦఆ͞Εͨղ౴༻ࢴ 1 ຕʹ‫ه‬ೖͤΑɽ(2) ໰୊ III–1 ͷղ౴ʹ͸ཪ໘Λ༻͍ͯ΋Α͍ɽ(3) ໰୊ III–2 ͸ಠཱͨ͠ 2 ͭͷখ໰ III–2AɼIII–2B ͔Βɼ໰୊ III–3 ͸ಠཱͨ͠ 2 ͭͷখ໰ III–3AɼIII–3B ͔ΒͳΔɽͦΕͧΕͷখ໰ͷղ౴͸ղ౴༻ࢴͷࢦఆ͞Εͨ৔ॴ (ཪ໘Λ‫ؚ‬Ή) ʹ‫ه‬ೖͤΑɽ(4) ֤ղ౴༻ࢴ͸ԣ௕ʹ࢖༻ͯ͠ɼදଆͷࠨ্෦ʢઢΑΓ্ʣʹड‫ݧ‬൪߸ɼࢯ໊
Λ‫ه‬ೖͤΑɽղ౴༻ࢴͷଞͷ෦෼ʹड‫ݧ‬൪߸ɼࢯ໊Λॻ͍ͯ͸͍͚ͳ͍ɽղ
౴༻ࢴ্෦ͷઢΑΓ্ͷཝʹ͸දɼཪͱ΋ղ౴Λॻ͍ͯ͸͍͚ͳ͍ɽ
(5) ղ౴༻ࢴ͸ 3 ໰ʢ‫ ܭ‬3 ຕʣ͢΂ͯఏग़͢Δ͜ͱɽͳ͓ɼ໰୊࡭ࢠ͓ΑͼԼॻ͖༻ࢴ͸ճऩ͠ͳ͍ɽ
(6) ໰୊࡭ࢠ͸දࢴΛ‫ؚ‬Ίͯ 9 ϖʔδ·Ͱ͋Δɽ

1

III-1 (ྗֶ) (100 ఺)ࣗવ௕ aɼόωఆ਺ k ͷόωʹͭͳ͕Εͨෳ਺ͷ࣭఺ͷӡಈΛߟ࡯͢Δɽ࠷ॳʹ 3 ‫ݸ‬ͷ࣭఺ͷӡಈΛߟ͑Δɽ࣭఺͸ɼਤ̍ͷΑ͏ʹ x ࣠ํ޲ʹͷΈಈ͘΋ͷͱ͠ɼ྆୺ͷόωͷ୺͸‫ݻ‬ఆ͞Ε͍ͯΔɽ·ͨɼ‫ݻ‬ఆ୺ؒͷ‫཭ڑ‬͸ L = 4a Ͱ͋Δɽ࣭఺ͷҐஔΛࠨ͔Β a + x1 ɼ2a + x2 ɼ3a + x3 ͱ͓͘ɽ͜͜Ͱɼx1 ɼx2 ɼx3 ͸ͦΕͧΕͷ࣭఺ͷฏߧ఺͔ΒͷมҐΛද͢ɽதԝͷ࣭఺ͷ࣭ྔΛ M ͱ͠ɼͦͷଞpͷ࣭఺ͷ࣭ྔ͸౳͘͠ m ͱ͢ΔɽҎԼɼω0 = 2k/m ͱ͢Δɽਤ̍(1) ࣭ྔ m ͷ 2 ͭͷ࣭఺ͷҐஔΛ‫ݻ‬ఆ͠ɼͦΕͧΕ x1 = 0ɼx3 = 0 ͱ͓͘ɽத
ԝͷ࣭఺ͷ࣌ࠁ t ʹ͓͚ΔมҐ x2 (t) Λ‫ٻ‬ΊΑɽͨͩ͠ɼt = 0 ʹ͓͚Δॳ‫ظ‬৚݅Λ x2 = X ɼdx2 /dt = V ͱͤΑɽҎԼͰ͸ɼখ໰ (1) Ͱ‫ݻ‬ఆͨ͠ 2 ͭͷ࣭఺Λࣗ༝ʹಈ͚ΔΑ͏ʹ͠ɼ࣭఺‫ܥ‬ͷ࿈੒ৼಈͷ‫ن‬४ৼಈΛ‫ٻ‬ΊΑ͏ɽ‫ن‬४ৼಈʹ͸̏ͭͷϞʔυ͕͋Γɼͦͷ‫ن‬४֯ৼಈ਺ͷେ͖͞Λখ͍͞ํ͔Βॱʹ ωs ɼωc ɼωl ͱ͢Δɽ(2) ࣭఺ͷมҐ xj (t)(j = 1, 2, 3) ʹର͢ΔӡಈํఔࣜΛॻ͚ɽ·ͨɼ xj (t) =Aj cos ωt ͱ͓͍ͯɼωs2 ɼωc2 ɼωl2 Λ ω0 ͱ࣭ྔൺ β = m/M Λ༻͍ͯදͤɽͨ
ͩ͠ɼAj ͸࣮ͷఆ਺Ͱ͋Δɽ(3) ͦΕͧΕͷ‫ن‬४ৼಈʹରͯ͠ɼβ = 1 ͷ৔߹ʹ A2 ͱ A3 Λ A1 Λ༻͍ͯදͤɽ
(4) தԝͷ࣭఺ͷ࣭ྔ M ΛมԽͤ͞Δɽ͜ͷͱ͖ɼωs2 /ω02 ɼωc2 /ω02 ɼωl2 /ω02 ͷมԽΛਤʹදͤɽͨͩ͠ɼԣ࣠ʹ͸ β Λ༻͍ͯɼ0 ≤ β ≤ 2 ͷൣғͰඳ͖ɼβ = 0ͱ β = 1 Ͱͷ஋͕෼͔ΔΑ͏ʹ͢Δ͜ͱɽ

2

࣍ʹɼे෼ʹଟ਺ͷ࣭఺Λલ໰ͱಉҰͷόωͰҰ௚ઢঢ়ʹͭͳ͛Δɽ͜ΕΒͷ࣭఺ͷ͏ͪ 1 ͭͷ࣭఺ͷ࣭ྔΛ M ͱ͠ɼͦͷଞͷ࣭఺ͷ࣭ྔΛ m ͱ͢Δɽ؆୯ͷͨΊɼਤ̎ͷΑ͏ʹ࣭ྔ͕ M Ͱ͋Δ࣭఺Λ 0 ൪ͱ͢ΔॱংΛಋೖ͢Δɽn ൪໨ͷ࣭఺ʹର͢Δฏߧ఺Λ x = na ͱ͠ɼͦͷ఺͔ΒͷมҐΛ xn ͱ͢Δɽਤ̎͜ͷ࣭ྔ M ͷ࣭఺Λ‫੍ڧ‬తʹ֯ৼಈ਺ ω Ͱৼಈͤ͞Δɽ(5) n ൪໨ͷ࣭఺ͷมҐΛ xn = Re[Bn eiωt ] ͱ͓͘ɽͨͩ͠ɼBn = B0 bn ͱ͠ɼB0ͱ b ͸ෳૉఆ਺Ͱ͋Δɽ͜ͷมҐΛ࣭ྔ m ͷ࣭఺ͷӡಈํఔࣜʹ୅ೖ͠ɼ bΛ ω ͱ ω0 Λ༻͍ͯදͤɽ·ͨɼ|b| 6= 1 ͱͳΔ৚݅Λ‫ٻ‬ΊΑɽ
|b| = 1 ͷ৔߹͸ਐߦ೾ʹɼ|b| < 1 ͷ৔߹͸‫ࡏہ‬೾ͱͳΔɽ࣭ྔ M ͷ࣭఺ͷӡಈΛ߆ଋ͢ΔͷΛ΍Ίͯɼଞͷ࣭఺ͱಉ༷ʹࣗ༝ʹӡಈͤ͞Δɽ
M < m ͷ৔߹ʹۭؒతʹ‫نͨ͠ࡏہ‬४ৼಈ͕ଘࡏ͢ΔɽҎԼɼ͜ͷ‫ن‬४ৼಈΛ‫ٻ‬ΊΑ͏ɽͨͩ͠ɼখ໰ (5) ͱಉ༷ʹ n ൪໨ͷ࣭఺ͷมҐ͸ xn = Re[Bn eiωt ] ͱ͓͘ɽ(6) ࣭ྔ M ͷ࣭఺ʹରͯ͠ɼͦͷ྆ଆʹଓ࣭͘఺ͷมҐ xn ͕ |n| Λେ͖͘͢ΔͱͦΕͧΕ‫ݮ‬ਰ͢ΔΑ͏ʹɼখ໰ (5) Ͱಘͨ b ͷ஋͔Βద੾ͳ΋ͷΛબͿɽ࣭ྔ M ͷ࣭఺ͷ྆ྡʹ͋Δ࣭఺ͷมҐ x−1 ͱ x+1 ΛɼબΜͩ b ͱ x0 Λ༻͍ͯදͤɽ͜͜ͰಘͨมҐΛ࣭ྔ M ͷ࣭఺ͷӡಈํఔࣜʹ୅ೖ͠ɼ͞Βʹ b Λফ‫͢ڈ‬Δ͜ͱͰɼ‫ن‬४֯ৼಈ਺ ω Λ‫ٻ‬ΊΑɽͨͩ͠ɼ ω0 ͱ࣭ྔൺ β = m/M Λ༻͍ͯ ω Λදͤɽ·ͨɼm → ∞ ͷ‫͚͓ʹݶۃ‬Δ ω ͷ઴ۙ஋Λ‫ٻ‬ΊΑɽখ໰ (6) Ͱ‫ٻ‬Ίͨ‫ن‬४ৼಈ͸ɼ࣭ྔ M ͷ࣭఺ͷৼ෯͕࠷େͰɼ੄͕ࢦ਺ؔ਺తʹ‫ݮ‬ਰৼಈ͢Δ‫ͨ͠ࡏہ‬ఆࡏ೾Ͱ͋Δɽ

3

III-2A (ి࣓‫( )ֶؾ‬50 ఺)ਅۭͷ༠ి཰Λ ε0 ͱͯ͠ҎԼͷ໰͍ʹ౴͑Αɽͳ͓ɼղ౴ʹ͸ SI ୯Ґ‫ܥ‬Λ༻͍Αɽ(1) ૯ిՙ͕ e ʹ౳͘͠ɼ಺෦͕Ұ༷ʹଳిͨ͠൒‫ ܘ‬a ͷ‫͕͋ٿ‬Δͱ͢Δɽ‫ٿ‬ͷத৺Λ‫ͯ͠ͱ఺ݪ‬ɼ‫͔఺ݪ‬Βͷ‫ ཭ڑ‬r Λ༻͍ͯɼ͜ͷ‫͕ͭ͘ٿ‬Δి৔Λදͤɽ(2) খ໰ (1) ͷ‫͕ٿ‬΋ͭ੩ిΤωϧΪʔ U0 Λ‫ٻ‬ΊΑɽπ தؒࢠʹ͸ɼதੑͷ΋ͷͱɼૉిՙ e Λ࣋ͭ΋ͷ͕͋Γɼՙి π தؒࢠ͸தੑ πதؒࢠΑΓΘ͔ͣʹΤωϧΪʔ͕ߴ͍ (͢ͳΘͪɼ࣭ྔ͕Θ͔ͣʹେ͖͍)ɽ͜ͷΤωϧΪʔࠩΛ੩ిΤωϧΪʔͷࠩͰઆ໌Ͱ͖Δͱ͢Δɽ͜͜ͰɼҎԼͷ 2 ͭͷϞσϧΛൺֱ͢Δɽ
Ϟσϧ I : π தؒࢠ͸൒‫ ܘ‬a ͷ‫͋Ͱٿ‬Δɽՙి π தؒࢠ͸಺෦͕Ұ༷ʹଳి͍ͯ͠
Δͷʹର͠ɼதੑ π தؒࢠ͸·ͬͨ͘ଳి͍ͯ͠ͳ͍ɽ
Ϟσϧ II: π தؒࢠ͸ɼ͍ͣΕ΋ಉ͡‫ ཭ڑ‬b ཭Εͨ 2 ͭͷ఺ిՙͷର͔ΒͳΔɽՙి π தؒࢠ͸ిՙ 13 e ͱ 23 e Λ࣋ͭ఺ిՙର͔ΒͳΔɽҰํɼதੑ π தؒࢠ͸ɼ
ిՙ 31 e ͱ − 13 e Λ࣋ͭ఺ిՙର͔ΒͳΔঢ়ଶ A ͱɼిՙ 32 e ͱ − 32 e Λ࣋ͭ఺ిՙର͔ΒͳΔঢ়ଶ B ͷॏͶ߹ΘͤͰද͞ΕΔɽதੑ π தؒࢠͷ੩ిΤωϧΪʔ͸ɼ֤ঢ়ଶ A, B ͷ੩ిΤωϧΪʔͷ૬Ճฏ‫Ͱۉ‬༩͑ΒΕΔɽͳ͓ɼԾ
૝తʹ b Λແ‫ݶ‬େͱͨ͠৔߹ɼՙి͓Αͼதੑ π தؒࢠͷ੩ిΤωϧΪʔʹࠩ͸ͳ͍΋ͷͱ͢Δɽ(3) ՙి π தؒࢠͱதੑ π தؒࢠͷ੩ిΤωϧΪʔͷ͕ࠩϞσϧ I ͱϞσϧ II Ͱ౳͍͠ͱͨ͠ͱ͖ɼൺ a/b ͷ஋Λ‫ٻ‬ΊΑɽ

4

III-2B (౷‫( )ֶྗܭ‬50 ఺)N ‫ݸ‬ͷཁૉ͕௚ઢঢ়ʹ࿈ͳͬͨ࠯ঢ়෼ࢠ͕͋Δɽ֤ཁૉ i ͸ͦΕͧΕಠཱʹ σi = +1
·ͨ͸ −1 ͷ͍ͣΕ͔ͷঢ়ଶΛऔΔ͜ͱ͕Ͱ͖ɼ௕͞ l + σi aɼΤωϧΪʔ  + σi ∆Λ΋ͭ΋ͷͱ͢Δɽͨͩ͠ɼl, a, , ∆ ͸ఆ਺Ͱ͋Δɽ͜ͷ࠯ঢ়෼ࢠΛҰఆͷ֎ྗf ͰҾͬுΔͱ͖ͷશ‫ܥ‬ͷΤωϧΪʔ͸E({σi }) =NX( + σi ∆) − f L

(A)

i=1

Ͱ༩͑ΒΕΔɽͨͩ͠ɼ

L=

N
X

(l + σi a)i=1͸࠯ঢ়෼ࢠશମͷ௕͞Ͱ͋ΔɽҎԼͷઃ໰ʹ͓͍ͯɼA ͸෺ཧྔ A ͷԹ౓ T ͷΧϊχΧϧ෼෍Ͱͷ‫ظ‬଴஋Λද͢ɽϘϧπϚϯఆ਺͸ kB ͱ͢Δɽ(1) Թ౓ T ʹ͓͚ΔΧϊχΧϧ෼෍Ͱ σi = 1 Ͱ͋Δ֬཰Λ‫ٻ‬ΊΑɽ(2) σi ͷ‫ظ‬଴஋ σi ͱΏΒ͗ͷେ͖͞ (σi − σi )2 Λ‫ͤࢉܭ‬Αɽ(3) Թ౓ҰఆͰ֎ྗΛ f ͔Β f + δf ʹมԽͤ͞Δͱɼ෼ࢠͷ௕͞ͷ‫ظ‬଴஋͸ L
͔Β L + δL ʹมԽͨ͠ɽ͜ͷͱ͖ɼδf ͕খ͚͞Ε͹ɼۙࣅతʹδL ' kδf(B)ͱද͞ΕΔɽ͜ͷ܎਺ k Λ‫ٻ‬ΊΑɽ(4) ࣜ (A) ΛҰൠԽͯ͠ɼશ‫ܥ‬ͷΤωϧΪʔ͕E({σi }) = E0 ({σi }) − f L!
2ͷ‫ʹܗ‬༩͑ΒΕΔͱ͖ɼࣜ (B) Ͱఆٛ͞ΕΔ k Λ෼ࢠͷ௕͞ͷΏΒ͗ L − LΛ༻͍ͯදͤɽͳ͓ɼE0 ({σi }) ͸ f ʹґଘ͠ͳ͍΋ͷͱ͢Δɽ

5

III-3A (ӳ‫( )ޠ‬60 ఺)
The excerpt below has been adapted from David Lee’s 1996 Nobel Prize in Physics
lecture. Read the passage carefully and answer questions [Q-a]-[Q-j] below inclear, easy-to-read English or symbols where appropriate. The questions can be
answered without advanced physics knowledge.
The major breakthrough in our understanding aaa(a) the revolution a superconductivity
in field of vast a(b) congregate a
a
a
a
a
a
(c)
macroscopic a(d1) BCS theorya(d2) metal a(d3) electrons a (d4) metal a(d5) pairs. aa(d6)
single ground state a(d7) order parameter a(d8) Pauli principle aaa(e) discrete
diatomic molecules. a
a
a
a
a
a
a
a
(f) marching in lock step, aa (g).
Why do electrons form pairs? (h) aaeven a (i) one. a
a
a
a
a
(j).

6

Answer the following questions concerning the passage above.
[Q-a] The underlined words after (a) have been scrambled. Put them in the correct
order. Change the capitalization and include punctuation as necessary.
[Q-b] Which of the following best describes the meaning of the underlined text
after (b) in this sentence? Write one of b1, b2, b3, or b4 in the answer space.
b1) distributeb3) accumulateb2) congressb4) concrete[Q-c] Which of the following words or phrases can replace the underlined text after(c) without changing the meaning of the sentence? Write one of c1, c2, c3,
or c4 in the answer space.
c1) slowc3) limitedc2) frequentc4) large scale[Q-d] Fill in the correct article (“the”, “an”, or “a”) in the underlined spaces (d1)(d8) in the sentences. Capitalize your answers as necessary and if no article
is needed write “φ” in the answer space.
[Q-e] Which of the following best describes the meaning of the underlined text
after (e)? Write one of e1, e2, e3, or e4 in the answer space.
e1) two atomse3) single correlated atomse2) individual pairs of atomse4) pairs of two molecules[Q-f ] Which of the following best describes the meaning of the underlined text
after (f)? Write one of f1, f2, f3, or f4 in the answer space.
f1) move togetherf3) flow in linesf2) move in discrete stepsf4) flow freely[Q-g] Using several sentences and your own expressions explain why the analogy
in the section of the text ending with (g) is suitable to describe the behaviorof Cooper pairs.
[Q-h] Based on the passage after (h), explain why it is surprising that electrons
form pairs using around 20 or 30 words.
[Q-i] Complete the sentence by providing an appropriate word to fill in the spacemarked by (i).
[Q-j] Complete the sentence by providing an appropriate phrase to fill in the spacemarked by (j).

7III-3B (ӳ‫( )ޠ‬40 ఺)
Read the passage carefully and answer questions [Q-1]-[Q-6] below. The questions
can be answered without advanced physics knowledge.
Operational amplifiers, often called op-amps, are active linear circuit elementswith a variety of applications in experimental physics. Real op-amps have a large
internal input resistance, Rint , and a single output voltage, Vo . The output is the
product of a large number, which is called the gain G, and the voltage difference
between its two inputs, V+ and V− . An example circuit diagram showing an opamp’s internal structure is shown in Figure 1.
The behavior of an ideal op-amp can be summarized with the following two
golden rules. First, no current flows into its inputs. Second, when connected ina feedback loop, such as the one connecting Cd and the terminal producing Vo in
Figure 2, the op-amp will produce whatever output reduces the voltage difference
between its input terminals to zero.

Figure 1. Op-amp circuit diagram.

Figure 2. Charge sensitive amplifier.

The circuit in Figure 2 is an example of a charge sensitive amplifier. Such amplifiers are used to amplify the signal from a particle detector. Charge collected in
the detector accumulates in the op-amp’s feedback capacitor, Cf , inducing a voltage which in turn causes the op-amp’s output voltage, Vo , to increase sharply. Its
amplitude will be proportional to the integrated charge deposited in the detector
provided the circuit’s time constant, Rf Cf , is sufficiently longer than the durationof the input pulse. The output will decay exponentially, eventually returning to
its nominal value if the amplifier receives no other input.

8

Answer the following questions concerning the passage above.
[Q-1] Write an expression for the output voltage Vo of the op-amp in Figure 1 interms of its input voltages, V+ and V− , and gain G.
[Q-2] Write an appropriate value for an ideal op-amp’s internal input resistance,
Rint .
[Q-3] Op-amps are powered externally via two additional source inputs, Vs+ and
Vs− , which are drawn as vertical lines connected to opposite sides of the
op-amp symbol between its input and output ends. Draw an example of an
op-amp symbol showing all of its inputs and outputs but without its internal
structure. The positive source, Vs+ , should appear on the bottom of the
op-amp symbol.
[Q-4] Make a rough sketch of the output voltage as a function of time for a charge
sensitive amplifier that has received an input signal that is much shorter
than its time constant.
[Q-5] As in [Q-4] make a rough sketch of the output for two short input signals
with the same integrated charge but separated by a time interval about aslong as the time constant of the amplifier.
[Q-6] A voltage sensitive amplifier can be made by replacing the coupling capacitor, Cd , and feedback system in Figure 2 with two resistors, RA and RB ,respectively. Draw an example of such an amplifier.

9