令和５年度
筑波大学大学院 入学試験
理工情報生命学術院 数理物質科学研究群
物理学学位プログラム 試験問題

専門科目

注意事項（選択、解答についての必要な指示）
１．5 つの問題 (I~V) がある。問題 I では最初の説明文を良く読んでから解答せよ。問題 II~V では、基礎問題と応用問題 ([A], [B]) があり、全ての問題に解答せよ。問題文は、最初に日本語で、次に英語で書かれている。問題の内容は同じものである。
There are five problems (I~V). For the problem I, read carefully the
explanation in the first page. Each of problems II~V consists of basic
and advanced problems ([A], [B]). Answer both of them. All the problems
are given first in Japanese, and then in English. The contents of the
problems are the same.
２．それぞれの問題につき一枚の解答用紙を用いよ。また、問題番号を明記せよ。
Use one sheet of answer paper separately for each problem. Write theproblem number at the top of the sheets.
３．下書き用紙は採点の対象としない。
Draft sheets will not be marked.

I
以下の 10 問のうち 5 問を選択し、問題番号を明記の上、解答せよ。但し、[A]、[B] の 2つの問題群の中から、それぞれ最低 2 問を選択すること。i は虚数単位を表すものとする。

[A]
問１．次の行列式を計算せよ。

!
! 1+a
!
! 1
!
! 1

!
1
1 !!
1+b
1 !!
1
1+c !

問２．次の行列の固有値と固有ベクトルを求めよ。a、b は実数とする。

"

問３．複素数√i − 1 を実数 a、b を用いて

a b
−b a

√

#

i − 1 = a + bi と書くとき、a と b を求めよ。

問４．w = z−iz+i により、z の複素平面の上半面 Imz > 0 は w の複素平面のどのような領域に写像されるか。

問５．複素平面上で |z| = 1 の円周上を反時計回りに１周する経路を C とするとき、次の複素積分を計算せよ。

$

cos z
dz
z
C

[B]
問６．次の微分方程式の一般解を求めよ。

dy2+ y = 8xdx x

(次頁につづく）

問７．!
r = (x, y, z) を３次元空間の位置ベクトルとし、!a = (ax , ay , az ) を定点の位置ベクトルとする。次の関係が、!
r != !a となる全ての点 !r において成り立つことを示せ。

!2∇! は次のように定義される。
∇

!

! =
∇

1
|!r − !a|

!

"

=0

∂ ∂ ∂
,
,
∂x ∂y ∂z"!
問８．A(r)を、位置ベクトル !
r の大きさ r = |!r| だけに依存する 3 次元空間のベクトル場とする。このとき、次の関係を示せ。

#$! · A(r)!
∇× !r = 0

問９．２つの実関数 f (x) と g(x) の区間 [a, b] の積分について、次のシュワルツの不等式%&bf (x)g(x)dx
a

'2

≤

& b

2

f (x) dxa& bg(x)2 dxaが成り立つことを示し、等号が成り立つ条件を述べよ。

問１０．３次元空間のベクトル !
r = (x, y, z) の関数 f (!r) に対するフーリエ変換 F (!k) を、F (!k) =

& ∞

−∞

dx

& ∞

−∞

dy

& ∞

!

dz eik·!r f (!r)−∞

により定義する。関数 f (!
r) が !r の大きさ r = |!r| だけに依存する関数であるとき、F (!k) は !k の大きさ k = |!k| だけに依存する関数となることを示せ。

I
Answer five out of the following ten questions choosing at least two questions ineach of the groups [A] and [B]. Write the question number clearly on the answer sheet.
The imaginary unit is represented by i.

[A]
Q1. Calculate the determinant of the following matrix.
!
!
! 1+a
!
1
1
!
!
! 1
1+b
1 !!
!
! 1
1
1+c !

Q2. Find all the eigenvalues and the eigenvectors of the following matrix where aand b are real numbers.

Q3. The complex number

√

"

a b
−b a

i − 1 is expressed as#√a and b. Find the values of a and b.

i − 1 = a + bi using real numbers

Q4. Consider the upper-half of the complex plane for complex variable z, Imz > 0.
Describe the region in the complex plane occupied by the complex variable w,
given the mapping w = z−iz+i .
Q5. Evaluate the integral

$

cos z
dz,
z
C

where C is a contour counter-clockwise along the circle |z| = 1.

[B]
Q6. Find the general solution for the following differential equation.
dy2+ y = 8x
dx x(Continued on the next page)Q7. Let !r = (x, y, z) be a position vector in three-dimensional space and !a =(ax , ay , az ) be the position vector of a fixed point. Show that the following
relation holds at every point !r except at !r = !a.
!
"
1
2
!
∇
=0
|!r − !a|! is defined by
∇

! =
∇

!

∂ ∂ ∂
,
,
∂x ∂y ∂z"!
Q8. Let A(r)be a vector field that depends on r, the magnitude of the position
vector !r in three-dimensional space. Show that the following relation is true.
#$! · A(r)!
∇× !r = 0
Q9. Show that the following Schwarz inequality holds,%&bf (x)g(x)dx
a

'2

≤

& b

2

f (x) dxa& bg(x)2 dx,afor the integrals involving two real-valued functions of f (x) and g(x) over theinterval [a, b]. Find the condition for equality.
Q10. Define the Fourier transform F (!k) of a function f (!r) that depends on the vector!r = (x, y, z) in three-dimensional space throughF (!k) =

& ∞

−∞

dx

& ∞

−∞

dy

& ∞

!

dz eik·!r f (!r).

−∞

Show that F (!k) depends only on k = |!k|, the magnitude of the vector !k, if thefunction f (!r) depends only on r = |!r|, the magnitude of the vector !r.

II[A]໰ 1. Ґஔ !r = (x, y, z) ʹ͋Δ࣭ྔ m ͷཻࢠʹ‫͢΅ٴ‬ສ༗Ҿྗ͸ɺ
!
Gmρ(!q ) !r − !q!
F (!r) = − d3 q|!r − !q |2 |!r − !q |Ͱ༩͑ΒΕΔɻୠ͠ɺρ(!r) ͸ͦͷ·ΘΓͷີ౓෼෍Ͱ͋ΓɺG ͸ສ༗Ҿྗఆ਺Ͱ͋Δɻ!
q ͸ҐஔΛද͢ੵ෼ม਺Ͱ͋Δɻ͜Εͱ౳Ձͳɺສ༗Ҿྗͷ๏ଇͷඍ෼‫( ܗ‬F! ͱρ ͷඍ෼ํఔࣜ) Λφϒϥԋࢉࢠ! =( ∂ , ∂ , ∂ )∇∂x ∂y ∂zΛ༻͍ͯॻ͚ɻ! ͱ͢Δɻۭؒ಺ʹɺ
໰ 2. ີ౓෼෍͕ ρ(!r) ͷ෺ମ͕࣭ྔ m ͷཻࢠʹ‫͢΅ٴ‬ສ༗ҾྗΛ F
͋ΔྖҬ V Λߟ͑Δɻͦͷ‫ڥ‬քͷด‫ۂ‬໘Λ S ͱॻ͘ɻྖҬ V ͷ಺෦ʹ͋Δશ࣭ྔͱɺS ্ͷྗͷؔ܎ࣜ (ສ༗Ҿྗͷ๏ଇͷੵ෼‫ )ܗ‬Λॻ͚ɻ
໰ 3. ສ༗ҾྗͰ͸ਅۭத (෺࣭͕ଘࡏ͠ͳ͍ྖҬ) ʹ҆ఆͳ௼Γ߹͍ͷҐஔ͸࣮‫ͳ͖Ͱݱ‬͍ɻ͜ͷࣄΛɺສ༗Ҿྗͷ๏ଇʹ‫͍ͯͮج‬આ໌ͤΑɻສ༗Ҿྗͷʰ௼Γ߹͍ͷҐஔʱͱ͸ɺ෺࣭͔Βͷສ༗Ҿྗͷ߹ྗ͕θϩͰɺͦͷҐஔ͔Βগ͠཭ΕͨҐஔͰ͸௼Γ߹͍ͷҐஔͷํ޲ʹ෮‫͕ྗݩ‬ಇ࣌͘ɺͦͷ௼Γ߹͍ͷҐஔ͸ʰ҆ఆʱͰ͋Δͱ‫͏ݴ‬ɻ

（次頁につづく）
次頁につづく[B]൒‫ ܘ‬Rɺີ౓ ρ ͷҰ༷ͳ‫ٿ‬ମͷ಺෦ʹे෼ࡉ͍τϯωϧʢ͜ΕʹΑΓ‫ٿ‬ମ͔Βഉআ͞ΕΔྖҬ͕ແࢹͰ͖Δʣ͕͋Γɺ࣭ྔ m ͷ࣭఺͕τϯωϧ಺Λӡಈ͢Δ (ਤ 1)ɻτϯωϧͷ
‫ܦ‬࿏͸ɺ‫ٿ‬໘্ͷ̎఺Λ݁ͿͳΊΒ͔ͳઢͰ͋Δɻ࣭఺ʹಇ͘ྗ͸ɺ‫ٿ‬ମͷ֤෦͔Βͷສ༗ҾྗͱɺӡಈΛτϯωϧ಺ʹཹΊΔ߆ଋྗͷΈͰ͋Γɺτϯωϧͱͷຎࡲ͸ແ͍΋ͷͱ
͢Δɻ‫ٿ‬ମͷ࣭ྔ͸࣭఺ͷ࣭ྔʹൺ΂ͯѹ౗తʹେ͖͘ɺ‫ٿ‬ମ͸੩ࢭ͍ͯ͠ΔͱΈͳͯ͠Α͍ɻສ༗Ҿྗఆ਺Λ G ͱͯ͠ɺҎԼͷ໰ʹ౴͑Αɻ
໰ 1. ࣭఺ʹಇ͘‫ٿ‬ମ͔Βͷສ༗ҾྗΛɺ‫ٿ‬ମͷத৺Λ‫͢ͱ఺ݪ‬Δ࣭఺ͷҐஔϕΫτϧ "rΛ༻͍ͯදͤɻ
໰ 2. τϯωϧ಺ͷ࣭఺ͷӡಈΛཧղ͢Δҝɺ·ͣɺ໰ 1 ͷྗͷΈΛड͚ͯӡಈ͢Δ࣭఺
Λߟ͑Δ (ਤ 2)ɻ࣭఺͸‫఺ݪ‬Λத৺ͱ͢Δ൒‫ ܘ‬R ͷ‫ٿ‬ͷ಺ଆྖҬΛࣗ༝ʹӡಈͰ͖ɺͦͷྖҬͷ֎ʹඈͼग़͢͜ͱ͸ͳ͍΋ͷͱ͢Δɻ(a) ͜ͷ࣭఺ͷӡಈͰ͸ྗֶతΤωϧΪʔͱ‫఺ݪ‬ͷ·ΘΓͷ֯ӡಈྔ͕อଘ͢Δɻ֤ʑͷදࣜΛॻ͚ɻ(b) ࣭఺ͷӡಈ͸͋Δฏ໘಺ʹཹ·Δ͜ͱΛࣔͤɻ(c) ࣭఺ͷ‫ٿ‬ମ಺ͷ‫ي‬ಓͰɺ‫ٿ‬໘্ͷҰ఺ "r = (x, y, z) = (R, 0, 0) Λ௨Γɺx − y ฏ໘಺ʹ͋Δ΋ͷΛ‫ٻ‬ΊΑɻx, y, z ͸‫ٿ‬ͷத৺Λ‫͢ͱ఺ݪ‬Δ௚ަ࠲ඪͰ͋Δɻඞ
ཁͰ͋Ε͹ɺ‫఺࣭ͱ఺ݪ‬ͷ࠷খ‫ ཭ڑ‬A Λ༻͍ͯ౴͑Αɻ
໰ 3. ࣍ʹɺτϯωϧ಺ͷ࣭఺ͷӡಈΛߟ͑ΔɻτϯωϧͷҰ୺ʹ࣭఺Λ੩͔ʹ (ॳ଎θϩͰʣஔ͍ͨͱ͜Ζɺ࣭఺͸τϯωϧ಺Λӡಈ͠ɺτϯωϧͷଞ୺ʹୡͨ͠ɻͦͷ࣌ͷ࣭఺ͷ଎͞͸τϯωϧͷ‫ܦ‬࿏ʹΑΒͣҰఆͰ͋Δɻ͜ͷࣄΛɺτϯωϧ಺ͷӡಈͰ੒ΓཱͭอଘଇΛ༻͍ͯઆ໌͠ɺଞ୺Ͱͷ࣭఺ͷ଎͞Λ‫ٻ‬ΊΑɻอଘྔͷදࣜΛॻ͖ɺͦΕ͕อଘ͢Δཧ༝Λ؆ܿʹड़΂Δ͜ͱɻ໰ 4. τϯωϧ͸ɺ‫ٿ‬໘্ͷ྆୺Λ݁Ϳ௚ઢʢͷ‫ٿ‬ମ಺ͷ෦෼) ͱ͢Δ (ਤ 3)ɻҰ୺͔Βॳ଎θϩͰτϯωϧʹೖ࣭ͬͨ఺͕ଞ୺ʹୡ͢Δ·Ͱͷ࣌ؒΛ‫ٻ‬ΊΑɻ‫ٿ‬ମͷத৺ͱτϯωϧͱͷ‫཭ڑ‬Λ y0 ͱ͢ΔɻII
[A]
Q1. The gravitational force on a point mass particle with mass m at position !r = (x, y, z)
is given by the following integral equation :!
Gmρ(!q ) !r − !q!
F (!r) = − d3 q,|!r − !q |2 |!r − !q |where ρ(!r) is the density distribution of the ambient material and G is the gravitational constant and !q is position for the integral. Write down the equivalentdifferential form of the Newton’s law of gravitation (differential equation with F!
and ρ) using the following differential operator, nabla :! = ( ∂ , ∂ , ∂ ).
∇∂x ∂y ∂z
Q2. The density distribution ρ(!r) results in a gravitational force F! on a point mass
particle with mass m. Consider a volume V in 3D-space, bounded by surface S.
Write down the relation (integral formula of Newton’s law of gravitation) between
the total mass within the volume V and the force on the closed surface S.
Q3. Within a vacuum, a region devoid of matter, there exist no points of stable equiliblium in the presence of Newton’s gravitational forces. Explain this fact based on
Newton’s gravitational law. Note that any “stable equilibrium position” in the presence of Newton’s gravitational force, the total force from the surrounding materialmust be zero. At a slightly displaced position from the “stable equilibrium position”,
a restoring force must act in the direction of the “stable equilibrium position”.

Continued onon
the next(Continued
thepagenext page)[B]Inside a sphere with radius R and uniform density ρ, a point mass of mass m can go
through a small tunnel-like hole as shown in Fig.1. The tunnel is defined by smoothly
connecting any two different points on the surface of the sphere. The force acting on the
point mass is only the gravitational force from the sphere and the motion of the point
mass is restricted to the path of the tunnel. Any friction between the point mass and thetunnel can be ignored. We may also ignore the change in the gravity and mass of thesphere caused by the hole. The mass of the sphere is large enough compared to the point
mass, so that we can assume the sphere does not move. Answer the following questions
using the gravitational constant G.
Q1. Write down the gravitational force on the point mass from the sphere using theposition !r of the point mass, where the origin is the center of the sphere.
Q2. In order to understand the motion of the point mass within the tunnel, we firstconsider the motion of the point mass just by the force in Q1. without considering
the path of the tunnel as shown in Fig.2. Assume that the point mass can movefreely within the sphere of radius R and that it does not leave the sphere.
(a) As the point mass moves, its kinetic energy and its angular momentum aroundthe origin are conserved. Write down the conservation equation for each quantity.
(b) Show that the motion of the point mass is restricted to a plane.
(c) Find a trajectory of the point mass within the sphere that goes through a pointon the surface of the sphere at !r = (x, y, z) = (R, 0, 0) and stays on the x − yplane. x, y and z are Cartesian coordinates with the origin at the center of thesphere. Express the trajectory in terms of the minimum distance A between
the origin and the trajectory.
Q3. Let us now turn to the motion of the point mass through the tunnel. The point
mass is placed at rest on one end of the tunnel (with zero initial velocity). The
point mass has gone through the tunnel and reached the other end of the tunnel.
In this process, the velocity at the other end is a constant and does not depend on
the path of the tunnel. Explain this fact with a conservation law that applies tothe motion of the point mass within the tunnel and find the velocity of the point
mass at the other end of the tunnel. Write down the conservation equation for the
relevant conserved quantity and explain the reason why the quantity is conserved
briefly.
Q4. Consider the case in which the tunnel is a straight line between two points on thesurface of the sphere as shown in Fig.3. After entering one end of the tunnel withzero initial velocity, find the time needed for the point mass to reach the other end
Continued onon
the next(Continued
thepagenext page)of the tunnel, where the minimum distance between the center of the sphere andthe tunnel is y0 .

IIIh
! = 2π(h はプランク定数) として、以下の問 [A]，[B] に答えよ。

[A]
2 つの 12 スピン粒子のスピン演算子を !sˆ1 = (ŝx1 , ŝy1 , ŝz1 ), !sˆ2 = (ŝx2 , ŝy2 , ŝz2 ) とする。以下の問に答えよ。
問 1.

演算子 ŝz1 ŝz2 /!2 の最大固有値とそれを実現する状態の縮退度を答えよ。

問 3.

演算子 !sˆ1 · !sˆ2 /!2 + (ŝz1 + ŝz2 )/! の最大固有値とそれを実現する状態の縮退度を答

問 2.

演算子 (!
sˆ1 + !sˆ2 )2 /!2 の最大固有値とそれを実現する状態の縮退度を答えよ。
えよ。

[B]
p̂
2
質量 m と周波数 ω を持つ調和振動子のハミルトニアンを Ĥ = 2m
+ mω
2 x̂ とする (x̂22は座標演算子、p̂ は運動量演算子)。生成・消滅演算子 â† , â を

1
â = √
2!
†

!

√

p̂
mωx̂ − i √
mω

"

;

1
â = √
2!

とし、数演算子を n̂ = â† â とする。以下の問に答えよ。
問 1.
問 2.
問 3.

!

√

p̂
mωx̂ + i √mω"

ハミルトニアン Ĥ が Ĥ = !ω(n̂ + 1/2) となることを示せ。

真空状態 |φ0 # を â|φ0 # = 0 により定義する。真空状態の波動関数 φ0 (x) = $x|φ0 #を求めよ。規格化はしなくて良い。

数演算子 n̂ の関数 f (n̂) に対し、f (n̂)â = âf (n̂ − 1) 及び f (n̂)â† = â† f (n̂ + 1) が成り立つ。その理由を簡潔に説明し、この関係式を利用して、e ! Ĥt âe− ! Ĥt およびiie ! Ĥt â† e− ! Ĥt を c1 â + c2 â† の形で表わせ。ただし c1 , c2 は演算子を含まない複素
i

i

数とする。

(次頁に続く)（次頁につづく）

以下では、状態 |ψ! は (â cosh r + â† sinh r)|ψ! = 0 (r は実数) を満たす状態とし、状態 |ψ(t)! は、t = 0 における初期状態 |ψ(0)! = |ψ! からシュレーディンガー方程式∂
i! ∂t|ψ(t)! = Ĥ|ψ(t)! に従い時間発展をおこなった状態とする。

問 4.
問 5.
問 6.

状態 |ψ! に対する波動関数 ψ(x) = "x|ψ! を求めよ。規格化はしなくて良い。

状態 |ψ(t)! に対し、Â(t)|ψ(t)! = 0 を満たす演算子 Â(t) を見つけたい。Â(t) =e− ! Ĥt (â cosh r + â† sinh r)e ! Ĥt と選べることを示し、Â(t) を c1 â + c2 â† の形でii表わせ。ただし c1 , c2 は演算子を含まない複素数とする。

πππ
時刻 t = 2ω
における波動関数 ψ(x, t = 2ω
) = "x|ψ(t = 2ω)! を求めよ。規格化はしなくて良い。

IIIhAnswer the folloing questions [A] and [B]. We set ! = 2πwith the Planck constant h.

[A]Consider two spin-1/2 particles whose spin operators are !sˆ1 = (ŝx1 , ŝy1 , ŝz1 ) and !sˆ2 =(ŝx2 , ŝy2 , ŝz2 ). Answer the folloing questions.
Q1.

Answer the maximal eigenvalue of the operator ŝz1 ŝz2 /!2 and the degeneracy ofthe state(s) realizing it.

Q2.

Answer the maximal eigenvalue of the operator (!sˆ1 +!sˆ2 )2 /!2 and the degeneracyof the state(s) realizing it.

Q3.

Answer the maximal eigenvalue of the operator !sˆ1 · !sˆ2 /!2 + (ŝz1 + ŝz2 )/! and thedegeneracy of the state(s) realizing it.

[B]Consider the harmonic oscillator with mass m and frequency ω, whose Hamiltonian
2

2

p̂
2
+ mω
is given by Ĥ = 2m
2 x̂ (with the coordinate operator x̂ and the momentum

operator p̂). We introduce the creation and annihilation operators â† and â by
!
"
!
"
√
√
1
p̂
1
p̂
†
â = √
mωx̂ − i √; â = √mωx̂ + i √
,
mω
mω
2!
2!
as well as the number operator n̂ = â† â. Answer the following questions.
Q1.

Show the Hamiltonian Ĥ is equal to Ĥ = !ω(n̂ + 1/2).

Q2.

Let us define the vacuum state |φ0 # by â|φ0 # = 0. Answer the wavefunctionof the vacuum state φ0 (x) = $x|φ0 #. The normalization of the wavefunction is
not necessary.

(Continued on the next page)(Continued on the next page)Q3.

For an operator function f (n̂) of the number operator n̂, f (n̂)â = âf (n̂−1) andf (n̂)â† = â† f (n̂ + 1) are hold. Explain succinctly why these relations are hold.
i

i

i

i

And using the relations, express e ! Ĥt âe− ! Ĥt and e ! Ĥt â† e− ! Ĥt in a form ofc1 â + c2 â† , where c1 , c2 are complex numbers without operators.
In the following, let the state |ψ" satisfy (â cosh r + â† sinh r)|ψ" = 0, with a real
number r.

We consider the state |ψ(t)" that satisfies the Schrödinger equation∂i! ∂t|ψ(t)" = Ĥ|ψ(t)" with the initial state |ψ(0)" = |ψ" at t = 0.

Q4.

Answer the wavefunction of the state ψ(x) = #x|ψ". The normalization of the
wavefunction is not necessary.

Q5.

We like find an operator Â(t) characterized by Â(t)|ψ(t)" = 0. Show that youii

can choose Â(t) = e− ! Ĥt (â cosh r + â† sinh r)e ! Ĥt , and express Â(t) in a formof c1 â + c2 â† where c1 , c2 are complex numbers without operators.
Q6.

π
π
π
Answer the wavefunction ψ(x, t = 2ω
) = #x|ψ(t = 2ω)" at t = 2π
. Thenormalization of the wavefunction is no necessary.

IV.
[A]
問 1. 質量 m、電荷 q の荷電粒子が、一様な静磁場中を速さ v 、半径 r の円運動をしている。

(a) 磁束密度の大きさ B を求めよ。
! の向きに x 軸にとる。x 軸の正の向きに一様な静電場 E!
(b) 磁束密度ベクトル Bを印加したとき、荷電粒子の x 方向の変位が X となるまでに要する時間 t を求めよ。

問 2. 静電容量 C のコンデンサに、電圧 V = V0 sin ωt を印加した（ω は角振動数、t は時刻を表す）。このときの容量性リアクタンス XC を求めよ。

問 3. 真空中に電荷 q の点電荷を 1 個だけ置く。このとき、点電荷から距離 r だけ離れた
位置での電場をガウスの法則を用いて求めよ。なお、真空の誘電率を ε0 とする。

（次頁につづく）1

[B]
図１のように、半径 R の導体球の中心を原点 O とし、電荷 +Q を点 A(x, y, z) =(!, 0, 0) に置いた。導体球の外側における電場を鏡像法により求める。ここで、! > R
であり、真空中であるとして、その誘電率を ε0 とする。Q、R、! 及び ε0 を用いて、以下の問に答えよ。

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A

図1
問１. 導体球を接地した場合を考える。導体表面の電位は一定となる。この際、導体球の代わりに、点電荷 QB の仮想の電荷を点 B(!B , 0, 0) に置くことで、導体球の外側の任意の点において、導体球がある場合と同じ電位を与えることができる。QB と !B を求めよ。
問２. 導体球が接地されておらず、電荷を持たない場合を考える。問１の点電荷 QBに加え、点電荷 QC を点 C(!C , 0, 0) に置くことで、導体外部の電位を与えることができる。QC と !C を求めよ。また、点電荷 +Q を点 A から x 軸正方向の無限遠へと運ぶのに要する仕事を求めよ。

(次頁につづく）

次に、図２のように点 A に置かれた点電荷 +Q と、点 D(−!, 0, 0) に置かれた点電
荷 −Q について考える。原点 O から r の距離に点 P があり、線分 OP と x 軸がなす角を θ とする。

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図2# の極座標成分 Er と Eθ を求めよ。ただし、距離 r は ! よ問３. 点 P における電場 E
り十分大きいとして、(!/r) の２次以上の項は無視できるとする。この場合には、AP の距離 rAP と DP の距離 rDP は以下のように近似することができることを用いて良い。
rAP " r − ! cos θ, rDP " r + ! cos θ
問４. 次に、Q/!2 を一定に保ちながら、点 A と D に置かれた点電荷 +Q と点電荷
−Q を無限遠に置く極限 (! → ∞, Q → ∞) を考える。点電荷 +Q と点電荷 −Qが原点付近に作る電場の x 軸方向成分 E0 を求めよ。
問５. 問４の条件に加えて、図１と同様に、接地された半径 R の導体球も中心を原
点 O に一致させて置いた。このとき、中心 O から r の距離にある導体球外部# の極座標成分 Er と Eθ を求めよ。
の点 P における電場 E

IV.
[A]
Q1. There is a charged particle with a mass m and a charge q, which moves in acircular motion with a velocity v in a uniform magnetic field.
(a) Find the magnitude of the magnetic flux density B.
! and apply a(b) Let us define the x-axis along a magnetic flux density vector B,! Find the time t until the displacement of theuniform static electric field E.
charged particle along the x-axis reaches X.
Q2. A voltage V = V0 sin ωt is applied to a capacitor C, where ω and t denote theangular frequency and the time, respectively. Find the capacitive reactance XC .
Q3. Consider a point charge q in vacuum. Using the Gauss’s low, derive the electricfield at the distance r from the point charge. Here, a permittivity of vacuum isdenoted by ε0 .

1(Continued on the next page)[B]A spherical conductor of radius R is placed and its center is the origin O as shownin Figure 1. The electric field outside the conductor is calculated with the methodof the image charges. A point charge +Q is placed at point A (x, y, z) = (!, 0, 0),
where ! > R. The charge is placed in a vacuum, and the dielectric constant of thevacuum is ε0 . Answer the following questions with Q, R, !, and ε0 .

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Spherical conductor

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A

Figure 1
Q1. Suppose that the spherical conductor is grounded. The potential at the conductor surface is constant. In this case, a virtual charge QB can be placed at
the point B (!B , 0, 0) to give the same potential outside the conductor, insteadof placing the conductor. Find QB and !B .

Q2. Next, the spherical conductor is not grounded and has no charge. In addition
to the point charge QB obtained in Q1, a point charge QC can be placed at the
point C (!C , 0, 0) to give the same potential outside the conductor, instead of
placing the conductor. Find QC and !C . Also, find the work required to carry
the point charge +Q from point A to infinity on the x-axis positive direction.

(Continued on the next page)Now, suppose that a point charge −Q is placed at the point D (−!, 0, 0) in addition
to the point charge +Q at the point A as shown in Figure 2. The point P is at a
distance r from the origin O. The angle between the line OP and the x-axis is θ.

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Figure 2# at
Q3. Find the r-component Er and the θ-component Eθ of the electric field E
the point P. Suppose that the distance r is sufficiently larger than !, and thesecond order and higher-order terms of (!/r) can be neglected. In this case,distance rAP between AP and rDP between DP can be approximated as follows:rAP " r − ! cos θ, rDP " r + ! cos θ

Q4. Suppose that the charges +Q and −Q at the points A and D are placed at
infinity while keeping Q/!2 constant (! → ∞, Q → ∞). Find the x-axis
component E0 of the electric field around the origin O due to the charges +Q
and −Q.
Q5. Same as Q1, suppose that a grounded spherical conductor of radius R is also
placed at the origin O in Q4. Find the r-component Er and the θ-component# at the point P outside the conductor at a distance r
Eθ of the electric field Efrom the origin O.

V[A]
以下の問いにそれぞれ答えよ。ボルツマン定数を k とする。
問1. カノニカル集合において平衡状態を指定する熱力学的状態量を以下からすべて選べ。
選択肢： 粒子数、体積、内部エネルギー、温度、化学ポテンシャル、エントロピー
問2. 温度 T におけるカノニカル集合において、各微視的状態 r のエネルギーがEr で、分配関数（状態和）Z である系を考えたとき、微視的状態 r の出現
確率 pr を示せ。
問3. 温度 T におけるカノニカル集合において、ヘルムホルツ自由エネルギーFと分配関数（状態和）Z との関係を記せ。
問4. 以下の状態量から示強性状態量をすべて選べ。
選択肢： 粒子数、体積、内部エネルギー、温度、化学ポテンシャル、エントロピー（次⾴につづく）

[B]
温度 " 、体積 # の三次元の箱の中に質量 $ の気体分子が入っている状態を考える。気体分子は理想気体であるとき、以下の問いに答えよ。ただし、% = 1/(*")（* がボルツマン定数）を逆数温度、ℎ をプランク定数（ℏ = ℎ/(2/)）とする。
問1. 気体分子が 0 個の時、分配関数 1 が、
#!/"
#!
$
1=
3
4
0! 2/ℏ" %となることを示せ。ただし、以下の積分公式を使ってもよい。
(/!
(: > 0)5 6 %&' 78 = 9:%(
問2. 別の系との間で気体分子のやり取りがあるとすると、化学ポテンシャル =を用いて、大分配関数 Ξ と分配関数 1 の関係を示しなさい。
問3. 圧力 ? は大分配関数 Ξ と体積 # を用いて、
log Ξ
?=
%#
と表すことができる。この時、問 1 の分配関数 1 と問 2 の関係を用いることより、=, $, %, ℏ を使って理想気体の圧力 ? を求めよ。
次に、上述の分子の表面への吸着（吸着サイト系）について考える。表面には0) 個の分子が吸着できる吸着サイトがあり、各々の吸着サイトには、気体分子が一つだけくっつくか、何もついていないかのいずれかだとする。分子一つあたりのそれぞれ、化学ポテンシャルを = 、吸着エネルギーを −E < 0 とする。また、
吸着分子間の相互作用は無視できるものとする。この時、以下の問いについてそれぞれ答えよ。
問4. 吸着サイト系の大分配関数 Ξ が、!"

Ξ = G1 + 6 *(,-.) I
であることを示せ。

（次頁につづく）

問5. 表面に吸着している分子の総数 ⟨0⟩ が⟨0⟩ =0) 6 *(,-.)
1 + 6*(,-.)

であることを示せ。
問6. 吸着サイトのどの程度が分子でふさがっているかという被覆率をΘ ≡ ⟨0⟩/0) と定義すると、問 3 で求めた圧力 ? と問 5 で求めた⟨0⟩ を用いて、?
Θ=? + ?0 (%)となることを示せ。ここで、?0 (%) =
である。

#/"
6 %*,
$
3
4
% 2/ℏ" %V[A]
Answer the following questions. Let k be the Boltzmann constant.
Q1. Choose all thermodynamic properties that specify equilibrium in a canonicalensemble from the following:“particle number, volume, internal energy, temperature, chemical potential, entropy”
Q2. Write down the probability pr of a state r with energy Er in the canonical ensemble
with temperature T and partition function (sum over states) Z.
Q3. Write down the relation between the partition function (sum over states) Z and theHelmholtz free energy F in the canonical ensemble with temperature T.
Q4. Choose all intensive properties from the following:“particle number, volume, internal energy, temperature, chemical potential, entropy”(Continued on the next page)

[B]Consider an ideal gas consisting of independent molecules each with mass ! in a boxof volume # at temperature $. Let % = 1/(*$) (* is the Boltzmann constant) be thethermodynamic beta and ℎ be the Planck constant (ℏ = ℎ/(2/)). Answer the following
questions.
Q1. For 0 molecules, show that the partition function 1 is given by
1=

#!/"
#!
!
3
4
.
0! 2/ℏ" %You can use the following integral formula without proof:(/!
(; > 0).
6 7 %&' 89 = :;%(Q2. Write down the relation between the grand partition function Ξ and partition
function 1 by using the chemical potential ?.
Q3. The pressure @ can be expressed using the grand partition function Ξ and volume
# as
log Ξ
@=
.
%#
Find the pressure @ of an ideal gas in terms of ?, !, %, ℏ by using the results from
Q1 and Q2.
Next, consider the adsorption of the above-mentioned molecules on a surface. Let us
assume that the surface has adsorption sites that can adsorb 0) molecules, in which eachadsorption site has either one molecule attached to it or none. Let the chemical potential
and adsorption energy of each molecule be ? and −F < 0, respectively. Let us assume
that the intermolecular interactions of the adsorbed molecules are negligible. Answer thefollowing questions.
Q4. Show that the grand partition function Ξ in the adsorption system can be expressedas!"Ξ = H1 + 7 *(,-.)J .
(Continued on the next page)Q5. Show that the total number of molecules ⟨0⟩ adsorbed on the surface is⟨0⟩ =0) 7 *(,-.).
1 + 7*(,-.)

Q6. Let us define the coverage as the fraction of adsorption sites blocked by molecules,
that is Θ ≡ ⟨0⟩/0) . Show that the coverage can be expressed using the results of Q3
and Q5 as@Θ=,@ + @0 (%)
where@0 (%) =

#/"
7 %*,
!
3
4
.
% 2/ℏ" %