Department of Applied Physics
Entrance Examination Booklet
Physics(Answer the 4 Problems in this Booklet)
August 30 (Tuesday) 9:00 – 13:00, 2022

REMARKS
1. Do not open this booklet before the start is announced.
2. Inform the staff when you find misprints in the booklet.
3. Answer the four problems in this booklet.
4. Use one answer sheet for each problem (four answer sheets are given). You may
use the back side of each answer sheet if necessary.
5. Write down the number of the problem which you answer in the given space at thetop of the corresponding answer sheet.
6. You may use the draft sheets of this booklet to make notes, but you must not detachthem.
7. Any answer sheet with marks or symbols irrelevant to your answers will beconsidered invalid.
8. Do not take this booklet and the answer sheets with you after the examination.

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Problem 1
We consider a quantum mechanical particle conﬁned to an inﬁnite-depth square-well potential in
one or two dimensions. Let m represent the mass of the particle, i the imaginary unit, a a positive
constant, and ℏ the Planck constant divided by 2π. We neglect the spin degree of freedom, andwave functions are always normalized. You may use the following relations:∫ 1(π )
32x cosx sin(πx)dx =2
9π 2 ,−1∫ 1(π )
4x cos(πx)dx =cos2
3π ,−1∫ 1(π )
8x sin(πx)dx =sin23π .
−1[1] First, we consider the one-dimensional case. The time-independent Schrödinger equation iswritten asĤψn (x) = En ψn (x),where En (E1 < E2 < E3 · · · ) is the energy eigenvalue and ψn (x) is the correspondingeigenfunction. It is noted that n is an integer greater than or equal to 1. The HamiltonianĤ is expressed asℏ2 d 2Ĥ = −+ V (x),2
{ 2m dx
0 (|x| ≤ a)
V (x) =∞ (|x| > a).
[1.1] Find ψn (x) and En , where ψn (x) is a real function.
[1.2] Let P̂ be the operator which transforms the wave function ψ(x) to ψ(−x). Show thatψn (x) is an eigenstate of P̂ and ﬁnd the corresponding eigenvalue.

[2] Next, we consider the two-dimensional case. The Hamiltonian Ĥ is expressed as( 2)ℏ2∂∂2Ĥ = −++ V (x, y),2m ∂x2 ∂y 2{0 (|x| ≤ a and |y| ≤ a)
V (x, y) =∞ (|x| > a or |y| > a).
For integers nx and ny greater than or equal to 1, a wave function written as Ψnx ,ny (x, y) =ψnx (x)ψny (y) is an energy eigenstate of this system, where ψn (x) are taken from Question [1].
The ground state is Ψ1,1 (x, y), the ﬁrst excited states are linear combinations of Ψ2,1 (x, y)and Ψ1,2 (x, y), and the second excited state is Ψ2,2 (x, y).
[2.1] Let M̂ be the operator which transforms the wave function Ψ(x, y) to Ψ(y, x). Show
that the ground state and the second excited state are eigenstates of M̂ and ﬁnd thecorresponding eigenvalues.

[2.2] Among the ﬁrst excited states, Ψ2,1 (x, y) and Ψ1,2 (x, y) are not eigenstates of M̂ , but
appropriate linear combinations of these states can be simultaneous eigenstates of Ĥ andM̂ . Find all such simultaneous eigenstates and identify the corresponding eigenvalue ofM̂ for each of them. Here, express the simultaneous eigenstates by linear combinationsof Ψ2,1 (x, y) and Ψ1,2 (x, y), choosing the coeﬃcient of Ψ2,1 (x, y) to be a positive real
number.
[2.3] For each of the states found in Question
the expectation value of the( [2.2], calculate)∂∂angular momentum operator L̂ = −iℏ x ∂y − y ∂x .
[2.4] Let Ĉ4 be the operator which transforms the wave function Ψ(x, y) to Ψ(−y, x). Show
that the ground state and the second excited state are eigenstates of Ĉ4 and ﬁnd thecorresponding eigenvalues.
[2.5] Among the ﬁrst excited states, Ψ2,1 (x, y) and Ψ1,2 (x, y) are not eigenstates of Ĉ4 , but
appropriate linear combinations of these states can be simultaneous eigenstates of Ĥ andĈ4 . Find all such simultaneous eigenstates and identify the corresponding eigenvalue ofĈ4 for each of them. Here, express the simultaneous eigenstates by linear combinationsof Ψ2,1 (x, y) and Ψ1,2 (x, y), choosing the coeﬃcient of Ψ2,1 (x, y) to be a positive realnumber.
[2.6] For each of the states found in Question
the expectation value of the( [2.5], calculate)∂∂angular momentum operator L̂ = −iℏ x ∂y − y ∂x .

Problem 2
A parametric oscillation is an oscillatory phenomenon in the presence of a periodic modulationof a system parameter. We consider the system shown in Fig. 1 as a simple example for theparametric oscillation. A point mass m is suspended by a bar with length L and the supporting
point oscillates along the vertical direction with an amplitude a(≥ 0) and an angular frequencyΩ(> 0). The position of the mass at time t is (x, y) in the cartesian coordinate frame. We define
the origin of the coordinate frame as in Fig. 1. Note that the y-axis points downward in thevertical direction. The position of the supporting point is (0, a cos Ωt). Let the mass of the bar benegligible, let its tension be T (> 0), and let it never deform. The angle θ is measured from thevertical direction as in Fig. 1. The gravitational acceleration constant is denoted by g(> 0). We
neglect the friction at the supporting point and the air friction. In the following, the dot superscript
indicates the time derivative.
[1] Express the position (x, y) of the mass at time t using L, θ, a, Ω, and t.
[2] Express the equations of motion for (x, y) using ẍ, ÿ, m, T, θ, and g.
[3] By diﬀerentiating the results of Question [1], express ẍ and ÿ using L, θ, θ̇, θ̈, a, Ω, and t.
[4] Write down the equation of motion for θ. The tension T should not appear in the answer.
[5] Assuming that θ is small, we approximate sin θ as θ. We use this approximation in the following questions. Determine the solution θ0 (t) of the equation of motion derived in Question [4]when a = 0, θ0 (0) = A, and θ̇0 (0) = 0. Also, determine the characteristic angular frequency
ω0 (> 0) using g and L.
[6] We set the solution for small a to be θ(t) = θ0 (t) + aθ1 (t) using θ0 (t) derived in Question[5]. Assuming that the solution can be determined by perturbative expansion in a, find theequation for θ1 (t) in the first-order approximation.
[7] We want to obtain the solution θ1 (t) of the equation derived in Question [6]. We assume
the following form of a special solution with time-independent constants u1 and u2 and thecharacteristic angular frequency ω0 , which was obtained in Question [5]:
θ1 (t) = u1 cos[(Ω + ω0 )t] + u2 cos[(Ω − ω0 )t].
Express the condition for the existence of this special solution using Ω, g, and L. In addition,
determine u1 and u2 under this condition, and express them using only L, A, Ω, and g.

Figure 1

Problem 3
A container of a large heat capacity and a large mass has an internal cylindrical cavity, which holds
a gas of N (≫ 1) identical particles. Let L be the height of the cavity, R be its radius, and deﬁneν = N/L. We use a cartesian coordinate system xyz at rest with its origin O located at the centerof the cavity and its z axis aligned with the axis of rotational symmetry (Figure 1). The Boltzmann
constant and the Planck constant are denoted by k and h, respectively.
In the following, we refer to the z component of an angular momentum vector simply as angularmomentum. We refer to the pressure exerted on the internal side of the container as side pressure.
The natural logarithm of a positive number t is denoted by log t.
We represent the position of a particle by vector r = (x, y, z) and the corresponding momentum byvector p = (px , py , pz ). The Hamiltonian of one particle is given by H1 (r, p) = (p2x + p2y + p2z )/(2m),where m is its mass, and its angular momentum is given by M1 (r, p) = xpy −ypx . The Hamiltonianof N particles is given by the sum of one-particle Hamiltonians. Let ρ(r, p) denote the probability
density function for the distribution of one particle in phase space.
Treat the motion of the particles in classical mechanics and neglect the eﬀect of gravity on the∫∞√2particles. You may use the improper integral −∞ e−t dt = π.
[1] Consider the case where the container is at rest, ﬁxed to the ﬂoor, and is regarded as a heatbath of temperature T (Figure 2). The container and the gas are in thermal equilibrium. Theradius R is suﬃciently large and the motion of the N particles follows a canonical distribution.
The probability density function ρ(r, p) of one particle is then given by()1
H1 (r, p)ρ(r, p) =exp −,kT
h3 LZ̃1
where Z̃1 is a quantity that depends neither on r nor on p. Answer the following questions.
[1.1] The quantity Z̃1 can be regarded as a function of temperature T and radius R. Using
the normalization condition of the probability density function ρ(r, p),
∫ ∞−∞

∫ ∞
dpx−∞

∫ ∞
dpy−∞

∫ L/2dpz∫dz−L/2dxdy ρ(r, p) = 1,x2 +y 2 ≤R2calculate Z̃1 (T, R) and express it in terms of kT, R, and γ = π 5/2 (2m)3/2 h−3 .
[1.2] Let F be the Helmholtz free energy of this gas and deﬁne F̃ = F/L. By considering
the partition function of N particles in the limit of N, L → ∞ at ﬁxed ν = N/L, deriveF̃ and express it in terms of kT, ν, and Z̃1 (T, R). You may use the approximationlog N ! ≈ N log N − N .
[1.3] By considering the work required for a virtual inﬁnitesimal change in radius R of thecavity, calculate the side pressure P from the free energy F in (Question)[1.2] and express
it in terms of kT, R, ν, Z̃1 (T, R), and the partial derivative∂ Z̃1 (T,R).
∂RTIn addition,express P in terms of kT, R, and ν by replacing Z̃1 (T, R) with the answer to Question[1.1].

[2] Consider the case where the container is suspended at the center of its top surface and isallowed to rotate freely around the z axis without friction (Figure 3). Assume that thecontainer has rotational symmetry around the z axis. Suppose that the container is rotatingaround the z axis at a constant angular velocity ω and that its temperature is T . The
container and the gas are in thermal equilibrium and the probability density function ρ(r, p)of one particle is given by()H1 (r, p) − ωM1 (r, p)1
exp −ρ(r, p) =,kT
h3 LZ̃1
where Z̃1 is a quantity that depends neither on r nor on p. Answer the following questions.
[2.1] Noting that the position r is conﬁned to the cavity, derive the minimum value ofH1 (r, p) − ωM1 (r, p) in phase space and express it in terms of R, ω, and m.
[2.2] Let (vx (r), vy (r), vz (r)) be the expectation value of the velocity vector of a particle inthe vicinity of position r. Derive vx (r) and vy (r).
[2.3] The quantity Z̃1 can be regarded as a function of temperature T , radius R, and angularvelocity ω. Calculate Z̃1 (T, R, ω) in this case and express it in terms of kT, R, m, ω, andγ = π 5/2 (2m)3/2 h−3 .
[2.4] Express the side pressure P in terms of kT, R, ω, ν, and m. In addition, derive the valueof P in the limit of low temperature at ﬁxed angular velocity ω.
[2.5] Calculate the moment of inertia I of the gas around the z axis and express it in termsof kT, ω, N , and ε = mR2 ω 2 /2.
[2.6] Calculate the heat capacity Cω of the gas at constant angular velocity ω and express itin terms of k, T, N , and ε = mR2 ω 2 /2. In addition, derive the values of heat capacity
Cω in the limits of high temperature and of low temperature.

Container

Container

Angularvelocity 𝜔𝜔Cavity

CavityCavityFigure 1

Figure 3
Figure 2

Problem 4
We consider a classical charged particle with mass m and charge q (> 0) in vacuum, whose position
coordinate at time t is given by r(t) = (x(t), y(t), z(t)). We study ways to conﬁne it in a boundedregion around the origin (0, 0, 0) by using electromagnetic ﬁelds. Suppose that the velocity of thecharged particle is suﬃciently small compared to the speed of light, and the eﬀect of the electromagnetic ﬁeld generated by the charged particle itself can be neglected. Use i as the imaginary
unit and answer the following questions.
[1] First, suppose that a static electric ﬁeld is imposed so that the scalar potential at positionr = (x, y, z) is given by
ϕ(r) = ax2 + by 2 + cz 2 .

(1)Here, a, b, and c are constants. The electric ﬁeld is given by E(r) = −∇ϕ(r).
[1.1] The charged particle can be conﬁned in a bounded region if a, b, and c of the scalarpotential in Eq. (1) are all positive. However, it is impossible to impose a static electricﬁeld that makes all of a, b, and c positive since a + b + c = 0 holds in reality. Showa + b + c = 0 in Eq. (1).
Below, let us consider a way to conﬁne the charged particle in a bounded region by imposinga uniform static magnetic ﬁeld in addition to the scalar potential in Eq. (1). Suppose thatca = b = − < 0 in Eq. (1), and consider the situation where a static magnetic ﬁeld with2
magnetic ﬂux density B = (0, 0, B) is imposed.
[1.2] Write down the equation of motion for the position r(t) of the charged particle in terms
dr(t) d2 r(t)of,, m, q, ϕ (r(t)), and B by considering the force acting on the charged
dt
dt2
particle from the static electric ﬁeld and the static magnetic ﬁeld.
[1.3] The charged particle is conﬁned along the z axis and undergoes harmonic oscillations.
Express its angular frequency in terms of m, q, and c.
[1.4] By deﬁning u(t) = x(t) + iy(t), express the equation of motion for u(t) in terms of u(t),du(t) d2 u(t), m, q, c, and B.
,dt
dt2[1.5] Find the condition for m, q, c, and B under which the charged particle is conﬁned in abounded region along the x and y axes for any initial condition.
[2] Next, suppose that, instead of Eq. (1), the scalar potential at position r = (x, y, z) is givenbyϕ̃(r, t) = (αx2 + βy 2 + γz 2 ) cos ωt(2)and varies with time at angular frequency ω. Here, α, β, and γ are constants. In this case,under an appropriate condition, the charged particle can be conﬁned in a bounded region
along all three axes x, y, and z. Below, let us explain this phenomenon by focusing on thex component of the motion of the charged particle. Note that the magnetic ﬁeld can beneglected and the electric ﬁeld is given by Ẽ(r, t) = −∇ϕ̃(r, t).

[2.1] The x component of the position of the charged particle can be expressed as a functionωtof τ by x(τ ), where τ =. In this case, the equation of motion for x(τ ) can be written2
asd2 x(τ )+ 2λx(τ ) cos 2τ = 0,dτ 2(3)where λ is a constant. Find λ.
[2.2] Suppose that the solution of Eq. (3) is given byx(τ ) =∞∑Cn cos [(2n + Ω)τ ]

(4)

n=−∞

for an appropriate initial condition. Here, Ω and Cn (n is an integer) are independent of
τ , and 0 < Ω < 1 holds. Express the relation between Ω, Cn−1 , Cn , and Cn+1 in termsof λ and n.
[2.3] When 0 < λ ≪ 1 in Question [2.2], the contribution of Cn in Eq. (4) can be neglected
except for C−1 , C0 , and C1 . As an approximation, we use the relation in Question [2.2]for n = −1, 0, 1 only. Under this condition and under the assumption C0 ̸= 0, we obtain
C1C−1== ελ and Ω = δλ up to ﬁrst order in λ, where ε and δ are constants. Find εC0C0and δ.
[2.4] Express the solution x(τ ) of Eq. (3) in terms of C0 , λ, and τ , using the approximationsin Question [2.3]. When C0 > 0, draw a sketch of x(τ ) as a function of τ .