Department of Applied Physics
Entrance Examination Booklet
Physics(Answer the 4 Problems in this Booklet)
August 27 (Tuesday) 9:00 – 13:00, 2024

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.

Examinee number

No.

Write down your examinee number above

Problem 1
Consider the one-dimensional motion of a particle connected to one end of a spring. The particle
has mass m and the spring constant is mω02 (ω0 > 0). The other end of the spring is attached to an
immobile wall. The position x(t) of the particle is zero when the spring is of natural length. Thedx(t)particle is subject to a friction force −ζand an external force f (t). Here, ζ > 0 is the friction
dt
coeﬃcient and t is time. In the following questions, i represents the imaginary number unit.
[1] First, consider the case where the external force f (t) is a periodically oscillating force. Here,the position x(t) and the external force f (t) are expressed as complex numbers. In particular,
let f (t) be deﬁned as f (t) = F0 eiΩt with an angular frequency Ω > 0.
[1.1] Find x(t) in steady state.
[1.2] Deﬁning the response function ε = x(t)/f (t), ﬁnd the Ω dependences of the real part ε′and the imaginary part ε′′ of ε.
[1.3] When ζ ≪ mω0 and |Ω − ω0 | ≪ ω0 , draw the approximate shape of the Ω dependencesof ε′ and −ε′′ . In particular, annotate the angular frequencies where ε′ and −ε′′ reach
their extrema.
[2] Next, consider the case where the external force f (t) is a ﬂuctuating force. Here, the position
x(t) and the external force f (t) are expressed as real numbers. Let f (t) be represented by arandom function satisfying ⟨f (t)f (t + τ )⟩ = Iδ(τ ) and ⟨f (t)⟩ = 0. I is a positive constant.
For some physical quantity a, ⟨a⟩ represents the ensemble mean (the mean in an ensemble ofcountless identical systems). δ(τ ) is the Dirac delta function. In the following questions, weconsider the steady state of the motion of the particle when ζ < 2mω0 .
⟨(⟨() ⟩) ⟩dx(t) 2dx(t) 22[2.1] Whenis represented as v , namely,= v 2 , express the averdt
dt
age squared displacement of the particle ⟨x(t)2 ⟩ in terms of v 2 . Also, show the derivationof this result. You may use the fact that there is no correlation between f (t) and x(t):⟨f (t)x(t)⟩ = ⟨f (t)⟩⟨x(t)⟩.
∫ ∞[2.2] Express the Fourier transform X(ω) =x(t)e−iωt dt of x(t) in terms of the Fourier−∞transform F (ω) of f (t). In addition, express the power spectrum Ix (ω) = ⟨|X(ω)|2 ⟩ interms of the power spectrum If (ω) = ⟨|F (ω)|2 ⟩ of f (t).
[2.3] Find
the correlation function ⟨x(t)x(t + τ )⟩ of x(t), using the general relation Ia (ω) =∫∞−∞⟨a(t)a(t + τ )⟩e−iωτ dτ between the power spectrum Ia (ω) of the physical quantitya(t) and the correlation√function ⟨a(t)a(t+τ )⟩ of a(t). Here, a(t) is a real-valued function.
ζ2You may deﬁne ω1 = ω02 −and include it in your answer.
4m2[2.4] Express I in terms of v 2 , by setting τ = 0 in the expression for the correlation function
obtained in Question [2.3] and then comparing to the result of Question [2.1].

Problem 2Consider quantum mechanical particles scattered by a potential barrier in one dimension. The
time-independent Schrödinger equation is given by−ℏ2 d2 ψ(x)+ V (x)ψ(x) = Eψ(x),2m dx2(1)

where m is the mass of a particle, ℏ is Planck’s constant divided by 2π, V (x) is the potential,and E is the energy of the particle. Here, we consider the case where particles propagating in thepositive direction of the x-axis hit the potential barrier. Answer the following questions with i as
the imaginary number unit.
[1] Let us consider a potential barrier given by V (x) = V0 (|x| ≤ a) and V (x) = 0 (|x| > a).
When 0 < E < V0 , the wave function of the particle is written asikx−ikx (x < −a)Ae + Beψ(x) = Ceγx + De−γx (|x| ≤ a)(2)
 ikx
Ee(x > a),√2m2mE, γ =(V0 − E), and V0 and a are positive constants. Answer Queswhere k =2ℏℏ2tions [1.1] to [1.5] using k and γ but not E or V0 .
√[1.1] Using the boundary conditions that the wave function and its derivative function arecontinuous, ﬁnd two equations relating the coeﬃcients A, B, C and D.
[1.2] Similarly, ﬁnd two equations relating the coeﬃcients C, D and E.
[1.3] From the equations derived in Questions [1.1] and [1.2], we obtain
()()
()()
ik − γik + γik + γik − γB=e−2γa −e2γa .
E2ik
2γ2ik2γ(3)Using this relation, ﬁnd the transmission probability (transmittance) T1 of the particle
passing through the potential barrier. You may use the hyperbolic functions cosh(α) =eα + e−αeα − e−αand sinh(α) =.
22γ[1.4] Express T1 in terms of k and a when ≪ 1 and aγ ≪ 1.
k[1.5] Express T1 in terms of k, γ and a when aγ ≫ 1.
[2] Let us consider the case with E > V0 for the same potential barrier as in Question [1]. Answer
Questions [2.1] and [2.2] using E and V0 but not k or γ.
[2.1] Find the E dependence of the transmittance T2 in this case.
Eas the variable on the horizontal[2.2] Plot schematically the E dependence of T2 with
V0axis, noting the value of T2 when E ≫ V0 . Also, ﬁnd all conditions where T2 is at a localmaximum.
(Continue to the next page)[3] Next, we consider particles scattered by a potential barrier given by V (x) = 0 (x < −a),V (x) = V0 (|x| ≤ a) and V (x) = −V1 (x > a), where V1 > 0 and 0 < E < V0 .
[3.1] Express the transmittance√ T3 and the√reﬂectance R3 using k, γ√and k1 but not E, V0 or
2m2m2mV1 . Here, we deﬁne k =E, γ =(V0 − E), and k1 =(E + V1 ).
22
ℏ
ℏ
ℏ2
[3.2] Find the asymptotic value of T3 when increasing V1 while keeping E ﬁxed.

Problem 3We consider the Ising model, where N spins are each interacting with all the other spins with auniform coupling constant. Here, N is a natural number. The energy is given byE=−J ∑σα σβ ,N(1)α<βwhere σα and σβ are variables taking values of ±1, and J is a nonzero real number. We assume
that α and β represent natural numbers satisfying 1 ≤ α ≤ N and 1 ≤ β ≤ N . The sum is taken
for all combinations satisfying the condition α < β.
[1] First, we consider the cases with N = 2 and 3.
[1.1] Equation (1) for N = 2 is written asJ
E = − σ1 σ2 .
2(2)Determine all the energies E and the corresponding degeneracies W .
[1.2] Equation (1) for N = 3 is written as
E=−J(σ1 σ2 + σ2 σ3 + σ1 σ3 ) .
3(3)Determine all the energies E and the corresponding degeneracies W .
[1.3] Considering the cases of positive J and negative J separately, determine the degeneracy
W of the ground state for N = 3.
[2] Next, we consider the case where N is an arbitrary natural number. We deﬁne the total spinbyN∑σα = N − 2M,(4)α=1where M is the number of spins taking the value σα = −1.
[2.1] Determine E in terms of M .
[2.2] Considering the cases of positive J and negative J separately, determine the groundstate energy E and the corresponding degeneracy W .
(Continue to the next page)1

Next, we assume N ≫ 1 and J > 0. Let m = M/N be the fraction of spins with σα = −1 andconsider a thermal equilibrium state with the temperature T . Let S be its entropy and C be its
heat capacity. Let kB be the Boltzmann constant.
[3] We consider the thermal equilibrium state with 0 < m < 1/2.
[3.1] Determine E/N as a function of m.
[3.2] Determine S/N as a function of m. Use the Stirling formula
log n! ≃ n log n − n,(5)which is a good approximation for suﬃciently large n.
[3.3] Determine the temperature T as a function of m using the results of Questions [3.1] and[3.2], and the formula
dS
1= .
(6)
dE
T
[3.4] In the low temperature limit kB T ≪ J, the condition m ≪ 1 is satisﬁed. Determine thetemperature dependence of m.
[3.5] Using the result of Question [3.4], determine the temperature dependence of S/N in thelow temperature limit.
[3.6] Using the result of Question [3.5], determine the temperature dependence of C/N in thelow temperature limit.
[4] There is a critical temperature Tc above which m = 1/2 in thermal equilibrium. Determine
the temperature Tc by taking the limit m → 1/2 in Eq. (6).

2Problem 4
Suppose there is an electrically neutral and uniform plasma in which gas atoms are separated into
electrons and positive ions. Consider an electromagnetic wave of angular frequency ω(> 0), which
travels in the positive direction of the z-axis in this plasma. It is assumed that only electrons aremoved by the electromagnetic ﬁeld and that the movement of positive ions is negligible. We denotethe mass of an electron by m, the charge of an electron by −e, and the number density of electrons
in the plasma by N .
Note that the Coulomb interaction between particles, collisions between particles, and the eﬀectof gravity need not be considered. Assume that the thermal motion of the electrons is isotropic
and has no eﬀect on the electric current density. Assume that the speed of electrons is suﬃciently
small compared to the speed of light c in vacuum, and that the force exerted on electrons by themagnetic ﬁeld of electromagnetic waves is negligible. The electric ﬁeld E of the electromagnetic
wave, the velocity v of electrons, and the electric current density J are treated as having only xand y components and as being uniform in the plane perpendicular to the z-axis.
Under these assumptions, we consider the following wave equation for the electric ﬁeld E(z, t) of
the electromagnetic wave at time t,
1 ∂ 2 E(z, t)1 ∂J (z, t)∂ 2 E(z, t)−= 2.
222∂zc∂tc ε0 ∂t(1)Here, ε0 is the permittivity of vacuum.
Let us express the electric ﬁeld of an electromagnetic wave obeying this wave equation in thecomplex number representation as
E(z, t) = E0 exp[i(kz − ωt)],(2)

where E0 is a constant vector, i is the imaginary number unit, and√k is a complex wavenumber. In
answering the questions, you can use the plasma frequency, ωp =e2 N, as appropriate.
ε0 m[1] Write the equation of motion for an electron using the velocity v(z, t) of electrons and theelectric ﬁeld E(z, t) of the electromagnetic wave.
[2] Considering the answer to Question [1], express the time derivative of the current density,∂J (z, t), in terms of the electric ﬁeld E(z, t) of the electromagnetic wave.
∂t[3] Express k 2 as a function of the angular frequency ω. Also ﬁnd the condition for the angularfrequency ω so that the electromagnetic wave can travel for a long distance in this plasma.
（Continue to the next page）

A uniform static magnetic ﬁeld with magnetic ﬂux density Bz (> 0) is applied to the plasma along
the positive direction of the z-axis. Consider an electromagnetic wave with angular frequency ω,which travels in the positive direction of the z-axis in this plasma. Assume that the motion ofelectrons is caused only by the electric ﬁeld of the electromagnetic wave and by the static magnetic
ﬁeld. We consider that the electric ﬁeld E of the electromagnetic wave, the velocity v of electrons,
and the current density J all have only x and y components. We express the electric ﬁeld of theelectromagnetic wave as in Eq. (2). In the following, it is assumed that the angular frequency ω ofthe electromagnetic wave and the plasma frequency ωp are both larger than the cyclotron frequency
eBz
ωc =.
m
[4] The x and y components for the electric ﬁeld E(z, t) of the electromagnetic wave and thevelocity v(z, t) of electrons are written as()()
Ex (z, t)vx (z, t)E(z, t) =,v(z, t) =.
(3)
Ey (z, t)vy (z, t)
Write down the x and y components of the equation of motion for the electron separately.
[5] Assume the current density J (z, t) to be proportional to exp[i(kz − ωt)]. Then we express
the current density in terms of a 2-by-2 matrix σ as follows:()()
Jx (z, t)
Ex (z, t)J (z, t) ==σ.
(4)
Jy (z, t)
Ey (z, t)
Write down the matrix σ in terms of ε0 , ω, ωc , and ωp .
[6] From Eq. (1), we can derive an eigenequation described as()()
Ex (z, t)
2 Ex (z, t)
A=k.
Ey (z, t)
Ey (z, t)
Here A is a 2-by-2 matrix. When writing the matrix A as[()()]( ω )2
1 00 −iA=(1 − α)−β,c0 1i 0(5)(6)

express α and β in terms of ω, ωc , and ωp .
[7] Find all the eigenvalues k 2 and the corresponding eigenvectors of Eq. (5). Here, express k 2in terms of c, ω, ωc , and ωp . Also ﬁnd the condition for the angular frequency ω so that there
are two eigenmodes that can travel for a long distance in the positive direction of the z-axis
in this plasma.
[8] Consider the case where the angular frequency ω satisﬁes the condition derived in Question [7].
Let k+ and k− denote the wavenumbers of the two eigenmodes assuming k+ > k− > 0.
Assuming that the electric ﬁeld of the electromagnetic wave at position z = 0 is expressed as( )1exp(−iωt),(7)
E(0, t) = E00
where E0 is a constant, express the electric ﬁeld E(z, t) at position z(> 0) in terms of k+
and k− . Also explain how the polarization state of this electromagnetic wave changes as it
travels in the positive direction of the z-axis.