課程名稱︰統計物理一
課程性質︰物理所必修
課程教師︰林俊達
開課學院：理學院
開課系所︰物理所
考試日期（年月日）︰110/6/15
考試時限（分鐘）：8hrs Take home
試題 :
Your answers in English or Mandarin are both acceptable. Total 110 points
(including bonus 10 pts), but the maximum you can earn is 100 pts.
You must write down explicitly the following statement in English or Mandarin
before answering all the exam questions or your answers will not be graded
(BONUS 10PTS):
I affirm that I will not give or receive any unauthorized help on
this exam, and that all work will be my own. I will take full responsibility
for my actions.
我承諾在這場考試中不會提供或接受任何形式的幫助，以下的作答皆出自我個人的努力。
如有違規情事，我將會承擔應負之責任。
1. (35pts) Consider a gas of N identical non-interacting spin-0 bosons in d
dimensions, confined to a large volume V = L^d (you may assume periodic
boundary conditions) and held at temperature T. The dispersion relation,
expressing the single-particle energy ε in terms of the magnitude of its
→ s
momentum p = |p |, is given by ε = ap where a and s are both positive.
(a) Find the single-particle density of states g(ε) as a function of energy.
(b) Write an intergral expression for the total number of bosons N in the
limit of large volume V when the system is described by a chemical
potential μ (assuming there is no Bose-Einstein condensation). Evaluate
the integral explicitly in terms of the fugacity z = exp(βμ) and
temperature T using the formula given below.
(c) To determine whether Bose-Einstein condensation occurs, we have to examine
the expression for N in (b) in the limit μ→0. Explain why this is the
relevant limit and determine the condition satisfied by s and d for which
Bose-Einstein condensation occurs. Check that this relation holds for the
usual case of d = 3 and s = 2.
(d) When Bose-Einstein condensation occurs, find for T < Tc the relative
population of the ground state N_0/N as a function T/Tc.
(e) Find the Helmholtz free energy and the Gibbs free energy of a gas of free
spin-0 bosons in three dimensions. Show that, as the critical temperature
is approached from above, the isothermal compressibility κ_T diverges as
1
κ ~ ────
T T － Tc
You may find the following relation useful:
r-1
∞ x
∫ dx─────── ＝ Γ(r)F (z)
0 -1 x r
z e － 1
where Γ(r) is the gamma function and Fr(z) is the function defined by its
series expansion
∞ n r
F (z) ＝ Σ z / n
r n=1
Fr(1) diverges for r ≦ 1 and converges for r ＞ 1.
2. (15pts) Consider non-interacting spin-1/2 fermions in two dimensions (2D)
with a linear dispersion relation
→ →
ε ( k ) ＝ ± hbarν| k |
±
Positive energy states (with energy ε_+) define the conduction band and
negative energy states (with energy ε_-) define the valence band. Assume
→
that the allowed wavevectors k = {kx, ky} correspond to periodic boundary
conditions over a square region of area A.
At temperature T = 0 the valence band is completely filled and the
conduction band is completely empty. At finite T, excitations above this
ground state correspond to adding particles (occupied states) in the
conduction band or holes (unoccupied states) in the valence band.
(a) Find the single-particle density of states D(ε) as a function of the
energy in terms of hbar, ν, A.
(b) Using the Fermi-Dirac distribution, show that if μ(T) = 0 then the
probability of finding a particle at energy ε is equal to the probability
of finding a hole at energy -ε.
(c) Find the total internal energy of the excitations in excess of the T = 0
state, U(T) － U(0), expressed in terms of A, ν, hbar, k. Note that since
we are subtracting U(0), in the valence band you only need to count the
energy associated with holes.
You may find the following relation useful:
n
1 ∞ x -n
──∫ dx───── ＝ (1－2 )ζ(n+1)
n! 0 x
e ＋ 1
∞ -n-1
where ζ(n+1) = Σ k is the Riemann zeta function.
k=1
3. (20pts) Consider a two-dimensional gas of N non-relativistic fermions with
mass m and spin s moving in a square of area A.
(a) Evaluate the Fermi energy ε_F of the gas as a function of the density of
particles ρ = N/A.
(b) Calculate the total energy of the gas per particle E/N at temperature
T = 0 as a function of its density.
(c) Using your result in (b), determine the pressure of the gas P at T = 0 as
as a function of its density.
(d) A container is separated into two compartments by a sliding piston. Two
two-dimensional Fermi gases with spin 1/2 and 3/2 of the same mass are
placed in the left and right compartments, respectively. Find the ratio
between the densities of the two gases at equilibrium at T = 0.
4. (20pts) A linear chain consists of N+1 atoms of species A, whose mass is m,
and N+1 atoms of species B, whose mass is M. The two types of atom
alternate along the chain, that is, form an equally spaced pattern like
ABAB...AB with a separation a between neighboring atoms. Except that the
two end atoms are fixed, all atoms oscillate in the direction along the
chain (consider the longitudinal modes only). The harmonic forces
characterized by a "spring constant" K act between neighboring atoms.
(a) Find the dispersion relation for the normal modes of vibration of this
chain. You should find that is has a low-frequency branch (called the
acoustic branch) and a high-frequency branch (called the optical branch).
(b) Identify the circumstances under which the frequency of the optical branch
is almost independent of wavelength.
Under the conditions of (b), we construct a simplified version of the
excitation spectrum by assuming that all the optical modes have the same
frequency ω_0, while the acoustic modes can be treated in the Debye
approximation. Assuming that ω_0 ＞ ω_D, the density of states is
L/πc ω ＜ ω_D
g(ω) = {
Nδ(ω－ω_0) ω ＞ ω_D
where c is the speed of sound.
(c) Find the Debye frequency ω_D for this model.
(d) Find the dependence of the specific heat on temperature for kT ＜＜ hbarω_D
and kT ＞＞ hbarω_0.
5. (10pts) Consider a three-dimensional isotropic solid formed by N atoms as
an ensemble of harmonic oscillators, with a density of states given by the
Debye approximation.
(a) Show that the zero point energy of the solid is
9NkT
D
E ＝ ────
0 8
where T_D is the Debye temperature.
(b) Show that
∞
∫ dT[Cv(∞) － Cv(T)] ＝ E
0 0