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）︰111/4/11
考試時限（分鐘）：150
試題 :
You may write your answers in Chinese or English.
1. (30pts) Consider a gas consisting of N identical, indistinguishable spinless
particles of mass m. Each particle can move freely in the x-y plane within
a square of area L^2, but along the z direction it is subjected to a
harmonic potential mω^2z^2/2. Suppose the gas is at equilibrium at
temperature T.
(a) (20pts) Estimate the temperature T_1 above which the system can be treated
entirely classically. Calculate the classical partition function, average
energy, Helmholtz free energy, and entropy as a function of temperature.
Is the result consistent with the equipartition theorem?
(b) (5pts) Estimate the temperature T_2 below which quantum mechanical effects
become relevant in the x-y plane. Find the average energy U for
temperatures T such that T_2 << T < T_1, where T_1 is from (a).
(c) (5prs) Find the energy and entropy at T = 0.
2. (20pts) Consider a one-dimensional (say, in the x direction) polymer
consisting of N segments of length a linked end-to-end in a row. The angle
between two adjacent segments can be either 0° or 180°, and there is no
energy cost for a segment to flip from one angle to the other. The length
of the polymer is defined as L = |N_+ － N_-|a, where N_+ and N_- are the
numbers of the segments aligned and anti-aligned to the positive direction
of the x axis.
(a) (5pts) Show that, for a given L, the corresponding number of microstates
is given by
N!
Ω(N,n) = 2‧──────────
N＋n N-n
(───)! (───)!
2 2
where n = |N_+ － N_-|.
(b) (5pts) Assuming N >> n >> 1, use Boltzmann's formula S = klnΩ and the
Stirling's approximate formula to find S in terms of N and L.
(c) (5pts) Suppose f is the stretching force of the polymer. The work can be
expressed as -fdL, and further the first law and the second law yield
TdS = dU － fdL － μdN.
Express the stretching force as a function of N, L, and the temperature T.
(d) (5pts) Find the chemical potential μ in terms of N, L, and T.
3. (20pts) Consider a long vertical column of some specific substance that is
at temperature T everywhere. Below a certain height h(T) the substance is
solid whereas above h(T) is liquid due to the pressure caused by gravity.
Calculate the density difference Δρ between two phases at h(T) in terms
of the latent heat of fusion per unit mas L, the liquid density ρ_l,
dh/dT, T, and the gravitational field strength g. Assume |Δρ| << ρ_l.
4. (15pts) Define a quantity J
J = U － Nμ = TS － PV.
Show that for a system in the grand canonical ensemble
─── ───
2 2 ┌ ∂U ┐2 2
(ΔJ) = kT Cv ＋ │(──) － μ│ (ΔN)
└ ∂N T,V ┘
5. (15pts)
(a) (10pts) For a two-level system, its density matrix can be expressed as
╭ ╮
│ ρ ρ │
│ ee eg │
│ │
│ ρ ρ │
│ ge gg │
╰ ╯
Prove that the largest possible value of |ρ_eg| is 1/2.
(b) (5pts) Discuss (and explain) a scenario where a negative temperature can
emerge. In this case, explain what happens when such an object is in
contact with another object of normal temperature (T > 0).