課程名稱︰統計物理(一) Statistical Physics(I)
課程性質︰必修／物理學研究所；選修／天文物理所、應用物理所
課程教師︰蔡政達
開課學院：理學院
開課系所︰物理學研究所、天文物理所、應用物理所
考試日期（年月日）︰2018/06/25
考試時限（分鐘）：120 minutes
試題 :
Statistical Physics(I) Final Examination
1. Derive the density matrix ρ for (i) a free particle and (ii) a linear
harmonic oscillator in the momentum representation and study its main
property (the expectation value of the Hamiltonian: 〈H〉). [20 points]
2. Study the density matrix and the partition function of a system of free
particle, using the unsymmetrized wavefunction
N
ψ_E(q) = Π u_{εi} (q_i)
i=1
instead of the symmetrrized wavefunction
ψ_K(1, ..., N) = (N!)^{-1/2} Σ δ_P P{u_{k1}(1) ... u_{kN}(N)}.
P
Show that, following this procedure, one encounters neither the Gibbs'
correction factor (1/N!) nor a spatial correlation among the particles.
[20 points]
3. Show that the entropy of an ideal gas in thermal equilibrium is given by
the formula
S = k_B Σ [〈n_ε + 1〉 ln〈n_ε + 1〉 - 〈n_ε〉 ln〈n_ε〉]
ε
in the case of bosons and by the formula
S = k_B Σ [-〈1-n_ε〉 ln〈1-n_ε〉 - 〈n_ε〉 ln〈n_ε〉]
ε
in the case of fermions. [20 points]
4. Deduce the virial expansion
PV ∞ λ^3
──── = Σ a_l (───)^{l-1}
N k_B T l=1 v
from eqautions
P 1
─── = ─── g_{5/2}(z)
k_B T λ^3
and
N - N_0 1
──── = ─── g_{3/2}(z),
V λ^3
and verify the quoted values of the virial coefficients. [20 points]
5. Combining equations
h^2 N
T < T_c = ────── {─────}^{2/3}
2 π m k_B V ζ(3/2)
and
g_{3/2}(z) = (λ^3/v) < 2.612,
and making use of the first two terms of formula
Gamma(1 - ν) ∞ (-1)^i
g_ν(e^{-α}) = ─────── Σ ───── ζ(ν - i) α^i,
α^{1 - ν} i=0 i!
show that, as T approaches T_c from above, the parameter α(= -ln z) of
the ideal Bose gas assumes the form
1 3ζ(3/2) T - T_c
α ≒ ── (───────)^2 (─────)^2.
π 4 T_c
[20 points]