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] --

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