Physics Quiz 5 - Topics in Thermodynamics and Quantum Mechanics, Quizzes of Statistical mechanics

Download Physics Quiz 5 - Topics in Thermodynamics and Quantum Mechanics and more Quizzes Statistical mechanics in PDF only on Docsity! Physics 831 Practice Quiz #5 - Monday, Dec. 3 YOUR NAME: FUN FACTS TO KNOW AND TELL ∫ ∞ 0 dx xn−1 ex − 1 = Γ(n)ζ(n), ∫ ∞ 0 dx xn−1 ex + 1 = Γ(n)ζ(n) [ 1− (1/2)n−1 ] , ζ(n) ≡ ∞∑ m=1 m−n, Γ(n) ≡ (n− 1)!, ζ(3/2) = 2.612375..., ζ(2) = π2 6 , ζ(3) = 1.20205..., ζ(4) = π4 90 ,∫ ∞ −∞ dx e−x 2/2 = √ 2π, ∫ ∞ 0 dx xne−x = n! 1. Consider the virial expansion, P = ρT + ρT ∑ n=2,··· An ( ρ ρ0 )n−1 . For a gas of identical non-interacting bosons, A2 is: (circle one) greater than zero zero less than zero 2. A spherically symmetric molecule developed by super scientists at the prestigious University of Notre Dame has mass m and breathing mode excitations of nh̄ω, i.e., each mode is non- degenerate. Here, n = 0, 1, 2, · · ·. A dilute gas of such molecules is kept at temperature T and chemical potential µ. Calculate the number density in terms of µ, β = 1/T , m and h̄ω. 3. Suppose one has calculated a partition function, Z = Tr e−βH , H = SCF − µ~B · ~S, where SCF is some complicated function and ~S is the net spin of the system. Further assume that after performing all the fancy calculations that lnZ = N ln[2ez cosh(aβµB)], where a and z are functions of the temperature, and N is the number of sites. (a) Find the average spin per site 〈Sz〉 as a function of β, µB, a, z? Assume B points along the z axis. (b) What is the fluctuation of the spin per site 〈S2z 〉 when B = 0?