Download Physics Practice: Thermodynamics and Statistical Mechanics - Prof. Scott Pratt and more Study notes Statistical mechanics in PDF only on Docsity! PRACTICE When you are not practicing, remember, someone somewhere is practicing, and when you meet him he will win – E. Macauley THINGS TO MEMORIZE S = − ∑ ` p` ln(p`) TdS = dE + PdV − µdN, TS = E + PV − µN. dNfree 1−body states = (2J + 1) ddpddr (2πh̄)d = (2J + 1) ddkddr (2π)d f(p)bosons/fermions = e−β(−µ) 1∓ e−β(−µ) CV ≡ T dS dT ∣∣∣∣ V,N , CP ≡ T dS dT ∣∣∣∣ P,N 〈q∂H ∂q 〉 = T, 〈p∂H ∂p 〉 = T, 〈qṗ〉 = −〈pq̇〉. Don’t memorize factors, but know dependencies of more complicated expressions. For instance, you wouldn’t be expected to memorize all the 21/2 factors in: P = ρT [ 1 + ∞∑ n=2 An ( ρ ρ0 )n−1] , ρ0 ≡ B1 = (2j + 1) (2πh̄)3 ∫ d3p e−p/T = (mT )3/2 (2πh̄2)3/2 . ∆ dN d = 1 π ∑ ` (2`+ 1) dδ` d , A2 = −23/2 ∑ ` ∫ d (2`+ 1) π dδ d e−/T . δ = −ap/h̄, (a = scattering length), but you would be expected to know what happens to the second virial coefficient if a repulsive or attractive interaction is added, or in what way the pressure would change if a resonance was included. 1. Consider two neutrons (spin 1/2 particles). They occupy a two-level system where the single- particle energies are − and , which is thermalized at a temperature T . (a) What is the average energy 〈E〉 of the system as a function of T? (b) What is the chance that the ground state is occupied as a function of T? (c) What is the entropy S of the system as a function of T? 1 2. Beginning with the expression, TdS = dE + PdV − µdN derive the Maxwell relations 1 T 2 dT d(βµ) ∣∣∣∣ E = − dN dE ∣∣∣∣ µ/T . and −ρ2 d(S/N) dρ ∣∣∣∣ T = dP dT ∣∣∣∣ ρ 3. Using the last Maxwell relation from the previous problem, show that if P and S/N are functions of T and ρ, that dP dρ ∣∣∣∣ S/N = ∂P ∂ρ ∣∣∣∣ T + ( ∂P ∂T ∣∣∣∣ ρ )2 T ρ2CV . 4. Consider a system with an order parameter denoted by x along with a particle number N , energy E and volume V . The system will maximize entropy if dS dx ∣∣∣∣ E,V,N = 0. Show that if the system has a fixed volume and is connected to a bath at temperature TB and chemical potential µB that can can exchange both particles and energy, that the total entropy (of both the system and the bath) will be maximized if the pressure is maximized, i.e., dP dx ∣∣∣∣ T=TB ,µ=µB ,V = 0. You may use the identity, PV = TS − E + µN 5. Consider a particle moving in one-dimension according to the Hamiltionian H = √ p2 +m2 +Bx4 Using the equipartion, generalized equi-partition or virial theorems, find 〈x4〉. 6. Consider a thermalized two-dimensional gas of charged non-interacting massless spin-zero bosons, whose energies are given by: = pc. Find the density (number per area) required for Bose condensation. Give answer in terms of c, T and h̄. 2
