Physics Quiz 831 - Questions on Thermodynamics and Statistical Mechanics, Quizzes of Statistical mechanics

Download Physics Quiz 831 - Questions on Thermodynamics and Statistical Mechanics and more Quizzes Statistical mechanics in PDF only on Docsity! Physics 831 Quiz #1 - Friday, Sep. 7 1. Consider two single-particle levels of energy − and , which are populated by three electrons (each of which can be either spin-up or spin down). The system is then attached to a heat bath characterized by a temperature T . In terms of T and , find (a) the average energy (b) the T → 0 limit of (a) (c) the T →∞ limit of (a) (d) the entropy (e) the T → 0 limit of (d) (f) the T →∞ limit of (d) 2. Consider two single-particle levels of energy − and , which can be populated by indistin- guishable spin-zero bosons. The system is attached to a bath that can exchange particles and energy and is characterized by a temperature T and a chemical potential µ. In terms of µ,T and , find (assume µ < −) (a) an expression for the average number of particles. (b) the T → 0 limit of (a) (c) the T →∞ limit of (a) 3. Beginning with the grand canonical partition function ZGC(α = −βµ, β, V ), derive an ex- pression for the specific heat at constant volume, CV ≡ dE/dT |N,V in terms of derivatives of ZGC . 4. Beginning with: TdS = dE + PdV − µdQ, prove: ∂E ∂µ ∣∣∣∣ V,T = T ∂Q ∂T ∣∣∣∣ V,βµ