1 PHYS 414: Homework 3 This homework due Friday Feb 10 1. (7 points) A particle of mass m moves in an anharmonic oscillator potential V ( x ) = Ax 8 in one dimension. Estimate what the allowed quantum-mechanical energy levels of this system might be by considering the area of phase space trajectories. 2. System A has entropy that varies with energy E A as S A = k B ( E A /E 1 ) α . System B has entropy that varies with energy E B as S B = k B ( E B /E 2 ) γ . Here k B is the Boltzmann constant and E 1 and E 2 and α and γ are constants, and E 1 , E 2 , α , γ , and T are all positive. (a) (2 points) Now the two systems are placed in thermal contact and allowed to reach equilibrium. Express the energy E A of system A as a function of the energy E B of system B. (b) (2 points) What is the largest positive value that the constant α can have in a real physical system? (c) (2 points) What is the smallest value physically possible for the constant α ? (d) (2 points) The Helmholtz free energy
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