Question 1. A particle moving in a one dimensional harmonic oscillator potential is described by the Hamiltonian H = p²/2m + 1/2.mω²x². The particle is weakly coupled to a heat bath of temperature T with which it exchanges energy.

a) Calculate, using the correct quantum energies, the contribution to the Helmholtz free energy by calculating the appropriate sum.

b) Calculate, using the classical trace derived in class, the classical Helmholtz free energy of the oscillator.

c) Calculate the lowest - order surviving quantum correction to the classical result by performing the appropriate classical average.

d) Expand your quantum result of part a at high temperatures keeping the two lowest- order terms, and verify they agree with your results for parts b and c.

e) We argued in class that given that the commutator of position and momentum gave a factor of i?, that the quantum correction to the free energy from the Hamiltonian contained only even powers of ?. Show that that is true for the harmonic oscillator.

Question 2. A large number of N_p protons and N_e electrons described by quantum mechanics are placed in a (classical) static magnetic field B(r)=∇×A(r).

The Hamiltonian operator in the nonrelativistic approximation and Coulomb gauge ∇·A=0, is

H=(i=1)∑Ne[pie + e/c.A(rip)]2/2mp + (i=1)∑Np [pip - e/c.A(rip)]2/2mp + (i=2)∑Ne(j=1)∑i-1[e2/(|rie-rje|)+ (i=2)∑(N_p)(j=1)∑(i-1)[e2/(|rip-rjp |)-(i=1)∑(Ne)(j=1)∑Np e2/(|rie-rjp |] where rie is the position operator of the ith electron, rip is the position operator of the ith proton, pie and pip are the corresponding canonical momenta, and me and mp are their masses. In order to calculate various thermal averages, you will need to calculate ?O? = tr ρ O. Write the quantum operators you would need to trace with the density matrix to calculate:

A) The charge density at position r.

b) The charge current density at position r.

c) The number of particles in a sphere of radius a around the origin.

d) The total orbital angular momentum around the origin.