Solutions to Problem Set 6

6.1 Harmonic Oscillator Reif §6.1: A simple harmonic one-dimensional oscillator has energy levels given by E n = (n + 1 2)ω, where ω is the characteristic (angular) frequency of the oscillator and where the quantum number n can assume the possible integral values n = 0, 1, 2,. . .. Suppose that such an oscillator is in thermal contact with a heat reservoir at temperature T low enough so that kT /(ω) ≪ 1. (a) Find the ratio of the probability of the oscillator being in the first excited state to the probability of its being in the ground state. (b) Assuming that only the ground state and first excited state are appreciably occupied, find the mean energy of the oscillator as a function of the temperature T. (a) We have P 1 P 0 = exp[−βE 1 ] exp[−βE 0 ] = exp[−β(1 + 1/2)ω] exp[−β(0 + 1/2)ω] = e −βω (b) The average energy is given by