The Boltzmann Distribution Applied to a Harmonic Oscillator

In Chapter 3 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the Boltzmann distribution. If you have a system with energy levels of energy E_n populated by particles in thermal equilibrium, the Boltzmann distribution gives the probability of finding a particle in the n^th level.

A classic example of the Boltzmann distribution is for the energy levels of a harmonic oscillator. These levels are equally spaced starting from a ground state and increasing without bound. To see the power of the Boltzmann distribution, solve this new homework problem.

Section 3.7

Problem 29½. Suppose the energy levels, E_n, of a system are given by

E_n = n ε,     for  n = 0, 1, 2, 3, …

where ε is the energy difference between adjacent levels. Assume the probability of a particle occupying the n^th energy level, P_n, obeys the Boltzmann distribution

where A is a constant, T is the absolute temperature, and k is the Boltzmann constant.

(a) Determine A in terms of ε, k, and T. (Hint: the sum of the probabilities over all levels is one.)

(b) Find the average energy E of the particles. (Hint: E = ∑P_nE_n.)

(c) Calculate the heat capacity C of a system of N such particles. (Hint: U = N E and C = dU/dT.)

(d) What is the limiting value of C for high temperatures (kT >> ε)? (Hint: use the Taylor series of the exponential.)

(e) What is the limiting value of C for low temperatures (kT << ε)?

(f) Sketch a plot of C versus T.

You may need these infinite series

1 + x + x² + x³ + ⋯ = 1/(1−x) 

x + 2x² + 3x³ + ⋯ = x/(1−x)² 

This is a somewhat advanced problem in statistical mechanics, so I gave several hints to guide the reader. The calculation contains much interesting physics. For instance, the answer to part (e) is known as the third law of thermodynamics. Albert Einstein was the first to calculate the heat capacity of a collection of harmonic oscillators (a good model for a crystalline solid). There’s more physics than biology in this problem, because most of the interesting behavior occurs at cold temperatures but biology operates at hot temperatures.

If you’re having difficulty solving this problem, here’s one more hint:

e^nx = (e^x)ⁿ . 

Enjoy!