Blundell Thermal Physics · Problem 20.8

The partition function

Physics — Statistical Mechanics · hydrogen-atom-partition-function

Problem

The internal levels of an isolated hydrogen atom are given by

E = -R/n^2

where

R = 13.6\,\mathrm{eV}

. The degeneracy of each level is given by

2n^2

. (a) Sketch the energy levels. (b) Show that

Z = \sum_{n=1}^{\infty} 2n^2 \exp\left(\frac{R}{n^2 k_B T}\right). \tag{20.59}

Note that when

T \neq 0

, this expression for

Z

diverges. This is because of the large degeneracy of the hydrogen atom's highly excited states. If the hydrogen atom were to be confined in a box of finite size, this would cut off the highly excited states and

Z

would not then diverge. By approximating

Z

as follows:

Z \approx \sum_{n=1}^{2} 2n^2 \exp\left(\frac{R}{n^2 k_B T}\right), \tag{20.60}

i.e., by ignoring all but the

n=1

and

n=2

states, estimate the mean energy of a hydrogen atom at

300\,\mathrm{K}

Related theory

Partition Function For Two Independent Systems

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