Harmonic Oscillators

Our proof of the equipartition theorem depends crucially on the classical approximation. To see how quantum effects modify this result, let us examine a particularly simple system that we know how to analyze using both classical and quantum physics: namely, a simple harmonic oscillator. Consider a one-dimensional harmonic oscillator in equilibrium with a heat reservoir held at absolute temperature $T$ . The energy of the oscillator is given by

$$E = \frac{p^{ 2}}{2 m} + \frac{1}{2} \kappa x^{ 2},$$ (7.136)

where the first term on the right-hand side is the kinetic energy, involving the momentum, $p$ , and the mass, $m$ , and the second term is the potential energy, involving the displacement, $x$ , and the force constant, $\kappa$ . Each of these terms is quadratic in the respective variable. So, in the classical approximation, the equipartition theorem yields:

$$\frac{\overline{p^{ 2}}}{2 m} = \frac{1}{2} k T,$$ (7.137)

$$\frac{1}{2} \kappa \overline{x^{ 2}} = \frac{1}{2} k T.$$ (7.138)

That is, the mean kinetic energy of the oscillator is equal to the mean potential energy, which equals $(1/2) k T$ . It follows that the mean total energy is

$$\overline{E} = \frac{1}{2} k T + \frac{1}{2} k T = k T.$$ (7.139)

According to quantum mechanics, the energy levels of a harmonic oscillator are equally spaced, and satisfy

$$E_n = \left(n + \frac{1}{2}\right) \hbar \omega,$$ (7.140)

where $n$ is a non-negative integer, and

$$\omega = \sqrt{\frac{\kappa}{m}}.$$ (7.141)

(See Section C.11.) The partition function for such an oscillator is given by

$$Z = \sum_{n=0,\infty} \exp(-\beta E_n) = \exp\left(-\frac{1}{2}... ...eta \hbar \omega\right) \sum_{n=0,\infty} \exp(- n \beta \hbar \omega).$$ (7.142)

Now,

$$\sum_{n=0,\infty} \exp(- n \beta \hbar \omega) = 1 + \exp(-\beta \hbar \omega) + \exp(-2 \beta \hbar \omega) + \cdots$$ (7.143)

is simply the sum of an infinite geometric series, and can be evaluated immediately to give

$$\sum_{n=0,\infty} \exp(- n \beta \hbar \omega) = \frac{1}{1-\exp(-\beta \hbar \omega)}.$$ (7.144)

(See Exercise 1.) Thus, the partition function takes the form

$$Z = \frac{ \exp[-(1/2) \beta \hbar \omega]}{1-\exp(-\beta \hbar \omega)},$$ (7.145)

and

$$\ln Z = - \frac{1}{2} \beta \hbar \omega -\ln \left[1- \exp(-\beta \hbar \omega)\right].$$ (7.146)

The mean energy of the oscillator is given by [see Equation (7.35)]

$$\overline{E} = - \frac{\partial}{\partial \beta} \ln Z = - \left[... ...(-\beta \hbar \omega) \hbar \omega} {1-\exp(-\beta \hbar \omega)}\right],$$ (7.147)

or

$$\overline{E} = \hbar \omega \left[ \frac{1}{2} + \frac{1}{\exp( \beta \hbar \omega)-1} \right].$$ (7.148)

Consider the limit

$$\beta \hbar \omega = \frac{\hbar \omega}{k T} \ll 1,$$ (7.149)

in which the thermal energy, $k T$ , is large compared to the separation, $\hbar \omega$ , between successive energy levels. In this limit,

$$\exp( \beta \hbar \omega) \simeq 1 + \beta \hbar \omega,$$ (7.150)

so

$$\overline{E} \simeq \hbar \omega\left[\frac{1}{2} + \frac{1}{\be... ...omega}\right] \simeq \hbar \omega\left[ \frac{1}{\beta \hbar \omega}\right],$$ (7.151)

giving

$$\overline{E} \simeq \frac{1}{\beta} = k T.$$ (7.152)

Thus, the classical result, (7.139), holds whenever the thermal energy greatly exceeds the typical spacing between quantum energy levels.

Consider the limit

$$\beta \hbar \omega = \frac{\hbar \omega}{k T} \gg 1,$$ (7.153)

in which the thermal energy is small compared to the separation between the energy levels. In this limit,

$$\exp( \beta \hbar \omega) \gg 1,$$ (7.154)

and so

$$\overline{E} \simeq \hbar \omega \left[\frac{1}{2} + \exp(-\beta \hbar \omega)\right] \simeq \frac{1}{2} \hbar \omega.$$ (7.155)

Thus, if the thermal energy is much less than the spacing between quantum states then the mean energy approaches that of the ground-state (the so-called zero-point energy). Clearly, the equipartition theorem is only valid in the former limit, where $k T \gg \hbar \omega$ , and the oscillator possess sufficient thermal energy to explore many of its possible quantum states.