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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
which we know how to analyze using both classical and quantum physics: i.e.,
a
simple harmonic oscillator. Consider a onedimensional harmonic oscillator in equilibrium
with a heat reservoir at temperature . The energy of the oscillator is given by

(467) 
where the first term on the righthand side is the kinetic energy, involving the momentum
and mass , and the second term is the potential energy, involving the displacement
and the force constant . Each of these terms is quadratic in the respective
variable. So, in the classical approximation the equipartition theorem yields:
That is, the mean kinetic energy of the oscillator is equal
to the mean potential energy which
equals . It follows that the mean total energy is

(470) 
According to quantum mechanics, the energy levels of a harmonic oscillator are equally
spaced and satisfy

(471) 
where is a nonnegative integer, and

(472) 
The partition function for such an oscillator is given by

(473) 
Now,

(474) 
is simply the sum of an infinite geometric series, and can be evaluated immediately,

(475) 
Thus, the partition function takes the form

(476) 
and

(477) 
The mean energy of the oscillator is given by [see Eq. (399)]

(478) 
or

(479) 
Consider the limit

(480) 
in which the thermal energy is large compared to the separation
between the
energy levels. In this limit,

(481) 
so

(482) 
giving

(483) 
Thus, the classical result (470) holds whenever the thermal energy greatly exceeds the typical
spacing between quantum energy levels.
Consider the limit

(484) 
in which the thermal energy is small compared to the separation between
the energy levels. In this limit,

(485) 
and so

(486) 
Thus, if the thermal energy is much less than the spacing between quantum states then
the mean energy approaches that of the groundstate (the socalled zero point
energy).
Clearly, the equipartition theorem is only valid in the former limit, where
, and the oscillator possess sufficient thermal energy to explore many
of its possible quantum states.
Next: Specific heats
Up: Applications of statistical thermodynamics
Previous: The equipartition theorem
Richard Fitzpatrick
20060202