Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Section 7.5, Exercise 7.5.4.
One application of harmonic oscillator theory is in the behaviour of crystals as a function of temperature. A reasonable model of a crystal is of a number of atoms that vibrate as harmonic oscillators. From statistical mechanics, the probability of finding a system in a state is given by the Boltzmann formula
where , with being Boltzmann’s constant and the absolute temperature, and is the partition function
The thermal average energy of the system is then
For a classical harmonic oscillator, the energy is a continuous function of the position and momentum :
The classical partition function is then
The average energy of a classical oscillator thus depends only on the temperature, and not on the frequency .
For a quantum oscillator, the energies are quantized with values of
The quantum partition function is therefore
The sum is a geometric series, so we can use the standard result for :
The mean quantum energy is again found from 5, although this time the derivative is a bit messier, so is most easily done using Maple. However, by hand, you’d get
The average energy is the ground state energy plus a quantity that increases with increasing temperature (decreasing ). For small we have
since as , . Thus the quantum energy reduces to the classical energy 11 for high temperatures. The ‘high temperature’ condition is that
So far, we’ve considered the average behaviour of only one oscillator. Suppose we now have a 3-d crystal with atoms. Assuming small oscillations we can approximate its behaviour by a system of decoupled oscillators. In the classical case, the average energy is found from 11:
The heat capacity per atom is the amount of heat (energy) required to raise the temperature by , so
For the quantum system, we have from 20
The quantum heat capacity is therefore
We can define the Einstein temperature as
which gives the heat capacity as
For large temperatures, the exponent becomes small, so we have
For low temperatures so we have
The heat capacity again reduces to the classical value for high temperatures. The observed behaviour at low temperatures is that , so this simple model fails for very low temperatures. However, as is shown by Shankar’s figure 7.3 Einstein’s quantum model is actually quite good for all but the lowest temperatures.