A simple example - the quantum harmonic oscillator

As a simple example of the trace procedure, let us consider the quantum harmonic oscillator. The Hamiltonian is given by

and the eigenvalues of H are

Thus, the canonical partition function is

This is a geometric series, which can be summed analytically, giving

The thermodynamics derived from it as as follows:

1.

Free energy:

The free energy is

2.

Average energy:

The average energy is

3.

Entropy

The entropy is given by

Now consider the classical expressions. Recall that the partition function is given by

Thus, the classical free energy is

In the classical limit, we may take to be small. Thus, the quantum expression for A becomes, approximately, in this limit:

and we see that

The residual (which truly vanishes when ) is known as the quantum zero point energy. It is a pure quantum effect and is present because the lowest energy quantum mechanically is not E=0 but the ground state energy .