Quantum harmonic oscillator in thermal equilibrium

Exact Gibbs-state partition function, energy, entropy, and heat capacity for the quantum harmonic oscillator; classical and low-temperature limits.

Quantum harmonic oscillator in thermal equilibrium

Model and Gibbs state

Let $H$ be the quantum harmonic oscillator Hamiltonian with spectrum

E_n=\hbar\omega\left(n+\frac12\right),\qquad n=0,1,2,\dots

Thermal equilibrium at inverse temperature $\beta$ is described by the quantum Gibbs state

\rho_\beta = \frac{e^{-\beta H}}{Z(\beta)},

The quantum partition function is the geometric series

Z(\beta)=\sum_{n=0}^\infty e^{-\beta E_n} =\sum_{n=0}^\infty e^{-\beta\hbar\omega(n+1/2)} =\frac{e^{-\beta\hbar\omega/2}}{1-e^{-\beta\hbar\omega}} =\frac{1}{2\sinh(\beta\hbar\omega/2)}.

The free energy (compare Helmholtz free energy ) is

F(\beta)=-\beta^{-1}\ln Z(\beta) =\frac{\hbar\omega}{2}+\beta^{-1}\ln\!\bigl(1-e^{-\beta\hbar\omega}\bigr).

The mean energy is

U(\beta)=-\frac{\partial}{\partial\beta}\ln Z(\beta) =\frac{\hbar\omega}{2}+\frac{\hbar\omega}{e^{\beta\hbar\omega}-1}.

It is often convenient to introduce the Bose occupation number

\bar n(\beta)=\frac{1}{e^{\beta\hbar\omega}-1}, \qquad\text{so that}\qquad U=\hbar\omega\left(\bar n+\frac12\right).

The heat capacity at constant volume (see heat capacity at constant volume ) is

C_V=\frac{\partial U}{\partial T} = k_B(\beta\hbar\omega)^2\,\frac{e^{\beta\hbar\omega}}{(e^{\beta\hbar\omega}-1)^2}.

The thermodynamic entropy can be computed as the von Neumann entropy

S(\beta)=-k_B\,\mathrm{Tr}(\rho_\beta\ln\rho_\beta).

For a single oscillator this evaluates to the standard bosonic form

S(\beta)=k_B\Bigl[(\bar n+1)\ln(\bar n+1)-\bar n\ln \bar n\Bigr],

and it also agrees with the canonical identity $S=k_B(\ln Z+\beta U)$ .

High-temperature limit $\beta\hbar\omega\ll 1$ :
$Z(\beta)\sim \frac{1}{\beta\hbar\omega},\qquad U(\beta)\sim k_B T,\qquad C_V\sim k_B,$
matching the classical harmonic oscillator .
Low-temperature limit $\beta\hbar\omega\gg 1$ :
$U(\beta)\to \frac{\hbar\omega}{2},\qquad C_V \sim k_B(\beta\hbar\omega)^2 e^{-\beta\hbar\omega},$
showing exponential suppression of thermal excitations.