# What is the partition function of a classical harmonic oscillator?

A classical harmonic oscillator has energy given by $$\frac{1}{2m}p^2+\frac{1}{2}kx^2$$. This means its Boltzmann factor is

$$e^{-\frac{\beta p^2}{2m}}e^{-\frac{\beta k x^2}{2}}$$

where $$\vec{x}$$ and $$\vec{p}$$ are the continuous position and momentum vectors, respectively. The partition function should therefore be given by

$$Z=\int e^{-\frac{\beta p^2}{2m}}d^3\vec{p}\int e^{-\frac{\beta k x^2}{2}}d^3\vec{x},$$

but it is stated in my course homework that the partition function is instead

$$Z=\frac{1}{h^3}\int e^{-\frac{\beta p^2}{2m}}d^3\vec{p}\int e^{-\frac{\beta k x^2}{2}}d^3\vec{x}.$$

Some sources online have instead a factor $$\frac{1}{h}$$ but without any justification. Either way, I cannot see how $$h$$ enters into this calculation. Where does it come from?

• Isn't it just some dimensional analysis ? p*x having the dimension of h. So that'd go in the denominator of your d^3 p. – picop Oct 27 '20 at 11:24
• Sure, but there may be a factor there as well. – Pancake_Senpai Oct 27 '20 at 11:33

Classical partition function is defined up to an arbitrary multiplicative constant. dividing it by $$h$$ is done traditionally for the following reasons: