Mean energy of a 3D (quantum) harmonic oscillator

1. Mar 10, 2010

HappyJazz

This is not really homework, just a project I'm toying with in my sparetime. I'm doing some Path Integral Monte Carlo simulations, for now just for the 1D quantum harmonic oscillator. Anyways, currently I compare my results to the analytic mean energy of a 1D quantum harmonic oscillator, given by the Hamiltonion

$$\hat H = \frac{{{\hbar ^2}}}{{2m}}\frac{{{\partial ^2}}}{{\partial {x^2}}} + \frac{1}{2}m{\omega ^2}{x^2}$$

with the trace given as

$$\int\limits_{ - \infty }^\infty {\left\langle x \right|\exp \left( { - \beta \hat H} \right)\left| x \right\rangle } = \frac{{\exp \left( { - 0.5\beta \hbar \omega } \right)}}{{1 - \exp \left( { - 0.5\beta \hbar \omega } \right)}}$$.

Beta is representing inverse temperature. From the mean energy in terms of the trace we get

$$\left\langle H \right\rangle = - \frac{\partial }{{\partial \beta }}\ln \sum\limits_x {\left\langle x \right|\exp \left( { - \beta \hat H} \right)\left| x \right\rangle }$$,

which finally amounts to

$$\left\langle H \right\rangle = \frac{1}{2}\hbar \omega \coth \left( {\frac{1}{2}\beta \hbar \omega } \right)$$.

My next step is to expand my program to 3 dimensions. I know that the energy of the classical harmonic oscillator in 3D is simply the result from 1D times the degrees of freedom (3 translational, 2 rotational and 1 vibrational), but does this also hold for the Quantum version of the harmonic oscillator?

I.e. is

$$\left\langle {{H_{3D}}} \right\rangle = 3\hbar \omega \coth \left( {\frac{1}{2}\beta \hbar \omega } \right)$$

true?