# Density of states for 3D simple harmonic oscillator

I have the thermal partition function and the density of states for the 3D simple harmonic oscillator, which are given below $$Z(\beta) = \frac { 1 } { \left( 2 \sinh \left( \frac { \beta \omega } { 2 } \right) \right) ^ { 3 } }$$ and $$\rho ( E ) = \frac { \left( \frac { E } { \omega } - \frac { 1 } { 2 } \right) \left( \frac { E } { \omega } + \frac { 1 } { 2 } \right) } { 2 \omega }$$ where $$\beta$$ is the inverse temperature and $$\omega$$ is the natural frequency of the SHO.

Now, the partition function can also be written as the following integral $$Z(\beta) = \int_{0}^{\infty} dE \: \rho(E) e^{-\beta E}$$ which is a Laplace transform, hence we can calculate the asymptotic of the density of states by inverting this Laplace transform. The inverse Laplace transform can be done using the Bromwich contour integral which is $$\rho(E) = \frac{1}{2\pi i} \int_{\gamma - i \infty}^{\gamma + i \infty} d\beta \: Z(\beta) e^{\beta E}$$ where $$\gamma$$ is greater than the real part of all the singularities of $$Z(\beta)$$. Now, I tried doing this integral using the residue theorem by closing the contour on the left half of the complex plane. But somehow I am getting the correct density of states using the residue of just one pole, whereas there is an infinite number of poles on the imaginary axis due to the periodicity of the hyperbolic sine function (on the imaginary axis).

Can someone explain where I am wrong in this process?

P.S. We also know that $$n = \frac{E}{\omega} - \frac{3}{2}$$ is a positive integer.

• – user197851 Jan 4 at 8:00
• As I mentioned in my answer, your first expression for $\rho(E)$ is not the density of states for this system, it is just a (smooth) approximation valid at large $E$. The links in my answer explain where this kind of approximation comes from. Also, when discussing an integral over energy $E$, it is not possible to say (as you do in your "PS") that it only takes discrete values, unless you write it in terms of Dirac delta functions or similar. – user197851 Jan 4 at 8:22

Here again, the singularities in $$\Phi/\beta$$ are all poles located on the imaginary axis. Due to the term $$e^{\beta\epsilon}$$ in the integrand the residues at all poles except for the pole at the origin will contribute oscillating terms to the expression for $$W(\epsilon)$$. We may, therefore, reasonably expect to find a good smooth approximation to $$W(\epsilon)$$ by considering only the contribution from the residue at $$\beta=0$$.