# Calculating density of states given energy levels and degeneracy

In my statistical mechanics class, my professors did a problem in which he calculated the density of states, however I am having trouble justifying his approach. I did the problem beforehand in an entirely different way, and got the same answer, but his method is much quicker and I'd like to understand why it works.

Consider a gas of $$N$$ identical spin-$$0$$ bosons confined by an isotropic three-dimensional potential.The energy levels in this potential are $$ε_n= \dfrac{n}{\hbar \omega}$$ with $$n$$ a nonnegative integer and $$ω$$ the classical oscillation frequency. The degeneracy of level $$n$$ is $$g_n = \dfrac{(n+1)(n+2)}2$$.

Find a formula for the density of states, $$D(ε)$$ for an atom confined in this potential. Assume that $$n\gg1$$ and recall that the density of states $$D(ε)$$ is defined by the fact that $$D(ε)dε$$ is the number of orbitals of energy between $$ε$$ and $$ε+dε$$

My method:

The total number of energy levels below $$ε_n$$ will simply be:

$$N(ε_n) = \sum_{k = 0}^n g_k$$

Because $$n \gg 1$$, we can approximate this sum as an integral and $$g_n$$ as $$\frac{n^2}2$$

$$N(ε_n) \approx \int_0^n \frac{x^2}2 dx = \frac{n^3}6 = \frac{ε_n^3}{6 \hbar^3 \omega^3}$$ Taking the logarithm of both sides:

$$\log(N(ε_n)) = 3 \log(ε_n) - \log(2 \hbar^3 \omega^3)$$ Differentiating:

$$\frac{dN}{N} = 3 \frac{dε}{ε} \implies D(ε) = \frac{dN}{dε} = \frac{3N}{ε} = \frac{ε^2}{2 \hbar^3 \omega^3}$$

His method:

$$n >> 1 \implies g_n \approx \frac{n^2}2$$ is the degeneracy of the energy level $$ε_n$$. The spacing between energy levels is $$ε_{n+1} - ε_n = \hbar \omega$$

Apply $$D(ε) = \frac{dN}{dε}$$ in a discrete setting:

$$D(ε) = \frac{n^2/2}{\hbar \omega} = \frac{\epsilon^2}{2 \hbar^3 \omega^3}$$

I can understand why $$dε$$ can be said to be $$ε_{n+1} - ε_n$$, but why does $$dN = g_n$$ here? It seems more reasonable, by this argument, that we would want $$g_{n+1} - g_n$$, which does not appear to yield the correct answer.

I believe it boils down to some confusion on what "the number of orbitals of energy between $$ε$$ and $$ε + dε$$" actually means. It's not an easy task to visualize when you are given discrete variables (despite them behaving somewhat continuously for $$n\gg1$$)

• In the main Problem shouldn't the energy be given by $\epsilon_n=n\hbar\omega$? Commented Mar 12, 2021 at 4:24

From first formula of your method for $n\gg 1$ we get $$dN = N(\varepsilon_{n+1}) - N(\varepsilon_n) = g_{n+1} \approx g_n$$.