If we consider a particle in a 2 dimensional harmonic oscillator potential with Hamiltonian $$H = \frac{\mathbf{p}^2}{2m} + \frac{m w^2 \textbf{r}^2}{2}$$ it can be shown that the energy levels are given by $$E_{n_x,n_y} = \hbar \omega (n_x + n_y + 1) = \hbar \omega (n + 1)$$ where $n = n_x + n_y$. Is it then true that the n$^{\text{th}}$ energy level has degeneracy $n-1$ for $n \geq 2$, and 1 for $0 \leq n \leq 1$?
How common is this scenario where it is possible to calculate the degeneracy of a "general" or "n$^\text{th}$" energy level? How common is this in more complicated quantum systems?
Best Answer
In the case of the n-dimensional harmonic oscillator, possibly the most elegant method is to recognize that the set of states with total number $m$ of excitation span the irrep $(m,0,\ldots,0)$ of $su(n)$. Thus the degeneracy is the dimension of this irrep.