Question

write down degeneracy for isotropic 1 d harmonic oscillator.

Answer 1

The degeneracy for an p-dimensional quantum harmonic oscillator is given as

$$g(n,p)=\frac{(n+p-1)!}{n!(p-1)!}$$

it corresponds to the number of weak compositions of the integer n into p integers.

The degeneracy for isotropic 1 d harmonic oscillator is

$$g(n)=\frac{(n+1-1)!}{n!(1-1)!}=\frac{(n)!}{n!(0)!}=1$$