Three-dimensional isotropic harmonic oscillator

Its energies are $E_N=\hslash \omega (N+3/2)$, and level $N$ has degeneracy $\left(\genfrac{}{}{0ex}{}{N+2}{2}\right)$.

From the normalized ground state annihilated by every $A_i$, the normalized Cartesian number states areThey have energy $\hslash \omega (n_1+n_2+n_3+3/2)$.

For the three-dimensional isotropic harmonic oscillator, substituting $X_i=\sqrt{\hslash /(2\mu \omega )}(A_i+A_i^{†})$ and $P_i=i\sqrt{\mu \hslash \omega /2}(A_i^{†}-A_i)$ into $L_i={\epsilon }_{ijk}{X}_j{P}_k$ gives $L_i=-i\hslash {\epsilon }_{ijk}{A}_j^{†}{A}_k$. On the first-excited Cartesian states $|j⟩=A_j^{†}|0⟩$, this is the three-dimensional vector representation and $L^2|j⟩=2{\hslash }^2|j⟩$, so the whole first-excited level has orbital quantum number $\ell =1$.