 Orbital angular momentum in oscillator ladder operators

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 $\ell =1$.