= Orbital angular momentum in oscillator ladder operators
{title2=$L_i=-i\hbar\varepsilon_{ijk}A_j^\dagger A_k$}
For the <three-dimensional isotropic harmonic oscillator>, substituting $X_i=\sqrt{\hbar/(2\mu\omega)}(A_i+A_i^\dagger)$ and $P_i=i\sqrt{\mu\hbar\omega/2}(A_i^\dagger-A_i)$ into $L_i=\varepsilon_{ijk}X_jP_k$ gives $L_i=-i\hbar\varepsilon_{ijk}A_j^\dagger A_k$. On the first-excited Cartesian states $|j\rangle=A_j^\dagger|0\rangle$, this is the three-dimensional vector representation and $L^2|j\rangle=2\hbar^2|j\rangle$, so the whole first-excited level has orbital quantum number $\ell=1$.
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