= Circular ladder operators of a two-dimensional oscillator
{title2=$L_z=\hbar(N_+-N_-)$}
For Cartesian <creation and annihilation operators> $a_x,a_y$, define
$$
a_+^\dagger=\frac{a_x^\dagger+ia_y^\dagger}{\sqrt2},\qquad
a_-^\dagger=\frac{a_x^\dagger-ia_y^\dagger}{\sqrt2},\qquad
N_\pm=a_\pm^\dagger a_\pm.
$$
Their <Hermitian adjoints> define $a_\pm$. The <canonical commutation relations> imply $[a_\pm,a_\pm^\dagger]=1$, with mixed <commutators> zero. Direct expansion gives
$$
H=\hbar\omega(N_++N_-+1),\qquad
L_z=-i\hbar(a_x^\dagger a_y-a_y^\dagger a_x)=\hbar(N_+-N_-).
$$
Each circular creation adds one <energy> quantum and <angular momentum> $+\hbar$ or $-\hbar$. At total occupation $N$, the allowed <angular momentum> values are $N\hbar,(N-2)\hbar,\ldots,-N\hbar$, each once in this <energy eigenspace>.
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