Circular ladder operators of a two-dimensional oscillator ($L_z=\hslash (N_{+}-N_{-})$)

For Cartesian creation and annihilation operators $a_x,a_y$, defineTheir Hermitian adjoints define $a_{\pm }$. The canonical commutation relations imply $[a_{\pm },a_{\pm }^{†}]=1$, with mixed commutators zero. Direct expansion givesEach circular creation adds one energy quantum and angular momentum $+\hslash$ or $-\hslash$. At total occupation $N$, the allowed angular momentum values are $N\hslash ,(N-2)\hslash ,\dots ,-N\hslash$, each once in this energy eigenspace.