Angular momentum ladder variable

The coordinate combinations $x_1\pm i{x}_2$ have $L_3$ eigenvalues $\pm \hslash$, so their powers carry magnetic quantum numbers $\pm m$.

If $\psi =z{e}^{-r^2/(2a^2)}$ in two dimensions, then $L_z\psi =\hslash \psi$ and $L_z{\psi }^{\ast }=-\hslash {\psi }^{\ast }$. Orthogonality therefore makes the angular-momentum expectation of $\alpha \psi +\beta {\psi }^{\ast }$ equal to $(|\alpha {|}^2-|\beta {|}^2)\hslash$ for normalized coefficients.