Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 15A / vi / Solution

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


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 Incoming links: Angular momentum of a complex-coordinate Gaussian state