Circular infinite quantum well ($E_{m,j}={\hslash }^2{j}_{|m|,j}^2/(2\mu a^2)$)

A particle confined by a Dirichlet boundary condition to a disc of radius $a$ has angular functions $e^{im\varphi }$, $m\in \mathbb{Z}$, and regular radial Bessel functions of the first kind $J_{|m|}(kr)$. Their boundary zeros give the displayed energies. The ground state belongs to $m=0$: nonzero angular modes add a nonnegative centrifugal term to the kinetic-energy form. At the origin exclude the singular radial solution, rather than accepting every square-integrable logarithmic function as a physical eigenstate.