 Circular infinite quantum well

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.