Separation of a two-dimensional central-potential eigenstate (${\psi }_k(r,\varphi )=(2\pi {)}^{-1/2}f(r)e^{ik\varphi }$)

An energy and orbital angular momentum eigenstate for a real two-dimensional central potential can be chosen with integer $k$ and real radial function $f$. Its radial equation isQuantum degeneracy permits other stationary states, including $f(r)\cos (k\varphi )/\sqrt{\pi }$, which are not single angular exponentials. Separation of variables supplies a basis rather than describing every superposition in a degenerate eigenspace.