= Separation of a two-dimensional central-potential eigenstate
{title2=$\psi_k(r,\phi)=(2\pi)^{-1/2}f(r)e^{ik\phi}$}
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 is
$$
-\frac{\hbar^2}{2m}\left(f''+r^{-1}f'-k^2r^{-2}f\right)+Vf=Ef,\qquad \int_0^\infty r f^2\,dr=1.
$$
<Quantum degeneracy> permits other <stationary states>, including $f(r)\cos(k\phi)/\sqrt\pi$, which are not single angular exponentials. <Separation of variables> supplies a basis rather than describing every superposition in a degenerate eigenspace.
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