Coulomb bound state ($E_N=-m{\gamma }^2/(2{\hslash }^2{N}^2)$)

A Coulomb bound state is a physical normalizable eigenfunction for an attractive potential $V(r)=-\gamma /r$, $\gamma >0$. For the usual regular boundary condition at the origin its energies are the displayed values, $N=1,2,\dots$. The spherically symmetric states have orbital angular momentum zero; series-termination quantization of the Coulomb radial equation constructs them as a polynomial times a decaying exponential. Square integrability on the punctured radial interval alone does not replace the physical condition at the origin.

For motion confined to a plane but retaining potential energy $-\kappa /r$, separation in polar coordinates gives angular wavefunctions $e^{im\varphi }$ with integer $m$. The regular radial factor behaves as $r^{|m|}$ at the origin. Removing this power and the decaying exponential function leaves an Generalized Laguerre polynomial differential equation. polynomial termination requires $N=n_r+|m|+1/2$, yielding the displayed energy. At fixed $q=n_r+|m|$, the spinless quantum degeneracy is $2q+1$. This is not the logarithmic potential of electrostatics formulated entirely in two spatial dimensions.

For a spherically symmetric bound state, write $\psi =e^{-br}\sum c_j{r}^j$. The radial Coulomb equation gives the displayed recurrence. Choosing $b=a/(2N)$ terminates it after degree $N-1$, producing a nonzero continuous polynomial times an exponentially decaying function. Its three-dimensional norm $4\pi {\int }_0^{\infty }{r}^2|\psi {|}^{2}dr$ is finite and can be normalized. The energy is $-{\hslash }^2{a}^2/(8m{N}^2)$. Continuity at the origin is compatible with the Coulomb cusp; no differentiability as a Cartesian radial function at the origin is asserted.