 Two-dimensional hydrogenic donor

An electron confined to a plane in the screened Coulomb potential $-\kappa /r$, $\kappa =e^2/(4\pi {ϵ}_0{ϵ}_r)$, has the radial ground state $\psi =A{e}^{-br}$. The planar radial Laplacian is $(b^2-b/r)\psi$. Cancelling the $1/r$ term in the Schrödinger equation gives $b=2m^{\ast }\kappa /{\hslash }^2$ and $E=-{\hslash }^2{b}^2/(2m^{\ast })$. The binding magnitude is four times the three-dimensional effective Rydberg. Normalization in the plane gives $A=b\sqrt{2/\pi }$. The ordinary three-dimensional electric field is retained; a logarithmic two-dimensional interaction is a different problem.