= Two-dimensional hydrogenic donor
{title2=$E_B=2m^*\kappa^2/\hbar^2,\quad b=2/a_B^*$}
An electron confined to a plane in the screened <Coulomb potential> $-\kappa/r$, $\kappa=e^2/(4\pi\epsilon_0\epsilon_r)$, has the radial ground state $\psi=Ae^{-br}$. The planar radial Laplacian is $(b^2-b/r)\psi$. Cancelling the $1/r$ term in the <Schrödinger equation> gives $b=2m^*\kappa/\hbar^2$ and $E=-\hbar^2b^2/(2m^*)$. 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.
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