= Solution
Put $\kappa=e^2/(4\pi\epsilon_0)$. Substitution of $\psi=R(r)\Phi(\phi)$ into the <time-independent Schrödinger equation>, followed by division by $R\Phi$ and multiplication by $r^2$, separates the angular dependence:
$$
\frac{r^2R''+rR'}R+\frac{2\mu}{\hbar^2}(Er^2+\kappa r)
=-\frac{\Phi''}{\Phi}=m^2.
$$
Thus $\Phi=e^{im\phi}/\sqrt{2\pi}$ and
$$
rR''+R'+\left[\frac{2\mu}{\hbar^2}(Er+\kappa)-\frac{m^2}{r}\right]R=0.
$$
The <single-valued wavefunction> condition, $\Phi(\phi+2\pi)=\Phi(\phi)$, requires \b[$m\in\mathbb Z$]. The regular <boundary condition> at the origin selects $R\sim r^{|m|}$. Accordingly use $s=|m|$ in the radial ansatz; the printed $x^m$ notation describes the $m\geq0$ representatives.
For the <two-dimensional Coulomb bound state>, substitution of $R=x^s e^{-x/2}G$ with the stipulated radial and <energy> scales gives
$$
xG''+(2s+1-x)G'+(N-s-\tfrac12)G=0.
$$
The coefficient of $x^n$ in the <power series> is
$$
(n+1)(n+2s+1)a_{n+1}
+(N-s-\tfrac12-n)a_n=0,
\qquad
\boxed{a_{n+1}=\frac{n+s+\tfrac12-N}{(n+1)(n+2s+1)}a_n}.
$$
The regular solution otherwise has an exponentially growing contribution at large $x$, incompatible with being <normalizable>. Termination at degree $n_r\geq0$ therefore requires
$$
\boxed{N=n_r+|m|+\tfrac12,\quad
E_{n_rm}=-\frac{\mu\kappa^2}{2\hbar^2(n_r+|m|+\tfrac12)^2}}.
$$
The resulting $G$ is proportional to a <Generalized Laguerre polynomial>.
Define $R_{\rm y}=\mu\kappa^2/(2\hbar^2)$, the hydrogenic Rydberg <energy> for the stipulated <mass>. The first three distinct two-dimensional <energies> are \b[$-4R_{\rm y},-4R_{\rm y}/9,-4R_{\rm y}/25$], whereas in three dimensions they are \b[$-R_{\rm y},-R_{\rm y}/4,-R_{\rm y}/9$]. Both series approach zero from below, but the planar <ground state> is four times more tightly bound. Without <spin> the respective <quantum degeneracies> are $1,3,5$ and $1,4,9$; if individual degenerate states are counted, the second and third planar states already share the first excited <energy>. This comparison retains the supplied $1/r$ interaction; it does not replace it by the logarithmic <electrostatic potential> of a genuinely two-dimensional electromagnetic theory.
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