Solution
= Solution
Substitute $g=r^ae^{-br}$ into the radial <Schrödinger equation> and equate the coefficients of $r^{-2}$, $r^{-1}$, and the constant term. Regularity selects
$$
a=l,
\qquad
b=\frac{me^2}{4\pi\varepsilon_0\hbar^2(l+1)}.
$$
The corresponding bound-state energy is
$$
\boxed{
E_l=-\frac{me^4}{2(4\pi\varepsilon_0)^2\hbar^2(l+1)^2}}.
$$