Radial Wave Equation

Also called “Associated Laguerre Equation” Solutions are products of decaying exponentials and Associated Laguerre polynomials. There are infinitely many of these polynomials for $n=[1,\infty ],l=[0,n-1]$

$$\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)+\frac{2\mu }{{\hslash }^2}\left[E+\frac{e^2}{4\pi {ϵ}_{0}r}-\frac{{\hslash }^2}{2\mu }\frac{l(l+1)}{r^2}\right]R=0$$

See Hydrogen Radial Wave Functions which are solutions to this

For example $n=1,l=0$ gives the solution the the ground state equation for Hydrogen.

$$a_0=\frac{4\pi {ϵ}_0{\hslash }^2}{\mu e^2}E=-\frac{{\hslash }^2}{2\mu a_0^2}$$

Same as Bohr Model’s results!