The Radial Equation *

After separation of variables, the radial equation depends on $\ell$.

$$\bgroup\color{black}{-\hbar^2\over 2\mu}\left[{1\over r^2}\le... ... r^2}\right] R_{E\ell}(r)+V(r)R_{E\ell}(r)=ER_{E\ell}(r)\egroup$$

It can be simplified a bit.

$$\bgroup\color{black}{-\hbar^2\over 2\mu}\left[{\partial^2\ove... ... r^2}\right]R_{E\ell}(r)+V(r)R_{E\ell}(r)=ER_{E\ell}(r) \egroup$$

The term due to angular momentum is often included with the potential.

$${-\hbar^2\over 2\mu}\left[{\partial^2\over\pa... ...+{\ell(\ell+1)\hbar^2\over 2\mu r^2}\right)R_{n\ell}(r)=ER_{n\ell}(r)$$

This pseudo-potential repels the particle from the origin.