The Radial Equation *

After separation of variables, the radial equation depends on \bgroup\color{black}$\ell$\egroup.

\begin{displaymath}\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\end{displaymath}

It can be simplified a bit.

\begin{displaymath}\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\end{displaymath}

The term due to angular momentum is often included with the potential.
\bgroup\color{black}$\displaystyle {-\hbar^2\over 2\mu}\left[{\partial^2\over\pa...
...r)+{\ell(\ell+1)\hbar^2\over 2\mu r^2}\right)R_{n\ell}(r)=ER_{n\ell}(r) $\egroup
This pseudo-potential repels the particle from the origin.



Jim Branson 2013-04-22