Behavior at the Origin *

The pseudo-potential dominates the behavior of the wavefunction at the originif the potential is less singular than ${1\over r^2}$.

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

For small $r$, the equation becomes

$$\bgroup\color{black}\left[{\partial^2\over\partial r^2}+{2\ov... ...ght]R_{n\ell}(r) -{\ell(\ell+1)\over r^2}R_{n\ell}(r)=0 \egroup$$

The dominant term at the origin will be given by some power of $r$

$$\bgroup\color{black}R(r)=r^s.\egroup$$

Higher powers of $r$ are OK, but are not dominant. Plugging this into the equation we get

$$\bgroup\color{black}\left[s(s-1)r^{s-2}+2sr^{s-2}\right]-\ell(\ell+1)r^{s-2}=0. \egroup$$

$$\bgroup\color{black} s(s-1)+2s=\ell(\ell+1) \egroup$$

$$\bgroup\color{black} s(s+1)=\ell(\ell+1) \egroup$$

There are actually two solutions to this equation, $s=\ell$ and $s=-\ell-1$. The first solution, $s=\ell$, is well behaved at the origin (regular solution). The second solution, $s=-\ell-1$, causes normalization problems at the origin (irregular solution).