Bound States in a Spherical Potential Well *

We now wish to find the energy eigenstates for a spherical potential well of radius $a$ and potential $-V_0$.

We must use the Bessel function near the origin.

$$\begin{eqnarray*} R_{n\ell}(r)=Aj_\ell(kr) \\ k=\sqrt{2\mu(E+V_0)\over\hbar^2} \\ \end{eqnarray*}$$

We must use the Hankel function of the first type for large $r$.

$$\begin{eqnarray*} \rho=kr\rightarrow i\kappa r \\ \kappa=\sqrt{-2\mu E\over\hbar^2} \\ R_{n\ell}=Bh_\ell^{(1)}(i\kappa r) \\ \end{eqnarray*}$$

To solve the problem, we have to match the solutions at the boundary. First match the wavefunction.

$$\bgroup\color{black}A\left[j_\ell(\rho)\right]_{\rho=ka}=B\left[h_\ell(\rho)\right]_{\rho=i\kappa a} \egroup$$

Then match the first derivative.

$$\bgroup\color{black}Ak\left[{dj_\ell(\rho)\over d\rho}\right]... ...left[{dh_\ell(\rho)\over d\rho}\right]_{\rho=i\kappa a} \egroup$$

We can divide the two equations to eliminate the constants to get a condition on the energies.

$$\bgroup\color{black}k\left[{{dj_\ell(\rho)\over d\rho}\over j... ...\over d\rho}\over h_\ell(\rho)}\right]_{\rho=i\kappa a} \egroup$$

This is often called matching the logarithmic derivative.

Often, the $\ell=0$ term will be sufficient to describe scattering. For $\ell=0$, the boundary condition is

$$\bgroup\color{black}k\left[{{\cos\rho\over\rho}-{\sin\rho\ove... ...}\over {e^{i\rho}\over i\rho}}\right]_{\rho=i\kappa a} .\egroup$$

Dividing and substituting for $\rho$, we get

$$\begin{eqnarray*} k\left(\cot(ka)-{1\over ka}\right)=i\kappa\left(i-{1\over i\ka... ...}\right) .\\ ka\cot(ka)-1=-\kappa a-1 \\ k\cot(ka)=-\kappa \\ \end{eqnarray*}$$

This is the same transcendental equation that we had for the odd solution in one dimension.

$$\bgroup\color{black} -\cot\left(\sqrt{2\mu(E+V_0)\over\hbar^2}a\right)=\sqrt{-E\over V_0+E} \egroup$$

The number of solutions depends on the depth and radius of the well. There can even be no solution.