Particle in A Spherically Symmetric Potential - Solutions For Potentials of Interest

Solutions For Potentials of Interest

Five special cases arise, of special importance:

1. V(r) = 0, or solving the vacuum in the basis of spherical harmonics, which serves as the basis for other cases.
2. (finite) for and 0 elsewhere, or a particle in the spherical equivalent of the square well, useful to describe scattering and bound states in a nucleus or quantum dot.
3. As the previous case, but with an infinitely high jump in the potential on the surface of the sphere.
4. V(r) ~ r2 for the three-dimensional isotropic harmonic oscillator.
5. V(r) ~ 1/r to describe bound states of hydrogen-like atoms.

We outline the solutions in these cases, which should be compared to their counterparts in cartesian coordinates, cf. particle in a box. This article relies heavily on Bessel functions and Laguerre polynomials.