A.6 Constant spherical potentials

This addendum describes the solutions of the Hamiltonian eigenvalue problem in spherical coordinates if the potential is constant.

These solutions are important for many reasons. For example, you might want to create a simplified model for the hydrogen atom that way. To do so, you could, for example, assume that the potential energy has a constant negative value up to say the Bohr radius and is zero beyond it. That is not really a very good model for the hydrogen atom. However, it works much better for nucleons in atomic nuclei, chapter 14.12.

The solutions in this note are also important for describing experiments in which particles are scattered from some target, {A.30}. And more fundamentally, they give the energy states of definite angular momentum for particles in empty space.

A.6.1 The eigenvalue problem

The Hamiltonian eigenvalue problem is

In this note it is assumed that the potential is a constant in the radial region of interest.

To clean the problem up a bit, take the potential energy term to the
other side, and also the coefficient of the Laplacian. That produces

where

The constant is what classical physics would take to be the linear momentum of a particle with total energy and potential energy .

A.6.2 The eigenfunctions

Because the potential is spherically symmetric like for the hydrogen
atom, the eigenfunctions are of similar form:

The radial functions in the eigenfunctions are different from those of the hydrogen atom. Depending on whatever is easiest in a given application, they can be written in two ways, {D.16}.

The first way is as

Expressions for these Bessel functions can be found in advanced
mathematical handbooks, [1]:

The spherical Bessel functions are often convenient in a region of
constant potential that includes the origin, because the Bessel
functions of the first kind give the solutions that are finite
at the origin. (To see that, note that the Taylor series of
divided by is a power series in , and that
.) In particular for small :

The Bessel functions of the second kind are singular at the origin and normally do not appear if the origin is part of the considered region.

Also, the spherical Bessel functions are real for real . However, in a region where the potential is larger than the energy of the particles, the argument of the Bessel functions in (A.18) will be imaginary.

The other way to write the radial functions is as

where and are called the “spherical Hankel functions.”

The spherical Hankel functions can again be found in advanced
mathematical handbooks, [1]:

The given expressions in terms of the spherical Bessel functions are readily inverted to give the Bessel functions in terms of the Hankel functions,

For large the spherical Hankel functions can be approximated as

A.6.3 About free space solutions

The most important case for which the energy eigenfunctions of the previous subsection apply is for particles in empty space. They describe energy states with definite square and angular momentum. However, sometimes particles in empty space are better described by states of definite linear momentum. And in some cases, like in scattering problems, you need both types of solution. Then you also need to convert between them.

The energy states in empty space with definite square and angular
momentum are

A state that has definite linear momentum purely in
the -direction has an energy eigenfunction

It is sometimes necessary to write a linear momentum eigenfunction of
the form (A.27) in terms of angular momentum ones of the
form (A.26). Rayleigh worked out the correct combination a
very long time ago, {D.16}: