Chapter 9 The Two-Body Problem

1.7. Motion of a Particle Inside a Sphere
The first problem that we will study is the three-dimensional motion of a particle confined inside a sphere of radius without the presence of any other force. The Hamiltonian for this motion is simple;

As in the case of the motion of a particle in a box, (Sec. 8.4), we choose the lowering operator as

and as the Hermitian adjoint of . For zero angular momentum, , Eq. (9.168) has exactly the same form as the Hamiltonian for a particle in a box, . In this special case, , we can use the method of factorization outlined Sec. 8.4 to find the eigenvalues,

When we use the operator

The Hamiltonian (9.168) can be written in terms of this operator and as

Then the eigenvalue equation is

where is the energy eigenvalue.
The two operators and have the following additional properties:

and

These results can be derived directly from the definition of , Eq. (9.171), and its Hermitian conjugate.
Next we determine which is the solution of or

If we substitute for and from (9.169) in the Hamiltonian (9.168) and set the coefficient of equal to zero we find and

with the energy eigenvalues

Now by eliminating between (9.169) and (9.171) we have

From (9.177) and (9.179) we find ;

The general form of eigenvector is found from (9.177) turns out to be

where is obtained from the recurrence relation

In the range of

the term in the parenthesis in (9.182) varies between and and at the boundary becomes infinite. So at this point the coefficient of , i.e. has to be zero. The smallest root of gives us the lowest eigenvalue with the smallest energy given by (9.178). We note that Eq. (9.182) in the absence of the last term, , is the recurrence relation for the spherical Bessel function of the the order ;