Shankar, R. (1994), *Principles of Quantum Mechanics*, Plenum Press. Chapter 12, Exercise 12.6.6.

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In solving the Schrödinger equation for spherically symmetric potentials, we found that we could reduce the problem to the equation

where is related to the radial function by

For a free particle, and , so we have

Defining

we convert the equation to

This equation can be solved by a method similar to that for the harmonic oscillator and its raising and lowering operators. The entire solution is fairly involved, so we’ll start out here by showing how the new raising and lowering operators are defined.

We define

The adjoint is

To see where the minus sign comes from on the RHS, we need to recall that the momentum operator is defined in one dimension as

Since is an observable, it is hermitian, so that . Under the hermitian operation , so we must also have . Thus the first derivative with respect to a position variable is anti-hermitian. If this doesn’t convince you, you can also work out the integral:

Under the usual assumption that at the limits, the integrated term is zero and we have

In bracket notation, this is

which shows that is an anti-hermitian operator.

Returning to 7 and 8, we have

Comparing with 6 we see that

We can also show that

Starting from 18 we multiply on the left by to get

Comparing this with 24 we see that

where is a constant.

Thus is a raising operator, in that it raises the angular momentum number by 1 when it acts on . By convention, (any adjustments to the constant can be made when normalizing).

We can start the process by looking at 6 with which is

This has the two solutions

The minus sign in front of is just conventional. Since we require , is unacceptable if the region we’re considering include , so we have

For the general case that excludes , we must include the cosine term as well.

From here, we can generate solutions for higher values of by applying 26. Actually, the radial function that appears in the wave function is given by 2, so it is that we really want. That is, we want

As with the constant in 26, we can absorb into normalization to be done later, so we can generate functions

Applying 26 we have

We can convert this into a general formula by writing

Starting at , we have

For the next step, we have

Thus in general

Note that

since the factor of has to be included when taking the derivative.

We’ll explore the nature of these solutions in the next post.