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
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.
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
since the factor of has to be included when taking the derivative.
We’ll explore the nature of these solutions in the next post.