Shankar, R. (1994), *Principles of Quantum Mechanics*, Plenum Press. Section 7.5, Exercises 7.5.1 – 7.5.3.

Earlier, we found the position space energy eigenfunctions of the harmonic oscillator to be

where in the first equation is shorthand for

It turns out that an alternative method for deriving these functions uses the lowering operator . Shankar gives the derivation of in his section 7.5, but we can use the same technique to derive the momentum space functions. We start with the ground state and use

In terms of and , we have

To find the momentum space functions, we need to express and in terms of :

We thus have

If we define the auxiliary variable

we get

This has the solution

for some normalization constant . Thus in terms of we have

Normalizing in the usual way, making use of the Gaussian integral, we have

This agrees with the earlier result which was obtained by solving a second-order differential equation.

We can also use and to verify a couple of recursion relations for Hermite polynomials. Reverting back to position space we have

so 5 becomes

Also from 5 we have, since and are both hermitian operators

Defining

we have

We also recall the normalization conditions on the raising and lowering operators:

Applying 23 to 1 we have, after cancelling common factors from each side:

Another recursion relation for Hermite polynomials can be found as follows. We start with 22 to get

We now apply 23 and 24 to 1. We can cancel common factors, including , from both sides to get