Shankar, R. (1994), *Principles of Quantum Mechanics*, Plenum Press. Chapter 12, Exercise 12.6.11.

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We’ve solved the 3-d isotropic harmonic oscillator before, so we’ve already solved most of Shankar’s exercise 12.6.11. We can quote the results here. The solution has the form

The earlier solution uses notation from Griffiths’s book, but as the end result is the same, it’s not worth going through the derivation again using Shankar’s notation.

The potential is

The radial equation to be solved is

If we define

Taking the asymptotic behaviour of the radial function for small and large into account leads us to a solution of form

Note that Griffiths’s is not the same as Shankar’s , the latter of which is defined by Shankar’s equation 12.6.49.

This gives a differential equation for Griffiths’s

The function can be solved as a power series, giving

Substituting into 8 leads to the recursion relation

with , so that for all odd . The requirement that the series terminates at some finite value of leads to the quantization condition on :

or, defining ,

We worked out the degeneracies in the earlier post as well, so that the degeneracy of is

To complete Shankar’s exercise, we need to work out the eigenfunctions for and . For , , so only and we have

where in the fourth line we used

Normalizing this requires that

This is a standard Gaussian integral and can be done using software or tables so we get

This gives a wave function of

which agrees with the earlier result.

For , the degeneracy is, from 13

The three possibilities are which are reflected in the three spherical harmonics . The radial function is the same in all cases, and is obtained from , . From 7, this gives

Again, we work out by imposing normalization. For example

The normalization integral is

I used Maple to do the integrals. This gives a wave function of

We can work out the other two wave functions the same way (I used Maple, so I won’t go into the details):

The here is the same as in our rectangular solution set. The other two are linear combinations of and from our rectangular set, which were (the suffixes in these 2 equations stand for , and , and not , and ):

We have