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 ):