Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 10, Exercises 10.2.2 – 10.2.3.
We’ve seen that the 3-d isotropic harmonic oscillator can be solved in rectangular coordinates using separation of variables. The Hamiltonian is
The solution to the Schrödinger equation is just the product of three one-dimensional oscillator eigenfunctions, one for each coordinate. That is
Each one-dimensional eigenfunction can be expressed in terms of Hermite polynomials as
with the functions for and obtained by replacing by or and by or . We also saw earlier that in the 3-d oscillator, the total energy for state is given in terms of the quantum numbers of the three 1-d oscillators as
and that the degeneracy of level is .
Since the Hermite polynomial has parity (that is, odd (even) polynomials are odd (even) functions), the 3-d wave function has parity .
We can write the one state and three states in spherical coordinates using the standard transformation
Using the notation , we have, using and :
We can check that these are the correct spherical versions of the eigenfunctions by using the Schrödinger equation in spherical coordinates, which is
The spherical laplacian operator is
You can grind through the derivatives by hand if you like, but I just used Maple to do it, giving the results
In two dimensions, the analysis is pretty much the same. In the more general case where the masses are equal, but , the Hamiltonian is
A solution by separation of variables still works, with the result
The total energy is
For a given energy level , there are ways of forming out of a sum of 2 non-negative integers, so the degeneracy of level is .
The one state and two states are
To translate to polar coordinates, we use the transformations
so we have
Again, we can check this by plugging these polar formulas into the polar Schrödinger equation, where the 2-d Laplacian is
The results are