The 3D Harmonic Oscillator

The 3D harmonic oscillator can also be separated in Cartesian coordinates. For the case of a central potential, , this problem can also be solved nicely in spherical coordinates using rotational symmetry. The cartesian solution is easier and better for counting states though.

Lets **assume the central potential** so we can compare to our later solution.
We could have three different spring constants and the solution would be as simple.
The Hamiltonian is

The problem separates nicely, giving us **three independent harmonic oscillators**.

This was really easy.

This problem has a different Fermi surface in
-space than did the particle in a box.
The boundary between filled and unfilled energy levels is a plane defined by

Jim Branson 2013-04-22