# Isotropic Harmonic Oscillator

### From FSUPhysicsWiki

We now solve the isotropic harmonic oscillator using the formalism that we have just developed. While it is possible to solve it in Cartesian coordinates, we gain additional insight by solving it in spherical coordinates, and it is easier to determine the degeneracy of each energy level.

The radial part of the Schrödinger equation for a particle of mass in an isotropic harmonic oscillator potential is given by:

Let us begin by looking at the solutions in the limits of small and large

As , the equation reduces to

The only solution of this equation that does not diverge as is

In the limit as on the other hand, the equation becomes

whose solution is given by

We may now assume that the general solution to the equation is given by

Substituting this expression into the original equation, we obtain

We now use a series solution for this equation:

Substituting this solution into the reduced form of the equation, we obtain

which reduces to

For this equation to hold, the coefficients of each of the powers of must vanish seperately. Doing this for the positive powers of yields the following recursion relation:

In addition, we have an term; for it to vanish, we must set This, combined with the above recursion relation, means that the function contains only even powers of In other words,

By a similar argument as the one that we employed for the one-dimensional harmonic oscillator, we find that, unless the series for terminates, the resulting full wave function will diverge as Because the series must only contain even powers of the resulting quantization condition on the energy is

or

where

The degeneracy corresponding to the level may be found to be We see that energy levels with even correspond to even values of while those with odd have odd values of

The total wave function of the isotropic harmonic oscillator is thus given by

One may show that, in fact, is an associated Laguerre polynomial in