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