3D Symmetric HO in Spherical Coordinates

We have already solved the problem of a 3D harmonic oscillator by
separation of variables in Cartesian coordinates.
It is instructive to
**solve the same problem in spherical coordinates**and compare the results.
The potential is

Our radial equation is

Write the equation in terms of the dimensionless variable

Plugging these into the radial equation, we get

Now find the behavior for large .

Also, find the behavior for small .

Explicitly put in this behavior and use a power series expansion to solve the full equation.

We'll need to compute the derivatives.

We can now plug these into the radial equation.

Each term will contain the exponential , so we can factor that out. We can also run a single sum over all the terms.

The terms for large which go like and some of the terms for small which go like should cancel if we did our job right.

Now as usual, the coefficient for each power of
must be zero
for this sum to be zero for all
.
Before shifting terms, we must examine the first few terms of this
sum to learn about conditions on
and
.
The first term in the sum runs the risk of giving us a power of
which cannot be canceled by the second term if
.
For
, there is no problem because the term is zero.
For
the term is
which cannot be made
zero unless

This indicates that all the odd terms in the sum will be zero, as we will see from the recursion relation.

Now we will do the usual shift of the first term of the sum so that everything has a in it.

For large
,

Which will cause the wave function to diverge. We must terminate the series for some , by requiring

These are the same energies as we found in Cartesian coordinates. Lets plug this back into the recursion relation.

To rewrite the series in terms of and let take on every integer value, we make the substitutions and in the recursion relation for in terms of .

The table shows the quantum numbers for the states of each energy for our separation in spherical coordinates, and for separation in Cartesian coordinates. Remember that there are states with different components of angular momentum for the spherical coordinate states.

00 | 000 | 1 | 1 | |

01 | 001(3 perm) | 3 | 3 | |

10, 02 | 002(3 perm), 011(3 perm) | 6 | 6 | |

11, 03 | 003(3 perm), 210(6 perm), 111 | 10 | 10 | |

20, 12, 04 | 004(3), 310(6), 220(3), 211(3) | 15 | 15 |

Jim Branson 2013-04-22