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 coordinatesand compare the results.
The potential is
We'll need to compute the derivatives.
We can now plug these into the radial equation.
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
The first term in the sum runs the risk of giving us a power of
which cannot be canceled by the second term if
, there is no problem because the term is zero.
the term is
which cannot be made
Now we will do the usual shift of the first term of the sum so that everything has a in it.
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.
|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