## SHM p.10

### Degeneracy

Here are the collection of energy levels produced by solving
the 2-d oscillator in the *x-y* coordinate system:

Here are the collection of energy levels produced by solving
the 2-d oscillator in the *r-* coordinate system:

The wavefunctions for these levels look different:
the *x-y* wavefunctions have square symmetry while the
*r-* wavefunctions have circle-symmetry.
Nevertheless, notice that for any value of *E'* the number
of levels ("degeneracy") is the same in either system. For example,
*E'*=10 includes 5 |*n*_{r}*m*> wavefunctions:
|0,-4>, |1,-2>, |2,0>, |1,2>, and |0,4> or includes 5
|*n*_{x}n_{y}> wavefunctions:
|0,4>, |1,3>, |2,2>, |3,1>, and |4,0> .
In fact the set of functions spanned by forming all possible
linear combinations of the 5 |*n*_{r}m> wavefunctions
is identical to that formed by all possible
linear combinations of the 5 |*n*_{x}n_{y}>.
In particular any |*n*_{r}m> wavefunction
can be expressed as a linear combination of the co-degenerate
|*n*_{x}n_{y}> wavefunctions
(and visa-versa). The degenerate *x-y* wavefunctions
and the *r-* wavefunctions are thus
different basis vectors for the same subspace.

Here is a start on finding the sum of |*n*_{x}n_{y}>
which equals any |*n*_{r}m> for *E'*=10.

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