Next: 11.7 General solutions
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The key issue about three-dimensional motion in a spherical potential is
angular momentum. This is true classically as well as in quantum theories.
The angular momentum in classical mechanics is defined as the vector (outer)
product of r and p,
This has an easy quantum analog that can be written as
After exapnsion we find
= - i y
- z , z
- x , x
- y
 |
(11.28) |
This operator has some very interesting properties:
[ , ] = 0. |
(11.29) |
Thus
[ , ] = 0! |
(11.30) |
And even more surprising,
Thus the different components of L are not compatible (i.e., can't be
determined at the same time). Since L commutes with H we can diagonalise
one of the components of L at the same time as H. Actually,
we diagonalsie
,
and H at the same time!
The solutions to the equation
YLM( , ) = L(L + 1)YLM( , ) |
(11.32) |
are called the spherical harmonics.
Question: check that
is independent of r!
The label M corresponds to the operator
,
YLM( , ) = MYLM( , ). |
(11.33) |
Next: 11.7 General solutions
Up: 11. 3D Schrödinger equation
Previous: 11.5 Now where does