Next: Mathematical Interlude: The Legendre Up: Angular Momentum Previous: Eigenvalues for the Orbital
Next: Mathematical
Interlude: The Legendre Up: Angular
Momentum Previous: Eigenvalues
for the Orbital
We now determine the eigenfunctions for the operators 
and 
. The conventional way is to write out these operators in spherical polar
coordinates:
from which we can derive:
Then the
operator in spherical polar coordinates is:
while the raising/lowering operators are:
truein
in spherical polar coordinates. (Hint: You could use the result proven
earlier, i.e. 
). truein
Since we have already shown that 
is a simultaneous eigenfunction of
and
, from the above results it is clear that the easiest way to determine
is by using the method of separation of variables:
so that
implies
which is a first-order differential equation in 
. The normalised solutions are:
Therefore
To solve for 
we proceed as follows. We use the fact that the maximum value of m
is l such that:
Expanding this out gives:
whose solution is
Starting from this, we can determine all the other states by applying the lowering operator. This means that an arbitrary state can be represented by:
where C is a constant.
truein
on the ``top" state. First verify the following identity:
Then show that by acting on 
with
we get:
where C', C'' are constants.
truein
such that 
then the general form for
is
These eigenfunctions (the Spherical Harmonics) are to be normalised
over the unit sphere where the range of integration is 
and 
. This means
