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for the Orbital Up: Angular
Momentum Previous: Orbital
AM
The results of the previous section imply that 
is compatible with each component of 
and so it is possible to find simultaneous eigenstates of
and 
for example (there is nothing special about the choosing the z-component.
We could just as well have chosen the x or y component). We then have:
and
where Y(x,y,z) is an eigenfunction of both
and
.
To determine the eigenvalues, we use the ``ladder operator" technique, similar to the one used to study the Harmonic Oscillator. We define:
which satisfy the following commutation relation
and
From the result above, we have
So if Y is an eigenfunction of
and
then so is 
. Now consider the following:
This means that 
is also an eigenfunction of
with a new eigenvalue 
. 
is called a ``raising" operator because it increases the value of
by 
, whereas 
is called a ``lowering " operator since it decreases the eigenvalue
of
by
. Therefore starting from a given value of 
one obtains a ``ladder" of states with each ``rung" separated
from its neighbours by one unit of
in the eigenvalue of
. Then to go up the ladder, one applies
while to go down,
. This process cannot go on forever though: Eventually we are going to
reach a state for which the z-component of its AM exceeds the total
AM of that state and this cannot be!
truein
Problem :5 Prove that if Y is simultaneously an eigenfunction
of
and
then the square of the eigenvalue of
cannot exceed the eigenvalue of
. (Hint: Examine the expectation value of
).
truein
, such that
Let the eigenvalue of
for this ``top" state to be 
. Then we have
truein
truein
and hence
which tells us the eigenvalue of
in terms of the maximum eigenvalue of
. Following the same reasoning, there is also a ``bottom" rung, 
, such that
with 
the eigenvalue of
. Following the same method above, we deduce that
truein
Problem :7 Verify (49). Hence deduce
that the only possible value of 
is
truein
are 
, where m is an integer going from +l to -l in N
integer steps. This implies that 
so that l must be an integer or half-integer. The eigenfunctions
of
and
are characterised by the numbers l and m, where:
Note that for a given value of l, there are 2l+1 different values for m. truein
What is the value of 
, if the eigenfunctions 
are to be normalised?
Answer
can take the values:
Note that this result will be of use later when we study the addition of two angular momenta.
