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Next: Spin
Angular Momentum Up: Mathematical
Interlude: The Legendre Previous: The
Legendre Polynomials
The Spherical Harmonics, 
, that are conventionally used are:
which is valid for 
, while the eigenfunctions corresponding to negative values of m
are obtained from:
The functions 
are the associated Legendre polynomials defined by:
which is valid for
, and the values for negative m given by:
(Remember 
). truein
and 
, determine the values of 
. Write out the quantities 
in terms of the above
(and their complex conjugates). (For example, 
where 
.) Hence show that 
is an eigenfunction of 
and 
is an eigenfunction of 
. What are their respective eigenvalues? truein
When m=0, (79) simplifies to a value
independent of the azimuthal angle 
where 
are the Legendre polynomials.
At the poles 
and 
the azimuthal angle
are indistinguishable so
cannot depend on
at these angles. Therefore the only non-zero spherical harmonics for these
two cases has to have m=0 i.e.
and
Consider two arbitrary directions in space defined respectively by the
angles 
and 
and call the angle between them 
. The Spherical Harmonic Addition Theorem states that
truein
be the distance between two points in space so that the 
direction is 
and 
direction is 
. Prove that
where 
stands for the smaller of the two distances 
and 
, and 
is the larger of the two distances. truein
Note that the
form a complete set of orthonormal functions, i.e.
and
This means that any function of 
and
can be expanded as follows:
where
Furthermore, if 
is the angular wavefunction of some state, normalised such that
then the 
are the probabilities that simultaneous measurement of 
and 
on the state described by
yields 
and 
respectively.
In the Dirac notation, an arbitrary state 
can be expanded as:
where with the help of the orthonormality condition
we get
These eigenstates are also complete so:
truein
and
calculate the matrix representation of 
and
for the AM state 3/2. Check that the commutation relations hold.
truein
Find the eigenvalues of 
if (a) the AM of the system is 1, (b) the AM of the system is 2.
truein
