Next: Exercises
Up: Orbital Angular Momentum
Previous: Eigenvalues of
Spherical Harmonics
The simultaneous eigenstates,
, of and
are known as the spherical harmonics. Let us investigate their
functional form.
Now, we know that

(591) 
since there is no state for which has a larger value than .
Writing

(592) 
[see Eqs. (570) and (574)], and making use of
Eq. (555), we obtain

(593) 
This equation yields

(594) 
which can easily be solved to give

(595) 
Hence, we conclude that

(596) 
Likewise, it is easy to demonstrate that

(597) 
Once we know , we can obtain by operating
on with the lowering operator . Thus,

(598) 
where use has been made of Eq. (555).
The above equation yields

(599) 
Now,

(600) 
where is a general function. Hence, we can write

(601) 
Likewise, we can show that

(602) 
We can now obtain by operating on with the
lowering operator. We get

(603) 
which reduces to

(604) 
Finally, making use of Eq. (600), we obtain

(605) 
Likewise, we can show that

(606) 
A comparison of Eqs. (596), (601), and (605)
reveals the general functional form of the spherical harmonics:

(607) 
Here, is assumed to be nonnegative. Making the substitution , we can also write

(608) 
Finally, it is clear from Eqs. (597), (602), and (606)
that

(609) 
Figure 18:
The
plotted as a functions of . The solid, shortdashed, and longdashed curves correspond to
, and , and , respectively.

We now need to normalize our spherical harmonic functions so as to ensure that

(610) 
After a great deal of tedious analysis, the normalized spherical
harmonic functions are found to take the form

(611) 
for , where the are known as associated Legendre
polynomials, and are written

(612) 
for . Alternatively,

(613) 
for .
The spherical harmonics characterized by
can be calculated from those characterized by via the identity

(614) 
The spherical harmonics are orthonormal: i.e.,

(615) 
and also form a complete set. In other words,
any function of and can be represented as
a superposition of spherical harmonics. Finally, and most importantly,
the spherical harmonics are the simultaneous eigenstates of and
corresponding to the eigenvalues and
,
respectively.
Figure 19:
The
plotted as a functions of . The solid, shortdashed, and longdashed curves correspond to
, and , and , respectively.

All of the , , and spherical harmonics are listed below:



(616) 



(617) 



(618) 



(619) 



(620) 



(621) 
The variation of these functions is illustrated in Figs. 18 and
19.
Subsections
Next: Exercises
Up: Orbital Angular Momentum
Previous: Eigenvalues of
Richard Fitzpatrick
20100720