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# 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 non-negative. Making the substitution , we can also write (608)

Finally, it is clear from Eqs. (597), (602), and (606) that (609) 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. 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 2010-07-20