Eigenvalues of

We can satisfy the orthonormality constraint (558) provided that

Note, from Eq. (553), that the differential operator which represents
only depends on the azimuthal angle , and is independent
of the polar angle . It therefore follows from Eqs. (553), (556), and (570)
that

(573) |

Here, the symbol just means that we are neglecting multiplicative constants.

Now, our basic interpretation of a wavefunction
as a quantity whose modulus squared represents the probability density
of finding a particle at a particular point in space suggests that a
physical wavefunction must be *single-valued* in space. Otherwise, the probability density at a given point would not, in general, have a unique value, which does not
make physical sense.
Hence, we demand that the wavefunction (574)
be single-valued: *i.e.*,
for all . This immediately implies that the quantity is *quantized*.
In fact, can only take *integer values*. Thus, we conclude that the eigenvalues
of are also quantized, and take the values , where is an integer. [A more rigorous argument is that
must be continuous in order to ensure that is an Hermitian operator, since the proof of
hermiticity involves an integration by parts in that has canceling contributions from and .]

Finally, we can easily normalize the eigenstate (574) by making use of the
orthonormality constraint (572). We obtain

(575) |