Next: Spherical Harmonics Up: Orbital Angular Momentum Previous: Eigenvalues of

# Eigenvalues of

Consider the angular wavefunction . We know that
 (576)

since is a positive-definite real quantity. Hence, making use of Eqs. (194) and (539), we find that
 (577)

It follows from Eqs. (541), and (556)-(558) that
 (578)

We, thus, obtain the constraint
 (579)

Likewise, the inequality
 (580)

 (581)

Without loss of generality, we can assume that . This is reasonable, from a physical standpoint, since is supposed to represent the magnitude squared of something, and should, therefore, only take non-negative values. If is non-negative then the constraints (579) and (581) are equivalent to the following constraint:

 (582)

We, thus, conclude that the quantum number can only take a restricted range of integer values.

Well, if can only take a restricted range of integer values then there must exist a lowest possible value it can take. Let us call this special value , and let be the corresponding eigenstate. Suppose we act on this eigenstate with the lowering operator . According to Eq. (560), this will have the effect of converting the eigenstate into that of a state with a lower value of . However, no such state exists. A non-existent state is represented in quantum mechanics by the null wavefunction, . Thus, we must have

 (583)

Now, from Eq. (540),
 (584)

Hence,
 (585)

or
 (586)

where use has been made of (556), (557), and (583). It follows that
 (587)

Assuming that is negative, the solution to the above equation is
 (588)

We can similarly show that the largest possible value of is
 (589)

The above two results imply that is an integer, since and are both constrained to be integers.

We can now formulate the rules which determine the allowed values of the quantum numbers and . The quantum number takes the non-negative integer values . Once is given, the quantum number can take any integer value in the range

 (590)

Thus, if then can only take the value , if then can take the values , if then can take the values , and so on.

Next: Spherical Harmonics Up: Orbital Angular Momentum Previous: Eigenvalues of
Richard Fitzpatrick 2010-07-20