Consider a particle whose position is described by the spherical polar coordinates . The classical momentum conjugate to the azimuthal angle is the -component of angular momentum, . According to Section 2.5, in quantum mechanics we can always adopt the Schrödinger representation, for which ket space is spanned by the simultaneous eigenkets of the position operators , , and , and takes the form
We can do this because there is nothing in Section 2.5 which specifies that we have to use Cartesian coordinates--the representation (165) works for any well-defined set of coordinates.
Consider an operator that rotates the system through an angle about the -axis. This operator is very similar to the operator , introduced in Section 2.8, which translates the system a distance along the -axis. We were able to demonstrate in Section 2.8 that
where is the linear momentum conjugate to . There is nothing in our derivation of this result which specifies that has to be a Cartesian coordinate. Thus, the result should apply just as well to an angular coordinate. We conclude that
According to Equation (348), we can write
in the limit . In other words, the angular momentum operator can be used to rotate the system about the -axis by an infinitesimal amount. We say that is the generator of rotations about the -axis. The above equation implies that
which reduces to
Note that has all of the properties we would expect of a rotation operator: i.e.,
Suppose that the system is in a simultaneous eigenstate of and . As before, this state is represented by the eigenket , where the eigenvalue of is , and the eigenvalue of is . We expect the wavefunction to remain unaltered if we rotate the system degrees about the -axis. Thus,
We conclude that must be an integer. This implies, from the previous section, that must also be an integer. Thus, an orbital angular momentum can only take integer values of the quantum numbers and .
Consider the action of the rotation operator on an eigenstate possessing zero angular momentum about the -axis (i.e., an state). We have
Thus, the eigenstate is invariant to rotations about the -axis. Clearly, its wavefunction must be symmetric about the -axis.
There is nothing special about the -axis, so we can write
by analogy with Equation (351). Here, denotes an operator that rotates the system by an angle about the -axis, etc. Suppose that the system is in an eigenstate of zero overall orbital angular momentum (i.e., an state). We know that the system is also in an eigenstate of zero orbital angular momentum about any particular axis. This follows because implies , according to the previous section, and we can choose the -axis to point in any direction. Thus,
Clearly, a zero angular momentum state is invariant to rotations about any axis. Such a state must possess a spherically symmetric wavefunction.
Note that a rotation about the -axis does not commute with a rotation about the -axis. In other words, if the system is rotated an angle about the -axis, and then about the -axis, it ends up in a different state to that obtained by rotating an angle about the -axis, and then about the -axis. In quantum mechanics, this implies that , or , [see Equations (357)-(359)]. Thus, the noncommuting nature of the angular momentum operators is a direct consequence of the fact that rotations do not commute.
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)