Quantum Mechanics for Scientists and Engineers

Thus far, we have dealt primarily with energy, position, and linear momentum and have proposed operators for each of these. One other quantity that is important in classical mechanics, angular momentum, is particularly important also in quantum mechanics. Here, we introduce angular momentum, its operators, eigenvalues, and eigenfunctions. If this discussion seems somewhat abstract, the reader can be assured that the concepts of angular momentum will become very concrete in the discussion of the hydrogen atom.

One aspect of angular momentum that is different from the quantities and operators discussed previously is that its operators always have discrete eigenvalues. Whereas linear momentum is associated with eigenfunctions that are functions of position along a specific spatial direction, angular momentum is associated with eigenfunctions that are functions of angle or angles about a specific axis. The fact that the eigenvalues are discrete is associated with the fact that for a single-valued spatial function, once we have gone an angle 2π about a particular axis, we are back to where we started. The wavefunction is presumably continuous and single-valued and, hence, we must therefore have integral numbers of periods of oscillation with angle within this angular range; this requirement of integer numbers of periods leads to the discrete quantization of angular momentum.

Another surprising aspect of angular momentum operators is that the operators corresponding to angular momentum about different orthogonal axes (e.g., and) do not commute with one another (in contrast, e.g., to the linear momentum operators for the different orthogonal coordinate directions).