Introduction to Nanomaterials and Devices

Appendix C

Angular Momentum

Angular momentum is a very important problem in many fields, including semiconductor materials. One may encounter angular momentum when dealing with doping in semiconductors, solving Schrödinger equation of semiconductor energy bands, and many other cases. In this appendix, we present the most important properties of the angular momentum without going through derivations. In dealing with angular momentum, one needs to distinguish between the spin, the orbital angular momentum, and total angular momentum. For the orbital angular momentum of a spinless particle, we have three observables Lx, Ly, and Lz, which are the components of the orbital angular momentum operator L. The three components can be written as

C.1

where X, Y, and Z are the position observables and Px, Py, and Pz are the momentum observables. It was shown in Chapter 1 that [X, Px] = [Y, Py] = [Z, Pz] = ih. Using this relation, we can write

C.2

Similarly, the components of the total angular momentum J can be expressed as

C.3

It can be shown that [J2, J] = 0. It is customary to use the following raising and lowering operators

C.4

The eigenvalues of J2 usually takes the following form j(j + ...