Commutators, matrices, and 3-dimensional Schrödinger equation
PROFESSOR: This is very important. This is the beginning of the uncertainty principle, the matrix formulation of quantum mechanics, and all those things. I want to just tabulate the information of matrices. We have an analog, so we have operators. And we think of them as matrices. Then in addition to operators, we have wave functions. And we think of them as vectors. The operators act on the wave functions or functions, and matrices act on vectors. We have eigenstate sometimes and eigenvectors. So matrices do the same thing. They don't necessarily commute. There are very many examples of that. I might as well give you a little example that is famous in the theory of spin, spin 1/2. There is the Pauli matrices. Sigma 1 is equal to 1, 1, 0, 0. Sigma 2 is 0 minus i, i 0, and sigma 3 is 1 minus 1, 0, 0. And a preview of things to come-- the spin operator is actually h bar over 2 sigma. And you have to think of sigma as having three components. That's where it is. Spin...