The Angular Momentum Matrices

An important case of the use of the matrix form of operators is that of
Angular Momentum
Assume we have an atomic state with
(fixed) but
free.
We may use the eigenstates of
as a basis for our states and operators.
Ignoring the (fixed) radial part of the wavefunction,
our state vectors for
must be a linear combination of the

where , for example, is just the numerical coefficient of the eigenstate.

We will write our **3 component vectors** like

The angular momentum operators are therefore 3X3 matrices. We can easily derive the matrices representing the angular momentum operators for .

The matrices must satisfy the same **commutation relations** as the differential operators.

We verify this with an explicit computation of the commutator.

Since these matrices represent physical variables, we expect them to be **Hermitian.**
That is, they are equal to their conjugate transpose. Note that they are also
**traceless.**

As an example of the use of these matrices,
let's compute an **expectation value** of
in the matrix representation for the general state
.

Jim Branson 2013-04-22