An important case of the use of the matrix form of operators is that of
Assume we have an atomic state with
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
We will write our 3 component vectors like
The matrices must satisfy the same commutation relations as the differential operators.
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