in a spinorial basis jm (2j+1 dimensional)

A 3-d representation of of spinors is a family of cones.

This satisfies:

The cone has an x-y length of and a hypoteneuse length of

Relation to in coordinate space:

You can represent in and

Since Then use lowering operator to get other .

For rotation of an wavefunction, .

Rotations generated by total angular momentum: for the pauli spin
matrices. This means . For electrons , commute with everything,
so CSCO is . This means a general eigenfunction is . and . . rotating
about you get

source

psfile jl@crush.caltech.edu index

tensor_operator_rotation

orbital_angular_momentum

rotation_matrices

operator_rotation

angular_momentum

particle_spin