Spin (physics) - Mathematical Formulation of Spin - Spin Operator

Spin Operator

Spin obeys commutation relations analogous to those of the orbital angular momentum:

where is the Levi-Civita symbol. It follows (as with angular momentum) that the eigenvectors of S2 and S_z (expressed as kets in the total S basis) are:

The spin raising and lowering operators acting on these eigenvectors give:

, where

But unlike orbital angular momentum the eigenvectors are not spherical harmonics. They are not functions of θ and φ. There is also no reason to exclude half-integer values of s and m.

In addition to their other properties, all quantum mechanical particles possess an intrinsic spin (though it may have the intrinsic spin 0, too). The spin is quantized in units of the reduced Planck constant, such that the state function of the particle is, say, not, but where is out of the following discrete set of values:

One distinguishes bosons (integer spin) and fermions (half-integer spin). The total angular momentum conserved in interaction processes is then the sum of the orbital angular momentum and the spin.