**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 *S*2 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.

Read more about this topic: Spin (physics), Mathematical Formulation of Spin

### Famous quotes containing the word spin:

“In tragic life, God wot,

No villain need be! Passions *spin* the plot:

We are betrayed by what is false within.”

—George Meredith (1828–1909)