Because spin is a type of built-in angular momentum, spin operators have a lot in common with orbital angular momentum operators. As your quantum physics instructor will tell you, there are analogous spin operators, S^{2} and S* _{z}*, to orbital angular momentum operators L

^{2}and L

_{z}. However, these operators are just operators; they don’t have a differential form like the orbital angular momentum operators do.

In fact, all the orbital angular momentum operators, such as L* _{x}*, L

*, and L*

_{y}*, have analogs here: S*

_{z}*, S*

_{x}*, and S*

_{y}*. The commutation relations among L*

_{z}*, L*

_{x}*, and L*

_{y}*are the following:*

_{z}And they work the same way for spin:

The L^{2} operator gives you the following result when you apply it to an orbital angular momentum eigenstate:

And just as you’d expect, the S^{2} operator works in an analogous fashion:

The L* _{z}* operator gives you this result when you apply it to an orbital angular momentum eigenstate:

And by analogy, the S* _{z}* operator works this way:

What about the raising and lowering operators, L_{+} and L_{–}? Are there analogs for spin? In angular momentum terms, L_{+} and L_{–} work like this:

There are spin raising and lowering operators as well, S_{+} and S_{–}, and they work like this: