Representation of Angular Momentum

It is most convenient to perform our investigation using conventional
*spherical polar coordinates*: *i.e.*, , , and . These are
defined with respect to our usual Cartesian coordinates as follows:

It follows, after some tedious analysis, that

Making use of the definitions (527)-(529), (534), and (538), the fundamental representation (478)-(480) of the operators as spatial differential operators, the Eqs. (545)-(550), and a great deal of tedious algebra, we finally obtain

as well as

(554) |

We, thus, conclude that all of our angular momentum operators can be represented as differential operators involving the