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Representation of angular momentum
Now, we saw earlier, in Sect. 7.2, that the operators, , which represent
the cartesian components of linear momentum in quantum mechanics, can be represented
as the spatial differential operators
.
Let us now investigate whether angular momentum operators can similarly
be represented as spatial differential operators.
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:
Making use of the definitions (509)(511), (516), and (520), the fundamental representation (460)(462) of the operators as spatial differential operators, the definitions (527)(529), and a great deal of tedious analysis, we finally obtain
as well as

(533) 
and

(534) 
We, thus, conclude that all of our angular momentum operators can be represented
as differential operators involving the angular spherical
coordinates, and , but not involving the radial coordinate,
.
Next: Eigenstates of angular momentum
Up: Orbital angular momentum
Previous: Angular momentum operators
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Richard Fitzpatrick
20061212