Physics Friday 52

Quantum Mechanics and Momentum
Part 9: Angular Momentum as an Operator

Classically, the angular momentum L about the origin for a mass with linear momentum p at displacement r is given by . The Cartesian components are thus:
,
,
.
To turn this into a quantum operator, we replace the position and momentum r and p with the corresponding quantum operators. The components of the operator L are thus
,
,
.
The commutator of the x and y components is:
.
Similar relations hold for the cyclic permutation of the coordinates:

.
And thus we also have


.
From our previous discussion of commutators, we then see that two different components of the angular momentum cannot be simultaneously fixed for a quantum wavefunction.

Now, we examine the square norm of the angular momentum operator:
.
Finding it’s commutator with one of the components, and using several of the commutator relations we noted previously:
.
Similarly,
.
Thus, we can simultaneously specify the magnitude of the angular momentum and (any) one of it’s individual components, but no more components than that one. Usually, we choose to label the fixed component as our z.