## 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.

### 2 Responses to “Physics Friday 52”

1. Physics Friday 53 « Twisted One 151’s Weblog Says:

[…] Friday 53 By twistedone151 Quantum Mechanics and Momentum Part 10: Ladder Operators  Last week, we introduced the quantum angular momentum operator L, and found some properties of its magnitude […]

2. Physics Friday 77 « Twisted One 151’s Weblog Says:

[…] of freedom. Recall our discussion of the angular momentum in quantum mechanics, particularly the operators and their commutation relations, and the eigenvalue ladder. Spin is often described as being like the particles rotating around […]