## Angular Momentum Algebra: Raising and Lowering Operators

We have already derived the commutators of the angular momentum operators

We have shown that angular momentum is quantized for a rotor with a single angular variable. To progress toward the possible quantization of angular momentum variables in 3D, we define the operator and its Hermitian conjugate .

Since commutes with and , it commutes with these operators.

The commutator with is.

From the commutators and , we can derive the effect of the operators on the eigenstates , and in so doing, show that is an integer greater than or equal to 0, and that is also an integer

Therefore, raises the component of angular momentum by one unit of and lowers it by one unit. The raising stops when and the operation gives zero, . Similarly, the lowering stops because .

Angular momentum is quantized. Any measurement of a component of angular momentum will give some integer times . Any measurement of the total angular momentum gives the somewhat curious result

where is an integer.

Note that we can easily write the components of angular momentum in terms of the raising and lowering operators.

We will also find the following equations useful (and easy to compute).

* Example: What is the expectation value of in the state ?*

* Example: What is the expectation value of in the state ?*

Jim Branson 2013-04-22