How to Find Commutators of Angular Momentum, L
Derive the Formula for the Rotational Energy of a Diatomic Molecule
Pauli Matrices

Find the Eigenvalues of the Raising and Lowering Angular Momentum Operators

In quantum physics, you can find the eigenvalues of the raising and lowering angular momentum operators, which raise and lower a state’s z component of angular momentum.

Start by taking a look at L+, and plan to solve for c:

L+| l, m > = c | l, m + 1 >

So L+ | l, m > gives you a new state, and multiplying that new state by its transpose should give you c2:

image0.png

To see this equation, note that

image1.png

On the other hand, also note that

image2.png

so you have

image3.png

What do you do about L+ L? Well, you assume that the following is true:

image4.png

So your equation becomes the following:

image5.png

Great! That means that c is equal to

image6.png

So what is

image7.png

Applying the L2 and Lz operators gives you this value for c:

image8.png

And that’s the eigenvalue of L+, which means you have this relation:

image9.png

Similarly, you can show that L gives you the following:

image10.png
  • Add a Comment
  • Print
  • Share
blog comments powered by Disqus
Find the Eigenfunctions of Lz in Spherical Coordinates
Find the Missing Spot with the Stern-Gerlach Experiment
How to Find Angular Momentum Eigenvalues
How to Change Rectangular Coordinates to Spherical Coordinates
Fermions and Bosons
Advertisement

Inside Dummies.com