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:

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To see this equation, note that

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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
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