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

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so you have

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

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So your equation becomes the following:

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Great! That means that c is equal to

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So what is

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

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

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

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