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How to Find Angular Momentum Eigenvalues

When you have the eigenvalues of angular momentum states in quantum mechanics, you can solve the Hamiltonian and get the allowed energy levels of an object with angular momentum. The eigenvalues of the angular momentum are the possible values the angular momentum can take.

Here’s how to derive eigenstate equations with

image0.png

Note that L2 – Lz2 = Lx2 + Ly2, which is a positive number, so

image1.png

That means that

image2.png

And substituting in

image3.png

and using the fact that the eigenstates are normalized, gives you this:

image4.png

So there’s a maximum possible value of

image5.png

which you can call

image6.png

You can be clever now, because there has to be a state

image7.png

such that you can’t raise

image8.png

any more. Thus, if you apply the raising operator, you get zero:

image9.png

Applying the lowering operator to this also gives you zero:

image10.png

And because

image11.png

that means the following is true:

image12.png

Putting in

image13.png

gives you this:

image14.png

At this point, it’s usual to rename

image15.png

You can say even more. In addition to a

image16.png

there must also be a

image17.png

such that when you apply the lowering operator, L, you get zero, because you can’t go any lower than

image18.png

And you can apply L+ on this as well:

image19.png

From

image20.png

you know that

image21.png

which gives you the following:

image22.png

And comparing this equation to

image23.png

gives you

image24.png

Note that because you reach

image25.png

by n successive applications of

image26.png

you get the following:

image27.png

Coupling these two equations gives you

image28.png

Therefore,

image29.png

can be either an integer or half an integer (depending on whether n is even or odd).

Because

image30.png

and n is a positive number, you can find that

image31.png

So now you have it:

  • The eigenstates are | l, m >.

  • The quantum number of the total angular momentum is l.

  • The quantum number of the angular momentum along the z axis is m.

    image32.png

For each l, there are 2l + 1 values of m. For example, if l = 2, then m can equal –2, –1, 0, 1, or 2.

image33.png

You can see a representative L and Lz in the figure.

L and L<i><sub>z</sub></i>.
L and Lz.

L is the total angular momentum and Lz is the projection of that total angular momentum on the z axis.

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