##### Quantum Physics For Dummies 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 Note that L2 – Lz2 = Lx2 + Ly2, which is a positive number, so That means that And substituting in and using the fact that the eigenstates are normalized, gives you this: So there’s a maximum possible value of which you can call You can be clever now, because there has to be a state such that you can’t raise any more. Thus, if you apply the raising operator, you get zero: Applying the lowering operator to this also gives you zero: And because that means the following is true: Putting in gives you this: At this point, it’s usual to rename You can say even more. In addition to a there must also be a such that when you apply the lowering operator, L, you get zero, because you can’t go any lower than And you can apply L+ on this as well: From you know that which gives you the following: And comparing this equation to gives you Note that because you reach by n successive applications of you get the following: Coupling these two equations gives you Therefore, can be either an integer or half an integer (depending on whether n is even or odd).

Because and n is a positive number, you can find that 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. 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. You can see a representative L and Lz in the figure. L and Lz.

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