How to Create Angular Momentum Eigenstates

You can create the actual eigenstates, | l, m >, of angular momentum states in quantum mechanics. When you have the eigenstates, you also have the eigenvalues, and when you have the eigenvalues, you can solve the Hamiltonian and get the allowed energy levels of an object with angular momentum.

Don't make the assumption that the eigenstates are | l, m >; rather, say they're

where the eigenvalue of

So the eigenvalue of

Similarly, the eigenvalue of

To proceed further, you have to introduce raising and lowering operators. That way, you can solve for the ground state by, for example, applying the lowering operator to the ground state and setting the result equal to zero — and then solving for the ground state itself.

In this case, the raising operator is L₊ and the lowering operator is L_–. These operators raise and lower the L_z quantum number. You can define the raising and lowering operators this way:

• Raising: L₊ = L_x + iL_y

• Lowering: L_– = L_x – iL_y

These two equations mean that

You can also see that

That means the following are all equal to L²:

You can also see that these equations are true:

Okay, now you can put all this to work. You're getting to the good stuff.

Take a look at the operation of

To see what

is, start by applying the L_z operator on it like this:

From

you can see that

And because

you have the following:

This equation means that the eigenstate

is also an eigenstate of the L_z operator, with an eigenvalue of

Or in a more comprehensible way:

where c is a constant.

So the L₊ operator has the effect of raising the

quantum number by 1. Similarly, the lowering operator does this:

Now take a look at what

equals:

Because L² is a scalar, it commutes with everything. L² L₊ – L₊ L² = 0, so this is true:

And because

you have the following equation:

Similarly, the lowering operator, L_–, gives you this:

So the results of these equations mean that the

operators don't change the

at all.