Spin One-Half Matrices
How to Find Angular Momentum Eigenvalues
How Spin Operators Resemble Angular Momentum Operators

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

image0.png

where the eigenvalue of

image1.png

So the eigenvalue of

image2.png

Similarly, the eigenvalue of

image3.png

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 Lz quantum number. You can define the raising and lowering operators this way:

  • Raising: L+ = Lx + iLy

  • Lowering: L = LxiLy

These two equations mean that

image4.png

You can also see that

image5.png

That means the following are all equal to L2:

image6.png

You can also see that these equations are true:

image7.png

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

Take a look at the operation of

image8.png

To see what

image9.png

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

image10.png

From

image11.png

you can see that

image12.png

And because

image13.png

you have the following:

image14.png

This equation means that the eigenstate

image15.png

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

image16.png

Or in a more comprehensible way:

image17.png

where c is a constant.

So the L+ operator has the effect of raising the

image18.png

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

image19.png

Now take a look at what

image20.png

equals:

image21.png

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

image22.png

And because

image23.png

you have the following equation:

image24.png

Similarly, the lowering operator, L, gives you this:

image25.png

So the results of these equations mean that the

image26.png

operators don't change the

image27.png

at all.

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