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Spin One-Half Matrices

In quantum physics, when you look at the spin eigenstates and operators for particles of spin 1/2 in terms of matrices, there are only two possible states, spin up and spin down.

The eigenvalues of the S2 operator are

image0.png

and the eigenvalues of the Sz operator are

image1.png

You can represent these two equations graphically as shown in the following figure, where the two spin states have different projections along the z axis.

Spin magnitude and <i>z</i> projection.
Spin magnitude and z projection.

In the case of spin 1/2 matrices, you first represent the eigenstate

image3.png

like this:

image4.png

And the eigenstate

image5.png

looks like this:

image6.png

Now what about spin operators like S2? The S2 operator looks like this in matrix terms:

image7.png

And this works out to be the following:

image8.png

Similarly, you can represent the Sz operator this way:

image9.png

This works out to

image10.png

Using the matrix version of Sz, for example, you can find the z component of the spin of, say, the eigenstate

image11.png

Finding the z component looks like this:

image12.png

Putting this in matrix terms gives you this matrix product:

image13.png

Here’s what you get by performing the matrix multiplication:

image14.png

And putting this back into ket notation, you get the following:

image15.png

How about the raising and lowering operators S+ and S? The S+ operator looks like this:

image16.png

And the lowering operator looks like this:

image17.png

Here it is in matrix terms:

image18.png

Performing the multiplication gives you this:

image19.png

Or in ket form, it’s

image20.png

Cool.

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