In quantum physics, when you work with spin eigenstates and operators for particles of spin 1/2 in terms of matrices, you may see the operators Sx, Sy, and Sz written in terms of Pauli matrices,
![image0.png](https://www.dummies.com/wp-content/uploads/395126.image0.png)
Given that the eigenvalues of the S2 operator are
![image1.png](https://www.dummies.com/wp-content/uploads/395127.image1.png)
and the eigenvalues of the Sz operator are
![image2.png](https://www.dummies.com/wp-content/uploads/395128.image2.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.](https://www.dummies.com/wp-content/uploads/395129.image3.jpg)
Spin magnitude and z projection.
Here’s what the Pauli matrices look like for the operators Sx, Sy, and Sz:
![image4.png](https://www.dummies.com/wp-content/uploads/395130.image4.png)
Now you can write Sx, Sy, and Sz in terms of the Pauli matrices like this:
![image5.png](https://www.dummies.com/wp-content/uploads/395131.image5.png)