Pauli Matrices

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 S_x, S_y, and S_z written in terms of Pauli matrices,

Given that the eigenvalues of the S² operator are

and the eigenvalues of the S_z operator are

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 z projection.

Here’s what the Pauli matrices look like for the operators S_x, S_y, and S_z:

Now you can write S_x, S_y, and S_z in terms of the Pauli matrices like this: