* Pauli matrices* are matrix representations of the spin operators , and .

To derive the Pauli matrices, we let and be basis vectors representing the electron spin eigenstates of and respectively, with the following assignments:

The general state of an electron can then be written as a linear combination of the two spin states:

where is the probability of finding the electron with the state , and is the probability of finding the electron with the state , with

Having defined the two eigenstates, we can work out the corresponding matrix representation of using eq169 by letting :

So,

Similarly, using eq168 for , we have

For and , we make use of eq170 where

Since , we have or . Similarly, . Using eq175 and eq176,

In an earlier article, when we constructed quantum orbital angular momentum component operators , and , we replaced the position and linear momentum components of the classical angular momentum components , and with their corresponding operators. We also suggested that we could have constructed an angular momentum operator using , such that , but for certain reasons chose to construct instead. For electron spin, it is useful to construct the spin operator using that suggestion:

where , with

, and are called Pauli matrices. They represent observables and are Hermitian (the complex conjugate of each matrix returns the same matrix). Eq179 and eq180 are used to analyse the results of successive Stern-Gerlach experiments.

###### Question

Show that and but .

Answer

For , substitute in and and expand the expression. From eq102, commutes with all components of ; and also all components of because orbital angular momentum operators and spin angular momentum operators act on different vector spaces. Thus, . Similarly, from eq165, eq166 and eq167, commutes with all components of and and so .

Now,

Substituting eq181 in and expanding the expression, we find, after some algebra, that

Similarly,