Pauli Matrices as Spin Operators

Spin-1/2 is the simplest quantum mechanical degree of freedom. A spin-1/2 particle has only two spin states along any axis: spin-up and spin-down. The mathematical machinery for all three spin components is encoded in three $2 \times 2$ matrices called the Pauli matrices:

\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

The physical spin operator along each axis is obtained by scaling:

S_x = \frac{\hbar}{2}\,\sigma_x, \qquad S_y = \frac{\hbar}{2}\,\sigma_y, \qquad S_z = \frac{\hbar}{2}\,\sigma_z.

The factor $\hbar/2$ is not arbitrary: it ensures that measuring $S_z$ yields one of the two values $\pm\hbar/2$ , the experimentally observed spin eigenvalues. In many calculations it is convenient to work with the dimensionless Pauli matrices directly and restore factors of $\hbar/2$ at the end.

Eigenstates and eigenvalues

An eigenstate of an operator $A$ with eigenvalue $\lambda$ satisfies $A|\psi\rangle = \lambda|\psi\rangle$ . For $\sigma_z$ the two eigenstates are the standard basis vectors:

\sigma_z |0\rangle = |0\rangle, \qquad \sigma_z |1\rangle = -|1\rangle,

where $|0\rangle = \binom{1}{0}$ and $|1\rangle = \binom{0}{1}$ . This can be verified by direct matrix multiplication:

\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} = +1 \cdot |0\rangle.

So $|0\rangle$ is the spin-up eigenstate of $\sigma_z$ with eigenvalue $+1$ , and $|1\rangle$ is the spin-down eigenstate with eigenvalue $-1$ .

For $\sigma_x$ the eigenstates are the diagonal states:

|{+x}\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix}, \qquad |{-x}\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\-1\end{pmatrix}.

One can check: $\sigma_x |{+x}\rangle = \begin{pmatrix}0&1\\1&0\end{pmatrix}\frac{1}{\sqrt{2}}\binom{1}{1} = \frac{1}{\sqrt{2}}\binom{1}{1} = +1\cdot|{+x}\rangle$ , confirming eigenvalue $+1$ .

Algebraic properties

The Pauli matrices satisfy three important relations. First, each squares to the identity:

\sigma_x^2 = \sigma_y^2 = \sigma_z^2 = I.

Second, they anticommute in pairs:

\sigma_x\sigma_y + \sigma_y\sigma_x = 0, \quad \sigma_y\sigma_z + \sigma_z\sigma_y = 0, \quad \sigma_z\sigma_x + \sigma_x\sigma_z = 0.

Third, they satisfy cyclic commutation relations:

[\sigma_x, \sigma_y] = 2i\,\sigma_z, \quad [\sigma_y, \sigma_z] = 2i\,\sigma_x, \quad [\sigma_z, \sigma_x] = 2i\,\sigma_y.

These three commutators encode the non-commutativity of spin: measuring $S_x$ and then $S_y$ is not the same as measuring $S_y$ and then $S_x$ . Physically, this means spin components along different axes cannot both have definite values simultaneously.

Expectation values

The expectation value of an observable $A$ in state $|\psi\rangle$ is the average measurement outcome, computed as the inner product:

\langle A \rangle = \langle\psi|A|\psi\rangle.

For example, the expectation value of $\sigma_z$ in the state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ is

\langle\sigma_z\rangle = \langle\psi|\sigma_z|\psi\rangle = (\alpha^*, \beta^*)\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}\alpha\\\beta\end{pmatrix} = |\alpha|^2 - |\beta|^2.

This result has a clean interpretation: $|\alpha|^2$ is the probability of finding spin-up (contributing $+1$ ) and $|\beta|^2$ is the probability of finding spin-down (contributing $-1$ ), so the expectation value is the probability-weighted average of the two outcomes.

Try it

This is a numerical exercise — return a number. Using matrix arithmetic, compute the expectation value $\langle{+x}|\sigma_x|{+x}\rangle$ where $|{+x}\rangle = \frac{1}{\sqrt{2}}\binom{1}{1}$ is the spin-up eigenstate of $\sigma_x$ .

Run your code to see the quantum state.