The Pauli matrices

Intuition

For spin one-half the three components of spin are two-by-two matrices, and taking out the factor $\hbar/2$ leaves three matrices with remarkable properties, Pauli’s. Each squares to the identity; any two anticommute; the product of two is $i$ times the third. Together with the identity they are a basis for every two-by-two matrix, which is why they turn up wherever something has two states.

The imaginary unit is one square root of $-1$ . Multiplied by $i$ , the three Pauli matrices are three more, which do not commute with one another and multiply in a cycle: the product of two gives the third.

In the basis $\ket{\uparrow},\ket{\downarrow}$ , the spin operators are $\hat{\mathbf{S}}=\tfrac{\hbar}{2}\vec{\sigma}$ with the three Pauli matrices below.

\sigma_{x}=\begin{pmatrix}0&1\\1&0\end{pmatrix},\ \ \sigma_{y}=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\ \ \sigma_{z}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}

Properties

Each is Hermitian and unitary, with trace zero and eigenvalues $\pm1$ : $\sigma_{i}^{2}=1$ .
$\sigma_{x}\sigma_{y}=i\sigma_{z}$ , , ; reversing the order changes the sign.

The product rule

\sigma_{x}\sigma_{y}=i\sigma_{z}\ \Longrightarrow\ [\hat{S}_{x},\hat{S}_{y}]=i\hbar\hat{S}_{z}

Multiply the matrices in both orders: the products are $i\sigma_{z}$ and $-i\sigma_{z}$ . Their difference is $2 i σ_{}$ , and multiplying by gives the angular momentum commutator for .

Proof steps

\sigma_{x}\sigma_{y}=\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}0&-i\\i&0\end{pmatrix}=\begin{pmatrix}i&0\\0&-i\end{pmatrix}=i\sigma_{z}

Multiply the matrices.

\sigma_{y}\sigma_{x}=\begin{pmatrix}-i&0\\0&i\end{pmatrix}=-i\sigma_{z}

The other order gives the opposite sign.

[\sigma_{x},\sigma_{y}]=2i\sigma_{z}

Subtract.

[\hat{S}_{x},\hat{S}_{y}]=\frac{\hbar^{2}}{4}\,2i\sigma_{z}=i\hbar\hat{S}_{z}

Multiply by $(\hbar/2)^{2}$ and use $\hat{S}_{z}=\tfrac{\hbar}{2}\sigma_{z}$ .

Applications

Practice

Three Matrices

The Pauli matrices are the spin components divided by $\hbar/2$ . $\sigma_{x}$ swaps up and down, $\sigma_{z}$ changes the sign of down, and swaps them with factors of .

\sigma_{y}=\begin{pmatrix}0&-i\\i&0\end{pmatrix}

Try it

The matrix $\sigma_{y}$ is shown, rows and columns in the order $\ket{\uparrow},\ket{\downarrow}$ . Press the entry $\bra{\downarrow}\sigma_{y}\ket{\uparrow}$ .

$\sigma_{y}=\begin{pmatrix}0 & -i \\ i & 0\end{pmatrix}$