03 Jun 2026
Pauli Spin Matrices and Spin Eigenvectors
Spin one-half operators, Pauli matrices, spin eigenvectors, and basic spin measurement probabilities.
Spin is intrinsic angular momentum. Unlike orbital angular momentum, it is not produced by motion around an origin. Experiments such as Stern-Gerlach show that a spin-one-half particle gives only two possible results when one component of spin is measured.
Therefore the spin state is represented by a two-component vector, called a spinor. The spin operators acting on this two-dimensional space are represented by the Pauli matrices.
Spin operators
For spin one-half,
\[\mathbf S=\frac{\hbar}{2}\boldsymbol{\sigma},\]where
\[\boldsymbol{\sigma}=(\sigma_x,\sigma_y,\sigma_z).\]The Pauli matrices are
\[\boxed{ \sigma_x= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \qquad \sigma_y= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \qquad \sigma_z= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}. }\]Thus
\[S_x=\frac{\hbar}{2}\sigma_x,\qquad S_y=\frac{\hbar}{2}\sigma_y,\qquad S_z=\frac{\hbar}{2}\sigma_z.\]The factor $\hbar/2$ is fixed by the fact that a spin-one-half particle has possible measured values $+\hbar/2$ and $-\hbar/2$ for any chosen spin component.
Algebra of Pauli matrices
The Pauli matrices satisfy
\[\boxed{ [\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k }\]and
\[\boxed{ \{\sigma_i,\sigma_j\}=2\delta_{ij}I. }\]Therefore the spin operators satisfy the angular momentum algebra
\[[S_i,S_j]=i\hbar\epsilon_{ijk}S_k.\]Also,
\[\sigma_x^2=\sigma_y^2=\sigma_z^2=I.\]Spin eigenvectors along $z$
The eigenvectors of $S_z$ are
\[\chi_+= \begin{pmatrix} 1\\ 0 \end{pmatrix}, \qquad \chi_-= \begin{pmatrix} 0\\ 1 \end{pmatrix}.\]They satisfy
\[S_z\chi_+=\frac{\hbar}{2}\chi_+, \qquad S_z\chi_-=-\frac{\hbar}{2}\chi_-.\]These are often written as
\[\chi_+=\lvert +z\rangle, \qquad \chi_-=\lvert -z\rangle.\]Spin eigenvectors along $x$
The eigenvectors of $S_x$ are
\[\lvert +x\rangle = \frac{1}{\sqrt2} \begin{pmatrix} 1\\ 1 \end{pmatrix}, \qquad \lvert -x\rangle = \frac{1}{\sqrt2} \begin{pmatrix} 1\\ -1 \end{pmatrix}.\]In the $z$-basis,
\[\lvert +x\rangle = \frac{1}{\sqrt2}(\lvert +z\rangle+\lvert -z\rangle).\]Therefore a particle prepared in $\lvert +x\rangle$ has equal probability of giving $+\hbar/2$ or $-\hbar/2$ when $S_z$ is measured.
Spin eigenvectors along $y$
The eigenvectors of $S_y$ are
\[\lvert +y\rangle = \frac{1}{\sqrt2} \begin{pmatrix} 1\\ i \end{pmatrix}, \qquad \lvert -y\rangle = \frac{1}{\sqrt2} \begin{pmatrix} 1\\ -i \end{pmatrix}.\]General spinor
A general spin-one-half state can be written as
\[\chi= \begin{pmatrix} a\\ b \end{pmatrix} =a\lvert +z\rangle+b\lvert -z\rangle,\]with normalization
\[|a|^2+|b|^2=1.\]If $S_z$ is measured, the probabilities are
\[P\left(+\frac{\hbar}{2}\right)=|a|^2, \qquad P\left(-\frac{\hbar}{2}\right)=|b|^2.\]The complex numbers $a$ and $b$ are probability amplitudes. Their moduli squared give probabilities, while their relative phase affects measurements along axes other than $z$.
Verification of a commutator
Using the Pauli matrices,
\[\sigma_x\sigma_y= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} = \begin{pmatrix} i&0\\ 0&-i \end{pmatrix} =i\sigma_z.\]Similarly,
\[\sigma_y\sigma_x=-i\sigma_z.\]Therefore
\[\boxed{ [\sigma_x,\sigma_y]=2i\sigma_z. }\]Multiplying by $\hbar/2$ gives
\[[S_x,S_y]=i\hbar S_z.\]Measurement example
Let
\[\chi= \begin{pmatrix} 1/\sqrt3\\ \sqrt{2/3} \end{pmatrix}.\]The state is normalized because
\[\frac13+\frac23=1.\]If $S_z$ is measured,
\[P\left(+\frac{\hbar}{2}\right)=\left|\frac{1}{\sqrt3}\right|^2=\frac13,\]and
\[P\left(-\frac{\hbar}{2}\right)=\left|\sqrt{\frac23}\right|^2=\frac23.\]Main points
- Pauli matrices represent spin-one-half operators.
- Spin components satisfy the same angular momentum algebra.
- Spin eigenvectors form two-component spinors.
- A spinor contains probability amplitudes for different spin measurement results.
Practice questions
- Verify that $[\sigma_x,\sigma_y]=2i\sigma_z$.
- Find the eigenvectors of $\sigma_x$.
- Show that $S^2=\frac{3\hbar^2}{4}I$ for spin one-half.
- If $\chi=(1/\sqrt3,\sqrt{2/3})^T$, find the probabilities for measuring $S_z$.
- Express $\lvert +x\rangle$ in the $S_z$ basis.
Discussion