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 in ordinary space. For a spin-one-half particle, the spin space is two-dimensional, and the spin operators are represented by 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
\[S_x=\frac{\hbar}{2}\sigma_x,\qquad S_y=\frac{\hbar}{2}\sigma_y,\qquad S_z=\frac{\hbar}{2}\sigma_z.\]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
\[P\left(+\frac{\hbar}{2}\right)=|a|^2, \qquad P\left(-\frac{\hbar}{2}\right)=|b|^2.\]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