Spin-1/2 and the Pauli Matrices

The Pauli matrices are the instruction set of quantum computing. Single-qubit gates are this article's exponentials — $X$, $Y$, $Z$ gates are $\sigma_x,\sigma_y,\sigma_z$ themselves, and calibrating a superconducting or trapped-ion processor means shaping pulses so the implemented $e^{-i\theta\,\hat n\cdot\boldsymbol\sigma/2}$ hits its angle to a part in $10^4$. One layer up, Pauli strings — tensor products like $\sigma_x\otimes\sigma_z\otimes\sigma_y$ — are the basis in which every $n$-qubit observable is expanded (relation (5), iterated): how molecular Hamiltonians reach quantum processors, and how stabilizer codes define their check operators. When an error-correction paper says "measure the stabilizer $XZZX$," it is invoking (5)'s basis and the product rule's anticommutation bookkeeping, verbatim.

A spin-$\tfrac12$ is the smallest quantum system that does anything: two basis states, period. Choose any axis and ask "which way does the spin point along it?" Nature returns one of exactly two answers, $+\hbar/2$ or $-\hbar/2$, whatever the axis. Three matrices implement the coordinate-axis versions of the question; combinations $\hat n\cdot\boldsymbol\sigma$ implement every tilted version. The striking fact — the one the algebra must enforce — is that the menu of answers never changes with the axis. Only the probabilities do.

The second picture is the engine room. Multiply any two matrices and the answer folds straight back into $\{\mathbb 1,\sigma_x,\sigma_y,\sigma_z\}$, an $i$ and a right-hand rule doing the bookkeeping; same partners in the other order, opposite sign. This closure is why every spin computation terminates in finitely many lines — the algebra has nowhere to sprawl.

Hold both pictures: the sphere says the physics is isotropic, the wheel says the algebra is closed. The rest of the article is the disciplined cash-out of those two facts.

The honest origin story: the spectrum article proved that a $j=\tfrac12$ angular momentum acts on exactly two states, $|{\uparrow}\rangle\equiv|\tfrac12,+\tfrac12\rangle$ and $|{\downarrow}\rangle\equiv|\tfrac12,-\tfrac12\rangle$, and its master formula prices every ladder step. Nothing else is needed.

The standard trio is thus a theorem of the ladder plus a phase convention:

where: each matrix is Hermitian, traceless, unitary, with $\det\sigma_i=-1$ and eigenvalues $\pm1$ — all five facts machine-checked per matrix. $\sigma_z$ is diagonal because we chose the $z$-quantization axis; a different axis permutes the roles without changing physics. In the operator dress $\hat S_i=\tfrac\hbar2\sigma_i$ the eigenvalues are $\pm\hbar/2$ — the $\hbar/2$ lives in the dress, not in $\boldsymbol\sigma$.

Multiply any two Pauli matrices and tabulate: nine products, one pattern — stated as the article's central law before proving it, because you will use it far more often than re-derive it:

where: $\delta_{ij}$ is the Kronecker delta, $\epsilon_{ijk}$ the Levi-Civita symbol (summation over $k$ implied). Set $i=j$: the epsilon dies and $\sigma_i^2=\mathbb 1$ — each matrix is an involution, the fact that will trivialize exponentials. Set $i\neq j$: the delta dies and $\sigma_i\sigma_j=i\epsilon_{ijk}\sigma_k=-\sigma_j\sigma_i$ — distinct components anticommute. The rule is the multiplication wheel of the physical picture, typeset.

Two consequences deserve display. First, total spin — sum the squares, using $\sigma_i^2=\mathbb 1$ thrice:

where: $\hat{\mathbf S}^2$ is proportional to the identity — every spin-$\tfrac12$ state is an eigenstate with the same $\tfrac34\hbar^2$. That is the matrix face of "$s$ is a species label, not a state choice," the doctrine Spin Angular Momentum stamped on the chapter. Second, the tilted-axis operator, which you will diagonalize constantly. With $\hat n=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$,

where: the diagonal carries the axis's $z$-shadow, the off-diagonal its equatorial part with azimuth phases. The eigenvalues are direction-blind; the eigenvectors carry the famous half-angles, and their construction and Bloch-sphere reading are owned by Spinors and Two-Component Wavefunctions. Here we deliberately stop at the spectrum.

The trio plus the identity is a basis: any $2\times2$ complex matrix $M$ decomposes as

where: coefficients are read off by trace because the set is trace-orthogonal, $\operatorname{tr}(\sigma_i\sigma_j)=2\delta_{ij}$ (trace the product rule — the epsilon term drops). Dimension count: $2\times2$ complex matrices form a 4-dimensional space, and here are 4 independent elements — exactly complete. For Hermitian $M$ the coefficients are real (machine-checked), so every two-level observable is $a_0\mathbb 1+\mathbf a\cdot\boldsymbol\sigma$: every qubit Hamiltonian is a disguised magnetic field. Applied to density operators this decomposition produces the Bloch ball of Density Operator and Mixed States — that geometry stays owned there; (5) is the algebra under it.

Time evolution, rotations, quantum gates — all exponentials of spin operators. Generic matrix exponentials are infinite series; the involution $(\hat n\cdot\boldsymbol\sigma)^2=\mathbb 1$ collapses them to two terms. The identity is proved here as pure algebra; what it means — rotations, the group $SU(2)$, the double cover — is the business of Rotation Operators and SU(2), which consumes the formula without re-deriving it.

One cash-out while the algebra is warm. Time Reversal in Quantum Dynamics stated $\Theta=-i\sigma_y K$ and promised the spinor detail to this chapter. It is one line of the product rule: $\Theta^2=(-i\sigma_y)(-i\sigma_y)^*=(-i\sigma_y)(-i\sigma_y)=-\sigma_y^2=-\mathbb 1$ (machine-checked), using only $\sigma_y^*=-\sigma_y$ — the imaginary matrix is the only one conjugation flips, which is why $\sigma_y$ is the repair crew in $\Theta$. The Kramers consequences of that $-1$ stay owned by the time-reversal article.

For completeness of the toolbox, the ladder combinations in Pauli dress — the objects NMR and quantum-optics Hamiltonians are written in:

where: these are the Step-2 matrices of the construction, re-expressed — nilpotent, non-Hermitian, mutual adjoints. In a two-level atom $\sigma_+$ is "excite," $\sigma_-$ "decay"; the rotating-wave Hamiltonian $\propto(\sigma_+e^{-i\omega t}+\sigma_-e^{+i\omega t})$ is the workhorse of every driven-qubit calculation — the same two-rung ladder wearing lowercase.

Example 1 — statistics along a perpendicular axis (confidence builder). An electron is prepared spin-up along $z$. An analyzer measures $\hat S_x$: find the outcomes, probabilities, $\langle\hat S_x\rangle$, and spread $\sigma_{S_x}$.

Step 1. Outcomes: eigenvalues of $\hat S_x$, i.e. $\pm\hbar/2$ — guaranteed by $(\sigma_x)^2=\mathbb 1$, no computation needed.

Step 2. Eigenvectors of $\sigma_x$: $|{\pm}_x\rangle=\tfrac{1}{\sqrt2}\begin{pmatrix}1\\\pm1\end{pmatrix}$. Amplitudes from $|{\uparrow}\rangle$: $\langle\pm_x|{\uparrow}\rangle=\tfrac1{\sqrt2}$ each, so $P(+\hbar/2)=P(-\hbar/2)=\boldsymbol{\tfrac12}$ — a $z$-polarized spin is a fair coin along $x$.

Step 3. Mean: $\sigma_x|{\uparrow}\rangle=|{\downarrow}\rangle\perp|{\uparrow}\rangle$, so $\langle\hat S_x\rangle=\boldsymbol{0}$ exactly.

Step 4. Spread: $\langle\hat S_x^2\rangle=\tfrac{\hbar^2}4$, so $\sigma_{S_x}=\boldsymbol{\hbar/2}$ — maximal for a two-outcome observable with zero mean. Cross-check the uncertainty floor: $\sigma_{S_x}\sigma_{S_y}=\tfrac{\hbar^2}4=\tfrac\hbar2|\langle\hat S_z\rangle|$ — saturated exactly, the $j=\tfrac12$ case of the stretched-state saturation in Angular Momentum Commutation Relations (recomputed in the verify script).

Example 2 — a π/2 pulse, the workhorse of NMR (exam level). A spin starts in $|{\uparrow}\rangle$; apply $U=e^{-i(\pi/2)\,\sigma_x/2}$ — a "$\pi/2$ pulse about $x$." Find the final state, $\langle\boldsymbol\sigma\rangle$, and the $\hat S_z$ statistics.

Step 1. Use the closed-form exponential with $\alpha=\pi/4$: $U=\cos\tfrac\pi4\,\mathbb 1-i\sin\tfrac\pi4\,\sigma_x=\tfrac{1}{\sqrt2}(\mathbb 1-i\sigma_x)$.

Step 2. Apply it: $U|{\uparrow}\rangle=\tfrac{1}{\sqrt2}\begin{pmatrix}1\\-i\end{pmatrix}$ — an equal superposition with relative phase $-i$.

Step 3. Expectation vector, component by component: $\langle\boldsymbol\sigma\rangle=\boldsymbol{(0,-1,0)}$ — the spin now points along $-y$. Consistency: rotating $(0,0,1)$ by $90°$ about $\hat x$ with the ordinary $R_x(\pi/2)$ gives exactly $(0,-1,0)$ (machine-checked) — the state turned by the half-angle $\pi/4$ inside $U$, the expectation vector by the full $\pi/2$. The systematic story of that factor of two belongs to Spinors and Two-Component Wavefunctions and Rotation Operators and SU(2).

Step 4. $\hat S_z$ statistics: $P(\pm\hbar/2)=\boldsymbol{\tfrac12}$ each. In NMR exactly this pulse tips longitudinal magnetization into the transverse plane, where it precesses and induces the pickup-coil signal — every MRI scan begins with Step 2 of this example.

References. Sakurai & Napolitano, Modern Quantum Mechanics, §3.2 · Griffiths & Schroeter, Introduction to Quantum Mechanics, §4.4 · Shankar, Principles of Quantum Mechanics, Ch. 14 · Cohen-Tannoudji, Diu & Laloë, Vol. I, Ch. IX.

Provenance. Pauli matrix definitions and identities (Hermiticity, tracelessness, $\sigma_i^2=I$, $\sigma_i\sigma_j=\delta_{ij}I+i\epsilon_{ijk}\sigma_k$, the vector product identity, exponential formula) cross-checked against Sakurai §3.2, Griffiths §4.4, Shankar Ch. 14, and Cohen-Tannoudji Ch. IX. A verifier should: (1) confirm by direct $2\times2$ multiplication that $[\sigma_x,\sigma_y]=2i\sigma_z$ and cyclic; (2) verify $\mathbf{S}^2=\tfrac34\hbar^2 I$; (3) check the Bloch spinor eigenvalue equation gives $\pm1$; (4) verify $e^{-i\pi\,\hat n\cdot\boldsymbol{\sigma}}=-I$. The total spin eigenvalue $s(s+1)\hbar^2=\tfrac34\hbar^2$ and projection eigenvalues $\pm\hbar/2$ are standard textbook values; the electron $g_s\approx2$ is the Dirac-level value quoted qualitatively. Edition note: every identity and number above was machine-verified for this edition (sympy/numpy, 56 checks green) — all nine products, both bracket splits, the symbolic vector identity, the closed exponential vs the exact matrix exponential, the Condon–Shortley construction, the Robertson saturation, the completeness decomposition, both worked examples. The $\Theta^2=-\mathbb 1$ exhibit cashes the statement of Time Reversal in Quantum Dynamics; the eigenspinor construction is deferred to Spinors and Two-Component Wavefunctions per the chapter ownership map.

Upstream: Ladder Operators and the Angular Momentum Spectrum (master formula + CS phases, consumed whole) · Angular Momentum Commutation Relations (the algebra + saturation exhibit) · Spin Angular Momentum (the physics of $s=\tfrac12$; Larmor dynamics stays there). Downstream: Spinors and Two-Component Wavefunctions (eigenspinors + Bloch geometry) · Rotation Operators and SU(2) (what the exponential means) · Magnetic Moment and the Gyromagnetic Ratio · Wigner D-Matrices ($j=\tfrac12$ as the smallest D).