Spin-½, the Pauli Matrices, and Stern–Gerlach

The angular-momentum algebra admits half-integer quantum numbers that no wavefunction on the sphere can carry. Orbital motion realizes only integer $\ell$, because $Y_{\ell }^m$ must be single-valued under $\varphi \to \varphi +2\pi$. The missing half-integers are physical, and the electron carries the smallest one, $s=\frac{1}{2}$. This degree of freedom has no position-space representation: it is not the electron spinning on an axis in any literal sense, and treating it that way gives a surface speed exceeding $c$. Spin is an intrinsic two-valued observable, defined operationally by what a magnet does to a beam of atoms.

The Stern–Gerlach experiment

A neutral atom with magnetic moment $\vec{\mu }$ sitting in a field $\vec{B}$ has energy $H=-\vec{\mu }\cdot \vec{B}$. If the field varies in space, the atom feels a force

Arrange the pole pieces so that near the beam $\vec{B}$ points along $z$ and its magnitude changes rapidly with $z$ while the transverse variation is small. The $z$-component of the force is then

proportional to the projection ${\mu }_z$ of the moment along the field. The deflection at the screen measures ${\mu }_z$, and through $\vec{\mu }=\gamma \vec{S}$ it measures $S_z$.

The 1922 experiment used silver. Silver's ground-state electron configuration closes every shell except a single $5s$ electron, so the total orbital angular momentum is zero and the atomic moment comes entirely from that one electron's spin.¹ Classically ${\mu }_z=|\vec{\mu }|\cos \theta$ takes every value in $[-|\vec{\mu }|,+|\vec{\mu }|]$ as the moment's orientation ranges over the sphere, so the screen should show a continuous vertical smear. It does not. The beam splits into two sharp spots, symmetric about the undeflected line.

Two spots means the observable $S_z$ takes exactly two values. Measuring them from the deflection and the known field gradient gives $\pm \frac{1}{2}\hslash$. The apparatus is a two-outcome measuring device, and the atom's spin along $z$ is a two-valued quantity. No continuous internal orientation survives contact with the data.

The two-dimensional state space

A single spin-½ has a two-dimensional state space. Take the $S_z$ eigenstates as the basis,

with $⟨+|-⟩=0$ and each normalized. A general spin state is a superposition

written as a two-component column, the spinor

Born's rule reads off the outcome statistics: a measurement of $S_z$ yields $+\hslash /2$ with probability $|a{|}^2$ and $-\hslash /2$ with probability $|b{|}^2$. After the measurement the state collapses onto the corresponding eigenspinor.

The three spin components inherit the angular-momentum algebra $[S_i,S_j]=i\hslash {\epsilon }_{ijk}{S}_k$. In the $S_z$ basis they are represented by $2\times 2$ Hermitian matrices. Writing $S_i=\frac{\hslash }{2}{\sigma }_i$ defines the Pauli matrices ${\sigma }_i$.

The Pauli matrices

The matrix elements follow from the angular-momentum results for $j=\frac{1}{2}$.² The ladder operators $S_{\pm }=S_x\pm i{S}_y$ act by $S_{+}|-⟩=\hslash |+⟩$ and $S_{-}|+⟩=\hslash |-⟩$, with the top and bottom rungs annihilated. Solving for $S_x,S_y$ gives

These three matrices, with the identity $I$, span the real vector space of $2\times 2$ Hermitian matrices. Their algebraic properties are the entire toolkit for spin-½.

Each ${\sigma }_i$ has eigenvalues $\pm 1$ (so each $S_i$ has eigenvalues $\pm \hslash /2$), determinant $-1$, and trace $0$. The single product rule above generates every identity one needs. Two consequences are used constantly. First, for any two ordinary vectors $\vec{a},\vec{b}$,

Second, setting $\vec{a}=\vec{b}=\hat{n}$ a unit vector gives $(\vec{\sigma }\cdot \hat{n}{)}^2=I$, which lets any function of $\vec{\sigma }\cdot \hat{n}$ collapse to a linear expression. That fact drives spin rotations in the next lesson.

Spin along an arbitrary axis

Point a Stern–Gerlach magnet along an arbitrary unit vector $\hat{n}=(\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta )$. It measures the component

Because $(\vec{\sigma }\cdot \hat{n}{)}^2=I$, the eigenvalues are again $\pm 1$, so every direction is a two-outcome measurement with results $\pm \hslash /2$. Any axis is as good as $z$; nothing distinguishes a preferred direction. The eigenspinors follow from diagonalizing the $2\times 2$ matrix.

The half-angle is the source of all the interesting statistics. Prepare an atom in $|+⟩$ (spin up along $z$) and measure $S_n$. The amplitude for the $+\hslash /2$ outcome is $⟨{\chi }_{+}(\hat{n})|+⟩=\cos (\theta /2)$, so

The expectation value tracks the classical projection $|\vec{S}|\cos \theta$, but every individual measurement returns only $\pm \hslash /2$. The classical cosine reappears as an average over the two quantized outcomes, not as a value any single atom carries.

Sequential Stern–Gerlach filters

Cascading magnets exposes the collapse. Block one output of a first magnet and feed the survivors into a second oriented along a different axis. The atoms have already been filtered, yet the second magnet still splits them.

Read the cascade left to right. The first $z$-magnet passes only $|+{⟩}_z$. The $x$-magnet decomposes that state in the $S_x$ basis, $|+{⟩}_z=\frac{1}{\sqrt{2}}(|+{⟩}_x+|-{⟩}_x)$, so it splits into two beams of equal intensity; keep $|+{⟩}_x$. That survivor, expressed back in the $z$ basis, is $|+{⟩}_x=\frac{1}{\sqrt{2}}(|+{⟩}_z+|-{⟩}_z)$. The final $z$-magnet therefore splits it again, half into each $z$ outcome, even though the very first magnet had already discarded $|-{⟩}_z$.

The intermediate $x$-measurement destroyed the $z$-information. $S_x$ and $S_z$ do not commute, $[S_x,S_z]=-i\hslash S_y\ne 0$, so no state carries a definite value of both. Filtering along $x$ prepares a definite $S_x$ and necessarily makes $S_z$ maximally uncertain. Remove the middle magnet and the two $z$-magnets in series pass every atom through the upper channel with no second split; insert it and the beam is regenerated in both channels. Measurement is not passive readout; it resets the state.

The spinor and the density of a spin state

Any normalized spinor can be written, up to an overall phase, as ${\chi }_{+}(\hat{n})$ for a unique $\hat{n}$: solve $⟨\chi |\vec{\sigma }|\chi ⟩=\hat{n}$. The vector $\vec{P}=⟨\vec{\sigma }⟩$ is the polarization, and for a pure state $|\vec{P}|=1$, so every pure spin-½ state points somewhere on the unit sphere. This is the seed of the Bloch-sphere picture developed later in the module. The map pure state $\leftrightarrow$ direction $\hat{n}$ is two-to-one at the level of $SU(2)$ versus $SO(3)$: the $\theta /2$ in the eigenspinor means a spatial rotation by $2\pi$ multiplies the spinor by $-1$, and only a $4\pi$ rotation returns it to itself. That sign is not a mathematical artifact; neutron interferometry measures it directly by rotating a spin in one arm and reading the interference against the other.³

What spin adds

The electron's full state is a wavefunction times a spinor, $\Psi (\vec{r})\otimes \chi$, an element of $L^2({\mathbb{R}}^3)\otimes {\mathbb{C}}^2$. When the Hamiltonian ignores spin the two factors decouple and every spatial level carries a two-fold spin degeneracy. Coupling appears through the magnetic moment (the Zeeman and spin–orbit terms) and through the exchange symmetry of identical particles, where the antisymmetry of the total state ties the spatial and spin parts together. The two-state structure isolated here — a qubit with the $SU(2)$ algebra of the Pauli matrices — is also the smallest nontrivial quantum system and the template for every two-level problem in the next two lessons.