Stern-Gerlach and spin-1/2

Intuition [Beginner]

Send a silver atom through a region of space where a strong magnet's field is non-uniform — stronger on one side than the other. A silver atom has a tiny magnetic moment, like a microscopic bar magnet, so the magnetic field pushes it. The direction of the push depends on which way the atom's little magnet points. If the atom's magnet is aligned with the field gradient, it gets pulled one way; if anti-aligned, the other way; if perpendicular, hardly at all.

You'd expect a continuum of outcomes. The atom's tiny magnet should be able to point in any direction. The push should vary smoothly with the angle. The pattern of atoms striking a screen on the far side should be a smear, with most landing in the middle and fewer at the extremes.

That is not what happens. Stern and Gerlach ran this experiment in 1922 and found only two spots on the screen — one above the beam axis, one below, with nothing in between. Every atom ended up in one of two places. The microscopic magnet behaves as if it is forced to point either fully up or fully down with respect to the magnet's axis, never in between.

This is one of the cleanest demonstrations that the microscopic world does not obey the rules of the everyday one. The property responsible is called spin — an intrinsic angular-momentum-like quantity attached to elementary particles. Spin-1/2 means the property has only two possible measured values along any axis, conventionally called "up" and "down," or "$+$" and "$-$." The silver atom's outermost electron is what carries the spin; the rest of the atom is, for this purpose, a passive carrier.

Why "spin-1/2"? The number labels how the particle's state transforms under rotation, and the rule is: rotating the apparatus by a full 360 degrees does not return a spin-1/2 state to itself — only 720 degrees does. That fact alone tells you spin is not a hidden classical rotation. It is something genuinely new. The Master tier shows how this 720-degree quirk is the signature of the spin double cover of the rotation group.

Now the strange part. Send your two-spot output through a second Stern-Gerlach apparatus, this one with its magnet rotated to point horizontally. Take just the "spin up" beam from the first apparatus — atoms that have already been measured to be spin-up along the vertical axis — and feed them in. You expect: they're all spin-up vertically, so they should all behave the same in the horizontal measurement. They don't. Half come out spin-up horizontally, half come out spin-down horizontally. The horizontal measurement seems to re-randomise the vertical measurement that had already been done.

Worse: take the spin-up-horizontal output from the second apparatus and send it through a third vertical apparatus. The atoms that were "spin up vertically" and then "spin up horizontally" are, when measured vertically again, half spin-up and half spin-down. The intermediate horizontal measurement has destroyed the vertical-up status that the first apparatus selected for.

There is no consistent classical picture of what the atom "really has" for spin in all three directions at once. Measurement along one axis forces a discrete outcome and erases any prior assignment along an incompatible axis. The Intermediate tier reframes this as the non-commutativity of the spin operators $S_x$, $S_y$, $S_z$ — and from there, all the standard features of quantum mechanics (superposition, collapse, incompatible observables, complementarity) follow.

Why bother with this single example so heavily? Because Stern-Gerlach is the smallest informative quantum system: two outcomes, no spatial wavefunction to worry about, just a complex two-dimensional state space. Every defining quantum-mechanical move — superposition, measurement projection, eigenstates, basis change, time evolution — appears in this system in its purest possible form. Once you have spin-1/2 fluent, the rest of QM is bookkeeping on top of the same moves.

Visual [Beginner]

Imagine the Stern-Gerlach apparatus as a box with three openings: an inlet (atoms enter), and two outlets stacked vertically (atoms exit, sorted by spin). The box has an internal magnet whose field strongly gradients in the vertical direction; you can rotate the entire box to change which spatial axis becomes the measurement axis.

A useful piece of geometric scaffolding: the set of all possible spin-1/2 states (more carefully, all pure spin-1/2 states modulo a global phase) is a sphere — the Bloch sphere. The north pole is spin-up-z, the south pole is spin-down-z. The equator carries the spin-up-x, spin-down-x, spin-up-y, spin-down-y states at the four cardinal points. Any direction on the sphere corresponds to a spin state polarised along that axis. Rotating an SG apparatus rotates the choice of two antipodal points on the Bloch sphere — measurement always picks one of two antipodes, projecting the Bloch-sphere state onto whichever pole is closer along that axis.

Worked example [Beginner]

Set up the three-stage Stern-Gerlach experiment described above and predict, by counting, what fraction of atoms come out where.

Start with $N$ unpolarised silver atoms entering apparatus 1 (axis $z$). Half go to the spin-up-$z$ output, half to spin-down-$z$. The spin-up-$z$ beam has $N/2$ atoms.

Take just the spin-up-$z$ beam and send it into apparatus 2 (axis $x$). The rule of quantum mechanics for incompatible axes is: a state that is definitely spin-up in the $z$-direction has no definite value in the $x$-direction. The measurement is then 50/50, splitting $N/2$ atoms into $N/4$ spin-up-$x$ atoms and $N/4$ spin-down-$x$ atoms.

Take the $N/4$ spin-up-$x$ atoms and send them into apparatus 3 (axis $z$ again). After apparatus 2 the atoms are definitely spin-up in $x$. By the same rule, they have no definite value in $z$ — so the $z$-measurement re-randomises, 50/50. Final outputs: $N/8$ spin-up-$z$ and $N/8$ spin-down-$z$.

Compare to the classical prediction. Classically, an atom that was first measured spin-up-$z$ and then survived the $x$-measurement is still "the same atom that was spin-up-$z$" — running another $z$-measurement should leave it spin-up-$z$ with certainty. The classical prediction is $N/4$ spin-up-$z$ and $0$ spin-down-$z$.

The experiment matches the quantum count, not the classical one. What this tells us: the second apparatus didn't just read off an existing spin-$x$ value — it imposed one, and in doing so it erased the previously imposed spin-$z$ value. Measurement is not passive. There is no underlying state with definite values for both $S_z$ and $S_x$ at once. The Intermediate tier promotes this picture to the formalism of non-commuting operators, where "no joint definite value" becomes the precise statement "the operators do not share an eigenbasis."

Check your understanding [Beginner]

Formal definition [Intermediate+]

A spin-1/2 quantum system is described by the two-dimensional complex Hilbert space $\mathcal{H}={\mathbb{C}}^2$, equipped with its standard Hermitian inner product $⟨\cdot ,\cdot ⟩$ (linear in the second slot, conjugate-linear in the first — the physics convention). A pure state is a unit vector $|\psi ⟩\in \mathcal{H}$ modulo overall phase; we write kets $|\psi ⟩$ for vectors and bras $⟨\psi |$ for the dual covectors. We adopt the Dirac bracket convention: $⟨\varphi |\psi ⟩$ denotes the inner product of $|\varphi ⟩$ and $|\psi ⟩$.

Fix an orthonormal basis labelled $\{|+z⟩,|-z⟩\}\subset \mathcal{H}$ — the eigenstates of "spin along the $z$-axis." Concretely identify

$$|+z⟩=\left(\begin{array}{c}1\\ 0\end{array}\right),|-z⟩=\left(\begin{array}{c}0\\ 1\end{array}\right).$$

A general pure state is $|\psi ⟩=a|+z⟩+b|-z⟩$ with $a,b\in \mathbb{C}$ and $|a{|}^2+|b{|}^2=1$.

The Pauli matrices ${\sigma }_x,{\sigma }_y,{\sigma }_z$ are the three Hermitian, traceless $2\times 2$ matrices

$${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$$

They generate the complex Clifford algebra ${\mathrm{Cl}}_3^{\mathbb{C}}\cong M_2(\mathbb{C})$ acting on $\mathcal{H}$ 03.09.02; the Clifford relation is the anticommutator identity

$$\{{\sigma }_i,{\sigma }_j\}:={\sigma }_i{\sigma }_j+{\sigma }_j{\sigma }_i=2{\delta }_{ij}1,$$

with $i,j\in \{x,y,z\}$ (or $1,2,3$). Combined with the commutator identity

$$[{\sigma }_i,{\sigma }_j]:={\sigma }_i{\sigma }_j-{\sigma }_j{\sigma }_i=2i{ϵ}_{ijk}{\sigma }_k,$$

(summation on $k$; ${ϵ}_{ijk}$ the Levi-Civita symbol with ${ϵ}_{xyz}=+1$) the Pauli matrices satisfy

$${\sigma }_i{\sigma }_j={\delta }_{ij}1+i{ϵ}_{ijk}{\sigma }_k.$$

The three matrices together with $1$ form a Hermitian basis of the real vector space of $2\times 2$ Hermitian matrices.

The spin operators are

$$S_i:=\frac{\hslash }{2}{\sigma }_i,i\in \{x,y,z\},$$

with $\hslash$ the reduced Planck constant. They are the components of the spin angular-momentum vector operator $\mathbf{S}=(S_x,S_y,S_z)$. From the Pauli commutator identity above, they obey

$$\boxed{[S_i,S_j]=i\hslash {ϵ}_{ijk}{S}_k}$$

— the canonical angular-momentum commutation relations, which define a Lie-algebra representation of $\mathfrak{su}(2)$ 07.06.01 on $\mathcal{H}$. The spin-1/2 representation is the fundamental (defining) representation: $\mathcal{H}={\mathbb{C}}^2$ is the smallest faithful complex representation of $\mathfrak{su}(2)$, and equivalently of the group $\mathrm{SU}(2)$ 07.07.01, whose Lie algebra is $\mathfrak{su}(2)$.

Each $S_i$ is Hermitian with eigenvalues $\pm \hslash /2$. Its eigenstates are the spin states along the $i$-axis: for $i=z$ they are $|+z⟩,|-z⟩$ by construction; for $i=x,y$ they are

$$|\pm x⟩=\frac{1}{\sqrt{2}}(|+z⟩\pm |-z⟩),|\pm y⟩=\frac{1}{\sqrt{2}}(|+z⟩\pm i|-z⟩).$$

A direct check: ${\sigma }_x|+x⟩=|+x⟩$, ${\sigma }_x|-x⟩=-|-x⟩$, so $S_x|\pm x⟩=\pm (\hslash /2)|\pm x⟩$. Analogously for $y$. The general spin state polarised along the unit vector $\hat{n}=(\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta )$ — the Bloch-sphere parameterisation — is

$$|+\hat{n}⟩=\cos (\theta /2)|+z⟩+e^{i\phi }\sin (\theta /2)|-z⟩,|-\hat{n}⟩=-e^{-i\phi }\sin (\theta /2)|+z⟩+\cos (\theta /2)|-z⟩,$$

with $|\pm \hat{n}⟩$ the eigenstates of $\hat{n}\cdot \mathbf{S}$ at eigenvalue $\pm \hslash /2$.

The measurement postulate for an observable $A$ with spectral decomposition $A={\sum }_{a}a{P}_a$ (where $P_a$ is the projector onto the eigenspace of eigenvalue $a$) says: in state $|\psi ⟩$, the outcome of measuring $A$ is the eigenvalue $a$ with probability $\mathrm{Prob}(a|\psi )=\Vert P_a|\psi ⟩{\Vert }^2=⟨\psi |P_a|\psi ⟩$. After the outcome $a$ is recorded, the state collapses to $P_a|\psi ⟩/\Vert P_a|\psi ⟩\Vert$. For a Stern-Gerlach apparatus measuring $S_z$, the projectors are $P_{\pm }=|\pm z⟩⟨\pm z|$ and the probabilities are $|⟨\pm z|\psi ⟩{|}^2$.

A Stern-Gerlach apparatus oriented along $\hat{n}$ implements the measurement of $\hat{n}\cdot \mathbf{S}$, projecting onto $|\pm \hat{n}⟩⟨\pm \hat{n}|$. Sequential SG along incompatible axes ${\hat{n}}_1,{\hat{n}}_2$ projects the state successively, and the joint statistics depend on $|⟨+{\hat{n}}_2|+{\hat{n}}_1⟩{|}^2$. We adopt the standard sign convention $\hat{n}\cdot \mathbf{S}$ for the SG-along-$\hat{n}$ observable; the apparatus's force-direction sign (which way the spin-up beam deflects) depends on the sign of the magnetic moment $\mu =g_s(q/2m)\mathbf{S}$ of the particle (negative for the electron), and is the convention component most easily mis-stated. State the convention before extracting deflection directions.

Counterexamples to common slips

• The state $|+z⟩$ has no $S_x$-value at all, not "an unknown $S_x$-value." A measurement does not reveal a pre-existing value; it imposes one, with the probability rule above. Treating QM as if every observable always has a definite value (the value-definiteness assumption) is the gateway error that hidden-variable interpretations attempt to repair — see Bell's theorem in the dedicated unit on quantum nonlocality.
• The Pauli matrices are not the spin operators. They differ by the factor $\hslash /2$. Confusing ${\sigma }_z$ with $S_z$ produces angular-momentum commutators that are off by a factor of $\hslash$ and is the single most common notation error in spin calculations.
• The "spin direction" $\hat{n}$ in $|+\hat{n}⟩$ does not mean that the spin angular-momentum vector $(S_x,S_y,S_z)$ points along $\hat{n}$ in any classical sense. The three components do not commute and have no joint definite values. What is true is that $\hat{n}$ is the direction along which a measurement of $\hat{n}\cdot \mathbf{S}$ returns $+\hslash /2$ with certainty.
• A rotation by $2\pi$ does not send a spin-1/2 state to itself: rotation by $\theta$ around axis $\hat{n}$ acts as $U(\theta ,\hat{n})=\exp (-i\theta \hat{n}\cdot \mathbf{S}/\hslash )=\cos (\theta /2)1-i\sin (\theta /2)\hat{n}\cdot \mathbf{\sigma }$, which at $\theta =2\pi$ equals $-1$, not $1$. Only $\theta =4\pi$ recovers the identity. This is the signature of the spin double cover $\mathrm{SU}(2)\to \mathrm{SO}(3)$.

Key theorem with proof [Intermediate+]

Theorem (Pauli matrices as the basis of Hermitian traceless $2\times 2$ matrices). The real vector space ${\mathcal{H}}_0$ of Hermitian traceless complex $2\times 2$ matrices has dimension 3, and $\{{\sigma }_x,{\sigma }_y,{\sigma }_z\}$ is an orthogonal basis for it under the Hilbert-Schmidt inner product $⟨A,B{⟩}_{\mathrm{HS}}:=\frac{1}{2}\mathrm{tr}(A^{†}B)$. In particular, every Hermitian traceless $A\in M_2(\mathbb{C})$ admits a unique decomposition $A=a_x{\sigma }_x+a_y{\sigma }_y+a_z{\sigma }_z$ with real coefficients $a_i=\frac{1}{2}\mathrm{tr}({\sigma }_{i}A)$.

Proof. A general $2\times 2$ complex matrix has eight real parameters. Imposing Hermiticity $A=A^{†}$ identifies the off-diagonal entries as complex conjugates and forces the diagonal entries to be real, reducing to four real parameters $(d_1,d_2,\mathrm{Re}c,\mathrm{Im}c)$ where

$$A=\left(\begin{array}{cc}{d}_1& c\\ \bar{c}& d_2\end{array}\right),d_1,d_2\in \mathbb{R},c\in \mathbb{C}.$$

Imposing tracelessness $d_1+d_2=0$ removes one further parameter, leaving three: $d_1$, $\mathrm{Re}c$, $\mathrm{Im}c$. So ${\dim }_{\mathbb{R}}{\mathcal{H}}_0=3$, matching the claim.

The three Pauli matrices lie in ${\mathcal{H}}_0$ (each is Hermitian by inspection; each has trace $0$). They are pairwise Hilbert-Schmidt-orthogonal: a direct computation using ${\sigma }_i{\sigma }_j={\delta }_{ij}1+i{ϵ}_{ijk}{\sigma }_k$ gives

$$\mathrm{tr}({\sigma }_i{\sigma }_j)=\mathrm{tr}({\delta }_{ij}1)+i{ϵ}_{ijk}\mathrm{tr}({\sigma }_k)=2{\delta }_{ij},$$

since $\mathrm{tr}({\sigma }_k)=0$. Thus $⟨{\sigma }_i,{\sigma }_j{⟩}_{\mathrm{HS}}={\delta }_{ij}$ — the Pauli matrices are orthonormal under the Hilbert-Schmidt inner product. Three orthonormal vectors in a three-dimensional real inner-product space form an orthonormal basis. The coefficient formula $a_i=\frac{1}{2}\mathrm{tr}({\sigma }_{i}A)$ is the standard inner-product expansion in the basis. ∎

Corollary (Bloch decomposition for density matrices). Every $2\times 2$ Hermitian matrix decomposes as $A=a_{0}1+\mathbf{a}\cdot \mathbf{\sigma }$ with $a_0=\frac{1}{2}\mathrm{tr}(A)$ and $a_i=\frac{1}{2}\mathrm{tr}({\sigma }_{i}A)$. A density matrix — positive semidefinite Hermitian with unit trace — is $\rho =\frac{1}{2}(1+\mathbf{r}\cdot \mathbf{\sigma })$ with the Bloch vector $\mathbf{r}\in {\mathbb{R}}^3$ satisfying $|\mathbf{r}|\le 1$; pure states are exactly the boundary $|\mathbf{r}|=1$ — the Bloch sphere. ∎

Bridge. The same three matrices that diagonalise spin along the three coordinate axes are the algebraic generators of ${\mathrm{Cl}}_3^{\mathbb{C}}\cong M_2(\mathbb{C})$ via ${\sigma }_i{\sigma }_j+{\sigma }_j{\sigma }_i=2{\delta }_{ij}1$ — that is, the Pauli matrices are a faithful matrix representation of the Clifford algebra ${\mathrm{Cl}}_{0,3}^{\mathbb{C}}$ of $({\mathbb{R}}^3,q)$ with $q$ positive-definite 03.09.02. The spin double cover $\mathrm{SU}(2)\to \mathrm{SO}(3)$ is then the restriction of the twisted adjoint action of $\mathrm{Spin}(3)$ on ${\mathrm{Cl}}_3$ to the unit-norm-vector products inside ${\mathrm{Cl}}_3^0\cong \mathrm{SU}(2)$. The "quantum mystery" of spin-1/2 sitting in a 2-complex-dimensional space rather than a 3-real-dimensional one is, structurally, the Clifford-algebraic fact that $\mathrm{Spin}(3)$ is a double cover of $\mathrm{SO}(3)$.

Exercises [Intermediate+]

A graded set covering Pauli algebra, sequential SG, basis change, and Larmor precession.

Lean formalization [Intermediate+]

The companion module Codex.Quantum.SternGerlachSpinHalf (at lean/Codex/Quantum/SternGerlachSpinHalf.lean) builds on Mathlib's Matrix, Mathlib.Analysis.InnerProductSpace, and Matrix.SpecialUnitaryGroup. The module:

• Defines the three Pauli matrices as concrete $2\times 2$ complex matrices.
• States and proves the anticommutator identity $\{{\sigma }_i,{\sigma }_j\}=2{\delta }_{ij}1$ and the commutator identity $[{\sigma }_i,{\sigma }_j]=2i{ϵ}_{ijk}{\sigma }_k$.
• Defines the spin operators $S_i=(\hslash /2){\sigma }_i$ (with $\hslash$ treated as a formal symbol via noncomputable def) and re-exports the $\mathfrak{su}(2)$ commutation relation $[S_i,S_j]=i\hslash {ϵ}_{ijk}{S}_k$.
• Declares (with sorry) the measurement-postulate map on density matrices $\rho \mapsto P_i\rho P_i/\mathrm{tr}(P_i\rho )$. Mathlib does not formalise this as a primitive; the partial formalisation supplies the projector decomposition and leaves the trace-normalisation step as a contribution candidate.
• States Bloch-sphere parameterisation lemmas: every pure state in ${\mathbb{C}}^2$ modulo phase corresponds to a point $\hat{n}\in S^2$ and a unique unit vector $|+\hat{n}⟩$ up to phase.

example (i j : Fin 3) (h : i ≠ j) :
    σ i * σ j + σ j * σ i = 0 := by
  fin_cases i <;> fin_cases j <;> simp_all [σ] <;> ring_nf

lean_status: partial reflects that the algebraic content compiles cleanly; the measurement-postulate normalisation and the entanglement / partial-trace material are stub-gated and human-reviewer-attested.

SU(2) representation theory and the spin double cover [Master]

Spin-1/2 is the fundamental (defining) two-dimensional complex representation of $\mathrm{SU}(2)$. Concretely, $\mathrm{SU}(2)$ consists of $2\times 2$ complex unitary matrices of determinant 1; as a real manifold it is diffeomorphic to $S^3$, and it acts on $\mathcal{H}={\mathbb{C}}^2$ by matrix multiplication. The Lie algebra $\mathfrak{su}(2)$ is the real vector space of $2\times 2$ skew-Hermitian traceless matrices, spanned by $\{-i{\sigma }_i/2{\}}_{i=1}^3$, with Lie bracket $[A,B]=AB-BA$ satisfying $[-i{\sigma }_i/2,-i{\sigma }_j/2]={ϵ}_{ijk}(-i{\sigma }_k/2)$ — the Lie-algebra version of the spin-operator commutator relations 07.06.01.

The finite-dimensional irreducible representations of $\mathrm{SU}(2)$ are indexed by a half-integer $s\in \{0,1/2,1,3/2,2,\dots \}$ (the spin quantum number), of dimension $2s+1$, with $s=0$ the one-dimensional invariant representation, $s=1/2$ the fundamental two-dim representation discussed in this unit, $s=1$ the three-dim adjoint representation, and so on. The classification reduces to the highest-weight theory of $\mathfrak{su}(2)$ via the ladder operators $S_{\pm }:=S_x\pm i{S}_y$, satisfying $[S_z,S_{\pm }]=\pm \hslash S_{\pm }$ and $[S_{+},S_{-}]=2\hslash S_z$; the construction of higher-spin irreps from products of spin-1/2 irreps is the Clebsch-Gordan decomposition

$${\mathrm{D}}^{(s_1)}\otimes {\mathrm{D}}^{(s_2)}\cong \underset{s=|s_1-s_2|}{\overset{s_1+s_2}{⨁}}{\mathrm{D}}^{(s)},$$

derived in Sakurai-Napolitano Ch. 3 §3.7 ^{[Sakurai-Napolitano Ch. 3]} and treated in the dedicated forthcoming unit 12.05.01.

The covering map $\rho :\mathrm{SU}(2)\to \mathrm{SO}(3)$ is constructed as follows. For $U\in \mathrm{SU}(2)$ and $\mathbf{v}\in {\mathbb{R}}^3$, identify $\mathbf{v}$ with the Hermitian traceless matrix $\mathbf{v}\cdot \mathbf{\sigma }$ (the Theorem of the Intermediate tier guarantees this identification is a real-linear isomorphism). The adjoint action $\mathbf{v}\cdot \mathbf{\sigma }\mapsto U(\mathbf{v}\cdot \mathbf{\sigma })U^{-1}$ preserves Hermiticity and tracelessness, and a short computation (using $\mathrm{tr}((\mathbf{v}\cdot \mathbf{\sigma }{)}^2)=2|\mathbf{v}{|}^2$) shows it preserves the Euclidean length of $\mathbf{v}$. Hence $U(\mathbf{v}\cdot \mathbf{\sigma })U^{-1}=(\rho (U)\mathbf{v})\cdot \mathbf{\sigma }$ for a unique $\rho (U)\in \mathrm{O}(3)$, which is in fact in $\mathrm{SO}(3)$ by connectedness (the map is continuous and $\rho (1)=1$). The kernel of $\rho$ is the centre $\{\pm 1\}\subset \mathrm{SU}(2)$, since $\rho (U)=1$ forces $U\mathbf{v}\cdot \mathbf{\sigma }=\mathbf{v}\cdot \mathbf{\sigma }U$ for all $\mathbf{v}$, hence $U$ commutes with all of $M_2(\mathbb{C})$ and is a scalar (in $\mathrm{SU}(2)$ that means $U=\pm 1$). The short exact sequence

$$1⟶\mathbb{Z}/2\mathbb{Z}⟶\mathrm{SU}(2)⟶\mathrm{SO}(3)⟶1$$

is the spin double cover in dimension 3 [03.09.02 Theorem on the spin sequence]. The fundamental group ${\pi }_1(\mathrm{SO}(3))=\mathbb{Z}/2$ is detected exactly by the $2\pi$-rotation phase $-1$ in $\mathrm{SU}(2)$.

The connection to Clifford algebra: $\mathrm{SU}(2)\cong \mathrm{Spin}(3)$, with the isomorphism realised by the even part ${\mathrm{Cl}}_3^0$ of the real Clifford algebra of $({\mathbb{R}}^3,+)$ — explicitly, ${\mathrm{Cl}}_{0,3}^0\cong \mathbb{H}$ and the unit-norm quaternions form a copy of $\mathrm{SU}(2)$ acting by left multiplication on $\mathbb{H}\cong {\mathbb{R}}^4$. The Pauli matrices furnish an alternative explicit faithful representation of ${\mathrm{Cl}}_3^{\mathbb{C}}\cong M_2(\mathbb{C})$, in which the spin double cover acts on the natural Clifford module $\Delta ={\mathbb{C}}^2$ — the spinor representation [03.09.02 Connection to physics].

Spin precession in a magnetic field [Master]

A particle with magnetic moment $\mathbf{\mu }$ in an external magnetic field $\mathbf{B}$ has interaction Hamiltonian $H=-\mathbf{\mu }\cdot \mathbf{B}$. For a spin-1/2 particle with $\mathbf{\mu }=\gamma \mathbf{S}$ (gyromagnetic ratio $\gamma$), the Hamiltonian is $H=-\gamma \mathbf{B}\cdot \mathbf{S}$. In a uniform field $\mathbf{B}=B\hat{z}$, $H=-\gamma B{S}_z=(\hslash \omega /2){\sigma }_z$ with Larmor frequency $\omega :=-\gamma B$.

Time evolution is governed by the Schrödinger equation $i\hslash {\partial }_t|\psi ⟩=H|\psi ⟩$, with solution $|\psi (t)⟩=U(t)|\psi (0)⟩$ for $U(t)=\exp (-iHt/\hslash )=\exp (-i(\omega t/2){\sigma }_z)$. By the same identity used in Exercise 6, $U(t)=\cos (\omega t/2)1-i\sin (\omega t/2){\sigma }_z$, which acts on the basis as $U(t)|+z⟩=e^{-i\omega t/2}|+z⟩$ and $U(t)|-z⟩=e^{+i\omega t/2}|-z⟩$.

The Bloch vector $\mathbf{r}(t)$ of the evolved state evolves by classical-style precession around $\mathbf{B}$:

$$\frac{d\mathbf{r}}{dt}=\gamma \mathbf{r}\times \mathbf{B}.$$

This is the Ehrenfest theorem for spin — the expectation-value motion of $⟨\mathbf{S}⟩=(\hslash /2)\mathbf{r}$ obeys classical Newtonian mechanics for a magnetic moment in a field, while the underlying quantum state evolves by the unitary $U(t)$ above. The fact that the expectation obeys classical mechanics but the state requires the half-angle phases is again the spin-1/2 doubling: a $2\pi$ rotation of the Bloch vector around $\hat{z}$ corresponds to $\omega t=2\pi$, at which the underlying state is $-|\psi (0)⟩$, not $|\psi (0)⟩$.

Magnetic resonance (NMR, ESR) exploits Larmor precession by adding a small rotating transverse field ${\mathbf{B}}_1(t)=B_1(\cos \omega t,\sin \omega t,0)$ tuned to the Larmor frequency, producing Rabi oscillations of the spin between $|+z⟩$ and $|-z⟩$; the resulting rotation angle is ${\omega }_{R}t$ with ${\omega }_R=\gamma B_1$ (Rabi frequency). The full theory is the subject of dedicated units 12.07.02 (time-dependent perturbation theory, Rabi formula) and the chem-side spectroscopy unit on NMR.

Density matrices and mixed states [Master]

A density matrix $\rho$ is a positive semidefinite Hermitian operator on $\mathcal{H}$ with $\mathrm{tr}(\rho )=1$. Pure states are the rank-one density matrices $\rho =|\psi ⟩⟨\psi |$; mixtures $\rho ={\sum }_i{p}_i|{\psi }_i⟩⟨{\psi }_i|$ with $p_i\in [0,1]$, $\sum p_i=1$ represent classical-probability ensembles of pure states. For spin-1/2 the Bloch decomposition $\rho =\frac{1}{2}(1+\mathbf{r}\cdot \mathbf{\sigma })$ with $\mathbf{r}\in {\mathbb{R}}^3$, $|\mathbf{r}|\le 1$ parameterises the full state space; pure states are $|\mathbf{r}|=1$ (the Bloch sphere), the maximally mixed state is $\mathbf{r}=0$ (the centre).

Expectation values: for any observable $A$, $⟨A{⟩}_{\rho }=\mathrm{tr}(\rho A)$; for spin-1/2 this gives $⟨\mathbf{S}{⟩}_{\rho }=(\hslash /2)\mathbf{r}$ — the Bloch vector is precisely (twice) the expectation-value of the spin operator divided by $\hslash$. The probability of finding spin-up along $\hat{n}$ is $\mathrm{tr}(\rho P_{\hat{n},+})=\frac{1}{2}(1+\mathbf{r}\cdot \hat{n})$.

For a bipartite system ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$ with joint state ${\rho }_{AB}$, the reduced density matrix of subsystem $A$ is ${\rho }_A:={\mathrm{tr}}_B({\rho }_{AB})$, defined by ${\mathrm{tr}}_B(O_A\otimes O_B)=O_A\mathrm{tr}(O_B)$ and extended linearly. The partial trace is the unique operation such that $⟨O_A{⟩}_{{\rho }_A}=⟨O_A\otimes 1_B{⟩}_{{\rho }_{AB}}$ for every operator $O_A$ on subsystem $A$ alone. For the singlet state $|{\Psi }^{-}⟩$ of Exercise 7, ${\rho }_A=\frac{1}{2}{1}_A$ — maximally mixed despite $|{\Psi }^{-}⟩$ being pure. Entanglement is the property of having ${\rho }_A$ mixed when ${\rho }_{AB}$ is pure; the von Neumann entropy $S({\rho }_A)=-\mathrm{tr}({\rho }_A\log {\rho }_A)$ quantifies how much. For the singlet $S({\rho }_A)=\log 2$ — maximal entanglement of two qubits.

Spin-statistics and relativistic extension [Master]

The classification of irreducible representations of the connected component of the Poincaré group, due to Wigner ^{[Wigner 1939 *Ann. Math.* 40, 149]}, assigns to every elementary particle a mass and a spin, where spin labels the irrep of the little group of the particle's rest frame. For massive particles in $(3+1)$-dimensional Minkowski space the little group is $\mathrm{SO}(3)$, with double cover $\mathrm{SU}(2)\cong \mathrm{Spin}(3)$; spin takes values in $\{0,1/2,1,3/2,\dots \}$. Spin-1/2 particles are fermions; they obey the spin-statistics theorem: the multi-particle wavefunction is antisymmetric under particle exchange, equivalently fermionic creation operators anticommute, $\{a_p^{†},a_q^{†}\}=0$. The full proof requires QFT — specifically, the requirement that anticommuting field operators commute at spacelike separation (microcausality) for half-integer spin, while commuting field operators do so for integer spin; otherwise either the Hamiltonian is unbounded below or local observables fail to commute at spacelike separation. The Pauli-Lüders-Schwinger proof is treated at the QFT level in unit 12.12.04 (forthcoming) and in Peskin-Schroeder Ch. 3 ^{[Peskin-Schroeder]}.

Relativistically, a spin-1/2 particle is described by a Dirac spinor $\psi (x)\in {\mathbb{C}}^4$ — a section of the spinor bundle of Minkowski space — satisfying the Dirac equation $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$, where the ${\gamma }^{\mu }$ are the gamma matrices generating ${\mathrm{Cl}}_{1,3}^{\mathbb{C}}$. In the Weyl (chiral) basis, ${\gamma }^{\mu }$ block-decompose with the spatial gammas containing the Pauli matrices ${\gamma }^i=\left(\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right)$ [03.09.02 §Connection to physics]. The Dirac spinor decomposes into two two-component Weyl spinors, left- and right-chiral, each transforming under one of the two two-dimensional irreps of $\mathrm{SL}(2,\mathbb{C})\cong {\mathrm{Spin}}^{+}(1,3)$ — the relativistic spin doubling that reduces, in the non-relativistic limit, to the single two-component Pauli spinor of this unit. The Dirac equation is the canonical first-order elliptic operator on the spinor bundle and is the object of unit 03.09.08 Dirac operator.

In condensed-matter and quantum-information contexts the non-relativistic spin-1/2 description is exact; the relativistic Dirac extension is needed for atomic fine-structure, the gyromagnetic factor $g_s=2$ (predicted by Dirac, with the QED correction $g_s-2$ predicted to 12 significant figures and measured to 12), the Lamb shift, and high-energy scattering. Spin-1/2's appearance throughout particle physics — every fermion in the Standard Model is spin-1/2 — is, at root, the statement that the smallest nontrivial irrep of the universal cover of the Lorentz group is two-dimensional, an algebraic-geometric fact that the Clifford-algebraic and Lie-algebraic apparatus of 03.09.02 and 07.06.01–07.07.01 makes precise.

Decoherence and the limits of the discrete-output picture [Master]

The "discrete two-output" character of the Stern-Gerlach measurement requires a particular environmental condition: that the spatial part of the silver-atom wavepacket, after passing through the inhomogeneous field, decoheres rapidly into two macroscopically distinguishable spatial branches before the atom reaches the screen. The interaction Hamiltonian $H_{\mathrm{int}}=-\mathbf{\mu }\cdot \mathbf{B}(\mathbf{r})$ couples the spin degree of freedom to the centre-of-mass position; for a field gradient $\partial B_z/\partial z$, the spin-up and spin-down components experience opposite forces and their wavepackets separate. The separation, combined with thermal interactions and detector backreaction, produces classical-probabilistic outcomes from quantum-mechanical superpositions.

In the absence of decoherence — say, in a coherent-superposition regime maintained by careful isolation — the post-SG state of the atom is an entangled spin-position state $\alpha |+z⟩|{\psi }_{+}(\mathbf{r})⟩+\beta |-z⟩|{\psi }_{-}(\mathbf{r})⟩$ in which the two spatial branches can in principle be recombined and interfered. The Bohm-Aharonov split-beam reunion experiments (1957) and modern atom-interferometric variants demonstrate that the "two spots" of textbook SG are a consequence of detection, not of the SG apparatus itself.

The Stern-Gerlach setup is therefore the canonical experimental system for the measurement problem: at what stage in the chain (apparatus, environment, detector, observer) does the quantum superposition $\alpha |+⟩+\beta |-⟩$ become the classical mixture "either $|+⟩$ with probability $|\alpha {|}^2$ or $|-⟩$ with probability $|\beta {|}^2$"? Decoherence theory localises the transition to the environmental-coupling step but does not resolve the interpretive question of why a particular outcome is observed in a given run rather than another. This is the entry point for the philosophical units on quantum measurement; see 20.03.01 (forthcoming).

The phenomenological scope of the discrete-output picture is therefore: pure spin-1/2 dynamics is exactly two-dimensional and exactly described by the Pauli machinery above; SG measurement outcomes are exactly discrete; the link between the two is via spin-environment coupling and decoherence, the rigorous account of which uses open-quantum-system master-equation techniques (Lindblad form) developed in unit 12.07.05 (forthcoming) and treated in Breuer-Petruccione ^{[Breuer-Petruccione *The Theory of Open Quantum Systems*]}.

Connections [Master]

• Clifford algebra 03.09.02. The Pauli matrices are an explicit faithful representation of ${\mathrm{Cl}}_3^{\mathbb{C}}\cong M_2(\mathbb{C})$; the spin operators $S_i$ generate the spin-1/2 representation of $\mathrm{Spin}(3)\cong \mathrm{SU}(2)$ inside ${\mathrm{Cl}}_3^0$. The anticommutator $\{{\sigma }_i,{\sigma }_j\}=2{\delta }_{ij}1$ is the defining Clifford relation in matrix form. The "720° to return" property is the spin double cover.

• Dirac operator 03.09.08. Relativistic spin-1/2 is described by the Dirac spinor, a section of the spinor bundle of Minkowski space, on which the Dirac operator $/\partial ={\gamma }^{\mu }{\partial }_{\mu }$ acts as the canonical first-order elliptic operator. The four-component Dirac spinor splits as two two-component Weyl spinors, each transforming under one of the chiral irreps of ${\mathrm{Spin}}^{+}(1,3)$; the non-relativistic reduction gives the single two-component Pauli spinor of this unit.

• SU(2) representations 07.06.01, 07.07.01. Spin-1/2 is the fundamental two-dimensional complex representation of $\mathrm{SU}(2)$ and of $\mathfrak{su}(2)$. Higher spins are obtained by tensor products and Clebsch-Gordan decomposition; the full classification of irreps of $\mathrm{SU}(2)$ is via highest-weight theory (07.06.06) applied to $\mathfrak{su}(2)$.

• Atomic structure (forthcoming, 12.06.03 hydrogen fine structure; 14.04.01 multi-electron atoms). Electrons in atoms carry spin-1/2; the Pauli exclusion principle (spin-statistics) is what forces shell filling and gives chemistry its discrete structure. The chem-side cite from the multi-electron atom unit takes this unit's spin-statistics statement as its formal input.

• Bell inequalities and quantum nonlocality (forthcoming, 12.09.04). Pairs of spin-1/2 particles in the singlet state exhibit perfect anti-correlation along arbitrary axes — a correlation pattern Bell showed cannot be reproduced by any local hidden-variable model. The Bell-test experiments (Aspect 1982; loophole-free 2015) confirm the quantum prediction; the singlet state of Exercise 7 is the canonical state.

• Open systems and decoherence (forthcoming, 12.07.05). The link between the unitary Pauli dynamics of an isolated spin and the classical-probabilistic Stern-Gerlach outcome statistics is supplied by decoherence — environmental coupling that destroys off-diagonal coherence terms in the density matrix and produces effectively-classical mixtures. The transition is rigorous within open-system QM; the interpretive content (why a particular outcome is observed) is the subject of the phil unit on the measurement problem.

• Measurement problem 20.03.01 (proposed phil unit). Stern-Gerlach is the canonical instance: a finite-dimensional system, two outcomes, no spatial-wavefunction complications, in which every interpretive question (Copenhagen, many-worlds, Bohmian, QBist) can be cleanly posed.

• Bosonic Fock space and second quantisation 12.03.01 pending. Forward reference. The Section 3 Fock-space construction puts the present unit's spin-1/2 Hilbert space ${\mathbb{C}}^2$ in a wider context: ${\mathbb{C}}^2$ is a finite-dimensional Hilbert space with the spectrum $\{0,1\}$ of $S_z+1/2$, contrasting with the one-mode bosonic Fock space ${\mathcal{F}}_s(\mathbb{C})={⨁}_n\mathbb{C}\cdot |n⟩$, which is infinite-dimensional with the ${\mathbb{N}}_0$ spectrum of the number operator $N$. The contrast is the foundational contrast between fermionic and bosonic occupation statistics: in the spin-1/2 case the Pauli matrices ${\sigma }_{+},{\sigma }_{-}$ satisfy the anticommutator relation $\{{\sigma }_{+},{\sigma }_{-}\}=1$ and ${\sigma }_{\pm }^2=0$, so a single site can hold at most one excitation — the smallest possible CAR representation, foreshadowing the full fermionic Fock-space construction. The bosonic side replaces the anticommutator by a commutator, $[a,a^{\ast }]=1$, and removes the squaring obstruction, giving the infinite-dimensional ladder. The chapter seed for the formalism develops here ultimately propagates into both the bosonic and fermionic many-body theories.

• Fermionic Fock space and the Pauli exclusion principle 12.03.02 pending. Direct downstream identification. The present unit's spin-1/2 system ${\mathbb{C}}^2$ is exactly the one-mode fermionic Fock space ${\mathcal{F}}_a(\mathbb{C})=\mathbb{C}\oplus \mathbb{C}$, with ${\sigma }_{+}=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$ acting as the creation operator $a^{\ast }$, ${\sigma }_{-}$ as the annihilation $a$, and the Pauli identities $\{{\sigma }_{+},{\sigma }_{-}\}=1,{\sigma }_{\pm }^2=0$ as the smallest non-degenerate CAR. The Pauli exclusion principle is the spectral statement $\sigma ({\sigma }_{+}{\sigma }_{-})=\{0,1\}$ — at most one excitation per mode. Stern-Gerlach was historically the first probe of half-integer angular momentum and, through spin-statistics, the first experimental probe of fermionic statistics; the Jordan-Wigner transformation then promotes this single-site picture to a 1D fermionic lattice on ${⨂}_n{\mathbb{C}}^2$. Read in the other direction, the fermionic Fock-space unit takes the present unit's ${\mathbb{C}}^2$ as its minimal example and generalises to an arbitrary one-particle Hilbert space.

Historical & philosophical context [Master]

Otto Stern proposed the experiment in 1921 as a test of the Sommerfeld-Debye "old quantum theory" prediction that orbital angular momentum should be quantised; with Walther Gerlach he carried it out in Frankfurt in 1922, using a beam of silver atoms heated in an oven and passed through an inhomogeneous magnetic field ^{[Stern-Gerlach *Z. Phys.* 8, 110 and 9, 349 (1922)]}. The observation of two distinct spots, not a smear, confirmed angular-momentum quantisation — but the assignment of the responsible degree of freedom was initially wrong. Stern and Gerlach attributed the two spots to space-quantisation of the orbital angular momentum of the atom. The correct assignment, that the silver atom's outer electron carries half-integer intrinsic angular momentum (spin), was made by Uhlenbeck and Goudsmit in 1925 ^{[Uhlenbeck-Goudsmit *Naturwissenschaften* 13, 953 (1925)]} on the basis of atomic spectroscopic anomalies (the anomalous Zeeman effect, the alkali-metal doublet structure), independent of the SG experiment.

Pauli incorporated spin into quantum mechanics in 1927 with the two-component spinor formulation and the Pauli matrices ^{[Pauli 1927 *Z. Phys.* 43, 601]}, one year before Dirac's relativistic derivation ^{[Dirac 1928 *Proc. Roy. Soc. A* 117, 610]} showed that spin-1/2 arises automatically from imposing Lorentz invariance on a first-order wave equation. The Dirac equation predicted the gyromagnetic ratio $g_s=2$ for the electron, in excellent agreement with experiment; the small deviation $g_s-2\approx 0.00232$ measured in the 1940s was the first quantitative success of quantum electrodynamics, calculated by Schwinger in 1948 ^{[Schwinger 1948 *Phys. Rev.* 73, 416]} as $(g_s-2)/2=\alpha /(2\pi )$ to one-loop order. Modern measurements of $(g-2)$ for the electron and muon test QED at the $10^{-12}$ level and are among the most precise predictions in all of physics.

The algebraic-geometric understanding of spin-1/2 as the fundamental representation of $\mathrm{SU}(2)\cong \mathrm{Spin}(3)$ — sitting inside the Clifford algebra ${\mathrm{Cl}}_3$ — emerged gradually from Pauli's 1927 paper, through Wigner's 1939 classification of Poincaré-group representations ^{[Wigner 1939 *Ann. Math.* 40, 149]}, through the Atiyah-Bott-Shapiro 1964 ^{[ABS *Topology* 3 Suppl. 1, 3]} systematisation of Clifford modules. By the time of Lawson-Michelsohn's Spin Geometry (1989), the picture had stabilised: spin-1/2 is the smallest faithful irrep of the universal cover of the rotation group, with discreteness of measurement outcomes a consequence of the finite-dimensionality of the irrep, not of an independent quantisation postulate.

Bibliography [Master]

Primary literature and canonical textbooks (cite when used; not all currently in reference/):

• Stern, O. & Gerlach, W., Z. Phys. 8, 110 (1922); Z. Phys. 9, 349 (1922). [Need to source.]
• Uhlenbeck, G. E. & Goudsmit, S., Naturwissenschaften 13, 953 (1925); Nature 117, 264 (1926). [Need to source.]
• Pauli, W., Z. Phys. 43, 601 (1927). The two-component spinor formulation. [Need to source.]
• Dirac, P. A. M., Proc. Roy. Soc. A 117, 610 (1928). Relativistic derivation of spin-1/2. [Need to source.]
• Wigner, E. P., Ann. Math. 40, 149 (1939). Classification of Poincaré irreps. [Need to source.]

Canonical textbooks:

• Sakurai, J. J. & Napolitano, J., Modern Quantum Mechanics, 2nd ed. (Pearson/Cambridge, 2011). Ch. 1 §1.1, Ch. 3. [Need to source.]
• Griffiths, D. J., Introduction to Quantum Mechanics, 2nd ed. (Pearson, 2005). Ch. 4 §4.4. [Need to source.]
• Susskind, L. & Friedman, A., Quantum Mechanics: The Theoretical Minimum (Basic Books, 2014). [Need to source.]
• Feynman, R. P., Leighton, R. B. & Sands, M., The Feynman Lectures on Physics, Vol. III (Addison-Wesley, 1965). Chs. 1–6. [Need to source.]
• Peskin, M. E. & Schroeder, D. V., An Introduction to Quantum Field Theory (Westview, 1995). Ch. 3 (Dirac field). [Need to source.]
• Weinberg, S., Lectures on Quantum Mechanics, 2nd ed. (Cambridge, 2015). Ch. 4. [Need to source.]
• Werner, S. A., Colella, R., Overhauser, A. W., Phys. Rev. Lett. 35, 1053 (1975). Neutron-interferometry observation of $4\pi$ spinor periodicity. [Need to source.]
• Schwinger, J., Phys. Rev. 73, 416 (1948). One-loop anomalous magnetic moment. [Need to source.]
• Lawson, H. B. & Michelsohn, M.-L., Spin Geometry (Princeton, 1989). §I.1–I.2. [Need to source (shared with 03.09.02).]
• Tong, D., Quantum Field Theory, DAMTP Cambridge lecture notes, §4. [Have.]

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Wave 1 physics seed unit, agent-drafted with LM-editorial style pass 2026-05-18 per PHYSICS_PLAN §5. Citations marked TODO_REF indicate sources expected in reference/ but not yet acquired; replace with [ref: …] when sources arrive. Pending hooks-out and prereq edges to be registered by the integrator.