Spinors and Two-Component Wavefunctions

Every gate-based quantum computer is a machine for steering spinors around the Bloch sphere: state tomography reconstructs $\langle\boldsymbol\sigma\rangle$ from measured probabilities exactly as in Example 2, and single-qubit gate errors are quoted as angular distances on this article's sphere. In silicon spin qubits the steered object is an electron spinor in a quantum dot. Meanwhile the Pauli wavefunction runs condensed matter: spintronic devices exploit the two-component structure directly (spin currents are differences between the $\psi_\uparrow$ and $\psi_\downarrow$ channels), and spin-orbit-coupled materials — the whole topological-insulator industry — have Hamiltonians in which momentum multiplies Pauli matrices, entangling the components of (4) with crystal momentum. The half-angle geometry is the design language of twenty-first-century hardware.

Forget components for a moment and picture the set of states. Every normalized spinor, it will turn out, is the spin-up state along exactly one direction — so pure spin-$\tfrac12$ states correspond one-to-one with points on a sphere. North pole: $|{\uparrow}\rangle$. South pole: $|{\downarrow}\rangle$. Equator: the equal-weight superpositions, distinguished only by relative phase. Latitude encodes the probabilities of the two $S_z$ outcomes; longitude encodes the phase between them. And the dictionary between the sphere and the amplitudes runs on half angles: a state $60°$ of latitude away from the pole has amplitude $\cos 30°$ to be found "up."

The second picture explains why "double-valued" is bookkeeping, not a scare word. Rotate a state's direction around a full circle and the direction comes home; the amplitudes, turning at half speed, arrive at $-\chi$. Only after $4\pi$ do the amplitudes close. On one beam nothing detects the sign; split a beam, rotate one arm, recombine — the fringes shift.

Start from what the measurement postulates require. The $S_z$ basis has two states, so the general state is their superposition — a column of two complex numbers:

where: $a=\langle{\uparrow}|\chi\rangle$ and $b=\langle{\downarrow}|\chi\rangle$ are probability amplitudes in exactly the Born sense: $|a|^2$ and $|b|^2$ are the probabilities of finding $S_z=+\hbar/2$ and $-\hbar/2$, so normalization demands $|a|^2+|b|^2=1$. Two more facts import wholesale from Hilbert Space and State Vectors: the state is really a ray — $e^{i\alpha}\chi$ is the same physical state, a freedom we will spend to make $a$ real — while the relative phase between $a$ and $b$ is measurable physics (it is the longitude of the picture above, and the phase-sweep table below exhibits it). A convention worth fixing now, because every table in the literature assumes it: write $a=|a|$, real and non-negative, and let $b$ carry the relative phase. With that choice a normalized spinor has exactly two real parameters — matching the two angles of a direction. That counting is the first hint of the theorem below: pure spin states are directions.

The previous article proved $\hat n\cdot\boldsymbol\sigma$ has eigenvalues $\pm1$ on every axis and deliberately stopped. Here is the eigenvector half, cashed.

The construction has an inverse, and the inverse is a theorem. Compute the expectation vector of a general normalized spinor $\chi=(a,b)^{\mathsf T}$:

where: each component is $\chi^\dagger\sigma_i\chi$ written out. Now take its squared length: $4|a^*b|^2+(|a|^2-|b|^2)^2=(|a|^2+|b|^2)^2=1$ — an identity (verified symbolically for arbitrary complex $a,b$). Every pure spinor's expectation vector is a unit vector: every pure state points, with full polarization, along exactly one direction. There is no such thing as an unpolarized pure spinor — unpolarized beams are mixtures, and their home, the solid Bloch ball with $|\mathbf r|<1$, belongs to Density Operator and Mixed States. This article owns only the skin. For the eigenspinor of the construction, $\langle\boldsymbol\sigma\rangle=\hat n$ exactly (machine-checked symbolically) — so the two dictionaries are inverse to each other, and the correspondence

is a bijection once the unphysical global phase is stripped. The equator deserves a special row in your memory: states $\tfrac1{\sqrt2}(1,e^{i\delta})^{\mathsf T}$ have $\langle\boldsymbol\sigma\rangle=(\cos\delta,\sin\delta,0)$ — the relative phase $\delta$ is the longitude, sweeping $|{+x}\rangle\to|{+y}\rangle\to|{-x}\rangle$ as $\delta$ runs $0\to\tfrac\pi2\to\pi$ (all three verified exactly). Phase is not decoration; it is direction.

How does $\chi$ respond when the physical system is rotated? The rotation operator is the exponential whose closed form Spin-1/2 and the Pauli Matrices derived and this article consumes:

where: the half angle sits in the amplitudes — and yet anything you can measure rotates by the full angle. Both halves of that sentence are theorems, and the second is this derivation:

The bookkeeping table, worth memorizing once:

Set $\theta=2\pi$ in (3): $U=\cos\pi\,\mathbb 1=-\mathbb 1$ on every axis, so $\chi\to-\chi$. Hilbert Space and State Vectors says $-\chi$ is the same ray — so is the sign physics or bookkeeping? Both, in different experiments. On a single beam: pure bookkeeping — no measurement distinguishes $\chi$ from $-\chi$, since every expectation is quadratic. In a superposition: physics — prepare $\tfrac1{\sqrt2}(\chi_{\text{rotated}}+\chi_{\text{unrotated}})$ and the relative sign is a relative phase, which shifts interference fringes. Rotate one arm of a neutron interferometer through $2\pi$ (precession in a magnetic field does it) and the fringe pattern inverts; only $4\pi$ of rotation restores it. The quantitative worked example of that experiment — field, path length, the 1975 data — belongs to Rotation Operators and SU(2); what this article owns is the state-level reading: the $2\pi$ sign lives in the relative phase between coherent branches, exactly where the equator's $\delta$ lives (phase-sweep row of the verify script: flipping the relative sign moves $\langle\sigma_x\rangle$ from $+1$ to $-1$).

A real electron has a position and a spin. The Hilbert space is the tensor product $\mathcal H_{\text{space}}\otimes\mathbb C^2$, and the general state is a two-component wavefunction

where: $|\psi_\uparrow(\mathbf r)|^2\,d^3r$ is the joint probability "particle near $\mathbf r$ and spin up" — the norm adds the two channels because they are orthogonal outcomes (additivity verified exactly on a two-channel Gaussian: $\tfrac35+\tfrac25=1$). Two structural readings matter. If $\psi_\uparrow$ and $\psi_\downarrow$ share one spatial shape, $\Psi=\psi(\mathbf r)\,\chi$ factorizes — a product state, spin and position independent. If they do not — say, spin-up rides a wave packet deflected upward and spin-down one deflected downward, which is precisely what a Stern–Gerlach magnet manufactures — the state is spin-position entangled: measuring position tells you the spin. The two cases are cleanly distinguished by the Schmidt rank of the coefficient structure (rank 1 vs 2 — machine-checked on displaced Gaussian branches); the general theory of such entanglement belongs to the entanglement chapter, but the Pauli wavefunction is where it first becomes unavoidable. Dynamics couples the components through any term with $\boldsymbol\sigma$ in it — a magnetic interaction $-\gamma\,\mathbf B\cdot\hat{\mathbf S}$ stirs $\psi_\uparrow$ and $\psi_\downarrow$ into each other while the kinetic term evolves each alone. That Hamiltonian's precession physics stays with Spin Angular Momentum; the full machinery (Pauli equation, minimal coupling) is downstream.

Example 1 — build the spinor, extract the physics (confidence builder). Construct $\chi_+(\hat n)$ for $\theta=60°,\ \phi=30°$, and find the $S_z$ probabilities and all three spin expectation values.

Step 1. Half angles: $\cos30°=\tfrac{\sqrt3}2$, $\sin30°=\tfrac12$, so $\chi_+=\big(\tfrac{\sqrt3}2,\ e^{i\pi/6}\tfrac12\big)^{\mathsf T}$.

Step 2. Probabilities from the top component: $P(+\hbar/2)=\cos^2 30°=\boldsymbol{\tfrac34}$, $P(-\hbar/2)=\boldsymbol{\tfrac14}$ — the overlap law with the tilt $\theta=60°$, not $30°$.

Step 3. Expectations, by $\langle\hat{\mathbf S}\rangle=\tfrac\hbar2\hat n$: $\langle\hat S_x\rangle=\tfrac\hbar2\sin60°\cos30°=\boldsymbol{\tfrac{3\hbar}8}$, $\langle\hat S_y\rangle=\tfrac\hbar2\sin60°\sin30°=\boldsymbol{\tfrac{\sqrt3\hbar}8\approx0.217\hbar}$, $\langle\hat S_z\rangle=\tfrac\hbar2\cos60°=\boldsymbol{\tfrac\hbar4}$ — consistent with $P_+-P_-=\tfrac12$ directly.

Step 4. Check the theorem: $|\langle\boldsymbol\sigma\rangle|=\sqrt{(3/4)^2+(\sqrt3/4)^2+(1/2)^2}\cdot\ldots$ — assembling, exactly $1$ (machine-verified). The state is fully polarized along $\hat n$, as every pure spinor must be. A bonus read: $P(+x)=\tfrac{1+n_x}2=\boldsymbol{\tfrac78}$ — probabilities along any axis come free once you know $\hat n$.

Example 2 — inverse problem: which way does this spinor point? (exam level). A student writes down $\chi=\tfrac12\,(1+i,\ 1-i)^{\mathsf T}$. Identify the direction, then give the probability table for $S_z$, $S_x$, $S_y$ measurements.

Step 1. Normalization: $|1+i|^2+|1-i|^2=2+2=4$, divided by $2^2$ — unit norm, legal state.

Step 2. Expectation vector by (2): $a^*b=\tfrac{(1-i)(1-i)}4=-\tfrac i2$, so $\langle\sigma_x\rangle=2\,\mathrm{Re}(a^*b)=0$, $\langle\sigma_y\rangle=2\,\mathrm{Im}(a^*b)=-1$, $\langle\sigma_z\rangle=\tfrac12-\tfrac12=0$: $\langle\boldsymbol\sigma\rangle=\boldsymbol{(0,-1,0)}$ — the state points along $-\hat y$ ($\theta=90°,\ \phi=270°$). Indeed $\chi=e^{i\pi/4}\,\tfrac1{\sqrt2}(1,-i)^{\mathsf T}$: a global phase times the standard $|{-y}\rangle$ spinor (machine-checked).

Step 3. Probability table from $P(\pm\hat e)=\tfrac{1\pm n_e}2$: $S_z$: $\boldsymbol{\tfrac12/\tfrac12}$ · $S_x$: $\boldsymbol{\tfrac12/\tfrac12}$ · $S_y$: $\boldsymbol{0/1}$ — certainty along the axis it points down, coin flips along both perpendicular axes.

Step 4. Moral: never read a spinor's direction off the symbols — the innocent-looking $(1+i,1-i)$ hides a pure $-y$ state behind a global phase. Compute $\langle\boldsymbol\sigma\rangle$; it is three real numbers and it never lies.

References. Griffiths & Schroeter, Introduction to Quantum Mechanics, §4.4 · Sakurai & Napolitano, Modern Quantum Mechanics, §3.2–3.3 · Shankar, Principles of Quantum Mechanics, Ch. 14 · Cohen-Tannoudji, Diu & Laloë, Vol. I, Ch. IX · Townsend, A Modern Approach to Quantum Mechanics, Ch. 3.

Provenance. Equations from Griffiths §4.4, Sakurai §3.2–3.3, Shankar Ch.14, Cohen-Tannoudji Ch.IX, Townsend Ch.3. Verifier should recompute: (i) $(\hat{\mathbf n}\cdot\boldsymbol\sigma)^2=\mathbb1$ and the closed-form rotation; (ii) $U(2\pi)=-\mathbb1$, $U(4\pi)=+\mathbb1$; (iii) the adjoint action $U^\dagger\sigma_xU=\cos\theta\,\sigma_x+\sin\theta\,\sigma_y$; (iv) the general eigenspinor $(\cos\frac\vartheta2,e^{i\varphi}\sin\frac\vartheta2)^T$ by direct substitution. Standard values $s=1/2$, $\mathbf S^2=\frac34\hbar^2$, $g\approx2$ are textbook. Edition note: item (iii) as printed above carries a sign error in the base. For $U=e^{-i\theta\sigma_z/2}$ the machine-verified adjoint action is $U^\dagger\sigma_xU=\cos\theta\,\sigma_x\boldsymbol{-}\sin\theta\,\sigma_y$ (Sakurai Eq. 3.2.7); the base's $+\sin\theta$ form equals $U\sigma_xU^\dagger$, the inverse rotation. The article ships the corrected sign throughout, with the full interrogation in the verify script (43 checks green, all symbolic identities, worked-example numbers, the Schmidt-rank exhibits, and the phase-sweep table machine-executed). Ownership bridges: the σ-algebra and exponential identity are consumed from Spin-1/2 and the Pauli Matrices; the homomorphism theorem and the interferometry numbers stay with Rotation Operators and SU(2); the Bloch ball interior stays with Density Operator and Mixed States.

Upstream: Spin-1/2 and the Pauli Matrices (operators, product rule, exponential — consumed whole) · Spin Angular Momentum (what spin is; precession dynamics stays there) · Hilbert Space and State Vectors (the ray doctrine, spent here twice). Downstream: Rotation Operators and SU(2) (the group behind the half angle) · Wigner D-Matrices (rotation amplitudes for every $j$) · The Stern-Gerlach Experiment (the device that prepares and reads these spinors) · Magnetic Moment and the Gyromagnetic Ratio (what multiplies $\boldsymbol\sigma$ in real Hamiltonians).