Spin-½

A two-state quantum system

Spin-½ is the simplest non-trivial quantum system: a two-dimensional Hilbert space. Choose a quantization axis (call it z) and label its eigenstates |↑⟩ and |↓⟩ with S_z eigenvalues +ℏ/2 and −ℏ/2 respectively. Any pure state is a superposition:

|ψ⟩ = cos(θ/2) |↑⟩ + e^(iφ) sin(θ/2) |↓⟩

The angles (θ, φ) parametrize the Bloch sphere: the unit-sphere direction (sin θ cos φ, sin θ sin φ, cos θ) is exactly the expectation value of the spin direction ⟨S⟩/|⟨S⟩|. North pole = |↑⟩, south pole = |↓⟩, equator = (|↑⟩ ± |↓⟩)/√2 and similar superpositions.

Pauli matrices

The three spin operators on this two-dimensional space are S_i = (ℏ/2)σ_i with the Pauli matrices

σ_x = (0 1 / 1 0),  σ_y = (0 −i / i 0),  σ_z = (1 0 / 0 −1)

Their algebra encodes everything: [σ_i, σ_j] = 2iε_ijk σ_k (closure under commutator, generating SU(2)) and {σ_i, σ_j} = 2δ_ij I (anticommutation, related to Dirac's gamma algebra). Eigenvalues of any σ_i are ±1, so eigenvalues of S_i are ±ℏ/2 — the two-spot Stern-Gerlach result for any axis.

Why 720° to come home

A rotation by angle θ around axis n̂ acts on a spin-½ state by

U(θ, n̂) = exp(−iθ n̂·σ / 2) = cos(θ/2) I − i sin(θ/2) n̂·σ

Note the θ/2 — half the rotation angle. So at θ = 2π (one full turn), U = cos(π) I − i sin(π) n̂·σ = −I. The state |ψ⟩ becomes −|ψ⟩. The overall phase is unobservable in isolation but becomes observable in interference.

At θ = 4π (two full turns), U = +I and |ψ⟩ returns to itself. This is the geometric fact that SU(2) — the group of 2×2 unitary matrices with determinant 1 — double-covers SO(3): every spatial rotation corresponds to exactly two SU(2) elements ±U. The minus sign is real.

Worked example — neutron interferometry shows the −1

Rauch & Treimer (1974) and Werner et al. (1975) put a beam of cold neutrons through a Si-crystal interferometer with two paths. One path went through a uniform magnetic field B. A neutron with magnetic moment µ_n precesses through angle θ = γ_n B τ during transit time τ. Adjust B and τ so θ = 2π exactly. The neutron's spinor picks up the factor exp(−iπ σ·n̂) = −1.

• Neutron magnetic moment: µ_n ≈ −1.913 µ_N where µ_N = eℏ/(2m_p) is the nuclear magneton.
• Gyromagnetic ratio: γ_n = g_n µ_N / ℏ ≈ −1.832 × 10⁸ s⁻¹ T⁻¹.
• For a 64 mT field over 4 cm at 2 km/s velocity: precession ≈ −1.832 × 10⁸ × 0.064 × (0.04/2000) ≈ −0.234 rad — way too short; real experiments use stronger fields and longer paths to reach 2π.
• At θ = 2π precisely, interference between the two arms reverses (constructive ↔ destructive); at θ = 4π it restores. This is the −1 of the spinor double-cover, directly observed.

Spin-0, ½, 1, and higher

Spin	Hilbert dim	SO(3) representation	Stern-Gerlach split	Rotation period
0 (scalar)	1	Trivial	No split	360°
½ (spinor)	2	Fundamental of SU(2)	2 spots, ±ℏ/2	720°
1 (vector)	3	Adjoint of SO(3)	3 spots, 0, ±ℏ	360°
3/2	4	SU(2) higher	4 spots, ±ℏ/2, ±3ℏ/2	720°
2 (tensor)	5	SO(3) higher	5 spots	360°
Photon	2 helicities	Effective ±1	(massless, helicity)	360°

JavaScript — Pauli matrices, rotations, Bloch vector

// Complex numbers as [real, imag]
const C = (r, i = 0) => [r, i];
const cAdd = (a, b) => [a[0]+b[0], a[1]+b[1]];
const cMul = (a, b) => [a[0]*b[0] - a[1]*b[1], a[0]*b[1] + a[1]*b[0]];
const cScale = (a, s) => [a[0]*s, a[1]*s];

// 2x2 complex matrix utilities
const mat = (a,b,c,d) => [[a,b],[c,d]];
const mulMV = (M, v) => [cAdd(cMul(M[0][0], v[0]), cMul(M[0][1], v[1])), cAdd(cMul(M[1][0], v[0]), cMul(M[1][1], v[1]))];
const mulMM = (A, B) => mat(
  cAdd(cMul(A[0][0], B[0][0]), cMul(A[0][1], B[1][0])),
  cAdd(cMul(A[0][0], B[0][1]), cMul(A[0][1], B[1][1])),
  cAdd(cMul(A[1][0], B[0][0]), cMul(A[1][1], B[1][0])),
  cAdd(cMul(A[1][0], B[0][1]), cMul(A[1][1], B[1][1])),
);

// Pauli matrices
const I = mat(C(1), C(0), C(0), C(1));
const Sx = mat(C(0), C(1), C(1), C(0));
const Sy = mat(C(0), C(0,-1), C(0,1), C(0));
const Sz = mat(C(1), C(0), C(0), C(-1));

// Rotation U(θ, n̂) = cos(θ/2) I − i sin(θ/2) (nx Sx + ny Sy + nz Sz)
function rotU(theta, nx, ny, nz) {
  const c = Math.cos(theta / 2), s = Math.sin(theta / 2);
  const cosPart = mat(C(c),C(0),C(0),C(c));
  const minusI = mat(C(0,-s),C(0),C(0),C(0,-s));   // −i sin(θ/2) I, used as scalar then multiplied
  function scaleM(M, scalar) {
    return mat(cMul(scalar, M[0][0]), cMul(scalar, M[0][1]), cMul(scalar, M[1][0]), cMul(scalar, M[1][1]));
  }
  const nDotSigma = mat(
    cAdd(cMul(C(nz), Sz[0][0]), cAdd(cMul(C(nx), Sx[0][0]), cMul(C(ny), Sy[0][0]))),
    cAdd(cMul(C(nz), Sz[0][1]), cAdd(cMul(C(nx), Sx[0][1]), cMul(C(ny), Sy[0][1]))),
    cAdd(cMul(C(nz), Sz[1][0]), cAdd(cMul(C(nx), Sx[1][0]), cMul(C(ny), Sy[1][0]))),
    cAdd(cMul(C(nz), Sz[1][1]), cAdd(cMul(C(nx), Sx[1][1]), cMul(C(ny), Sy[1][1]))),
  );
  const sinPart = scaleM(nDotSigma, C(0, -s));
  return mat(cAdd(cosPart[0][0], sinPart[0][0]), cAdd(cosPart[0][1], sinPart[0][1]),
             cAdd(cosPart[1][0], sinPart[1][0]), cAdd(cosPart[1][1], sinPart[1][1]));
}

// Apply rotation to |↑⟩ = (1, 0)
const psi_up = [C(1), C(0)];
const after_2pi = mulMV(rotU(2 * Math.PI, 0, 0, 1), psi_up);
console.log('After 360° around z:', after_2pi);    // (-1, 0)  — picked up a minus sign!
const after_4pi = mulMV(rotU(4 * Math.PI, 0, 0, 1), psi_up);
console.log('After 720° around z:', after_4pi);    // (1, 0)  — back to itself

// Bloch vector r_i = ⟨ψ|σ_i|ψ⟩
function bloch(psi) {
  function expVal(M) {
    const Mpsi = mulMV(M, psi);
    // ⟨ψ| Mψ⟩ = ψ₀* Mψ₀ + ψ₁* Mψ₁
    const psiConj = [[psi[0][0], -psi[0][1]], [psi[1][0], -psi[1][1]]];
    return cAdd(cMul(psiConj[0], Mpsi[0]), cMul(psiConj[1], Mpsi[1]))[0]; // real part
  }
  return { x: expVal(Sx), y: expVal(Sy), z: expVal(Sz) };
}
console.log('Bloch vector of |↑⟩:', bloch(psi_up));  // (0, 0, 1)

// Stern-Gerlach probability: P(↑) for state cos(θ/2)|↑⟩ + sin(θ/2)|↓⟩ along z
function probZUp(theta) { return Math.cos(theta/2)**2; }
console.log('θ=0 (pure ↑):', probZUp(0));            // 1.000
console.log('θ=π/2 (equator):', probZUp(Math.PI/2));  // 0.500
console.log('θ=π (pure ↓):', probZUp(Math.PI));        // 0.000

Where spin-½ matters

• Atomic structure. Pauli exclusion (a corollary of spin-½ statistics) forces electrons into shells — the periodic table follows.
• Nuclear physics. Proton and neutron spin-½ build nuclear-spin multiplets, NMR, and MRI.
• Solid state. Spin-orbit coupling, exchange, and magnetism all reduce to coupled spin-½ systems on a lattice.
• Quantum information. A qubit IS a two-state quantum system — physically realised as electron spin, nuclear spin, polarization, or superconducting two-level systems.
• Particle physics. Every Standard-Model fermion is spin-½; Dirac equation, Pauli equation, and weak isospin all build on the spin-½ representation.
• Topology. Anyons in 2D, Majorana fermions, and topological superconductors generalize spin-½ statistics in non-trivial ways.

Common mistakes

• Picturing an electron as a literally spinning ball. A classical ball would need to rotate faster than light to have ℏ/2 of angular momentum given the electron's small radius bound. Spin is intrinsic, not orbital.
• Confusing |↑⟩ with a definite spatial direction. |↑⟩ is the +ℏ/2 eigenstate of S_z. A spin pointing in +x is (|↑⟩+|↓⟩)/√2 in the z-basis — the choice of basis is arbitrary.
• Treating the 360° minus sign as a global phase. Globally yes, but it becomes observable in interference experiments (Rauch 1974).
• Calling spin "magnetic moment". The magnetic moment µ = g (e/2m) S is proportional to spin but is a derived quantity. Spin exists for neutral particles too (e.g. neutrons, neutrinos).
• Thinking spin-½ implies a 720° rotation in space. Space is rotated by 360° = 2π; the rotation operator on the spinor accumulates phase that returns only after 720° has been applied to the operator.
• Forgetting non-commutation. [S_x, S_y] = iℏS_z. You cannot simultaneously assign definite values to S_x and S_z — measuring one disturbs the other.