Stern–Gerlach as an entangling device

The atom is neutral, so no Lorentz force acts; the only coupling is between its magnetic moment and the field. The state space is L²(ℝ³) ⊗ ℂ², and H = p̂²/2m ⊗ I − γ B(z) ⊗ Ŝ_z is an operator on both factors at once. Because B points along z in the idealised model, [Ŝ_z, H] = 0: the spin populations are constants of the motion, and H is block diagonal in the basis (|↑⟩, |↓⟩) that diagonalises Ŝ_z. Each block is an ordinary one-dimensional problem, p̂²/2m − γħmₛ(B₀ + bz) — a linear potential, which is free fall with effective acceleration γħmₛ b/m. The boundary condition is only normalisability on the line: no walls, no quantised levels, just a packet accelerating. For silver, with |γħ/2| = μ_B and b = 800 T m⁻¹, that acceleration is 4.14 × 10⁴ m s⁻², about 4200 g, and because γ < 0 it points toward −z for mₛ = +½ and toward +z for mₛ = −½.

Start from a product state, (α|↑⟩ + β|↓⟩) ⊗ |χ₀⟩. Each spin branch is dragged by its own constant force, so after time t the state is α|↑⟩⊗|χ₊(t)⟩ + β|↓⟩⊗|χ₋(t)⟩, where |χ_±⟩ are the same packet displaced by ±½at² and boosted by ±Ft. This is entangled — Schmidt rank two — as soon as αβ ≠ 0 and the two packets differ. Nothing has been measured. The evolution operator is unitary, the global state is still pure, and running the gradient backwards returns the original product state exactly. Structurally the magnet is a controlled-displacement gate: the spin is the control, the centre of mass is the target, and the same circuit drawn with two qubits is how a CNOT copies a bit without anybody reading it. That is the entire content of the beam splitting: a correlation was created, not an outcome.

Nobody watches the atom's centre of mass, so the spin's state is the partial trace: ρₛ carries |α|² and |β|² on the diagonal, unchanged for ever, and αβ*⟨χ₋|χ₊⟩ off it. Every coherence is multiplied by that one number c. The purity Tr ρₛ² = 1 − 2|α|²|β|²(1 − |c|²) runs from 1 when the packets coincide down to ½ for an equal superposition when they are orthogonal. For equal-width Gaussians separated in position only, |c| = exp(−Δz²/8σ²): 0.88 at one width, 0.61 at two, 0.32 at three, and 8.8 × 10⁻⁴ by seven and a half. Halfway, at |c| = 0.5 and still with equal amplitudes, the reduced state has eigenvalues 0.75 and 0.25 and an entanglement entropy of 0.81 bits — a partial record, which is exactly what a partly completed measurement should mean. Note that σ is the coherent packet width, not the collimated beam width: a slit-limited beam is an incoherent ensemble of far narrower packets.

The two branches differ in mean momentum as well as position, and for a minimum-uncertainty Gaussian both terms count: |c| = exp(−Δz²/8σ² − Δp²σ²/2ħ²). Momentum is the faster channel, and its width σₚ = ħ/2σ is fixed at the source and never changes under free flight or a uniform force. A thermal silver beam has a transverse coherence width near σ = 100 nm, giving σₚ = 5.3 × 10⁻²⁸ kg m s⁻¹, while Δp = 2Ft grows at 1.48 × 10⁻²⁰ kg m s⁻¹ per second of flight. The momentum exponent reaches 1 after t = 0.10 μs — 50 μm into a 40 mm magnet — while the centres have moved apart by only 0.42 nm, four thousandths of a packet width. Coherence is down to 1/e there and, since the exponent grows as t², to 10⁻³ by 132 μm. The which-path information exists long before there is anything to see. That is why the reversible Stern-Gerlach of Wigner, and of Scully, Englert and Schwinger, is so brutally hard: undoing the split means matching a momentum kick to better than ħ/σ.

B = bz ẑ has ∇⋅B = b, so it is not a magnetic field at all. The simplest legal first-order model near the pole gap is B = (B₀ + bz)ẑ − b x x̂, whose divergence is b − b = 0 and whose curl vanishes in the empty gap. The transverse piece tilts the field off the z axis by bx/B₀, and what rescues the simple picture is speed: the spin precesses about the local field at ω_L = |γ|B₀ = 1.8 × 10¹¹ rad s⁻¹ for B₀ = 1.0 T, while the direction seen by the atom turns at only v/L ≈ 1.3 × 10⁴ s⁻¹. With a ratio of 1.4 × 10⁷ the spin follows adiabatically, the energy is U = 2μ_B mₛ|B| ≈ 2μ_B mₛ(B₀ + bz + b²x²/2B₀), and the residual transverse force is bx/B₀ = 8% of the axial one at 0.10 mm off axis — inward for the low-field seeker mₛ = +½, outward for mₛ = −½. Remove the holding field and the argument dies: near a zero of |B| the spin cannot follow, and Majorana flips scramble the branches.

Integrating the constant force through the pole gap and the drift gives z = (μ_z b/mv²)(L²/2 + LD) — 1.79 mm for the silver numbers above — so the position of a spot fixes |μ_z| and nothing else. For silver in 1922 that came out at one Bohr magneton. It does not give the spin, because |μ_z| = gₛ μ_B mₛ is one equation in two unknowns. The second equation is the count: the beam divides into 2s + 1 spots, and two spots force s = ½, hence mₛ = ±½ and gₛ = 2. Stern and Gerlach read their two spots as space quantisation of an l = 1 orbital moment, which would in fact have produced three spots including an undeflected one; spin was not proposed until 1925. Notice too that z carries 1/v², so a thermal velocity spread smears each spot along the deflection axis — that width is kinematics, not quantum uncertainty.