When a magnetic moment is sent through an inhomogeneous field, can it point any direction (a continuous smear on the detector) — or only a discrete set of directions?
▶ Launch the interactive simulationSend a velocity-selected beam of silver atoms (one unpaired electron ⇒ net moment ≈ one Bohr magneton) through a field gradient G = ∂B_z/∂z. The force on a moment is F_z = μ_z·G, so each atom deflects by z = μ_z·G·L·(L/2+D)/(M v₀²) (a parabola through the magnet of length L, then a straight drift D to the detector). Forward-model two populations of 4000 atoms — quantum (μ_z = g_s·m_s·μ_B, m_s = ±½) and a classical control (μ_z = μ_B·cosθ, cosθ isotropic) — with a velocity passband and detector noise, seeded → reproducible. Measure the central-gap fraction and the mean deflection.
the spin quantum number s = ½ and the electron magnetic moment of one Bohr magneton (g_s ≈ 2). The beam does NOT paint a continuous band — it splits into exactly TWO spots, so 2s+1 = 2 ⇒ s = ½. The classical isotropic moment fills the centre (central-gap fraction ≈ 0.4); the quantum beam leaves a clean dead zone between the spots (≈ 0). From the mean deflection the lab recovers the Bohr magneton WITHOUT being told it: μ_B = ⟨|z|⟩·M·v₀²/(G·L·(L/2+D)) ≈ 9.27×10⁻²⁴ J/T, giving g_s = 2μ_z/μ_B ≈ 2 — twice the value orbital angular momentum would give, the anomaly that pointed at electron spin. Stern & Gerlach's 1922 result (Nobel 1943) — the first direct proof of space quantization and the eighth pillar of early quantum theory after Millikan's charge