Stern-Gerlach Beam Splitting

What is modelled

A neutral atom with a magnetic moment feels a force only where the field changes with height. In the wedge-and-groove magnet the field gradient is nearly constant, so an atom with moment component μ_z is pushed up or down with force μ_z dB/dz. A classical moment could point anywhere and would smear the beam; the measured atoms land in 2j+1 spots because the projection of the spin on the field axis can only take the values m = −j ... +j.

Equations

• Force F = μz dB/dz with μz = g m μB, g = 2, μ_B = 9.274×10⁻²⁴ J/T.
• Constant acceleration in the magnet and straight flight afterwards: z = (a/v²)(L²/2 + L D), a = g m μB (dB/dz)/M, M = 107.87 u.
• Oven flux speeds follow f(v) ∝ v³ e−Mv²/2kT, mean ¾√(2πkT/M).
• Prepared along an axis at angle θ from the field the outcome probability is P(m) = C(2j, j+m) cos2(j+m)(θ/2) sin2(j−m)(θ/2), so the mean projection is j cos θ.

Checks run in the page

• Step-by-step trajectory against the closed-form deflection.
• Probabilities sum to one, the mean projection and a 20 000-atom random sample.
• Spin 1/2 probability cos²(θ/2) and the spin 1 binomial weights.
• Classical edge equals the outermost quantum spot and the deflection is linear in the gradient.

Honest simplifications

The field is taken as a pure uniform gradient along z with no fringing, the atoms all start with the same axial velocity direction, the speed distribution is the ideal effusive one and the plate shows the landing height only. A real silver beam also has the nuclear spin and the large Larmor precession, which are ignored. The vertical axis of the stage is magnified so that the spots are visible.

• Press Classical prediction to see a continuous smear that ends at the same edge as the quantum spots.
• Press Spin 1 gives three spots, then drag the prepared angle to move weight between them.
• Raise the oven temperature: faster atoms spend less time in the magnet and the spots move closer together.