Stern-Gerlach Close Up

Explanation

This simulation shows an up-polarized Pauli spinor wave packet inside a hard-walled three-dimensional box, together with Bohmian particles guided by the Pauli current. The glowing cloud is the spinor density. The SG controls add the Stern-Gerlach z-gradient for manual splitting.

Wave

The wave function is a two-component spinor. Its density is

ρ (x, t) = Ψ^{†} Ψ

The wave evolves according to the Pauli equation inside the box. The x faces are periodic, while y and z remain hard-walled. The SG option applies the magnetic field through the spin Zeeman term for qualitative Stern-Gerlach deflection.

i ℏ \partial_{t} Ψ = [- \frac{ℏ^{2}}{2 m} \nabla^{2} - \frac{ℏ}{2 m} σ \cdot B] Ψ

Guidance

The Bohmian velocity is based on the Pauli probability current divided by density.

v_{B} = \frac{j}{ρ}

j = \frac{ℏ}{m} Im (Ψ^{†} \nabla Ψ)

The spin current uses the local Pauli spin density. For the default up-polarized packet it reduces to the familiar transverse curl around density gradients.

v = \frac{j}{ρ} + \frac{s ℏ}{m ρ} \nabla \times (Ψ^{†} σ Ψ)