Eigenstates in a 3D Box | Imagining Physics

Eigenstates in 3D Box

Explanation

This simulation shows a Pauli spinor inside a hard-walled three-dimensional box, together with Bohmian particles guided by the Pauli current. The glowing cloud is the spinor density. The magnetic-field controls apply a uniform field along a selected axis and can start the spatial wave in low particle-in-a-box energy eigenstates.

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. All six faces are hard-walled, and particles are reflected back into the box if numerical stepping reaches a wall. The uniform field enters through the spin Zeeman term and through the symmetric-gauge vector potential in the kinetic term.

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

Guidance

The Bohmian velocity includes the Pauli current, the spin-current curl, and the magnetic vector potential term from A5. In simulation units the coupling is q/c = 1, with the symmetric-gauge vector potential about the box center.

A = \frac{1}{2} (B \times r)

v_{conv} = \frac{j}{ρ}

j = \frac{ℏ}{m} Im (Ψ^{†} \nabla Ψ)

The spin current uses the local Pauli spin density. The spin basis is spatially constant here, so the gradient of chi term in A5 vanishes.

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

Contours

The floor curves are level sets of the logarithmic density on the middle horizontal slice. Brightness follows the steepness of the logarithmic density.

\ln ρ (x, y, z_{mid}) = constant

brightness \propto | \nabla \ln ρ |