In de Broglie–Bohm pilot-wave theory, a particle always has a real position; it is steered by the phase S of the ordinary wavefunction ψ = |ψ|e^(iS/ħ) through the guidance equation:
v = (ħ/m) ∇S (guidance equation)
ψ(y,z) = ψ_A(y,z) + ψ_B(y,z) (two coherent slit sources)
Each slit emits a diffracting Gaussian beam — the free-particle Schrödinger equation and the paraxial optical wave equation are the same equation with propagation distance z playing the role of time, so a slit of width w₀ spreads exactly like a spreading quantum wave packet:
w(z) = w₀√(1+(z/z_R)²), z_R = k·w₀²/2
ψ_i ∝ √(w₀/w(z))·exp[-(y-y_i)²/w(z)² + i(k z + k(y-y_i)²/(2R(z)) - Gouy term)]
This 2D top-down view plots the propagation axis z straight down the canvas and the transverse axis y across it — the natural plane the physics already lives in (the 3D version only added a perspective camera on top of the same flat computation). The simulator computes the exact analytic phase gradient ∂S/∂y of the two-beam sum at every point and integrates dy/dz = (1/k)·∂S/∂y for each trajectory, seeded with a Gaussian spread of initial positions matching |ψ(z=0)|² at each slit.
- The trajectories never cross — a strict theorem of Bohmian mechanics, visible directly in the render.
- Even though each particle takes a single deterministic path, the ensemble reproduces the Born-rule interference pattern (the bright/dark bands) exactly, because trajectories are pushed away from destructive-interference regions and bunched into constructive ones.
- λ / d / w₀ reshape the guiding wave and hence every trajectory; flow speed only changes how fast the reveal animates, not the physics.
- Drag to pan, scroll (or pinch) to zoom into any part of the interference pattern.
This is the same picture published by Philippidis, Dewdney & Hiley (1979) for the two-slit experiment, and is one of several serious interpretations of quantum mechanics alongside the Copenhagen and many-worlds views.