Pilot-Wave Trajectories Through a Double Slit

Wave Parameters

Wavelength λ 0.55

Slit separation d 1.00

Slit width w₀ 0.26

Flow speed 1.0×

Live Readout

Fringe spacing

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Front z

Trajectories

How it works

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)]

The two beams interfere; this simulator computes the exact analytic phase gradient ∂S/∂y of their 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 on the floor plane) 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.

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.

Bohmian trajectory |ψ|² intensity floor

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