Quantum Double-Slit

The equation

The two-dimensional time-dependent Schrödinger equation in natural units (ℏ = m = 1) is

i ∂_t ψ(x,y,t)  =  −½ (∂_x² + ∂_y²) ψ  +  V(x,y) ψ

The initial state is a minimum-uncertainty Gaussian, ψ₀ = A exp(−r²/2σ²) exp(i k₀ x), centred at x₀ ≈ 5.4, y₀ = 10 with mean momentum k₀ in the x-direction. The group velocity is v_g = k₀ and the kinetic energy is E = k₀²/2 — a particle in the quantum sense, but also a wave.

Split-operator method

In 2D, Crank-Nicolson requires solving large banded linear systems. The split-operator approach avoids this entirely by exploiting the fact that the kinetic operator T = −½∇² is diagonal in Fourier space:

ψⁿ⁺¹  =  e^{−iV dt/2} ·  ℱ⁻¹[e^{−i|k|²dt/2} ℱ[ψⁿ]] ·  e^{−iV dt/2}

Each factor is exactly unitary. The Trotter splitting error is O(dt²), and norm is conserved to floating-point noise inside the domain. The 2D FFT uses a standard Cooley–Tukey radix-2 algorithm applied row-by-row then column-by-column on a 128 × 128 grid, giving exact kinetic propagation at low computational cost.

Absorbing boundaries (a complex potential −iΓ f(r) at all four edges, folded into the position-space half-steps) prevent wrap-around and keep the norm readout honest: values below 1 mean amplitude has left the domain.

Fringe spacing

The central result of 2-slit interference: for slits separated by d, the bright fringe spacing at a screen distance L away is

Δy  =  λ L / d  =  (2π/k₀)  L / d

The Δy (fringe) readout computes this for the current parameters, using the right-edge of the domain as the screen. Watch it update as you move k₀ or d — and verify by eye that the simulation agrees. At k₀ = 6, d = 3, the expected spacing is about 2.8 units, roughly 18 grid cells, clearly visible as alternating bright bands.

What to look for

Probability view (|ψ|²). The inferno colormap maps zero to black and the peak to pale yellow. You see the packet approach, split at the barrier, and then interfere on the far side, building up a standing banded pattern of constructive and destructive interference. The fringes are not a recording artifact — they are real spatial modulations of the probability density.

Phase view. Hue encodes the complex argument of ψ and brightness encodes its amplitude. The phase field is smooth and slowly rotating inside each slit and on the far side, but the two contributions come in at different angles, creating the phase slip that produces the interference. Where the phase from slit 1 and slit 2 differ by π, the amplitude vanishes — a dark fringe.

Re(ψ) view. The real part of the wavefunction oscillates at the de Broglie wavelength. On the far side of the barrier you can see the oscillations organise into the familiar striped interference pattern, with nodes where the two paths cancel exactly.

Change d. Increasing slit separation compresses the fringes; the two paths pick up their relative phase difference over a shorter transverse distance. The formula Δy = λL/d is precise — use the readout to verify.

Change k₀. Higher momentum means shorter de Broglie wavelength, tighter fringes. The packet also crosses faster (v_g = k₀), so raise the speed slider to keep up.

Change w. The slit width controls single-slit diffraction: a narrow slit diffracts the wave over a wider angle (Heisenberg: Δy small → Δp_y large), illuminating more of the far field. A wide slit stays more collimated. At w ≫ λ, each slit starts to look like a geometric aperture.

Which path?

The interference pattern disappears the moment you can determine which slit the particle passed through — even in principle. This experiment simulates a closed system, so there is no which-path information anywhere: the particle exists in a coherent superposition of both paths simultaneously. The fringes are direct evidence of that superposition. Close one slit (drag d very large to separate them past the grid edge) and the pattern collapses to a single-slit diffraction envelope.

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