Wave Packet Interference: Colliding Quantum Amplitudes in a Box

A particle described by a smear, not a point

In quantum mechanics a localized particle is represented by a wave packet: a superposition of many plane waves of different momenta, weighted so that they interfere constructively in one region of space and cancel everywhere else. The simplest useful shape is a Gaussian wave packet, which has both a well-defined central position and a well-defined central momentum, spread by uncertainties Δx and Δk that are tied together by the Heisenberg relation Δx·Δk ≥ ½:

ψ(x, 0) = (2πσ²)^(−1/4) · exp(−(x−x₀)² / 4σ²) · exp(i k₀ x)

σ  = initial spatial width
k₀ = central wavenumber (momentum p₀ = ℏk₀)

Time evolution: the Schrödinger equation and dispersion

A free (or boxed) particle's wavefunction evolves under the time-dependent Schrödinger equation, iℏ ∂ψ/∂t = −(ℏ²/2m) ∂²ψ/∂x². Unlike a classical wave on a string, whose shape is preserved as it travels, a free quantum wave packet spreads as it moves, because it is built from a range of momentum components (a Fourier decomposition into plane waves) and different momenta carry different phase velocities — this spreading, not friction or any dissipative process, is why a released wave packet's width grows with time even in a perfectly empty, lossless box.

The split-operator method: how the simulation actually steps forward

The Schrödinger equation mixes a term that is simplest in position space (the potential energy) with a term that is simplest in momentum space (the kinetic energy, a second derivative), and no single fast update handles both directly. The standard numerical trick, the split-operator (or split-step Fourier) method, applies each half of the evolution operator where it is cheap and switches representations with a Fast Fourier Transform in between:

ψ(x, t+dt) = exp(−iV(x)dt/2ℏ) · IFFT[ exp(−iℏk²dt/2m) · FFT[ exp(−iV(x)dt/2ℏ) · ψ(x,t) ] ]

1. half-step potential kick in position space   (a simple pointwise multiply)
2. FFT to momentum space
3. full-step kinetic evolution in momentum space (another pointwise multiply — exact, not approximate)
4. IFFT back to position space
5. half-step potential kick in position space

Splitting the potential kick into two half-steps around one full kinetic step (a symmetric, second-order Strang splitting) makes the scheme accurate to O(dt³) per step rather than O(dt²), a meaningful gain for the same computational cost, and because the kinetic term is applied exactly in momentum space rather than by finite differences, the method conserves probability (total |ψ|²) far better than a naive finite-difference scheme at the same step size.

Interference when two packets meet

When two wave packets with different central momenta k₁ and k₂ overlap in the same region of the box, the total wavefunction is their sum, and the probability density |ψ₁+ψ₂|² is not simply the sum of the two individual densities — it contains a cross term 2·Re(ψ₁*ψ₂) that oscillates in space at a spatial frequency set by the momentum difference k₁ − k₂, producing visible interference fringes exactly like the classic double-slit pattern, except here the two "sources" are two localized wave packets rather than two slits. Where the fringe pattern is constructive the combined probability of finding the particle is enhanced above what either packet alone predicts; where it is destructive, it is suppressed — a direct, visual demonstration that quantum probability amplitudes interfere while classical probabilities simply add.

Beating and revival in a bounded box

Confined to a finite 1-D box, the exact energy eigenstates are discrete standing waves, and any wave packet is really a superposition of a finite comb of these discrete energy levels. Two dominant components with close but unequal energies E₁, E₂ produce quantum beating — a slow oscillation of the packet's shape or position at frequency |E₁−E₂|/ℏ, directly analogous to the beat frequency you hear from two slightly detuned musical notes. Because the box's energy levels are discrete (unlike the continuous spectrum of a free particle), the packet's spreading is not permanent: at long enough times the various phase relationships between energy components can realign closely enough to reconstruct something close to the original localized shape, a striking phenomenon called wave packet revival, first predicted theoretically and later observed directly in Rydberg atom experiments.

Frequently asked questions

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