t / T_rev 0T_rev 0FPS 0

Quantum Carpet: Wavepacket Revivals and Fractional Talbot Images in an Infinite Well

A particle in a box has energies E_n ∝ n². Every phase e^−iE_nt/ħ returns to 1 at the same time, so any state revives exactly at T_rev; at fractions of T_rev it splits into mirror and shifted copies.

Initial state

State

Particle and box

Particle

Time

Readouts

Energy-state weights |c_n|²

Uncertainty product vs time

Δx·Δp / (ħ/2) over one revival period; it never drops below 1 (dashed).

Experiment: when does the packet reappear?

For the current state, the overlap |⟨ψ(0)|ψ(t)⟩|² (autocorrelation) is computed from the eigenstate amplitudes, Σ|c_n|² e^−iE_nt, at fractions of T_rev. Measured = this sum; predicted = 1 at t = T_rev and a small value elsewhere.

Independent numerical checks

Controls

Top plot: the live wavefunction in the box; height is |ψ|², colour is the phase of ψ. Bottom plot: the space–time “carpet” of |ψ|² from t = 0 (top) to the chosen span (bottom), the white line marks now.
State: choose a Gaussian packet (position, width, momentum), the first two eigenstates, a square pulse or a pair of packets.
Particle and box: width L and particle mass only rescale time and energy to real units (hydrogen-atom-sized box for an electron by default).
Time: drag the slider to scrub, or let it run; the speed is in units of T_rev per second.

The science

Eigenstates. φ_n(x) = √(2/L) sin(nπx/L), E_n = n²π²ħ²/(2mL²). The state is ψ(x,t) = Σ c_n φ_n(x) e^−iE_nt/ħ with c_n = ∫φ_nψ(x,0)dx (computed here by Simpson integration with 160 terms).

Revivals. E_n t/ħ = n² (π²ħ t/(2mL²)); at T_rev = 4mL²/(πħ) every phase is a multiple of 2π, so ψ(x,T_rev) = ψ(x,0) exactly. At T_rev/2, e^−in²π = (−1)ⁿ and φ_n(L−x) = (−1)ⁿ⁺¹φ_n(x), so ψ(x,T_rev/2) = −ψ(L−x,0): a mirror image. At T_rev/4 and 3T_rev/4 a smooth packet splits into two copies shifted by L/2 (fractional revivals, the Talbot effect of optics). The picture drawn over x and t is the quantum carpet.

Uncertainty. Δx Δp ≥ ħ/2. ⟨p⟩ and ⟨p²⟩ are computed from finite differences of the grid wavefunction; for a Gaussian well inside the box, Δx Δp = ħ/2 at t = 0.

Simplifications. The truncation to 160 eigenstates smooths sharp features (a square pulse or a packet narrower than ~0.03 L shows Gibbs ripples); the walls are perfectly rigid and infinite; relativistic and many-body effects are ignored; the sketched real-time scale T_rev = 4mL²/(πħ) is exact for the model but a real nanometre well is not an infinite well. The displayed values for the "hydrogen-sized" box are for orientation only.