Quantum Wavepacket Revivals

Trap a quantum particle in a box, give it a localized wavepacket, and let it go. At first it behaves almost classically, sloshing back and forth and bouncing off the walls, but the packet soon smears out into an apparently structureless mess. Wait longer and something remarkable happens: the chaos is only apparent, and at sharp instants the packet reassembles into near-perfect copies of itself, sometimes one, sometimes several side by side, before dissolving again. This is because the energies of the box are exactly $E_n = n^2 E_1$, so every phase $e^{-iE_n t/\hbar}$ returns to one simultaneously at the revival time $T_\mathrm{rev}=2\pi\hbar/E_1$, and at simple fractions of it the sum rearranges into a handful of scaled-down replicas, the fractional revivals. Plotting the probability density $|\psi(x,t)|^2$ as an image over position (across) and time (down) turns this into a single picture, the quantum carpet, woven through with diagonal canals and a lattice of revival points that no single snapshot reveals. The strip at the top is the live density at the current instant, a horizontal slice of the carpet, and the yellow line marks where now sits. The lower panel is the survival probability, the overlap of the evolving state with its starting shape, which spikes to one at the full revival and to lower peaks at the fractions $1/2$, $1/3$, $1/4$ marked beneath it.