Quantum Revivals: When a Spreading Wave Packet Reassembles Itself

Why quantization forces a comeback

A wave packet is a superposition of energy eigenstates, Ψ(x,t) = Σₙ cₙ ψₙ(x) e^(−iEₙt/ℏ). Each term rotates in phase at its own rate ωₙ = Eₙ/ℏ. Because the packet is spatially localized, it contains a spread of eigenstates centered on some large quantum number n̄, and the interference of their phases is what produces — and destroys — the localized shape.

The crucial point is that the spectrum is discrete. If the energies happened to be evenly spaced, Eₙ = n·ℏω, every phase would return to its starting value after T = 2π/ω, and the packet would be perfectly periodic — this is exactly the quantum harmonic oscillator, which never spreads. Real bound systems have anharmonic spectra: the level spacing changes with n. That is what makes the dynamics rich.

Expand the energy about the central state n̄ in the quantum number:

• Eₙ ≈ E(n̄) + E′·(n − n̄) + ½ E″·(n − n̄)² + …
• The linear term E′ = dE/dn gives the classical period, T_cl = 2πℏ / |E′| — the time for one orbit or one vibration.
• The quadratic term E″ = d²E/dn² gives the revival time, T_rev = 4πℏ / |E″|.

The linear term makes the whole packet move rigidly; the quadratic term makes the different Fourier components drift out of step (spreading), then, after all of them have completed an integer number of extra cycles, back into step. That re-phasing is the revival.

The cleanest example: a particle in a box

The infinite square well of width L is the textbook case because the arithmetic is exact. Its energies are Eₙ = n²π²ℏ² / (2mL²), so the phase of the nth term is exactly proportional to n². The full time-dependent phase is e^(−iEₙt/ℏ) = e^(−in²π²ℏt/2mL²).

A phase that is 2π times an integer for every n restores the exact initial state. Since n² is always an integer, this happens whenever π²ℏt/(2mL²) = π · (integer), i.e. at

• T_rev = 4mL² / (πℏ) — every eigenstate simultaneously returns to phase, and Ψ(x, T_rev) = Ψ(x, 0) exactly.

Plug in real numbers for an electron. For a well L = 10 nm (a modest quantum-dot scale), T_rev = 4·(9.11×10⁻³¹ kg)·(10⁻⁸ m)² / (π·1.055×10⁻³⁴ J·s) ≈ 1.1 ps. Shrink the box to L = 1 nm and the phases evolve 100× faster: T_rev ≈ 11 fs. Grow it to L = 100 nm and T_rev ≈ 110 ps. The revival time scales as L², a huge lever — the same reason macroscopic systems never visibly revive.

Because the phase is purely quadratic in n, the square well also shows perfect fractional revivals at every rational fraction of T_rev — the quantum analogue of the optical Talbot effect, where a diffraction grating's image reappears at periodic distances downstream.

Fractional revivals: the packet clones itself

The most beautiful feature lives between full revivals. At times t = (p/q)·T_rev, with p/q a reduced fraction, the sum over the quadratic phases e^(−iπ(p/q)n²) collapses — via a Gauss-sum identity — into a small number of shifted copies of the original packet. At t = T_rev/2 you get one packet on the opposite side of the well; at t = T_rev/3 you get three equally spaced mini-packets; at t = T_rev/4 you typically get four, and so on.

Each clone is a scaled-down replica of the initial state, and they interfere coherently. This is genuinely counter-intuitive: at t = T_rev/3 a single electron's probability density has three sharp peaks orbiting in lockstep, even though nothing has divided the particle. It is one wavefunction wearing three faces.

• The number of sub-packets at t = p/q·T_rev is set by q (roughly q or q/2 depending on parity).
• Fractional revivals are the experimental signature people actually chase, because they appear long before the messy full revival and have unmistakable structure.
• They were first predicted for square wells and Rydberg atoms and are the direct kin of Talbot self-imaging and the Talbot/quantum carpet patterns seen in optics.

Rydberg atoms: revivals you can watch in a lab

The 1980s–90s breakthrough came from Rydberg wave packets. A short laser pulse excites an atom into a coherent superposition of high-lying states around principal quantum number n̄ ≈ 50–90. The electron's charge cloud briefly localizes into a blob that orbits the nucleus like a classical Kepler electron.

Hydrogenic energies are Eₙ = −Ry/n² (Ry = 13.6 eV). The derivatives give the scalings directly:

• Classical Kepler period: T_cl = 2πℏ / |dE/dn| = 2πn³ (in atomic units of time, 24.19 as).
• Revival time: T_rev = 4πℏ / |d²E/dn²| = (2n/3)·T_cl.

For n̄ = 65: T_cl ≈ 42 ps and T_rev ≈ 1.8 ns — a factor of about 43 apart. So the electron orbits ~43 times, dephases into a delocalized ring, then reassembles into the compact orbiting blob at 1.8 ns. The 2n/3 ratio means bigger atoms revive relatively later. Experimentalists mapped this with pump–probe ionization: a second delayed pulse ionizes efficiently only when the packet is localized near the core, so the ion yield oscillates at T_cl, fades, and roars back at T_rev.

Molecular vibrations: revivals in tens of picoseconds

Diatomic molecules give the most reproducible revivals because their vibrational spectra are well known. A Morse-like potential yields Eᵥ ≈ ℏωₑ(v+½) − ℏωₑxₑ(v+½)², where ωₑ is the harmonic frequency and ωₑxₑ the anharmonicity. The linear term sets the vibrational period; the tiny quadratic term sets the revival.

• Vibrational period: T_vib = 1/(ωₑ·c), with ωₑ in cm⁻¹.
• Revival period: T_rev = 1/(2·ωₑxₑ·c) — inversely proportional to the anharmonicity.

For molecular iodine (I₂: ωₑ ≈ 214.5 cm⁻¹, ωₑxₑ ≈ 0.614 cm⁻¹): T_vib ≈ 156 fs but T_rev ≈ 27 ps, a ratio of ~175. For the sodium dimer Na₂ (ωₑ ≈ 159 cm⁻¹, ωₑxₑ ≈ 0.73 cm⁻¹), T_vib ≈ 210 fs and T_rev ≈ 23 ps. These were among the first revivals imaged in real time with femtosecond pump–probe spectroscopy: the fluorescence signal beats at T_vib, decays as the packet spreads over the bond, and revives on the tens-of-ps scale — with clean half- and third-revivals in between.

The lesson is universal: the weaker the anharmonicity, the longer the revival. A perfectly harmonic bond (ωₑxₑ = 0) would never spread at all — T_rev → ∞ because there is no quadratic phase term to dephase.

Scales, decoherence, and where revivals hide

Two energy scales govern everything: E′ (level spacing) sets how fast the packet moves, and E″ (change of spacing) sets how fast it spreads and revives. Their ratio, T_rev/T_cl = 2·|E′|/|E″| (up to factors), tells you how many classical cycles fit inside one revival — 43 for n=65 Rydberg, ~175 for I₂. Higher derivatives (E‴, the cubic term) introduce a super-revival time T_sr, an even longer scale on which the fractional-revival pattern itself repeats.

Why don't we see revivals in everyday life? Three reasons:

• Mass and size. T_rev ∝ mL²/ℏ. For a 1 g bead in a 1 cm box, T_rev is ~10²⁷ s — longer than the age of the universe by about 10 orders of magnitude. Revivals are real for macroscopic systems; they are just unobservably slow.
• Level density. A continuous spectrum (a free particle, an ionized electron) never revives — you need discrete, commensurable levels.
• Decoherence. This is the practical killer. Any coupling to the environment scrambles the delicate relative phases before T_rev. Rydberg and molecular revivals are visible only because they occur in picoseconds to nanoseconds — faster than collisions or radiative decay can wash out the phase coherence.

Revivals are therefore a stringent probe of coherence: seeing a clean full revival certifies that a system has stayed phase-coherent for the entire dephasing-and-reassembly cycle.

Classical spreading versus quantum revival for a wave packet in a bound spectrum

Property	Classical / short-time	Quantum revival
What sets the motion	Group velocity along the orbit	Discrete energy phases e^(−iEₙt/ℏ)
Time scale	T_cl = 2πℏ / |dE/dn|	T_rev = 4πℏ / |d²E/dn²|
Ratio (Rydberg, n=65)	T_cl ≈ 42 ps	T_rev ≈ 1.8 ns (≈ 43 T_cl)
Ratio (I₂ molecule)	T_vib ≈ 156 fs	T_rev ≈ 27 ps (≈ 175 T_vib)
Packet shape	Localized, then dephases & spreads	Reassembles to near-original
In between	Irreversible-looking spread	Fractional revivals: 2, 3, 4 copies