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Quantum revival is the reappearance of an initially localized quantum wave packet after it has spread and evolved for a characteristic time. The effect occurs because the phases of the energy eigenstate components can realign, causing the probability distribution to resemble its earlier form.

Revivals are especially clear in systems with discrete, nonlinear energy spectra, such as particles in idealized wells, Rydberg atoms, molecular wave packets, and other bound quantum systems. Fractional revivals can occur at intermediate times, when the wave packet splits into several smaller copies. The phenomenon links wave-packet dynamics, interference, semiclassical motion, and the spectral structure of quantum systems.

Example - arbitrary truncated wave function of the quantum system with rational energies

Consider a quantum system with the energies $E_{i}$ and the eigenstates $ψ_{i}$

H ψ_{i} = E_{i} ψ_{i}

and let the energies be the rational fractions of some constant $C$

E_{i} = C \frac{M_{i}}{N_{i}}

(for example for hydrogen atom $M_{i} = 1$ , $N_{i} = i^{2}$ , $C = - 13.6 e V$ .

Then the truncated (till $ℕ_{m a x}$ of states) solution of the time dependent Schrödinger equation is

Ψ (t) = \sum_{i = 0}^{ℕ_{m a x}} a_{i} e^{- i \frac{E_{i}}{ℏ} t} ψ_{i}

Superrevival of the inversion (return of the full approximate revivals to the original shape) in Jaynes-Cummings model when the exact spectrum in resonance around the average number of photons $n_{0} = 100$ is approximated by the polynomial in the photon quantum number $n$ $E (n) = a δ n^{2} + b δ n + c$ , $δ n = n - n_{0}$

.

Let $L_{c m}$ be to lowest common multiple of all $N_{i}$ and $L_{c d}$ greatest common divisor of all $M_{i}$ then for each $N_{i}$ the $L_{c m} / N_{i}$ is an integer, for each $M_{i}$ the $M_{i} / L_{c d}$ is an integer, $2 π M_{i} L_{c m} / (N_{i} L_{c d})$ is the full multiple of $2 π$ angle and

Ψ (t) = Ψ (t + T)

after the full revival time time

T = \frac{2 π ℏ}{L_{c d} C} L_{c m}

.

For the quantum system as small as Hydrogen and $ℕ_{m a x}$ as small as 100 it may take quadrillions of years till it will fully revive. Especially once created by fields the Trojan wave packet in a hydrogen atom exists without any external fields stroboscopically and eternally repeating itself after sweeping almost the whole hypercube of quantum phases exactly every full revival time.

The striking consequence is that no finite-bit computer can propagate the numerical wave function accurately for the arbitrarily long time. If the processor number is n-bit long floating point number then the number can be stored by the computer only with the finite accuracy after the comma and the energy is (up to 8 digits after the comma) for example 2.34576893 = 234576893/100000000 and as the finite fraction it is exactly rational and the full revival occurs for any wave function of any quantum system after the time $t / 2 π = 100000000$ which is its maximum exponent and so on that may not be true for all quantum systems or all stationary quantum systems undergo the full and exact revival numerically.

In the system with the rational energies i.e. where the quantum exact full revival exists its existence immediately proves the quantum Poincaré recurrence theorem and the time of the full quantum revival equals to the Poincaré recurrence time. While the rational numbers are dense in real numbers and the arbitrary function of the quantum number can be approximated arbitrarily exactly with Padé approximants with the coefficients of arbitrary decimal precision for the arbitrarily long time each quantum system therefore revives almost exactly. It also means that the Poincaré recurrence and the full revival is mathematically the same thing^[1] and it is commonly accepted that the recurrence is called the full revival if it occurs after the reasonable and physically measurable time that is possible to be detected by the realistic apparatus and this happens due to a very special energy spectrum having a large basic energy spacing gap of which the energies are arbitrary (not necessarily harmonic) multiples.

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Physics:Quantum revival

Example - arbitrary truncated wave function of the quantum system with rational energies

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