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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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 {{Short description|Quantum Collection topic on Quantum revival}}
 {{Quantum book backlink|Quantum dynamics and evolution}}
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 <div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
-during its time evolution. In between the fractional revivals occur when the scaled shape of the wave function replicates itself integer number of times over the well area.]]
+'''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.
-In [[Physics:Quantum mechanics|quantum mechanics]], the '''quantum revival'''<ref>
-{{cite journal
-|author1=J.H. Eberly |author2=N.B. Narozhny |author3=J.J. Sanchez-Mondragon  |name-list-style=amp | year        = 1980
-| title       = Periodic spontaneous collapse and revival in a simple quantum model
-| journal     = Phys. Rev. Lett.
-| volume      = 44
-| issue      = 20
-| pages       = 1323–1326
-| doi         = 10.1103/PhysRevLett.44.1323
-| bibcode=1980PhRvL..44.1323E
-}}
-</ref>
-is a periodic recurrence of the quantum wave function
-from its original form during the time evolution either many times in space as the multiple scaled fractions
-in the form of the initial wave function (fractional revival) or approximately or exactly to its original
-form from the beginning (full revival). The quantum wave function periodic in time exhibits therefore the full revival
-every period. The phenomenon of revivals is most readily observable for the wave functions being well localized wave packets at the beginning of the time evolution for example in the hydrogen atom. For Hydrogen, the fractional revivals show up
-as multiple angular Gaussian bumps around the circle drawn by the radial maximum of leading [[Physics:Hydrogen atom|circular state]] component (that with the highest amplitude in the eigenstate expansion)  of the
-original localized state  and the full revival as the original Gaussian.<ref>
-{{cite journal
-|author1=Z. Dacic Gaeta  |author2=C. R. Stroud, Jr.
- |name-list-style=amp | year        = 1990
-| title       = Classical and quantum mechanical dynamics of quasiclassical state of a hydrogen atom
-| journal     = Phys. Rev. A
-| volume      = 42
-| issue      = 11
-| pages       = 6308–6313
-| doi         = 10.1103/PhysRevA.42.6308
-|pmid=9903927
- | bibcode=1990PhRvA..42.6308G
-}}
-</ref>
-The full revivals are exact for the infinite quantum well, [[Physics:Quantum harmonic oscillator|harmonic oscillator]] or the [[Physics:Hydrogen atom|hydrogen atom]], while for shorter times are approximate
-for the hydrogen atom and a lot of quantum systems.<ref>{{cite journal |title= Nonsmooth and level-resolved dynamics illustrated with a periodically driven tight binding model |year=2014 |last1=Zhang |first1=Jiang-Min |last2=Haque |first2=Masudul |journal=Scienceopen Research |doi=10.14293/S2199-1006.1.SOR-PHYS.A2CEM4.v1 |arxiv = 1404.4280|s2cid=57487218 |doi-access=free }}</ref>
-The plot of  collapses and revivals of quantum oscillations of the JCM atomic inversion.<ref>{{cite journal|author1=A. A. Karatsuba |author2=E. A. Karatsuba | title=  A resummation formula for collapse and revival in the Jaynes–Cummings model| pages=195304, 16| journal= J. Phys. A: Math. Theor.| volume= 42| year= 2009|issue=19 | doi= 10.1088/1751-8113/42/19/195304|bibcode = 2009JPhA...42s5304K |s2cid=120269208 }}</ref>
-{{clear}}
+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.
 </div>

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

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