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{{Quantum book backlink|Quantum dynamics and evolution}}

[[File:Fullrevival.gif|thumb|right|Full and exact revival of the semi-Gaussian wave function in an infinite two-dimensional [[infinite potential well|potential well]] 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.]]
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|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 [[Physics:Trojan wave packet|well localized]] [[Physics:Wave packet|wave packet]]s 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>

[[File:ColRev3da10.tif|ColRev3a10|left|500px]]

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}}

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

Consider a quantum system with the energies <math>E_i</math> and the eigenstates <math>\psi_i</math>

:<math>H \psi_i = E_i \psi_i</math>

and let the energies be the [[Rational number|rational]] fractions of some constant <math>C</math>

:<math>E_i= C {M_i \over N_i}</math>

(for example for [[Physics:Hydrogen atom|hydrogen atom]] <math>M_i=1</math>, <math>N_i=i^2</math>, <math>C=-13.6 eV</math>.

Then the truncated (till <math>\mathbb{N}_{max}</math> of states) solution of the time dependent Schrödinger equation is

:<math>\Psi(t)=\sum_{i=0}^{\mathbb{N}_{max}}a_i e^{-i {{E_i} \over \hbar} t} \psi_i</math>

[[File:Superrevivaljc.jpg|thumb|right|250px|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 <math>n_0=100</math> is approximated by the polynomial in the photon quantum number <math>n</math> <math>E(n)=a \delta n^2 + b \delta n + c</math>, <math>\delta n = n - n_0</math>]].

Let <math>L_{cm}</math> be to lowest common multiple of all <math>N_i</math> and <math>L_{cd}</math> [[Greatest common divisor|greatest common divisor]] of all <math>M_i</math>
then for each <math>N_i</math> the <math>{L_{cm}}/ N_i</math> is an integer, for each <math>M_i</math> the <math>{M_{i}}/ L_{cd}</math> is an integer, <math>2 \pi M_i {L_{cm}}/(N_i L_{cd})</math> is the full multiple of <math>2 \pi</math> angle and

:<math>\Psi(t)=\Psi(t+T)</math>

after the full revival time time

:<math>T={2 \pi \hbar \over {L_{cd} C}} L_{cm}</math>.

For the quantum system as small as Hydrogen and <math>\mathbb{N}_{max}</math> as small as 100 it may take quadrillions of years till it will fully revive. Especially once created by fields the [[Physics:Trojan wave packet|Trojan wave packet]] in a
hydrogen atom exists without any external fields
[[Physics:Stroboscopic effect|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|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 <math>t/2 \pi=100000000</math> 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 set|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<ref>{{cite journal |first1=P. |last1=Bocchieri |first2=A. |last2=Loinger |title=Quantum Recurrence Theorem |journal=[[Physics:Physical Review|Phys. Rev.]] |volume=107 |issue=2 |pages=337–338 |year=1957 |doi=10.1103/PhysRev.107.337 |bibcode = 1957PhRv..107..337B }}</ref> 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.

==See also==
* [[Poincaré recurrence theorem]]

==References==
{{Reflist}}

[[Category:Quantum mechanics]]
[[Category:Quantum chaos theory]]

{{Sourceattribution|Quantum revival}}

Physics:Quantum revival - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template