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2013-12-20T15:02:42Z

DBNs

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[[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 [[quantum mechanics]], the '''quantum revival'''
<ref>
{{cite journal
| author = J.H. Eberly, N.B. Narozhny, and J.J. Sanchez-Mondragon
| 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
| url =
| 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 (physics)|period]]. The phenomenon of revivals is most readily observable for the wave functions being [[Trojan_wave_packet|well localized]] [[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 Gaussian bumps around the circle and the full revival as the original Gaussian
.<ref>
{{cite journal
| author = Z. Dacic Gaeta and C. R. Stroud, Jr.
| 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
| bibcode=1990PhRvA..42.6308G
}}
</ref>
The full revivals are exact for the infinite [[Infinite quantum well|quantum well]], [[quantum harmonic oscillator|harmonic oscillator]] or the [[hydrogen atom]], while for shorter times are approximate
for hydrogen atom and a lot of quantum systems.

==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 [[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>

Let <math>L_{cm}</math> be to [[lowest common multiple]] of all <math>N_i</math>
then for each <math>N_i</math> the <math>{L_{cm}}/ N_i</math> is an integer, <math>2 \pi M_i {L_{cm}}/N_i</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 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 [[Trojan wave packet]] in a
hydrogen atom exists without any external fields
[[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]] long [[floating point]] number then the energy is for example 2.34576893 = 234576893/100000000 and
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.

Despite of the [[General Relativity|general theory of the relativity]] and the [[Shape of the universe|cosmological models]] if there is a [[universal wavefunction]]
of the [[universe]] quantum theory predicts therefore the universe evolution as the repetitive and infinite appearance
of the [[big bang]] and [[big crunch]] ([[Big bounce]] and [[cyclic model]]) and so on as the global Poincaré recurrence.

==References==
{{Reflist}}

[[Category:Quantum mechanics]]

Restricted Boltzmann machine - Revision history

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