Physics:Quantum revival - Revision history

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	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		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.		spacing gap of which the energies are arbitrary (not necessarily harmonic) multiples.

	~~==See also==~~
	* Poincaré recurrence theorem

	== See also ==		== See also ==

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	:<math>E_i= C {M_i \over N_i}</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>.		(for example for [[Physics:Quantum atoms/hydrogen\|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		Then the truncated (till <math>\mathbb{N}_{max}</math> of states) solution of the time dependent Schrödinger equation is

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	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}
	{{Quantum article nav\|previous=Physics:Quantum speed limit\|previous label=Speed limit\|next=Physics:Quantum reflection\|next label=Reflection}}		{{Quantum article nav\|previous=Physics:Quantum speed limit\|previous label=Speed limit\|next=Physics:Quantum reflection\|next label=Reflection}}

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	{{Short description\|Quantum Collection topic on Quantum revival}}		{{Short description\|Quantum Collection topic on Quantum revival}}

	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}
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	{{Short description\|Quantum Collection topic on Quantum revival}}~~Book I~~		{{Short description\|Quantum Collection topic on Quantum revival}}

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	~~~~{{Short description\|Quantum Collection topic on Quantum revival}}		{{Short description\|Quantum Collection topic on Quantum revival}}Book I

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

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	{{Quantum book backlink\|Quantum dynamics and evolution}}		{{Quantum book backlink\|Quantum dynamics and evolution}}
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-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.
-{{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.
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	==See also==		==See also==
	* [[Poincaré recurrence theorem]]		* [[Poincaré recurrence theorem]]

			== See also ==
			{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}

	==References==		==References==
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	[[Category:Quantum chaos theory]]		[[Category:Quantum chaos theory]]

	{{Sourceattribution\|Quantum revival}}		{{Author\|Harold Foppele}}

			{{Sourceattribution\|Physics:Quantum revival\|1}}

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	{{Short description\|Quantum Collection topic on Quantum revival}}		{{Short description\|Quantum Collection topic on Quantum revival}}

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	}}		}}
	</ref>		</ref>
	is a periodic recurrence of the quantum ~~[[Wave function\|~~wave function]]		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		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		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		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		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		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>		original localized state and the full revival as the original Gaussian.<ref>
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	:<math>H \psi_i = E_i \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>		and let the energies be the rational fractions of some constant <math>C</math>

	:<math>E_i= C {M_i \over N_i}</math>		:<math>E_i= C {M_i \over N_i}</math>
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	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>]].		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>		Let <math>L_{cm}</math> be to lowest common multiple of all <math>N_i</math> and <math>L_{cd}</math> 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		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

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	:<math>T={2 \pi \hbar \over {L_{cd} C}} L_{cm}</math>.		:<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		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		hydrogen atom exists without any external fields
	~~[[Physics:Stroboscopic effect\|~~stroboscopically]] and eternally repeating itself		stroboscopically and eternally repeating itself
	after sweeping almost the whole hypercube of quantum phases exactly every full revival time.		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		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		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 <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.		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.		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		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		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		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		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=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		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		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
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	==See also==		==See also==
	* [[Poincaré recurrence theorem]]		* Poincaré recurrence theorem

	== See also ==		== See also ==