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{{Short description|Time evolution of quantum states and the conceptual role of time in quantum theory}}
{{Quantum book backlink|Quantum dynamics and evolution}}

The '''quantum time evolution''' of a system describes how its [[wavefunction]] or quantum state changes over time according to the laws of [[quantum mechanics]]. In standard formulations, time is treated as an external parameter that governs the evolution of physical states.

A central conceptual issue related to time in quantum theory is the '''problem of time'', which highlights a conflict between [[quantum mechanics]] and [[general relativity]]. Quantum mechanics treats time as universal and absolute, whereas relativity describes time as dependent on spacetime structure.<ref>{{Citation |last=Isham |first=C. J. |title=Canonical Quantum Gravity and the Problem of Time |date=1993 |work=Integrable Systems, Quantum Groups, and Quantum Field Theories |pages=157–287 |editor-last=Ibort |editor-first=L. A. |series=NATO ASI Series |place=Dordrecht |publisher=Springer Netherlands |language=en |doi=10.1007/978-94-011-1980-1_6 |isbn=978-94-011-1980-1 |editor2-last=Rodríguez |editor2-first=M. A. |arxiv=gr-qc/9210011 |s2cid=116947742 }}</ref><ref>{{cite magazine | title=Quantum Gravity's Time Problem | author=Wolchover, Natalie | date=December 1, 2016 | magazine=Quanta Magazine | url=https://www.quantamagazine.org/quantum-gravitys-time-problem-20161201/}}</ref>

[[File:Quantum_time_evolution.jpg|thumb|250px|right|'''Quantum time evolution of a wavefunction:''' an initial state <math>|\psi(0)\rangle</math> evolves under the unitary operator <math>U(t)=e^{-iHt/\hbar}</math> into the final state <math>|\psi(t)\rangle</math>, governed by the Schrödinger equation.]]

== Time evolution in quantum mechanics ==

In non-relativistic quantum mechanics, the time evolution of a state <math>|\psi(t)\rangle</math> is governed by the [[Schrödinger equation]]:

<math display="block">i\hbar \frac{\partial}{\partial t}\psi(t)=H\psi(t),</math>

where <math>H</math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] operator representing the total energy of the system. The formal solution is given by the unitary time-evolution operator

<math display="block">|\psi(t)\rangle=e^{-iHt/\hbar}|\psi(0)\rangle.</math>

This evolution preserves probability and provides the standard description of dynamics in quantum theory. In both [[classical mechanics]] and standard quantum mechanics, time plays a special role as an external parameter, and observables are defined at specific instants of time.<ref name=AndersonReview2012>{{Cite journal |last=Anderson |first=E. |date=2012-12-15 |title=Problem of time in quantum gravity |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.201200147 |journal=Annalen der Physik |language=en |volume=524 |issue=12 |pages=757–786 |doi=10.1002/andp.201200147 |issn=0003-3804|arxiv=1009.2157 |bibcode=2012AnP...524..757A }}</ref>{{rp|759}}

== Time in general relativity ==

In [[general theory of relativity|general relativity]], time is no longer a unique external background parameter, but part of the dynamical structure of spacetime. The field equations are formulated in terms of spacetime geometry rather than evolution with respect to a fixed time variable. These different roles of time in quantum theory and relativity are fundamentally incompatible.<ref name=AndersonReview2012/>

== Impact on quantum gravity ==

[[Quantum gravity]] attempts to reconcile or unify [[quantum mechanics]] and [[general relativity]], the current theory of gravity.<ref>{{Cite journal |last=Rovelli |first=Carlo |date=2008-05-23 |title=Quantum gravity |journal=Scholarpedia |volume=3 |issue=5 |page=7117 |language=en |doi=10.4249/scholarpedia.7117|doi-access=free |bibcode=2008SchpJ...3.7117R }}</ref> The problem of time is central to these efforts. It remains unclear how time is related to quantum probability, whether time is fundamental or emergent, and whether it is exact or approximate in a more complete theory.<ref name="Isham1993">{{Cite book |last=Isham |first=C. J. |url=https://link.springer.com/chapter/10.1007/978-94-011-1980-1_6 |title=Integrable Systems, Quantum Groups, and Quantum Field Theories |date=1993 |publisher=Springer Netherlands |isbn=978-94-011-1980-1 |editor-last=Ibort |editor-first=L. A. |location=Dordrecht |pages=157–287 |language=en |doi=10.1007/978-94-011-1980-1_6 |editor-last2=Rodríguez |editor-first2=M. A.}}</ref>

== The frozen formalism problem ==

One of the most discussed aspects of the problem of time is the frozen formalism problem. In ordinary quantum mechanics, the wavefunction evolves in time according to the Schrödinger equation. In canonical quantum gravity, however, the corresponding equation takes the form of the [[Wheeler–DeWitt equation]]:

<math display="block">\hat{H}(x)|\psi\rangle=0,</math>

where the Hamiltonian becomes a constraint. This suggests that the wavefunction of the universe is static or “frozen,” even though time-dependent behavior clearly appears at smaller scales in the physical world.<ref name=AndersonReview2012/>{{rp|762}}
== Wavepacket displacement ==
[[File:Directly detecting a geometric phase through wavepacket interference.png|thumb|From the study "Direct observation of geometric-phase interference in dynamics around a conical intersection"]]
A motional wavepacket is initially displaced to the minimum of the potential energy surface, after which it begins to encircle the conical intersection, denoted CI. b, Initial wavepacket density in 2D (left), and integrated over Q1 (right). c, After sufficient time evolution, the two components of the wavepacket destructively interfere due to geometric phase, giving a nodal line along Q2 = 0 (dotted line). d, Motional wavepacket density at the maximum interference time T . e, If the geometric phase were neglected, the two wavepacket components would interfere constructively. f, Density at t = T with geometric phase neglected. Contours in b, d, and f correspond to the potential energy surface E−. g, The Jahn-Teller Hamiltonian HJT is engineered in an ion-trap quantum simulator with a single 171Yb+ ion. The ion (white sphere) is confined in a Paul trap and HJT is realised using two simultaneous laser-induced interactions (purple and pink, corresponding to colour-coded terms in HJT)."
"The effects of geometric phase on dynamics around a conical intersection can be directly observed from the motional probability density, fig. 1a–d."

"As the initial wave-packet, we choose the ground state of the non-interacting vibrational Hamiltonian, H0 = ω(a†1a1 +a† 2a2), displaced to the potential-energy minimum at Q1 = −κ/ω, Q2 = 0 (fig. 1a–b). During the time evolution, the wavepacket splits into two components evolving in opposite directions around the conical intersection. The two components overlap at Q1 > 0, causing destructive interference at the nodal line Q2 = 0, where their equal and opposite geometric phases lead to a vanishing density (fig. 1c–d).

By contrast, if geometric phase were disregarded, the two wavepacket fragments would interfere constructively, reaching maximum amplitude at Q2 = 0 (fig. 1e–f).
== Proposed solutions ==

Several approaches have been proposed to address the problem of time. Work by [[Don Page (physicist)|Don Page]] and [[William Wootters]] suggests that time may emerge through entanglement between an evolving subsystem and an internal clock subsystem within a larger timeless universe.<ref>{{Cite journal |last1=Page |first1=Don N. |last2=Wootters |first2=William K. |date=1983-06-15 |title=Evolution without evolution: Dynamics described by stationary observables |url=https://link.aps.org/doi/10.1103/PhysRevD.27.2885 |journal=Physical Review D |volume=27 |issue=12 |pages=2885–2892 |doi=10.1103/PhysRevD.27.2885|bibcode=1983PhRvD..27.2885P |url-access=subscription }}</ref><ref>{{Cite journal |last=Rovelli |first=Carlo |date=1990-10-15 |title=Quantum mechanics without time: A model |url=https://link.aps.org/doi/10.1103/PhysRevD.42.2638 |journal=Physical Review D |volume=42 |issue=8 |pages=2638–2646 |doi=10.1103/PhysRevD.42.2638|pmid=10013133 |bibcode=1990PhRvD..42.2638R |url-access=subscription }}</ref><ref>{{Cite journal |last1=Giovannetti |first1=Vittorio |last2=Lloyd |first2=Seth |last3=Maccone |first3=Lorenzo |date=2015-08-26 |title=Quantum time |url=https://link.aps.org/doi/10.1103/PhysRevD.92.045033 |journal=Physical Review D |volume=92 |issue=4 |article-number=045033 |doi=10.1103/PhysRevD.92.045033|arxiv=1504.04215 |bibcode=2015PhRvD..92d5033G |hdl=1721.1/98287 |s2cid=85537706 }}</ref>

In 2013, Ekaterina Moreva and collaborators performed an experimental test of the Page–Wootters idea, showing for photons that time can appear for internal observers while remaining absent for external observers, in agreement with the timeless structure of the Wheeler–DeWitt equation.<ref name=Moreva2014>{{cite web |title=Quantum Experiment Shows How Time 'Emerges' from Entanglement |url=https://medium.com/the-physics-arxiv-blog/quantum-experiment-shows-how-time-emerges-from-entanglement-d5d3dc850933 |website=The Physics arXiv Blog |date=Oct 23, 2013 |archive-url=https://web.archive.org/web/20170603063357/https://medium.com/the-physics-arxiv-blog/quantum-experiment-shows-how-time-emerges-from-entanglement-d5d3dc850933 |archive-date=2017-06-03}}</ref><ref>{{cite journal |last1=Moreva |first1=Ekaterina |last2=Brida |first2=Giorgio |last3=Gramegna |first3=Marco |last4=Giovannetti |first4=Vittorio |last5=Maccone |first5=Lorenzo |last6=Genovese |first6=Marco |title=Time from quantum entanglement: An experimental illustration |journal=Physical Review A |date=20 May 2014 |volume=89 |issue=5 |article-number=052122 |doi=10.1103/PhysRevA.89.052122 |arxiv=1310.4691 |bibcode=2014PhRvA..89e2122M |s2cid=118638346 }}</ref><ref>{{cite journal |last1=Moreva |first1=Ekaterina |last2=Gramegna |first2=Marco |last3=Brida |first3=Giorgio |last4=Maccone |first4=Lorenzo |last5=Genovese |first5=Marco |title=Quantum time: Experimental multitime correlations |journal=Physical Review D |date=16 November 2017 |volume=96 |issue=5 |article-number=102005 |doi=10.1103/PhysRevD.96.102005 |arxiv=1710.00707 |bibcode=2017PhRvD..96j2005M |s2cid=119431509 }}</ref>

Another proposal is the ''thermal time hypothesis'', developed by [[Carlo Rovelli]] and [[Alain Connes]], in which time is associated with the thermodynamic or statistical state of a system rather than being fundamental.<ref>{{cite journal | last=Rovelli | first=C | title=Statistical mechanics of gravity and the thermodynamical origin of time | journal=Classical and Quantum Gravity | publisher=IOP Publishing | volume=10 | date=1993 | issue=8 | doi=10.1088/0264-9381/10/8/015 | pages=1549–1566| bibcode=1993CQGra..10.1549R }}</ref><ref>{{cite journal | last1=Connes | first1=A | last2=Rovelli | first2=C | title=Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories | journal=Classical and Quantum Gravity | publisher=IOP Publishing | volume=11 | issue=12 | date=1994-12-01 | issn=0264-9381 | doi=10.1088/0264-9381/11/12/007 | pages=2899–2917|arxiv=gr-qc/9406019| bibcode=1994CQGra..11.2899C }}</ref>

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
* [[Problem of time]]
* [[Schrödinger equation]]
* [[Quantum dynamics]]
* [[Quantum gravity]]
* [[Wheeler–DeWitt equation]]

== References ==
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Physics:Quantum Time evolution|1}}

[[Category:Quantum mechanics]]
[[Category:Quantum dynamics and evolution]]
[[Category:Quantum gravity]]
[[Category:Theoretical physics]]

Physics:Quantum Time evolution - Revision history

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