Editing Time evolution

{{Short description|Change of state over time, especially in physics}}

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{{More footnotes|date=September 2013}}
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'''Time evolution''' is the change of state brought about by the passage of [[time]], applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be [[discrete time|discrete]] or even [[wiktionary:finite|finite]]. In [[classical physics]], time evolution of a collection of [[rigid body|rigid bodies]] is governed by the principles of [[classical mechanics]]. In their most rudimentary form, these principles express the relationship between forces acting on the bodies and their acceleration given by [[Newton's laws of motion]]. These principles can be equivalently expressed more abstractly by [[Hamiltonian mechanics]] or [[Lagrangian mechanics]].

The concept of time evolution may be applicable to other stateful systems as well. For instance, the operation of a [[Turing machine]] can be regarded as the time evolution of the machine's control state together with the state of the tape (or possibly multiple tapes) including the position of the machine's read-write head (or heads). In this case, time is considered to be discrete steps.

Stateful systems often have dual descriptions in terms of states or in terms of [[observable]] values. In such systems, time evolution can also refer to the change in observable values. This is particularly relevant in [[quantum mechanics]] where the [[Schrödinger picture]] and [[Heisenberg picture]] are (mostly){{Clarification needed|date=December 2023}} equivalent descriptions of time evolution.

== Time evolution operators ==
Consider a system with state space ''X'' for which evolution is [[deterministic]] and [[reversible dynamics|reversible]]. For concreteness let us also suppose time is a parameter that ranges over the set of [[real number]]s '''R'''. Then time evolution is given by a family of [[Bijection|bijective]] state transformations

:<math>(\operatorname{F}_{t, s} \colon X \rightarrow X)_{s, t \in \mathbb{R}}</math>.

F<sub>''t'', ''s''</sub>(''x'') is the state of the system at time ''t'', whose state at time ''s'' is ''x''. The following identity holds

:<math> \operatorname{F}_{u, t} (\operatorname{F}_{t, s} (x)) = \operatorname{F}_{u, s}(x). </math>

To see why this is true, suppose ''x'' ∈ ''X'' is the state at time ''s''. Then by the definition of F, F<sub>''t'', ''s''</sub>(''x'') is the state of the system at time ''t'' and consequently applying the definition once more, F<sub>''u'', ''t''</sub>(F<sub>''t'', ''s''</sub>(''x'')) is the state at time ''u''. But this is also F<sub>''u'', ''s''</sub>(''x'').

In some contexts in mathematical physics, the mappings F<sub>''t'', ''s''</sub> are called ''propagation operators'' or simply [[propagator]]s. In [[classical mechanics]], the propagators are functions that operate on the [[phase space]] of a physical system. In [[quantum mechanics]], the propagators are usually [[unitary operator]]s on a [[Hilbert space]]. The propagators can be expressed as [[time-ordered]] exponentials of the integrated Hamiltonian. The asymptotic properties of time evolution are given by the [[S-matrix|scattering matrix]].<ref>{{cite AV media | people = | date =October 2, 2006 | title =Lecture 1 {{!}} Quantum Entanglements, Part 1 (Stanford) | medium =video | url =https://www.youtube.com/watch?v=0Eeuqh9QfNI&t=4421 | access-date =September 5, 2020 | archive-url =
 | archive-date = | format = | time = | location =Stanford, CA | publisher =Stanford | via=YouTube | quote =}}</ref>

A state space with a distinguished propagator is also called a [[dynamical system]].

To say time evolution is homogeneous means that

:<math> \operatorname{F}_{u, t} = \operatorname{F}_{u - t,0}</math> for all <math>u,t \in \mathbb{R}</math>.

In the case of a homogeneous system, the mappings G<sub>''t''</sub> = F<sub>''t'',0</sub> form a one-parameter [[group (mathematics)|group]] of transformations of ''X'', that is

:<math> \operatorname{G}_{t+s} = \operatorname{G}_{t}\operatorname{G}_{s}.</math>

For non-reversible systems, the propagation operators F<sub>''t'', ''s''</sub> are defined whenever ''t'' ≥ ''s'' and satisfy the propagation identity

:<math> \operatorname{F}_{u, t} (\operatorname{F}_{t, s} (x)) = \operatorname{F}_{u, s}(x)</math> for any <math>u \geq t \geq s</math>.

In the homogeneous case the propagators are exponentials of the Hamiltonian.

=== In quantum mechanics ===

In the [[Schrödinger picture]], the [[Hamiltonian (quantum mechanics)#Schrödinger equation|Hamiltonian operator]] generates the time evolution of quantum states. If <math> \left| \psi (t) \right\rangle</math> is the state of the system at time <math>t</math>, then

:<math> H \left| \psi (t) \right\rangle = i \hbar {\partial\over\partial t} \left| \psi (t) \right\rangle.</math>

This is the [[Schrödinger equation]]. 

==== Time-independent Hamiltonian ====
If <math>H</math> is independent of time, then a state at some initial time (<math>t = 0</math>) can be expressed using the [[unitary operator|unitary]] time evolution operator <math>U(t)</math> is the [[matrix exponential|exponential operator]] as

:<math> \left| \psi (t) \right\rangle = U(t)\left| \psi (0) \right\rangle = e^{-iHt/\hbar} \left| \psi (0) \right\rangle,</math>

or more generally, for some initial time <math>t_0</math>

:<math> \left| \psi (t) \right\rangle = U(t, t_0)\left| \psi (t_0) \right\rangle = e^{-iH(t-t_0)/\hbar} \left| \psi (t_0) \right\rangle.</math><ref>{{cite book |last1=Cohen-Tannoudji |first1=Claude |last2=Diu |first2=Bernard |last3=Laloë |first3=Franck |last4=Hemley |first4=Susan Reid |last5=Ostrowsky |first5=Nicole |last6=Ostrowsky |first6=D. B. |title=Quantum mechanics |date=2020 |publisher=Wiley-VCH Verlag GmbH & Co |location=Weinheim, Germany |isbn=9783527345533 |pages=313-315 |edition=Second}}</ref>

==See also==
*[[Arrow of time]]
*[[Time translation symmetry]]
*[[Hamiltonian system]]
*[[Propagator]]
*[[Hamiltonian (quantum mechanics)#Schrödinger equation|Time evolution operator]]
*[[Hamiltonian (control theory)]]

== References ==
{{Reflist}}

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[[Category:Dynamical systems]]

[[fr:Opérateur d'évolution]]