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<title>Time evolution</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Time evolution</span></span>

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<p><b>Time evolution</b> is the change of state brought about by the passage of <a href="Time" title="Time">time</a>, applicable to systems with internal state (also called <i>stateful systems</i>). In this formulation, <i>time</i> is not required to be a continuous parameter, but may be <a href="Discrete_time" class="mw-redirect" title="Discrete time">discrete</a> or even <a href="https://en.wiktionary.org/wiki/finite" class="extiw external" title="wiktionary:finite">finite</a>. In <a href="Classical_physics" title="Classical physics">classical physics</a>, time evolution of a collection of <a href="Rigid_body" title="Rigid body">rigid bodies</a> is governed by the principles of <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>. In their most rudimentary form, these principles express the relationship between forces acting on the bodies and their acceleration given by <a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a>. These principles can be equivalently expressed more abstractly by <a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a> or <a href="Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a>.

</p><p>The concept of time evolution may be applicable to other stateful systems as well. For instance, the operation of a <a href="Turing_machine" title="Turing machine">Turing machine</a> 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.

</p><p>Stateful systems often have dual descriptions in terms of states or in terms of <a href="Observable" title="Observable">observable</a> values. In such systems, time evolution can also refer to the change in observable values. This is particularly relevant in <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> where the <a href="Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger picture</a> and <a href="Heisenberg_picture" title="Heisenberg picture">Heisenberg picture</a> are (mostly) equivalent descriptions of time evolution.

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<div class="mw-heading mw-heading2"><h2 id="Time_evolution_operators">Time evolution operators</h2></div>

<p>Consider a system with state space <i>X</i> for which evolution is <a href="Deterministic" class="mw-redirect" title="Deterministic">deterministic</a> and <a href="Reversible_dynamics" class="mw-redirect" title="Reversible dynamics">reversible</a>. For concreteness let us also suppose time is a parameter that ranges over the set of <a href="Real_number" title="Real number">real numbers</a> <b>R</b>. Then time evolution is given by a family of <a href="Bijection" title="Bijection">bijective</a> state transformations

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<annotation encoding="application/x-tex">{\displaystyle (\operatorname {F} _{t,s}\colon X\rightarrow X)_{s,t\in \mathbb {R} }}</annotation>

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</math></span><img src="./eb511889f7dd3e84483511d3f210183a0e01de3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.327ex; height:3.009ex;" alt="{\displaystyle (\operatorname {F} _{t,s}\colon X\rightarrow X)_{s,t\in \mathbb {R} }}" loading="lazy"></span>.</dd></dl>

<p>F<sub><i>t</i>, <i>s</i></sub>(<i>x</i>) is the state of the system at time <i>t</i>, whose state at time <i>s</i> is <i>x</i>. The following identity holds

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {F} _{u,t}(\operatorname {F} _{t,s}(x))=\operatorname {F} _{u,s}(x).}</annotation>

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</math></span><img src="./67c7e1deb133eda3b112eb17d974fa0803eebf9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.065ex; height:3.009ex;" alt="{\displaystyle \operatorname {F} _{u,t}(\operatorname {F} _{t,s}(x))=\operatorname {F} _{u,s}(x).}" loading="lazy"></span></dd></dl>

<p>To see why this is true, suppose <i>x</i> ∈ <i>X</i> is the state at time <i>s</i>. Then by the definition of F, F<sub><i>t</i>, <i>s</i></sub>(<i>x</i>) is the state of the system at time <i>t</i> and consequently applying the definition once more, F<sub><i>u</i>, <i>t</i></sub>(F<sub><i>t</i>, <i>s</i></sub>(<i>x</i>)) is the state at time <i>u</i>. But this is also F<sub><i>u</i>, <i>s</i></sub>(<i>x</i>).

</p><p>In some contexts in mathematical physics, the mappings F<sub><i>t</i>, <i>s</i></sub> are called <i>propagation operators</i> or simply <a href="Propagator" title="Propagator">propagators</a>. In <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>, the propagators are functions that operate on the <a href="Phase_space" title="Phase space">phase space</a> of a physical system. In <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, the propagators are usually <a href="Unitary_operator" title="Unitary operator">unitary operators</a> on a <a href="Hilbert_space" title="Hilbert space">Hilbert space</a>. The propagators can be expressed as <a href="Time-ordered" class="mw-redirect" title="Time-ordered">time-ordered</a> exponentials of the integrated Hamiltonian. The asymptotic properties of time evolution are given by the <a href="S-matrix" title="S-matrix">scattering matrix</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>A state space with a distinguished propagator is also called a <a href="Dynamical_system" title="Dynamical system">dynamical system</a>.

</p><p>To say time evolution is homogeneous means that

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {F} _{u,t}=\operatorname {F} _{u-t,0}}</annotation>

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</math></span><img src="./d71c5b6c92e564ed05e88b2b67b0322ec2f76f86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.682ex; height:2.843ex;" alt="{\displaystyle \operatorname {F} _{u,t}=\operatorname {F} _{u-t,0}}" loading="lazy"></span> for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,t\in \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle u,t\in \mathbb {R} }</annotation>

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</math></span><img src="./063c25b501cb8901242d963b39f1b0965d950ac4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.722ex; height:2.509ex;" alt="{\displaystyle u,t\in \mathbb {R} }" loading="lazy"></span>.</dd></dl>

<p>In the case of a homogeneous system, the mappings G<sub><i>t</i></sub> = F<sub><i>t</i>,0</sub> form a one-parameter <a href="Group_(mathematics)" title="Group (mathematics)">group</a> of transformations of <i>X</i>, that is

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {G} _{t+s}=\operatorname {G} _{t}\operatorname {G} _{s}.}</annotation>

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</math></span><img src="./8bc0f8ccfedcc41d3b9077e860958f020b61de23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.31ex; height:2.509ex;" alt="{\displaystyle \operatorname {G} _{t+s}=\operatorname {G} _{t}\operatorname {G} _{s}.}" loading="lazy"></span></dd></dl>

<p>For non-reversible systems, the propagation operators F<sub><i>t</i>, <i>s</i></sub> are defined whenever <i>t</i> ≥ <i>s</i> and satisfy the propagation identity

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {F} _{u,t}(\operatorname {F} _{t,s}(x))=\operatorname {F} _{u,s}(x)}</annotation>

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</math></span><img src="./a498e99b33ffc72314885b478d37e801a66f52c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.418ex; height:3.009ex;" alt="{\displaystyle \operatorname {F} _{u,t}(\operatorname {F} _{t,s}(x))=\operatorname {F} _{u,s}(x)}" loading="lazy"></span> for any <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\geq t\geq s}">

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<annotation encoding="application/x-tex">{\displaystyle u\geq t\geq s}</annotation>

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</math></span><img src="./b86df45a1e1ded1e145f4cbb2b404831a111d055.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.457ex; height:2.176ex;" alt="{\displaystyle u\geq t\geq s}" loading="lazy"></span>.</dd></dl>

<p>In the homogeneous case the propagators are exponentials of the Hamiltonian.

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<div class="mw-heading mw-heading3"><h3 id="In_quantum_mechanics">In quantum mechanics</h3></div>

<p>In the <a href="Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger picture</a>, the <a href="Hamiltonian_(quantum_mechanics)#Schrödinger_equation" title="Hamiltonian (quantum mechanics)">Hamiltonian operator</a> generates the time evolution of quantum states. If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\psi (t)\right\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \left|\psi (t)\right\rangle }</annotation>

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</math></span><img src="./4ff277e550e74f3cd25a806b3684cee56703135d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.714ex; height:2.843ex;" alt="{\displaystyle \left|\psi (t)\right\rangle }" loading="lazy"></span> is the state of the system at time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span>, then

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<annotation encoding="application/x-tex">{\displaystyle H\left|\psi (t)\right\rangle =i\hbar {\partial \over \partial t}\left|\psi (t)\right\rangle .}</annotation>

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</math></span><img src="./21292945ea9b4f3e459e65fbb7abb16f4c4f298d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.5ex; height:5.509ex;" alt="{\displaystyle H\left|\psi (t)\right\rangle =i\hbar {\partial \over \partial t}\left|\psi (t)\right\rangle .}" loading="lazy"></span></dd></dl>

<p>This is the <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>.

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<div class="mw-heading mw-heading4"><h4 id="Time-independent_Hamiltonian">Time-independent Hamiltonian</h4></div>

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<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

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</math></span><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> is independent of time, then a state at some initial time (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}">

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<annotation encoding="application/x-tex">{\displaystyle t=0}</annotation>

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</math></span><img src="./43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy"></span>) can be expressed using the <a href="Unitary_operator" title="Unitary operator">unitary</a> time evolution operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(t)}">

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<annotation encoding="application/x-tex">{\displaystyle U(t)}</annotation>

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</math></span><img src="./666c639df532e88616357c4991cabce9a57b5611.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.431ex; height:2.843ex;" alt="{\displaystyle U(t)}" loading="lazy"></span> is the <a href="Matrix_exponential" title="Matrix exponential">exponential operator</a> as

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<annotation encoding="application/x-tex">{\displaystyle \left|\psi (t)\right\rangle =U(t)\left|\psi (0)\right\rangle =e^{-iHt/\hbar }\left|\psi (0)\right\rangle ,}</annotation>

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</math></span><img src="./5444b4167daa3022f78780ea42827e40f37f8962.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.183ex; height:3.343ex;" alt="{\displaystyle \left|\psi (t)\right\rangle =U(t)\left|\psi (0)\right\rangle =e^{-iHt/\hbar }\left|\psi (0)\right\rangle ,}" loading="lazy"></span></dd></dl>

<p>or more generally, for some initial time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>

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</math></span><img src="./02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle \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 .}</annotation>

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</math></span><img src="./1fc280873236aa6684a3efe9f05375ef93fdbcb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.557ex; height:3.343ex;" alt="{\displaystyle \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 .}" loading="lazy"></span><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></dd></dl>

<ul><li><a href="Arrow_of_time" title="Arrow of time">Arrow of time</a></li>

<li><a href="Time_translation_symmetry" class="mw-redirect" title="Time translation symmetry">Time translation symmetry</a></li>

<li><a href="Hamiltonian_system" title="Hamiltonian system">Hamiltonian system</a></li>

<li><a href="Propagator" title="Propagator">Propagator</a></li>

<li><a href="Hamiltonian_(quantum_mechanics)#Schrödinger_equation" title="Hamiltonian (quantum mechanics)">Time evolution operator</a></li>

<li><a href="Hamiltonian_(control_theory)" title="Hamiltonian (control theory)">Hamiltonian (control theory)</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite class="citation audio-visual cs1"><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=0Eeuqh9QfNI&t=4421"><i>Lecture 1 | Quantum Entanglements, Part 1 (Stanford)</i></a> (video). Stanford, CA: Stanford. October 2, 2006<span class="reference-accessdate">. Retrieved <span class="nowrap">September 5,</span> 2020</span> – via YouTube.</cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFCohen-TannoudjiDiuLaloëHemley2020" class="citation book cs1">Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck; Hemley, Susan Reid; Ostrowsky, Nicole; Ostrowsky, D. B. (2020). <i>Quantum mechanics</i> (Second ed.). Weinheim, Germany: Wiley-VCH Verlag GmbH & Co. pp. <span class="nowrap">313–</span>315. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783527345533</bdi>.</cite></span>

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<div class="mw-heading mw-heading3"><h3 id="General_references">General references</h3></div>

<ul><li><cite id="CITEREFAmannArendtNeubranderNicaise2008" class="citation cs2">Amann, H.; Arendt, W.; Neubrander, F.; Nicaise, S.; von Below, J. (2008), Amann, Herbert; Arendt, Wolfgang; Hieber, Matthias; Neubrander, Frank M; Nicaise, Serge; von Below, Joachim (eds.), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=HfJKFn73ySIC"><i>Functional Analysis and Evolution Equations: The Günter Lumer Volume</i></a>, Basel: Birkhäuser, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-7643-7794-6">10.1007/978-3-7643-7794-6</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-7643-7793-9</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2402015">2402015</a></cite>.</li>

<li><cite id="CITEREFJeromePolizzi2014" class="citation cs2">Jerome, J. W.; Polizzi, E. (2014), "Discretization of time-dependent quantum systems: real-time propagation of the evolution operator", <i>Applicable Analysis</i>, <b>93</b> (12): <span class="nowrap">2574–</span>2597, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1309.3587">1309.3587</a></span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00036811.2013.878863">10.1080/00036811.2013.878863</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17905545">17905545</a></cite>.</li>

<li><cite id="CITEREFLanford1975" class="citation cs2">Lanford, O. E. (1975), "Time evolution of large classical systems", in Moser J. (ed.), <i>Dynamical Systems, Theory and Applications</i>, Lecture Notes in Physics, vol. 38, Berlin, Heidelberg: Springer, pp. <span class="nowrap">1–</span>111, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-07171-7_1">10.1007/3-540-07171-7_1</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-37505-0</bdi></cite>.</li>

<li><cite id="CITEREFLanfordLebowitz1975" class="citation cs2">Lanford, O. E.; Lebowitz, J. L. (1975), "Time evolution and ergodic properties of harmonic systems", in Moser J. (ed.), <i>Dynamical Systems, Theory and Applications</i>, Lecture Notes in Physics, vol. 38, Berlin, Heidelberg: Springer, pp. <span class="nowrap">144–</span>177, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-07171-7_3">10.1007/3-540-07171-7_3</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-37505-0</bdi></cite>.</li></ul>

<ul><li><cite id="CITEREFLumer1994" class="citation cs2"><a href="G%C3%BCnter_Lumer" title="Günter Lumer">Lumer, Günter</a> (1994), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xv4vHAAACAAJ">"Evolution equations. Solutions for irregular evolution problems via generalized solutions and generalized initial values. Applications to periodic shocks models"</a>, <i>Annales Universitatis Saraviensis</i>, Series Mathematicae, <b>5</b> (1), <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1286099">1286099</a></cite>.</li></ul></div><div><div class="zim-footer">

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