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Time evolution is the change of state brought about by the passage oftime, applicable to systems with internal state (also calledstateful systems). In this formulation,time is not required to be a continuous parameter, but may bediscrete or evenfinite. Inclassical physics, time evolution of a collection ofrigid bodies is governed by the principles ofclassical mechanics. In their most rudimentary form, these principles express the relationship between forces acting on the bodies and their acceleration given byNewton's laws of motion. These principles can be equivalently expressed more abstractly byHamiltonian mechanics orLagrangian mechanics.

The concept of time evolution may be applicable to other stateful systems as well. For instance, the operation of aTuring 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 ofobservable values. In such systems, time evolution can also refer to the change in observable values. This is particularly relevant inquantum mechanics where theSchrödinger picture andHeisenberg picture are (mostly)^{[clarification needed]} equivalent descriptions of time evolution.

Consider a system with state spaceX for which evolution isdeterministic andreversible. For concreteness let us also suppose time is a parameter that ranges over the set ofreal numbersR. Then time evolution is given by a family ofbijective state transformations

${\displaystyle ({\mathrm{F}}_{t,s}:X\to X{)}_{s,t\in \mathbb{R}}}$.

F_t,s(x) is the state of the system at timet, whose state at times isx. The following identity holds

${\displaystyle {\mathrm{F}}_{u,t}({\mathrm{F}}_{t,s}(x))={\mathrm{F}}_{u,s}(x).}$

To see why this is true, supposex ∈X is the state at times. Then by the definition of F, F_t,s(x) is the state of the system at timet and consequently applying the definition once more, F_u,t(F_t,s(x)) is the state at timeu. But this is also F_u,s(x).

In some contexts in mathematical physics, the mappings F_t,s are calledpropagation operators or simplypropagators. Inclassical mechanics, the propagators are functions that operate on thephase space of a physical system. Inquantum mechanics, the propagators are usuallyunitary operators on aHilbert space. The propagators can be expressed astime-ordered exponentials of the integrated Hamiltonian. The asymptotic properties of time evolution are given by thescattering matrix.^[1]

A state space with a distinguished propagator is also called adynamical system.

To say time evolution is homogeneous means that

${\displaystyle {\mathrm{F}}_{u,t}={\mathrm{F}}_{u-t,0}}$ for all${\displaystyle u,t\in \mathbb{R}}$.

In the case of a homogeneous system, the mappings G_t = F_t,0 form a one-parametergroup of transformations ofX, that is

${\displaystyle {\mathrm{G}}_{t+s}={\mathrm{G}}_t{\mathrm{G}}_s.}$

For non-reversible systems, the propagation operators F_t,s are defined whenevert ≥s and satisfy the propagation identity

${\displaystyle {\mathrm{F}}_{u,t}({\mathrm{F}}_{t,s}(x))={\mathrm{F}}_{u,s}(x)}$ for any${\displaystyle u\ge t\ge s}$.

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

In theSchrödinger picture, theHamiltonian operator generates the time evolution of quantum states. If${\displaystyle |\psi (t)⟩}$ is the state of the system at time${\displaystyle t}$, then

${\displaystyle H|\psi (t)⟩=i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}|\psi (t)⟩.}$

This is theSchrödinger equation.

If${\displaystyle H}$ is independent of time, then a state at some initial time (${\displaystyle t=0}$) can be expressed using theunitary time evolution operator${\displaystyle U(t)}$ is theexponential operator as

${\displaystyle |\psi (t)⟩=U(t)|\psi (0)⟩=e^{-iHt/\hslash }|\psi (0)⟩,}$

or more generally, for some initial time${\displaystyle t_0}$

${\displaystyle |\psi (t)⟩=U(t,t_0)|\psi (t_0)⟩=e^{-iH(t-t_0)/\hslash }|\psi (t_0)⟩.}$^[2]

• Arrow of time
• Time translation symmetry
• Hamiltonian system
• Propagator
• Time evolution operator
• Hamiltonian (control theory)

• 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.),Functional Analysis and Evolution Equations: The Günter Lumer Volume, Basel: Birkhäuser,doi:10.1007/978-3-7643-7794-6,ISBN 978-3-7643-7793-9,MR 2402015.
• Jerome, J. W.; Polizzi, E. (2014), "Discretization of time-dependent quantum systems: real-time propagation of the evolution operator",Applicable Analysis,93 (12):2574–2597,arXiv:1309.3587,doi:10.1080/00036811.2013.878863,S2CID 17905545.
• Lanford, O. E. (1975), "Time evolution of large classical systems", in Moser J. (ed.),Dynamical Systems, Theory and Applications, Lecture Notes in Physics, vol. 38, Berlin, Heidelberg: Springer, pp. 1–111,doi:10.1007/3-540-07171-7_1,ISBN 978-3-540-37505-0.
• Lanford, O. E.; Lebowitz, J. L. (1975), "Time evolution and ergodic properties of harmonic systems", in Moser J. (ed.),Dynamical Systems, Theory and Applications, Lecture Notes in Physics, vol. 38, Berlin, Heidelberg: Springer, pp. 144–177,doi:10.1007/3-540-07171-7_3,ISBN 978-3-540-37505-0.

• Lumer, Günter (1994),"Evolution equations. Solutions for irregular evolution problems via generalized solutions and generalized initial values. Applications to periodic shocks models",Annales Universitatis Saraviensis, Series Mathematicae,5 (1),MR 1286099.

• Dynamical systems

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