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Time evolution is the change of state brought about by the passage of
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...
, 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 Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
or even
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...
. In
classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
, time evolution of a collection of
rigid bodies In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
is governed by the principles of
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classica ...
. 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 Newton's laws of motion are three basic Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at re ...
. These principles can also be equivalently expressed more abstractly by
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''moment ...
or
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-L ...
. The concept of time evolution may be applicable to other stateful systems as well. For instance, the operation of a
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algo ...
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 discrete. Stateful systems often have dual descriptions in terms of states or in terms of
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phys ...
values. In such systems, time evolution can also refer to the change in observable values. This is particularly relevant in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
where the
Schrödinger picture In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which may ...
and
Heisenberg picture In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, ...
are (mostly) equivalent descriptions of time evolution.


Time evolution operators

Consider a system with state space ''X'' for which evolution is
deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
and reversible. For concreteness let us also suppose time is a parameter that ranges over the set of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s R. Then time evolution is given by a family of
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
state transformations :(\operatorname_ \colon X \rightarrow X)_. F''t'', ''s''(''x'') is the state of the system at time ''t'', whose state at time ''s'' is ''x''. The following identity holds : \operatorname_ (\operatorname_ (x)) = \operatorname_(x). To see why this is true, suppose ''x'' ∈ ''X'' is the state at time ''s''. Then by the definition of F, F''t'', ''s''(''x'') is the state of the system at time ''t'' and consequently applying the definition once more, F''u'', ''t''(F''t'', ''s''(''x'')) is the state at time ''u''. But this is also F''u'', ''s''(''x''). In some contexts in mathematical physics, the mappings F''t'', ''s'' are called ''propagation operators'' or simply
propagator In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. I ...
s. In
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classica ...
, the propagators are functions that operate on the
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...
of a physical system. In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the propagators are usually
unitary operator In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the con ...
s on a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. The propagators can be expressed as
time-ordered In theoretical physics, path-ordering is the procedure (or a meta-operator \mathcal P) that orders a product of operators according to the value of a chosen parameter: :\mathcal P \left\ \equiv O_(\sigma_) O_(\sigma_) \cdots O_(\sigma_). ...
exponentials of the integrated Hamiltonian. The asymptotic properties of time evolution are given by the
scattering matrix In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More forma ...
. A state space with a distinguished propagator is also called a
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
. To say time evolution is homogeneous means that : \operatorname_ = \operatorname_ for all u,t \in \mathbb. In the case of a homogeneous system, the mappings G''t'' = F''t'',0 form a one-parameter group of transformations of ''X'', that is : \operatorname_ = \operatorname_\operatorname_. For non-reversible systems, the propagation operators F''t'', ''s'' are defined whenever ''t'' ≥ ''s'' and satisfy the propagation identity : \operatorname_ (\operatorname_ (x)) = \operatorname_(x) for any u \geq t \geq s. In the homogeneous case the propagators are exponentials of the Hamiltonian.


In quantum mechanics

In the
Schrödinger picture In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which may ...
, the
Hamiltonian operator Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonia ...
generates the time evolution of quantum states. If \left, \psi (t) \right\rangle is the state of the system at time t, then : H \left, \psi (t) \right\rangle = i \hbar \left, \psi (t) \right\rangle. This is the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
. Given the state at some initial time (t = 0), if H is independent of time, then the
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigrou ...
time evolution operator U(t) is the exponential operator as shown in the equation : \left, \psi (t) \right\rangle = U(t)\left, \psi (0) \right\rangle = e^ \left, \psi (0) \right\rangle.


See also

*
Arrow of time The arrow of time, also called time's arrow, is the concept positing the "one-way direction" or " asymmetry" of time. It was developed in 1927 by the British astrophysicist Arthur Eddington, and is an unsolved general physics question. Thi ...
*
Time translation symmetry Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the law that the laws of physics are unchanged (i ...
*
Hamiltonian system A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...
*
Propagator In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. I ...
*
Time evolution operator 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 disc ...
*
Hamiltonian (control theory) The Hamiltonian is a function used to solve a problem of optimal control for a dynamical system. It can be understood as an instantaneous increment of the Lagrangian expression of the problem that is to be optimized over a certain time period. Insp ...


References


General references

*. *. *. *. *. {{DEFAULTSORT:Lumer, Gunter Dynamical systems fr:Opérateur d'évolution