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{{Short description|Study of quantum systems changing with time}}
In physics, '''quantum dynamics''' is the quantum version of classical dynamics. Quantum dynamics deals with the motions, and energy and momentum exchanges of systems whose behavior is governed by the laws of [[Physics:Quantum mechanics|quantum mechanics]].<ref>{{Cite web
|url=http://www.quantiki.org/content/centre-quantum-dynamics-griffith-university
|title=Centre for Quantum Dynamics, Griffith University
|author=Joan Vaccaro
|date=2008-06-26
|work=Quantiki
|access-date=2010-01-25
|archive-url=https://web.archive.org/web/20091025092158/http://www.quantiki.org/content/centre-quantum-dynamics-griffith-university
|archive-date=2009-10-25
|url-status=dead
}}</ref><ref>
{{Cite book
| publisher = Springer
| isbn = 9780387229645
| last = Wyatt
| first = Robert Eugene
|author2=Corey J. Trahan
| title = Quantum dynamics with trajectories
| year = 2005
}}</ref> Quantum dynamics is relevant for burgeoning fields, such as [[Quantum computing|quantum computing]] and atomic optics.

In mathematics, '''quantum dynamics''' is the study of the mathematics behind [[Physics:Quantum mechanics|quantum mechanics]].<ref>
{{Cite book |last=Teufel |first=Stefan |title=Adiabatic perturbation theory in quantum dynamics |date=September 5, 2003 |publisher=Springer |isbn=9783540407232}}</ref> Specifically, as a study of ''dynamics'', this field investigates how quantum mechanical observables change over time. Most fundamentally, this involves the study of one-parameter automorphisms of the algebra of all bounded operators on the Hilbert space of observables (which are self-adjoint operators). These dynamics were understood as early as the 1930s, after Wigner, Stone, [[Biography:Erwin Hahn|Hahn]] and Hellinger worked in the field. Mathematicians in the field have also studied irreversible quantum mechanical systems on [[Von Neumann algebra|von Neumann algebra]]s.<ref name="Price 2003 p. ">{{cite book | last=Price | first=Geoffrey | title=Advances in quantum dynamics : proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Advances in Quantum Dynamics, June 16-20, 2002, Mount Holyoke College, South Hadley, Massachusetts | publisher=American Mathematical Society | publication-place=Providence, R.I | year=2003 | isbn=0-8218-3215-8 | oclc=52901091 | page=}}</ref>

== Fundamental Models of Time Evolution ==
The dynamics of a quantum system are governed by a specific equation of motion that depends on whether the system is considered '''closed''' (isolated from its environment) or '''open''' (coupled to an environment).

=== Closed Quantum Systems ===
A closed quantum system is one that is perfectly isolated from any external influence. The time evolution of such a system is described as [[Physics:Unitarity|unitary]], which means that the total probability is conserved and the process is, in principle, reversible. The dynamics of closed systems are described by two equivalent, fundamental equations.<ref name=":0">{{Cite book |last1=Griffiths |first1=David J. |url=https://doi.org/10.1017/9781316995433 |title=Introduction to Quantum Mechanics |last2=Schroeter |first2=Darrell F. |date=2018-08-16 |publisher=Cambridge University Press |doi=10.1017/9781316995433 |bibcode=2018iqm..book.....G |isbn=978-1-316-99543-3}}</ref>

The most common formulation of quantum dynamics is the time-dependent Schrödinger equation. It describes the evolution of the system's state vector, denoted as a ket <math>|\psi(t)\rangle</math>. The equation is given by:

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

Here, <math>i</math> is the imaginary unit, <math>\hbar</math> is the reduced Planck constant, <math>|\psi(t)\rangle</math> is the state of the system at time <math>t</math>, and <math>\hat{H}</math> is the Hamiltonian operator—the observable corresponding to the total energy of the system.

The Schrödinger equation is powerful but applies only to '''pure states'''. A more general description of a quantum system is the '''[[Density matrix|density matrix]]''' (or density operator), denoted <math>\rho</math>, which can represent both pure states and '''mixed states''' (statistical ensembles of quantum states). The time evolution of the density matrix is governed by the Liouville-von Neumann equation:

<math display="block">i\hbar\frac{d}{dt}\rho(t) = [\hat{H}, \rho(t)]</math>

where <math>[\hat{H}, \rho(t)] = \hat{H}\rho(t) - \rho(t)\hat{H}</math> is the commutator of the Hamiltonian with the density matrix. This equation is the quantum mechanical analogue of the classical Liouville's theorem. For a closed system, the Von Neumann equation is entirely equivalent to the Schrödinger equation,<ref name=":1">{{Cite book |last1=Sakurai |first1=J. J. |last2=Napolitano |first2=Jim |date=2020-09-17 |title=Modern Quantum Mechanics |url=https://www.cambridge.org/highereducation/books/modern-quantum-mechanics/DF43277E8AEDF83CC12EA62887C277DC |access-date=2025-08-27 |website=Cambridge Aspire website |language=en |doi=10.1017/9781108587280 |bibcode=2020mqm..book.....S |isbn=978-1-108-58728-0 }}</ref> but its framework is essential for understanding the dynamics of open systems.

=== Open Quantum Systems ===
In practice, no quantum system is perfectly isolated from its environment. A system that interacts with its surroundings (often called a "bath") is known as an [[Physics:Open quantum system|open quantum system]]. This interaction leads to a '''non-unitary''' evolution, where information and energy can be exchanged with the environment. <ref name=":2">{{Cite book |last1=Breuer |first1=Heinz-Peter |title=The theory of open quantum systems |last2=Petruccione |first2=Francesco |date=2009 |publisher=Clarendon Press |isbn=978-0-19-852063-4 |edition=1. publ. in paperback, [Nachdr.] |location=Oxford}}</ref>

This exchange causes uniquely quantum phenomena to decay, a process known as '''[[Physics:Quantum decoherence|decoherence]]''', where the clean superposition of states degrades into a classical mixture. It also leads to '''[[Dissipation|dissipation]]''', where the system loses energy to its environment. <ref name=":2" />

The dynamics of open quantum systems are typically modeled using [[Quantum master equation|quantum master equations]]. The most general form for a system whose environment has no memory (a Markovian system) is the [[Physics:Lindbladian|Lindblad equation]], also known as the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation:<ref>{{cite journal |last1=Chruściński |first1=Dariusz |title=A Brief History of the GKLS Equation |date=2017-11-04 |arxiv=1710.05993 |last2=Pascazio |first2=Saverio |journal=Open Systems & Information Dynamics |volume=24 |issue=3 |doi=10.1142/S1230161217400017 |bibcode=2017OSID...2440001C }}</ref>

<math display="block">\frac{d}{dt}\rho(t) = - \frac{i}{\hbar}[\hat{H}, \rho] + \sum_k \left( L_k \rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k, \rho\} \right)</math>

In this equation:

* The first term, <math>- \frac{i}{\hbar}[\hat{H}, \rho]</math>, describes the ordinary unitary evolution of the system, identical to the Von Neumann equation.
* The second term, often called the "dissipator" or "Lindbladian", describes the irreversible, non-unitary dynamics due to the environment.<ref>{{Cite book |last1=Nielsen |first1=Michael A. |url=https://doi.org/10.1017/cbo9780511976667 |title=Quantum Computation and Quantum Information |last2=Chuang |first2=Isaac L. |date=2012-06-05 |publisher=Cambridge University Press |doi=10.1017/cbo9780511976667 |isbn=978-1-107-00217-3}}</ref>
* The operators <math>L_k</math> are known as '''Lindblad operators''' or '''[[Physics:Quantum jump method|quantum jump]] operators'''.<ref>{{cite journal |last1=Plenio |first1=M. B. |title=The quantum-jump approach to dissipative dynamics in quantum optics |date=1997-02-01 |last2=Knight |first2=P. L. |journal=Reviews of Modern Physics |volume=70 |pages=101–144 |doi=10.1103/RevModPhys.70.101 |arxiv=quant-ph/9702007 }}</ref> They model the specific ways the system is coupled to the bath (e.g., through photon emission or thermal noise). The <math>\{A, B\} = AB + BA</math> is the anticommutator.

The study of open quantum systems is critical for understanding the quantum-to-classical transition and is essential for technologies like quantum computing, where decoherence is a primary engineering challenge.

== Relation to classical dynamics ==
While quantum dynamics is fundamentally different from classical dynamics, it is also a generalization of it. The principles of classical mechanics emerge from quantum mechanics as an approximation in the macroscopic limit, a concept known as the [[Physics:Correspondence principle|correspondence principle]].<ref name=":0" />

The primary departure from classical physics lies in the nature of physical variables. In classical dynamics, variables like position (<math>x</math>) and momentum (<math>p</math>) are simple numbers ([[Physics:C-number|c-number]]). In quantum dynamics, they are represented by operators ([[Physics:Quantum number|q-numbers]]) which, crucially, '''do not necessarily commute'''. For instance, the position operator <math>\hat{X}</math> and the momentum operator <math>\hat{P}</math> are related by the canonical commutation relation:

<math display="block">[\hat{X}, \hat{P}] = \hat{X}\hat{P} - \hat{P}\hat{X} = i\hbar</math>

This non-commutativity is the source of the Heisenberg uncertainty principle and fundamentally alters the dynamics of a system,<ref name=":1" /> making it impossible to simultaneously know the precise position and momentum of a particle. The relationship between the quantum commutator and the classical Poisson bracket, <math>[\hat{A}, \hat{B}] \leftrightarrow i\hbar\{A, B\}</math>, was a key insight in the development of quantum mechanics, first noted by Paul Dirac.<ref>{{Cite book |last=Dirac |first=P. A. M. |title=The principles of quantum mechanics |date=2010 |publisher=Clarendon Press, Oxford University Press |isbn=978-0-19-852011-5 |edition=4. ed. (rev.), repr |series=International series of monographs on physics |location=Oxford}}</ref>

Despite this difference, the role of the '''Hamiltonian''' remains central in both frameworks. Just as the classical Hamiltonian generates the time evolution of a system through Hamilton's equations, the quantum Hamiltonian operator <math>\hat{H}</math> dictates the evolution of the quantum state through the Schrödinger equation. For systems with large quantum numbers (i.e., on a macroscopic scale), the quantum evolution described by the Schrödinger equation will average out to produce the trajectory predicted by Newton's laws.<ref name=":0" />

== See also ==
*Quantum Field Theory
*[[Physics:Perturbation theory|Perturbation theory]]
*Semigroups
*Pseudodifferential operators
*[[Brownian motion]]
*Dilation theory
*[[Physics:Quantum probability|Quantum probability]]
*[[Free probability]]

== References ==
<references/>

{{Quantum mechanics topics}}
{{emerging technologies|quantum=yes|other=yes}}{{More categories|date=October 2024}}
[[Category:Quantum mechanics]]

{{Sourceattribution|Quantum dynamics}}

Physics:Quantum dynamics - Revision history

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