Revision as of 23:34, 23 May 2026

← Back to Quantum dynamics and evolution Book I

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 quantum mechanics.^[1]^[2] Quantum dynamics is relevant for burgeoning fields, such as quantum computing and atomic optics.

In mathematics, quantum dynamics is the study of the mathematics behind quantum mechanics.^[3] 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, Hahn and Hellinger worked in the field. Mathematicians in the field have also studied irreversible quantum mechanical systems on von Neumann algebras.^[4]

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 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.^[5]

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 $| ψ (t) ⟩$ . The equation is given by:

$i ℏ \frac{\partial}{\partial t} | ψ (t) ⟩ = \hat{H} | ψ (t) ⟩$

Here, $i$ is the imaginary unit, $ℏ$ is the reduced Planck constant, $| ψ (t) ⟩$ is the state of the system at time $t$ , and $\hat{H}$ 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 (or density operator), denoted $ρ$ , 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:

$i ℏ \frac{d}{d t} ρ (t) = [\hat{H}, ρ (t)]$

where $[\hat{H}, ρ (t)] = \hat{H} ρ (t) - ρ (t) \hat{H}$ 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,^[6] 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 open quantum system. This interaction leads to a non-unitary evolution, where information and energy can be exchanged with the environment. ^[7]

This exchange causes uniquely quantum phenomena to decay, a process known as decoherence, where the clean superposition of states degrades into a classical mixture. It also leads to dissipation, where the system loses energy to its environment. ^[7]

The dynamics of open quantum systems are typically modeled using quantum master equations. The most general form for a system whose environment has no memory (a Markovian system) is the Lindblad equation, also known as the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation:^[8]

$\frac{d}{d t} ρ (t) = - \frac{i}{ℏ} [\hat{H}, ρ] + \sum_{k} (L_{k} ρ L_{k}^{†} - \frac{1}{2} {L_{k}^{†} L_{k}, ρ})$

In this equation:

The first term, $- \frac{i}{ℏ} [\hat{H}, ρ]$ , 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.^[9]
The operators $L_{k}$ are known as Lindblad operators or quantum jump operators.^[10] They model the specific ways the system is coupled to the bath (e.g., through photon emission or thermal noise). The ${A, B} = A B + B A$ 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 correspondence principle.^[5]

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

$[\hat{X}, \hat{P}] = \hat{X} \hat{P} - \hat{P} \hat{X} = i ℏ$

This non-commutativity is the source of the Heisenberg uncertainty principle and fundamentally alters the dynamics of a system,^[6] 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, $[\hat{A}, \hat{B}] \leftrightarrow i ℏ {A, B}$ , was a key insight in the development of quantum mechanics, first noted by Paul Dirac.^[11]

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 $\hat{H}$ 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.^[5]

@@ Line 29: / Line 29: @@
 | title = Quantum dynamics with trajectories
 | year = 2005
-}}</ref> Quantum dynamics is relevant for burgeoning fields, such as [[Quantum computing|quantum computing]] and atomic optics.
+}}</ref> Quantum dynamics is relevant for burgeoning fields, such as quantum computing and atomic optics.
 In mathematics, '''quantum dynamics''' is the study of the mathematics behind [[Physics:Quantum mechanics|quantum mechanics]].<ref>
@@ Line 36: / Line 36: @@
 <div style="width:300px;">
-[[File:File not found.png|thumb|280px|No image available.]]
+[[File:Symbol_list_class.svg|thumb|280px|dynamics in the Quantum Collection.]]
 </div>
@@ Line 45: / Line 45: @@
 === 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>
+A closed quantum system is one that is perfectly isolated from any external influence. The time evolution of such a system is described as 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:
@@ Line 53: / Line 53: @@
 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:
+The Schrödinger equation is powerful but applies only to '''pure states'''. A more general description of a quantum system is the '''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>
@@ Line 60: / Line 60: @@
 === 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>
+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 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" />
+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''', 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>
+The dynamics of open quantum systems are typically modeled using quantum master equations. The most general form for a system whose environment has no memory (a Markovian system) is the 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>
@@ Line 77: / Line 77: @@
 == 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" />
+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 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:
+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 (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>
@@ Line 92: / Line 92: @@
 == References ==
 <references/>
-{{Quantum mechanics topics}}
 {{emerging technologies|quantum=yes|other=yes}}{{More categories|date=October 2024}}
 [[Category:Quantum mechanics]]
 {{Sourceattribution|Quantum dynamics}}

Anonymous

Search

Physics:Quantum dynamics: Difference between revisions

Revision as of 23:34, 23 May 2026

Fundamental Models of Time Evolution

Closed Quantum Systems

Open Quantum Systems

Relation to classical dynamics

See also

Table of contents (217 articles)

Index

Full contents

References

Navigation

Wiki tools

Page tools

Categories