Quantum dynamics

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. Recently, mathematicians in the field have studied irreversible quantum mechanical systems on von Neumann algebras.[4]

Relation to classical dynamics

Equations to describe quantum systems can be seen as equivalent to that of classical dynamics on a macroscopic scale, except for the important detail that the variables don't follow the commutative laws of multiplication.[5] Hence, as a fundamental principle, these variables are instead described as "q-numbers", conventionally represented by operators or Hermitian matrices on a Hilbert space.[6] Indeed, the state of the system in the atomic and subatomic scale is described not by dynamic variables with specific numerical values, but by state functions that are dependent on the c-number time. In this realm of quantum systems, the equation of motion governing dynamics heavily relies on the Hamiltonian, also known as the total energy. Therefore, to anticipate the time evolution of the system, one only needs to determine the initial condition of the state function |Ψ(t) and its first derivative with respect to time.[7]

For example, quasi-free states and automorphisms are the Fermionic counterparts of classical Gaussian measures[8] (Fermions' descriptors are Grassmann operators).[6]

See also

References
  1. Joan Vaccaro (2008-06-26). "Centre for Quantum Dynamics, Griffith University". Quantiki. Archived from the original on 2009-10-25. Retrieved 2010-01-25.
  2. Wyatt, Robert Eugene; Corey J. Trahan (2005). Quantum dynamics with trajectories. Springer. ISBN 9780387229645.
  3. Teufel, Stefan (1821-01-01). Adiabatic perturbation theory in quantum dynamics. Springer. ISBN 9783540407232.
  4. Price, Geoffrey (2003). 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. Providence, R.I: American Mathematical Society. ISBN 0-8218-3215-8. OCLC 52901091.
  5. "The physical interpretation of the quantum dynamics". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 113 (765): 621–641. 1927. doi:10.1098/rspa.1927.0012. ISSN 0950-1207.
  6. Kuypers, Samuel (2022). "The quantum theory of time: a calculus for q-numbers". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 478 (2263). doi:10.1098/rspa.2021.0970. ISSN 1364-5021. PMC 9326976. PMID 35909420.
  7. Tang, Chung Liang (2005). Fundamentals of quantum mechanics: for solid state electronics and optics. Cambridge: Cambridge Univ. Press. ISBN 978-0-521-82952-6.
  8. Alicki, Robert; Fannes, Mark (2001). Quantum dynamical systems (1. publ ed.). Oxford: Oxford University Press. pp. 103–121. ISBN 978-0-19-850400-9.
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