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 Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

. Quantum dynamics is relevant for burgeoning fields, such as

quantum computing A quantum computer is a computer that exploits quantum mechanical phenomena. On small scales, physical matter exhibits properties of wave-particle duality, both particles and waves, and quantum computing takes advantage of this behavior using s ...

and atomic optics. In mathematics, quantum dynamics is the study of the mathematics behind

. 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 algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann al ...

Relation to classical dynamics

Equations to describe quantum systems can be seen as equivalent to that of classical dynamics on a

macroscopic scale The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic. Overview When applied to physical phenom ...

, except for the important detail that the variables don't follow the commutative laws of multiplication. Hence, as a fundamental principle, these variables are instead described as " q-numbers", conventionally represented by operators or Hermitian matrices on a

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

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

References

{{More categories, date=October 2024 Quantum mechanics

Relation to classical dynamics

See also

References