# 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]