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OALib

Stochastic Dynamics of Quantum Physical Systems

Malkhaz Mumladze

Published Jan 1, 2016• 1 citation
Publisher

Abstract

In this article we define and build the one method of description of stochastic evolution of a physical quantum system.For each quantum state U E ω ∈ we construct the probability measure ω µ in the space ( ) , U P S , where U P is the space of the pure states of the quantum system, S the Borel σ -algebra in U P .Farther, for any Hermit's positive element with norm 1 u = , in the C * -algebra of observables U , we define the probability measure u µ on the set of states U E .If strongly conti- nuous group { } t α of * automorphisms on U describes the evolution of struc- ture of observables, according to this, we have a picture of evolution of distribution of states of quantum system relatively to each observable u .

DOI
10.4236/oalib.1103004

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In this article we define and build the one method of description of stochastic evolution of a physical quantum system.For each quantum state U E ω ∈ we construct the probability measure ω µ in the space ( ) , U P S , where U P is the space of the pure states of the quantum system, S the Borel σ -algebra in U P .Farther, for any Hermit's positive element with norm 1 u = , in the C * -algebra of observables U , we define the probability measure u µ on the set of states U E .If strongly conti- nuous group { } t α of * automorphisms on U describes the evolution of struc- ture of observables, according to this, we have a picture of evolution of distribution of states of quantum system relatively to each observable u .
In this article, we proposed a method for describing the evolution of quantum physical systems.We define the action integral on the functional space and the entropy of distribution of observable values on the set of quantum states.Dynamic of quantum system in this article is described as dynamical system represented by one parametric semi group which is extremal of this action integral.Evolution is a chain of change of distribution of observable values.In the closed system, it must increase entropy.Based on the notion of entropy of distribution energy, on the principle of maximum entropy production, we get a picture of evolution of closed quantum systems.
This introductory chapter sets out a general structure of physical theories which is applicable to both classical and quantum mechanics. We start with measurable properties of a given physical system, be it classical or quantum. These properties are called observables . We then introduce a definition of the state of a physical system in terms of measured values of a sufficiently large set of observables. A theory to describe the system should consists of four basic components: Basic mathematical framework This comprises a set of elements endowed with some specific mathematical structure and properties. In mathematics such a set is generally known as a space . Description of states States are described by elements of the space in the chosen mathematical framework. For this reason the space is called the state space of the system. Description of observables Observables are to be described by quantities defined on the state space. The description should yield all possible values of observables. The relationship between observables and states should be explicitly stated. The following two cases are of particular interest: 2 For a deterministic theory like classical mechanics a state should determine the values of all observables. For a probabilistic theory like quantum mechanics a state should determine the probability distribution of the values of all observables. Description of time evolution (dynamics).
For physical systems with Hamiltonians analytic in the position and momentum operators the diagonal matrix elements of the density operator in the coherent state representation may be considered as a phase-space probability measure in the stochastic formulation of quantum mechanics. The correspondence rule relating quantum-mechanical observables to ordinary functions in phase space is obtained. It is shown that the dynamics of the time-dependent phase-space probability densities is again representable in the form of a stochastic process.
From the analysis of the measurement process we make the hypothesis that we have to add to the quantum state psi a label z and a special function alpha in order to describe completely the preparation of a (pure) quantum system . Given such a terne every observable A receives a univocal value . Therefore all physical quantities of the system are well defined independently from the measuring process. Making the unknown parameter z to vary in its space we get the probabilities prescribed by quantum mechanics.We show also that every evolution of quantum states can be extended to an evolution of the ternes.
Beginning with the statistical description of the results of a duplicated experiment as a set of configurations of some observables tagether with frequencies of occurrence of each configuration it is shown that an evolution of the system by a stochastic process is obtained when the conditioning parameters of the experiment are altered.It is then shown that any such stochastic process has a representation in terms of linear operators in an abstract vector space with a state vector evolving by an isometric operator S and commuting Hermitian operators representing observables which evolve by another, unitary, operator U.This has the structure of conventional quantum theory as in the interaction, or Dirac picture, but here S is not unitary as in conventional theory.This is shown to yield a matrix of transition probabilities that is not doubly stochastic as in conventional theory and hence the Pauli master equation does not follow.All implications of this completely irreversible quantum theory have not yet been fully explored, but it points to a new viewpoint in irreversible thermod ynamics.
It is now well established that quantum tomography provides an alternative picture of quantum mechanics. It is common to introduce tomographic concepts starting with the Schrödinger–Dirac picture of quantum mechanics on Hilbert spaces. In this picture, states are a primary concept and observables are derived from them. On the other hand, the Heisenberg picture, which has evolved in the C ⋆ -algebraic approach to quantum mechanics, starts with the algebra of observables and introduces states as a derived concept. The equivalence between these two pictures amounts, essentially, to the Gelfand–Naimark–Segal construction. In this construction, the abstract C ⋆ -algebra is realized as an algebra of operators acting on a constructed Hilbert space. The representation that is defined may be reducible or irreducible, but in either case it allows us to identify a unitary group associated with the C ⋆ -algebra by means of its invertible elements. In this picture both states and observables are appropriate functions on the group; it also follows that quantum tomograms are strictly related with appropriate functions (positive-type) on the group. In this paper we present, using very simple examples, a tomographic description emerging from the set of ideas connected with the C ⋆ -algebra picture of quantum mechanics. In particular, we introduce the tomographic probability distributions for finite and compact groups, and formulate an autonomous criterion to recognize a given probability distribution as a tomogram of quantum state.