EXACTLY SOLVABLE MODELS OF STOCH- ASTIC QUANTUM MECHANICS WITHIN THE FRAMEWORK OF LANGEVIN- SCHROEDINGER TYPE EQUATIONS

Parujr Sevak

Published Jan 1, 2004• 0 citations

Abstract

A closed, uncountably infinite dimensional system of object + is mathematically stated in terms of a complex probabilistic process for the wave function, i.e. the solution of Langevin-Schrodinger (L-Sch)-type stochastic differential equation (SDE). The L-Sch SDE has been studied for nonstationary 1D random potential with quadratic space form. It was proved that when the random processes in poten- tial determined by δ-shaped correlators, then closed, exactly solvable nonrelativistic quantum mechanics may be constructed. In this case all physically measured parameters of the system are built as multiple integrals of the fundamental solution of some second-order partial dif- ferential equations (etalon equations). In the present work we obtain expressions for transition probabilities in a quantum subsystem. It is shown that depending on the coupling constant of the thermostat with the parametric quantum harmonic oscillator (PQHO), some phase tran- sitions of second kind may occur in the expressions for microscopic quan- tum transitions. A method of stochastic density matrix is developed for calculation of thermodynamic potentials of the quantum harmonic oscil- lator (QHO) immersed into a thermostat. The analytic expressions for the ground state energy level widening and shift of QHO are obtained. In other words the possibility of violating the second law of thermody- namics due to quantum fluctuations, (i.e., spontaneous transitions in QHO from vacuum to the excited states) is shown.

Related papers

Exactly Solvable Models of Stochastic Quantum Mechanics within the Framework of Langevin-Schroedinger Type Equations

Ashot S. Gevorkyan

A closed, uncountably infinite dimensional system of “quantum object + thermostat” is mathematically stated in terms of a complex probabilistic process for the wave function, i.e. the solution of Langevin-Schrödinger (L-Sch)-type stochastic differential equation (SDE). The L-Sch SDE has been studied for nonstationary 1D random potential with quadratic space form. It was proved that when the random processes in potential determined by δ-shaped correlators, then closed, exactly solvable nonrelativistic quantum mechanics may be constructed. In this case all physically measured parameters of the system are built as multiple integrals of the fundamental solution of some second-order partial differential equations (etalon equations). In the present work we obtain expressions for transition probabilities in a quantum subsystem. It is shown that depending on the coupling constant of the thermostat with the parametric quantum harmonic oscillator (PQHO), some phase transitions of second kind may occur in the expressions for microscopic quantum transitions. A method of stochastic density matrix is developed for calculation of thermodynamic potentials of the quantum harmonic oscillator (QHO) immersed into a thermostat. The analytic expressions for the ground state energy level widening and shift of QHO are obtained. In other words the possibility of violating the second law of thermodynamics due to quantum fluctuations, (i.e., spontaneous transitions in QHO from vacuum to the excited states) is shown.

Kluwer Academic Publishers eBooksFeb 4, 20060 citations

New Approach for Stochastic Quantum Processes, their Manipulation and Control

Ashot S. Gevorkyan

The dissipation and decoherence (for example, the effects of noise in quantum computations), interaction with thermostat or in general with physical vacuum, measurement and many other complicated problems of open quantum systems are a consequence of interaction of quantum system with the environment. These problems are described mathematically in terms of complex probabilistic process (CPP). Particularly, treating the environment as a Markovian process we derive an Langevin-Schroedinger type stochastic differential equation (SDE) for describing the quantum system interacting with environment. For the 1D randomly quantum harmonic oscillator (QHO) model L-Sh SDE is a solution in the form of orthogonal CPP. On the basis of orthogonal CPP the stochastic density matrix (SDM) method is developed and in its framework relaxation processes in the uncountable dimension closed system of "QHO+environment" is investigated. With the help of SDM method the thermodynamical potentials, like nonequilibrium entropy and the energy of ground state are exactly constructed. The dispersion for different operators are calculated. In particular, the expression for uncertain relations depending on parameter of interaction with environment is obtained. The Weyl transformation for stochastic operators is specified. Ground state Winger function is developed in detail.

arXiv:math-ph/0610033Oct 16, 20060 citations

Exactly constructing model of quantum mechanics with random environment

Ashot S. Gevorkyan

Dissipation and decoherence, interaction with the random media, continuous measurements and many other complicated problems of open quantum systems are a result of interaction of quantum system with the random environment. These problems mathematically are described in terms of complex probabilistic processes (CPP). Note that CPP satisfies the stochastic differential equation (SDE) of Langevin—Schrödinger(L—Sch)type, and is defined on the extended space R 1 ⊗ R {γ} , where R 1 and R {γ} are the Euclidean and the functional spaces, correspondingly. For simplicity, the model of 1D quantum harmonic oscillator (QHO) with the stochastic environment is considered. On the basis of orthogonal CPP, the method of stochastic density matrix (SDM) is developed. By S DM method, the thermodynamical potentials, such as the nonequilibrium entropy and the energy of the “ground state” are constructed in a closed form. The expressions for uncertain relations and Wigner function depending on interaction’s constant between 1D QHO and the environment are obtained.

Physics of Atomic NucleiFeb 1, 20105 citations

Quantum chaos in the framework of complex probability processes. Thermodynamics of nonrelativistic vaccum

А. В. Богданов, Ashot S. Gevorkyan

The problem of random motion of 1D quantum reactive harmonic oscillator (QRHO) is formulated in terms of a wave functional regarded as a complex probability process in an extended space. In the complex stochastic differential equation (SDE) for this process the variables are separated with the help of the Langevin-type model SDE introduced in the functional space. The complete positive Fokker-Plank measure of the functional space is obtained. The average wave function of roaming QRHO is obtained by means of functional integration over the process with the complete Fokker-Plank measure in the functional space. The local and averaged transition matrices of roaming QRHO were constructed. The thermodynamics of nonrelativistic vacuum is investigated in detail and expressions for the internal energy, Helmholtzian energy and entropy are obtained. The oscillator's ground state energy, its shift and broadening are calculated.

arXiv:quant-ph/9810079Oct 26, 19980 citations

Nonrelativistic quantum mechanics with consideration of influence of fundamental environment

Ashot S. Gevorkyan

Spontaneous transitions between bound states of an atomic system, the “Lamb Shift” of energy levels and many other phenomena in real nonrelativistic quantum systems are connected with the influence of the quantum vacuum fluctuations (fundamental environment (FE)), which are impossible to consider in the framework of standard quantum-mechanical approaches. The joint system quantum system (QS) and FE is described in the framework of the stochastic differential equation (SDE) of Langevin-Schrödinger type and is defined on the extended space ℝ3⊗Ξ n , where ℝ3 and Ξ n are the Euclidean and functional spaces, respectively. The method of stochastic density matrix is developed and the von Neumann equation for reduced density matrix of QS with FE is generalized. The entropy of QS entangled with FE is defined and investigated. It is proved that the interaction of QS with the environment leads to emerging structures of various topologies which present new quantum-field properties of QS. It is shown that when the physical system (irrelatively to its being micro ormacro) breaks up into two fragments by means of FE, there arises between these fragments a nonpotential interaction which does not disappear at large distances.

Physics of Atomic NucleiAug 1, 20130 citations

Exactly solvable models of quantum mechanics including fluctuations in the framework of representation of the wave function by random process

Ashot S. Gevorkyan, Udalov, A. A.

The problem of quantum harmonic oscillator with "regular+random" square frequency, subjected to "regular+random external force, is considered in framework of representation of the wave function by complex-valued random process. Average transition probabilities are calculated. Stochastic density matrix method is developed, which is used for investigation of thermodinamical characteristics of the system, such as entropy and average energy.

arXiv:quant-ph/0007108Jul 28, 20000 citations