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Physics of Atomic Nuclei

Exactly constructing model of quantum mechanics with random environment

Ashot S. Gevorkyan

Published Feb 1, 2010• 5 citations
Publisher

Abstract

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.

DOI
10.1134/s1063778810020171

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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.
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.
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.
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.
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.
The system of oscillator interacting with vacuum is considered as a problem of random motion of quantum reactive harmonic oscillator (QRHO). It is formulated in terms of a wave functional regarded as complex probability process in the extended space. This wave functional obeys some stochastic differential equation (SDE). Based on the nonlinear Langevin type SDE of second order, introduced in the functional space R{W(t)}, the variables in original equation are separated. The general measure in the space R{W(t)} of the Fokker-Plank type is obtained and expression for total wave function (wave mixture) of random QRHO is constructed as functional expansion over the stochastic basis set. The pertinent transition matrix S_br is constructed. For Wiener type measure W(t) of functional space the exact representation for ''vacuum-vacuum'' transition probability is obtained. The thermodynamics of vacuum is described in detail for the asymptotic space R1_as. The exact values for Energy, shift and expansion of ground state of oscillator and its Entropy are calculated.