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Physics of Atomic Nuclei

Nonrelativistic quantum mechanics with consideration of influence of fundamental environment

Ashot S. Gevorkyan

Published Aug 1, 2013• 0 citations
Publisher

Abstract

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.

DOI
10.1134/s1063778813080115

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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.
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 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 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.
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 evolution of the joint system (JS) - ``quantum system (QS)+thermal bath (TB)" is considered in the framework of a complex probabilistic processes that satisfies the stochastic differential equation of the Langevin-Schrödinger type. Two linearly coupled oscillators that randomly interact with the environment and with each other are selected as QS. In the case when the interactions obey the law of a white random process, all the construction of the statistical parameters of the QS and its environment are performed analytically in the form of double integrals and solutions of second-order partial differential equations. Expressions of time-dependent von Neumann entropy and its generalization are obtained, taking into account the self-organization and entanglement processes occurring in the JS. It is mathematically proved that as a result of the relaxation of JS in the TB, a small quantized environment is formed, which can be interpreted as a continuation of QS or its halo. Bell states formed as a result of the decay of coupled two linear oscillators are constructed taking into account the influence of the environment. The transitions between (in)(in)(in) and (out(out(out) asymptotic states of QS are studied in detail taking into account the influence of TB. Within the framework of the model problem, the possibility of constructing quantum thermodynamics from the first principle is proved without using any additional conditions.
Previous years researchers began to simulate open quantum system, taking into account the interaction between system and the environment. One approach to deal with this problem is to use the density matrix within the Liouville-von-Neumann formalism or the Markovian variant the Lindblad equations. Another way is to use a stochastic approach where a random force is added to the system. The benefit of the stochastic approach is to solve the dynamics of the system with less time and memory than the density matrix approaches. In this project we want to develop a stochastic approach that can deal with the stochastic wave functions approach. We did this on a 2-level system and found that it works well when comparing to a density matrix approach. Next, we tested a quantum particle connect to a bath of harmonic oscillators using the stochastic approach. We found that a friction term is necessary and applied it. Like in the classical Langevin equations the friction constant and the random force fluctuations are related by the fluctuation-dissipation constant. We showed that with friction the dynamics decays to an ensemble with energy of Egs+kBTE_{gs}+k_BTEgs​+kB​T. However, we also found here are problems. The system seems to absorb energy indefinitely if the temperature is higher than the zero point energy or if the system is a Morse oscillator. Thus more research is required to make this method work.