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Information Technologies in Education

NON-HAMILTONIAN QUANTUM MECHANICS AND THE NUMERICAL RESEARCHES OF THE ATTRACTOR OF A DYNAMICAL SYSTEM

Alexander J. Weissblut

Published Apr 12, 2012• 3 citations
Publisher

Abstract

This article – introduction to the structural theory of general view dynamical systems, based on construction of dynamic quantum models (DQM), offered by the author. This model is simply connected with traditional model of quantum mechanics (i.e. with the Schrodinger equation). At the same time obtained thus non – Hamiltonian quantum dynamics is easier than classical one: it allow building the clear structural theory and effective algorithms of research for concrete systems. This article is devoted mainly to such task. The algorithm of search for DQM attractors, based on this approach, is offered here.

DOI
10.14308/ite000303

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This article – introduction to the structural theory of general view dynamical systems, based on construction of dynamic quantum models (DQM), offered by the author. This model is simply connected with traditional model of quantum mechanics (i.e. with the Schrodinger equation). At the same time obtained thus non – Hamiltonian quantum dynamics is easier than classical one: it allow building the clear structural theory and effective algorithms of research for concrete systems. This article is devoted mainly to such task. The algorithm of search for DQM attractors, based on this approach, is offered here.
After S. Smale's works it became clear that in smooth dynamics the system of a general form is not structurally stable and therefore there is no strict mathematical basis for modeling and computational analysis of systems. The contradiction appeared in science: according to physicists dynamics is simple and universal. The solution to this problem was proposed based on the construc- tion of dynamic quantum models (DQM). From the assumption that quantum effects are caused by unrecoverable “white noise”, a certain mathematical mod- el of quantum mechanics already follows and is essentially unambiguous. On the other hand, in this model spectral problems are reduced tothe usual pertur- bation theory of smooth dynamical systems. Thus, the construction of such models can be considered as an asymptotic method for solving spectral prob- lems. But the definition of DQM is not formally related to Hamiltonian sys- tems. DQM is defined and constructed universally for both Hamiltonian sys- tems and systems with the truth function. As a result, for example, quantization with the Bohr-Sommerfeld condition also extends to systems with a truth func- tion. Hopefully DQM opens for new applications. The most important is to seek assistance and cooperation in future research.
We expose the Schrödinger quantum mechanics with traditional applications to Hydrogen atom. We discuss carefully the experimental and theoretical background for the introduction of the Schrödinger, Pauli and Dirac equations, as well as for the Maxwell equations. We explain in detail all basic theoretical concepts. We explain all details of the calculations and mathematical tools: Lagrangian and Hamiltonian formalism for the systems with finite degree of freedom and for fields, Geometric Optics, the Hamilton-Jacobi equation and WKB approximation, Noether theory of invariants including the theorem on currents, four conservation laws (energy, momentum, angular momentum and charge), Lie algebra of angular momentum and spherical functions, scattering theory (limiting amplitude principle and limiting absorption principle), the Lienard-Wiechert formulas, Lorentz group and Lorentz formulas, Pauli theorem and relativistic covariance of the Dirac equation, etc. We give a detailed oveview of the conceptual development of the quantum mechanics, and expose main achievements of the ``old quantum mechanics'' in the form of exercises. One of our basic aim in writing this book, is an open and concrete discussion of the problem of a mathematical description of the following two fundamental quantum phenomena: i) Bohr's quantum transitions and ii) de Broglie's wave-particle duality. Both phenomena cannot be described by autonomous linear dynamical equations, and we give them a new mathematical treatment related with recent progress in the theory of global attractors of nonlinear hyperbolic PDEs.
We review analyses of open quantum systems. We show how non-Hermiticity arises in an open quantum system with an infinite environment, focusing on the one-body problem. One of the reasons for taking the present approach is that we can solve the problem completely, making it easier to see the structures of problems involving open quantum systems. We show that this results in the discovery of a new complete set, which is one of the main topics of the present article. Another reason for focusing on the one-body problem is that the theory permits the strong coupling between the system and the environment. In the current research landscape, it is valuable to revisit the one-body problem for open quantum systems, which can be solved accurately for arbitrary strengths of the system-environment couplings. A rigorous understanding of the problem structures in the present approach will be helpful when we tackle problems with many-body interactions. First, we consider potential scattering and directly define the resonant state as an eigenstate of the Schrödinger equation under the Siegert outgoing boundary condition. We show that the resonant eigenstate can have a complex energy eigenvalue, even though the Hamiltonian is seemingly Hermitian. Second, we introduce the Feshbach formalism, which eliminates the infinite degrees of freedom of the environment and represents its effect as a complex potential. The resulting effective Hamiltonian is explicitly non-Hermitian. By unifying these two ways of defining resonant states, we obtain a new complete set of bases for the scattering problem that contains all discrete eigenstates, including resonant states. We finally mention the non-Markovian dynamics of open quantum systems. We emphasize the time-reversal symmetry of the dynamics that continuously connects the past and the future. We can capture it using the new complete set that we develop here.
Wave propagation algorithm is a common method to study electromagnetic field propagation. Quantum wave propagation algorithm is an improved algorithm based on wave propagation algorithm to calculate the evolution of schrodinger equation in quantum mechanics. This paper mainly introduces the general idea of quantum wave propagation algorithm, and presents the general analysis process by taking one-dimensional infinite deep potential well, one-dimensional square potential barrier and time-dependent Schrodinger equation as examples. By comparing with the strict analytical solution of Schrodinger equation, the evolution image of probability wave function of micro particles with time is given, which verifies the correctness and effectiveness of the algorithm. The programming of quantum wave algorithm is not complicated, and has the characteristics of simple structure and strong practicability.