Non-integrable quantum systems

We present a new effective numerical method to compute resonances of simple but non-integrable quantum systems, based on a combination of complex coordinate rotations with the finite element and the discrete variable method. By using model potentials we were able to compute atomic data for alkali systems. As an example we show some results for the radial Stark and the Stark effect and compare our values with recent published ones. 

In this article we will describe a modem computational method recently developed for calculating resonances in non-integrable quantum systems with a few degrees-of-freedom. Typical examples are given by simple atoms in external fields, where the phrase simple is a shorthand for systems, which can be accurately described by an effective one- or two-particle Hamiltonian. But note that the numerical methods presented in this paper are not only restricted to atomic systems and the adjective simple does not mean that computations are simple. 

Discrete Variable Methodfor Non-Integrable Quantum Systems... 

Discrete Variable techniques and Finite Element methods, if necessary combined with additional model dependent numerical techniques, turned out to be a useful, quick and accurate way for studying non-integrable quantum systems. By this methods we were... 

Discrete variable method for non-integrable quantum systems... 

MSN. 130. T. Petrosky and 1. Prigogine, Alternative formulation of classical and quantum dynamics for non-integrable systems, Physica A 175, 146-209 (1991). 

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MSN. 175. I. Prigogine, T. Petrosky, and G. Ordonez, Report The Extension of Classical and Quantum Mechanics, as Well as of Field Theory to Non-integrable Systems. EC report. Contract PSS 0992, 1999. 

The non-adiabatic quantum simulation procedures we employ have been well described previously in the literature, so we describe them only briefly here. The model system consists of 200 classical SPC flexible water molecules," and one quantum mechanical electron interacting with the water molecules via a pseudopotential. 2 The equations of motion were integrated using the Verlet algorithm with a 1 fs time step in the microcanonical ensemble, and the adiabatic eigenstates at each time step were calculated with an iterative and block Lanczos scheme. Periodic boundary conditions were employed using a cubic simulation box of side 18.17A (water density 0.997 g/ml). 

All the particles in Table 10.1 have spin. Quantum mechanical calculations and experimental observations have shown that each particle has a fixed spin energy which is determined by the spin quantum number s s = h for leptons and nucleons). Particles of non-integral spin are csWeA fermions because they obey the statistical rules devised by Fermi and Dirac, which state that two such particles cannot exist in the same closed system (nucleus or electron shell) having all quantum numbers the same (referred to as the Pauli principle). Fermions can be created and destroyed only in conjunction with an anti-particle of the same class. For example if an electron is emitted in 3-decay it must be accompanied by the creation of an anti-neutrino. Conversely, if a positron — which is an anti-electron — is emitted in the ]3-decay, it is accompanied by the creation of a neutrino. 

In both laboratory and natural plasmas electric fields play an important role in understanding the observed physical properties. Electric fields are not only due to external fields but in addition to ions and free electrons in the environment of the observed object. Electric fields allow the valence electron of an atom to ionize by tunneling through the combined electric and Coulomb potential. Hence in external electric fields all bound states become quasi-bound due to tunneling. Besides the fundamental questions associated with this problem, this work is strongly motivated by its application in astrophysics, in particular to the interpretation of spectra of white dwarf stars (F 10 10 V/m), and by its application in quantum chaology, especially related to questions with respect to open non-integrable systems. 

It should be noted that in the cases where y"j[,q ) > 0, the centroid variable becomes irrelevant to the quantum activated dynamics as defined by (A3.8.Id) and the instanton approach [37] to evaluate based on the steepest descent approximation to the path integral becomes the approach one may take. Alternatively, one may seek a more generalized saddle point coordinate about which to evaluate A3.8.14. This approach has also been used to provide a unified solution for the thennal rate constant in systems influenced by non-adiabatic effects, i.e. to bridge the adiabatic and non-adiabatic (Golden Rule) limits of such reactions. 

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