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Title: Detecting level crossings without solving the Hamiltonian. I. Mathematical background

Abstract

When the parameters of a physical system are varied, the eigenvalues of observables can undergo crossings and avoided crossings among themselves. It is relevant to be aware of such points since important physical processes often occur there. In a recent paper [M. Bhattacharya and C. Raman, Phys. Rev. Lett. 97, 140405 (2006)] we introduced a powerful algebraic solution to the problem of finding (avoided) crossings in atomic and molecular spectra. This was done via a mapping to the problem of locating the roots of a polynomial in the parameters of interest. In this article we describe our method in detail. Given a physical system that can be represented by a matrix, we show how to find a bound on the number of (avoided) crossings in its spectrum, the scaling of this bound with the size of the Hilbert space and the parametric dependencies of the Hamiltonian, the interval in which the (avoided) crossings all lie in parameter space, the number of crossings at any given parameter value, and the minimum separation between the (avoided) crossings. We also show how the crossings can reveal the symmetries of the physical system, how (avoided) crossings can always be found without solving for themore » eigenvalues, how they may sometimes be found even in case the Hamiltonian is not fully known, and how crossings may be visualized in a more direct way than displayed by the spectrum. In the accompanying paper [M. Bhattacharya and C. Raman, Phys. Rev. A 75, 033406 (2007)] we detail the application of these techniques to atoms and molecules.« less

Authors:
;  [1]
  1. School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332 (United States)
Publication Date:
OSTI Identifier:
20982353
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. A; Journal Volume: 75; Journal Issue: 3; Other Information: DOI: 10.1103/PhysRevA.75.033405; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS; ATOMS; EIGENFUNCTIONS; EIGENVALUES; ENERGY LEVELS; HAMILTONIANS; HILBERT SPACE; MATHEMATICAL SOLUTIONS; MATRICES; MOLECULES; POLYNOMIALS; SCALING; SPECTRA; SYMMETRY

Citation Formats

Bhattacharya, M., and Raman, C.. Detecting level crossings without solving the Hamiltonian. I. Mathematical background. United States: N. p., 2007. Web. doi:10.1103/PHYSREVA.75.033405.
Bhattacharya, M., & Raman, C.. Detecting level crossings without solving the Hamiltonian. I. Mathematical background. United States. doi:10.1103/PHYSREVA.75.033405.
Bhattacharya, M., and Raman, C.. Thu . "Detecting level crossings without solving the Hamiltonian. I. Mathematical background". United States. doi:10.1103/PHYSREVA.75.033405.
@article{osti_20982353,
title = {Detecting level crossings without solving the Hamiltonian. I. Mathematical background},
author = {Bhattacharya, M. and Raman, C.},
abstractNote = {When the parameters of a physical system are varied, the eigenvalues of observables can undergo crossings and avoided crossings among themselves. It is relevant to be aware of such points since important physical processes often occur there. In a recent paper [M. Bhattacharya and C. Raman, Phys. Rev. Lett. 97, 140405 (2006)] we introduced a powerful algebraic solution to the problem of finding (avoided) crossings in atomic and molecular spectra. This was done via a mapping to the problem of locating the roots of a polynomial in the parameters of interest. In this article we describe our method in detail. Given a physical system that can be represented by a matrix, we show how to find a bound on the number of (avoided) crossings in its spectrum, the scaling of this bound with the size of the Hilbert space and the parametric dependencies of the Hamiltonian, the interval in which the (avoided) crossings all lie in parameter space, the number of crossings at any given parameter value, and the minimum separation between the (avoided) crossings. We also show how the crossings can reveal the symmetries of the physical system, how (avoided) crossings can always be found without solving for the eigenvalues, how they may sometimes be found even in case the Hamiltonian is not fully known, and how crossings may be visualized in a more direct way than displayed by the spectrum. In the accompanying paper [M. Bhattacharya and C. Raman, Phys. Rev. A 75, 033406 (2007)] we detail the application of these techniques to atoms and molecules.},
doi = {10.1103/PHYSREVA.75.033405},
journal = {Physical Review. A},
number = 3,
volume = 75,
place = {United States},
year = {Thu Mar 15 00:00:00 EDT 2007},
month = {Thu Mar 15 00:00:00 EDT 2007}
}
  • A number of interesting phenomena occur at points where the energy levels of an atom or a molecule (anti) cross as a function of some parameter such as an external field. In a previous paper [M. Bhattacharya and C. Raman, Phys. Rev. Lett. 97, 140405 (2006)] we have outlined powerful mathematical techniques useful in identifying the parameter values at which such (avoided) crossings occur. In the accompanying article [M. Bhattacharya and C. Raman, Phys. Rev A 75, 033405 (2007)] we have developed the mathematical basis of these algebraic techniques in some detail. In this article we apply these level-crossing methodsmore » to the spectra of atoms and molecules in a magnetic field. In the case of atoms the final result is the derivation of a class of invariants of the Breit-Rabi Hamiltonian of magnetic resonance. These invariants completely describe the parametric symmetries of the Hamiltonian. In the case of molecules we present an indicator which can tell when the Born-Oppenheimer approximation breaks down without using any information about the molecular potentials other than the fact that they are real. We frame our discussion in the context of Feshbach resonances in the atom-pair {sup 23}Na-{sup 85}Rb which are of current interest.« less
  • The aim of this article is to discuss the distribution law of the gasification agent concentration in a deep-going way during underground coal gasification and the new method of solving the problem for the convection diffusion of the gas. In this paper, the basic features of convection diffusion for the gas produced in underground coal gasification are studied. On the basis of the model experiment, through the analysis of the distribution and patterns of variation for the fluid concentration field in the process of the combustion and gasification of the coal seams within the gasifier, the 3-D non-linear unstable mathematicalmore » models on the convection diffusion for oxygen are established. In order to curb such pseudo-physical effects as numerical oscillation and surfeit which frequently occurred in the solution of the complex mathematical models, the novel finite unit algorithm, the upstream weighted multi-cell balance method is advanced in this article, and its main derivation process is introduced.« less
  • In this paper we study the footprint of cosmic string as the topological defects in the very early universe on the cosmic microwave background radiation. We develop the method of level crossing analysis in the context of the well-known Kaiser-Stebbins phenomenon for exploring the signature of cosmic strings. We simulate a Gaussian map by using the best fit parameter given by WMAP-7 and then superimpose cosmic strings effects on it as an incoherent and active fluctuations. In order to investigate the capability of our method to detect the cosmic strings for the various values of tension, Gμ, a simulated puremore » Gaussian map is compared with that of including cosmic strings. Based on the level crossing analysis, the superimposed cosmic string with Gμ∼>4 × 10{sup −9} in the simulated map without instrumental noise and the resolution R = 1' could be detected. In the presence of anticipated instrumental noise the lower bound increases just up to Gμ∼>5.8 × 10{sup −9}.« less