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Title: Quantum mechanics of hyperbolic orbits in the Kepler problem

Abstract

The problem of deriving macroscopic properties from the Hamiltonian of the hydrogen atom is resumed by extending previous results in the literature, which predicted elliptic orbits, into the region of hyperbolic orbits. As a main tool, coherent states of the harmonic oscillator are used which are continued to imaginary frequencies. The Kustaanheimo-Stiefel (KS) map is applied to transform the original configuration space into the product space of four harmonic oscillators with a constraint. The relation derived between real time and oscillator (pseudo) time includes quantum corrections. In the limit ({h_bar}/2{pi}){yields}0, the time-dependent mean values of position and velocity describe the classical motion on a hyperbola and a circular hodograph, respectively. Moreover, the connection between pseudotime and real time comes out in analogy to Kepler's equation for elliptic orbits. The mean-square-root deviations of position and velocity components behave similarly in time to the corresponding ones of a spreading Gaussian wave packet in free space. To check the approximate treatment of the constraint, its contribution to the mean energy is determined with the result that it is negligible except for energy values close to the parabolic orbit with eccentricity equal to 1. It is inevitable to introduce a suitable scalar product inmore » R{sup 4} which makes both the transformed Hamiltonian and the velocity operators Hermitian. An elementary necessary criterion is given for the energy interval where the constraint can be approximated by averaging.« less

Authors:
;  [1]
  1. Institute of Physics, Carl von Ossietzky University Oldenburg, D-26111 Oldenburg (Germany)
Publication Date:
OSTI Identifier:
21544557
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. A; Journal Volume: 83; Journal Issue: 4; Other Information: DOI: 10.1103/PhysRevA.83.042101; (c) 2011 American Institute of Physics
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 74 ATOMIC AND MOLECULAR PHYSICS; ANNIHILATION OPERATORS; APPROXIMATIONS; EIGENSTATES; ELLIPTICAL CONFIGURATION; HAMILTONIANS; HARMONIC OSCILLATORS; HYDROGEN; HYPERBOLIC CONFIGURATION; MATHEMATICAL SPACE; ORBITS; QUANTUM MECHANICS; TIME DEPENDENCE; WAVE PACKETS; CALCULATION METHODS; CONFIGURATION; ELEMENTS; MATHEMATICAL OPERATORS; MECHANICS; NONMETALS; QUANTUM OPERATORS; SPACE

Citation Formats

Rauh, Alexander, and Parisi, Juergen. Quantum mechanics of hyperbolic orbits in the Kepler problem. United States: N. p., 2011. Web. doi:10.1103/PHYSREVA.83.042101.
Rauh, Alexander, & Parisi, Juergen. Quantum mechanics of hyperbolic orbits in the Kepler problem. United States. doi:10.1103/PHYSREVA.83.042101.
Rauh, Alexander, and Parisi, Juergen. Fri . "Quantum mechanics of hyperbolic orbits in the Kepler problem". United States. doi:10.1103/PHYSREVA.83.042101.
@article{osti_21544557,
title = {Quantum mechanics of hyperbolic orbits in the Kepler problem},
author = {Rauh, Alexander and Parisi, Juergen},
abstractNote = {The problem of deriving macroscopic properties from the Hamiltonian of the hydrogen atom is resumed by extending previous results in the literature, which predicted elliptic orbits, into the region of hyperbolic orbits. As a main tool, coherent states of the harmonic oscillator are used which are continued to imaginary frequencies. The Kustaanheimo-Stiefel (KS) map is applied to transform the original configuration space into the product space of four harmonic oscillators with a constraint. The relation derived between real time and oscillator (pseudo) time includes quantum corrections. In the limit ({h_bar}/2{pi}){yields}0, the time-dependent mean values of position and velocity describe the classical motion on a hyperbola and a circular hodograph, respectively. Moreover, the connection between pseudotime and real time comes out in analogy to Kepler's equation for elliptic orbits. The mean-square-root deviations of position and velocity components behave similarly in time to the corresponding ones of a spreading Gaussian wave packet in free space. To check the approximate treatment of the constraint, its contribution to the mean energy is determined with the result that it is negligible except for energy values close to the parabolic orbit with eccentricity equal to 1. It is inevitable to introduce a suitable scalar product in R{sup 4} which makes both the transformed Hamiltonian and the velocity operators Hermitian. An elementary necessary criterion is given for the energy interval where the constraint can be approximated by averaging.},
doi = {10.1103/PHYSREVA.83.042101},
journal = {Physical Review. A},
number = 4,
volume = 83,
place = {United States},
year = {Fri Apr 15 00:00:00 EDT 2011},
month = {Fri Apr 15 00:00:00 EDT 2011}
}
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  • We investigate the existence of orthogonality and completeness relations for the eigenvalue problem associated with the differential operator ..lambda.. = -Pi/sub ..mu../Pi/sub ..mu../ -iesigma x (E+iB), Pi/sub ..mu../ = -i partial/sub ..mu../-eA/sub ..mu../. The operator ..lambda.. acts on 2 x 1 Pauli-type spinor fields defined over all Minkowski space, and may be interpreted as the square of the mass of a charged Dirac particle moving in an external c-number electromagnetic field. We show that ..lambda.. is self-adjoint with respect to the not positive-definite inner product (phi/sub b/; phi/sub a/) = ..integral.. d/sup 4/x phi-bar/sub b/phi/sub a/, where phi-bar/sub b/ ismore » defined as phi-bar/sub b/ = phi/sup dagger//sub b/(-iPiX/sub 4/-sigmaxPiX). A proof is provided for the Coulomb case that the mass eigenfunctions form a complete set in spite of the indefinite metric in Hilbert space. The mass eigenfunction expansion of the propagator is worked out explicitly for the Kepler case. This mass eigenfunction expansion is expected to be quite useful for bound state calculations in quantum electrodynamics, since it involves the covariant denominators (m')/sup 2/-(m)/sup 2/.« less