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 KustaanheimoStiefel (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 timedependent 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 meansquareroot 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 »
 Authors:
 Institute of Physics, Carl von Ossietzky University Oldenburg, D26111 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. 2011.
"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 KustaanheimoStiefel (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 timedependent 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 meansquareroot 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 = 2011,
month = 4
}

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