Quantum mechanics of hyperbolic orbits in the Kepler problem
- Institute of Physics, Carl von Ossietzky University Oldenburg, D-26111 Oldenburg (Germany)
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.
- OSTI ID:
- 21544557
- Journal Information:
- Physical Review. A, Vol. 83, Issue 4; Other Information: DOI: 10.1103/PhysRevA.83.042101; (c) 2011 American Institute of Physics; ISSN 1050-2947
- Country of Publication:
- United States
- Language:
- English
Similar Records
Four-dimensional singular oscillator and generalized MIC-Kepler system
Variations on the Kepler problem
Related Subjects
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