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Physical Review A

Quantum mechanics of hyperbolic orbits in the Kepler problem

Alexander Rauh, J Iirgen Parisi

Published Apr 12, 2011• 8 citations
Publisher

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 $\ensuremath{\hbar}\ensuremath{\rightarrow}0$, 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 ${\mathbb{R}}^{4}$ 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

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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 \ensuremathℏ\ensuremath→0\ensuremath{\hbar}\ensuremath{\rightarrow}0\ensuremathℏ\ensuremath→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 R4{\mathbb{R}}^{4}R4 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.
A one-dimensional wave function is assumed whose logarithm is a quadratic form in the configuration variable with time-dependent coefficients. This trial function allows for general time-dependent solutions both of the harmonic oscillator (HO) and the reversed harmonic oscillator (RO). For the HO, apart from the standard coherent states, a further class of solutions is derived with a time-dependent width parameter. The width of the corresponding probability density fluctuates, or "breathes" periodically with the oscillator frequency. In the case of the RO, one also obtains normalized wave packets which, however, show diffusion through exponential broadening with time. At the initial time, the integration constants give rise to complete sets of coherent states in the three cases considered. The results are applicable to the quantum mechanics of the Kepler-Coulomb problem when transformed to the model of a four-dimensional harmonic oscillator with a constraint. In the classical limit, as was shown recently, the wave packets of the RO basis generate the hyperbolic Kepler orbits, and, by means of analytic continuation, the elliptic orbits are also obtained quantum mechanically.
The quantum-mechanical motion of Gaussian wave packets on elliptical and hyperbolic orbits is calculated and shown in plots. The approximation scheme makes explicit use of energy and angular-momentum conservation thus guaranteeing controlled accuracy of the approximation over long-time intervals. This allows the study of the revivals, a phenomenon of particular interest in Rydberg atoms on elliptical orbits. A comparison with the behavior of a classical phase-space distribution of the same position and momentum widths is carried out.
The Kustaanheino-Stiefel transformation is used to transform the Coulomb problem to a four-dimensional isotropic harmonic oscillator with a constraint. Ordinary coherent states are introduced over this oscillator and are shown to evolve in a fictitious time. When projected back into the physical space, the states follow the classical Kepler orbits in physical time. The resulting quasiclassical picture is obtained without the Bohr correspondence principle of taking only contributions from high principal quantum numbers.
Based on coherent states of the reversed harmonic oscillator, a wave function is proposed with the following three properties: (1) The mean values of position, r0, and velocity, v0, are parallel to the x axis; (2) The wave function is extended by a curve parameter w ≥ 0 such that, as a function of w, the mean values rw and vw describe a rectilinear orbit along the x axis with the mean initial values r0 and v0; (3) The mean energy is constant with respect to w by the leading order of the perturbation parameter ∈. For comparison, the classical one-dimensional equation of motion is integrated. Exact analytical equivalence between classical and the ∈ = 0 quantum limit is found, both for positive and negative mean energy. The curve parameter w is in 1-1 correspondence with time t ≥ 0.