Quantum mechanics of rectilinear orbits of the Coulomb-Kepler problem

Abstract

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, r 0 , and velocity, v 0 , 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 r w and v w describe a rectilinear orbit along the x axis with the mean initial values r 0 and v 0 ; (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.

Quantum mechanics of rectilinear orbits of the Coulomb-Kepler problem (II.)

“…In [20], these mean values were rigorously calculated, without asymptotic approximation. Thus, the results reported below are valid for all parameters…”

“…This work is to supplement the previous article [20] on the quantum mechanics of rectilinear orbits. Essentially, energy conservation will be implemented rigorously, which liberates us from resorting to an asymptotic approximation.…”

“…x 0 = X 0 {1, 0, 0}, v 0 = σV 0 {1, 0, 0}, σ = ±1, X 0 > 0, V 0 > 0. (1) As in related studies on the quantum mechanics of the Coulomb-Kepler problem, see [8,16,17,18,20] the wave function lives, at first, in the fourdimensional Kustaanheimo-Stiefel space u ∈ R 4 [14], abbreviated KS,…”

“…The question is, how far into the quantum region w remains a time-equivalent curve parameter. As a matter of fact, the study on quantum corrections to Kepler's equation [18] predicted a finite definition domain of w. Similarly, the asymptotically calculated mean rectilinear orbits [20] were defined over a finite interval only, which, in particular, did not cover the neighborhood of the Coulomb (gravitational) singularity. The motivation for the present paper was that, if energy conservation is taken into account rigorously, then the definition domain of w can be extended, possibly into the region where the potential is singular, a hope which came true partially.…”

“…The method, certainly, will not amount to a routine task, mainly due the presence of the Coulomb singularity. In [20] we mentioned that a regularized version [15] predicts an orbit which tunnels through the singularity rather than correctly describing backscattering. In addition, in the classical limit, the order of magnitudes differ extremely when the scales of a mean orbit and its quantum fluctuations are compared.…”