Stochastic mechanics and the Kepler problem.

Abstract

The stochastic mechanics of Nelson and Guerra is formulated for the hydrogen atom. We demonstrate that this simple quantum system can be described in terms of three independent Gaussian Markov processes which are driven (controlled) by the classical Kepler problem. It reveals a manifest connection between the classical and quantized versions of the Kepler problem. I. MOTIVATION The idea of stochastic quantization, as developed in Refs. 1 and 2, amounts to associating the stochastic processes to quantum states of the dynamical system. The procedure works quite successfully as long as ground states of the simplest models are considered; the determination of the Madelung fiuid representation for higher excited states is much more involved. In fact, the stochastic strategy works in full generality for an example of the harmonic oscillator and its most straightforward generalizations (see also the studies of its two-level, Fermi version ' ). However, the formulation of the stochastics mechanics for another simple quantum system, that of the hydrogen atom, except for the ground state, is yet to be accomplished. This fact is a bit puzzling since, like many other simple quantum systems, the quantized Kepler problem admits a realization in terms of (a quartet of) harmonic oscillators, and should in principle allow for the generalization of the arguments of Refs. 2 and 12. Moreover the concept of related coherent states was introduced in Ref. 11, and the construction of oscillator stochastic processes is most transparent with respect to the coherent basis. It is our aim to take advantage of the oscillator reconstruction of the Kepler problem, to formulate the stochastic mechanics of the latter. While working with the fouroscillator system, the functions of Madelung densityphase variables p;(x), S;(x), i=1,2,3,4, arise through computing the coherent-state expectation values (a ~ A ~ a) =Z(p, S) of operator-valued quantities. To recover the hydrogen problem, the constraints must be accounted for. As we demonstrate in the course of the paper, the (analytic) stochastic mechanics of the problem, if formulated in the Madelung [p(x),S(x)] parametrization, is in all respects equivalent to the standard classical mechanics of the singular (constrained) Hamiltonian system, whose phase manifold is parametrized by holom orphic coherent-state labels (a,a): 4 ~ a)=exp g (a;tt — a;tt;) ~ 0) . consequence, we identify the coherent-state domain for the Kepler problem, whose a~, az, a3, a4 parameters are completely determined in terms of the canonical variables of the standard classical Kepler problem. It allows for the final conclusion that the three independent GaussianMarkov processes can be associated with the hydrogen atom. Moreover, these processes are driven (controlled in the language of Ref. 12) by the classical Kepler motion. It establishes an apparent link between the quantized and classical versions of the Kepler problem, the connection which could hardly have been seen from the path-integral computation presented in Ref. 6.