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arXiv:0805.1343 · Journal of Mathematical Physics

On the divine clockwork: The spectral gap for the correspondence limit of the Nelson diffusion generator for the atomic elliptic state

Richard Durran, Andrew Neate, Aubrey Truman, Feng-Yu Wang

Published Oct 1, 2008• 5 citations
arXivHTMLPDF

Abstract

The correspondence limit of the atomic elliptic state in three dimensions is discussed in terms of Nelson’s stochastic mechanics. In previous work we have shown that this approach leads to a limiting Nelson diffusion, and here we discuss in detail the invariant measure for this process and show that it is concentrated on the Kepler ellipse in the plane z=0. We then show that the limiting Nelson diffusion generator has a spectral gap; thereby proving that in the infinite time limit the density for the limiting Nelson diffusion will converge to its invariant measure. We also include a summary of the Cheeger and Poincaré inequalities, both of which are used in our proof of the existence of the spectral gap.

DOI
10.1063/1.2988715

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The correspondence limit of the atomic elliptic state in three dimensions is discussed in terms of Nelson’s stochastic mechanics. In previous work we have shown that this approach leads to a limiting Nelson diffusion, and here we discuss in detail the invariant measure for this process and show that it is concentrated on the Kepler ellipse in the plane z=0. We then show that the limiting Nelson diffusion generator has a spectral gap; thereby proving that in the infinite time limit the density for the limiting Nelson diffusion will converge to its invariant measure. We also include a summary of the Cheeger and Poincaré inequalities, both of which are used in our proof of the existence of the spectral gap.
The correspondence limit of the atomic elliptic state in three dimensions is discussed in terms of Nelson’s stochastic mechanics. In previous work we have shown that this approach leads to a limiting Nelson diffusion, and here we discuss in detail the invariant measure for this process and show that it is concentrated on the Kepler ellipse in the plane ????=0. We then show that the limiting Nelson diffusion generator has a spectral gap; thereby proving that in the infinite time limit the density for the limiting Nelson diffusion will converge to its invariant measure. We also include a summary of the Cheeger and Poincaré inequalities, both of which are used in our proof of the existence of the spectral gap.
We consider the wave function for the atomic elliptic state using Nelson's stochastic mechanics. The Bohr correspondence limit is taken to reveal a limiting Nelson diffusion. We demonstrate that this limiting diffusion is a small random perturbation of a deterministic dynamical system in which trajectories converge to Keplerian motion on an ellipse. This shows how to derive Kepler's laws of motion in a quantum-mechanical setting. We show that the generator of the limiting Nelson diffusion has a spectral gap and thereby give an explicit rigorous estimate for the rate of convergence to the elliptical orbit. We discuss possible applications of these results to the long time limit of the quantum particle density for a semiclassical system in a Coulomb potential and also to the formation of planets from a protosolar nebula.
We consider the Bohr correspondence limit of the Schrödinger wave function for an atomic elliptic state. We analyze this limit in the context of Nelson’s stochastic mechanics, exposing an underlying deterministic dynamical system in which trajectories converge to Keplerian motion on an ellipse. This solves the long standing problem of obtaining Kepler’s laws of planetary motion in a quantum mechanical setting. In this quantum mechanical setting, local mild instabilities occur in the Keplerian orbit for eccentricities greater than 12 which do not occur classically.
We consider the Bohr correspondence limit of the Schrödinger wave function for an atomic elliptic state. We analyse this limit in the context of Nelson’s stochastic mechanics, exposing an underlying deterministic dynamical system in which trajectories converge to Keplerian motion on an ellipse. This solves the long standing problem of obtaining Kepler’s laws of planetary motion in a quantum mechanical setting. Local mild instabilities occur in the Keplerian orbit for eccentricities greater than 1 / √ 2. If the atomic elliptic state could be produced in the laboratory, this would give a possible experimental test of Nelson’s stochastic mechanics. These instabilities may also explain in some detail when and how the orbiting quantum particle changes its state by emitting or absorbing photons.
In stochastic mechanics, the Bopp-Haag-Dankel diffusions on R3×SO⁡(3){\mathbb {R}^3} \times \operatorname {SO} (3)R3×SO(3) are used to represent particles with spin. Bopp and Haag showed that in the limit as the particle’s moment of inertia III goes to zero, the solutions of the Bopp-Haag equations converge to that of the regular Pauli equation. Nelson has conjectured that in the same limit, the projections of the Bopp-Haag-Dankel diffusions onto R3{\mathbb {R}^3}R3 converge to a Markovian limit process. In this paper, we prove this conjecture for spin spin⁡  12\operatorname {spin} \;\tfrac {1} {2}spin21​ and regular potentials, and identify the limit process as the diffusion naturally associated with the solution to the regular Pauli equation.
An extension of Nelson's stochastic mechanics to the finite-temperature equilibrium states allows one to describe the quantal harmonic oscillator in equilibrium with a thermal bath in terms of a temperature-dependent diffusion process. The stochastic process is called a ``thermal-mixture diffusion process'' for the harmonic oscillator. In this paper, by the use of a regularization of the classical Kepler problem, a thermal-mixture diffusion process associated with the Coulomb potential in openRn{\mathit{openR}}^{\mathit{n}}openRn (n=2,3,5) is derived from one associated with the harmonic oscillator in openRN{\mathit{openR}}^{\mathit{N}}openRN (N=2,4,8), which reduces to Nelson's ground-state process for the quantal Kepler system at zero temperature.