Andrew Neate

The divine clockwork : Bohr's correspondence principle and Nelson's stochastic mechanics for the atomic elliptic state

We consider the Bohr correspondence limit of the Schrodinger 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.

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

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 Poincare inequalities, both of which are used in our proof of the existence of the spectral gap.

The correspondence limit of the atomic elliptic state

We consider the wave function for the atomic elliptic state using Nelsons 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 Keplers 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.

A one dimensional analysis of singularities and turbulence for the stochastic Burgers equation in d-dimensions.

The inviscid limit of the stochastic Burgers equation, with body forces white noise in time, is discussed in terms of the level surfaces of the minimising Hamilton-Jacobi function, the classical mechanical caustic and the Maxwell set, and their algebraic pre-images under the classical mechanical flow map. The problem is analysed in terms of a reduced (one-dimensional) action function. We give an explicit expression for an algebraic surface containing the Maxwell set and caustic in the polynomial case. Those parts of the caustic and Maxwell set which are singular are characterised. We demonstrate how the geometry of the caustic, level surfaces and Maxwell set can change infinitely rapidly causing turbulent behaviour which is stochastic in nature, and we determine its intermittence in terms of the recurrent behaviour of two processes.

Semiclassical wave functions and semiclassical dynamics for the Kepler/Coulomb problem

We investigate the semiclassical Kepler/Coulomb problem using the classical constants of the motion in the framework of Nelsons stochastic mechanics. This is done by considering the eigenvalue relations for a family of coherent states (known as the atomic elliptic states) whose wave functions are concentrated on the elliptical orbit corresponding to the associated classical problem. We show that these eigenvalue relations lead to identities for the semiclassical energy, angular momentum and Hamilton–Lenz–Runge vectors in the elliptical case. These identities are then extended to include the cases of circular, parabolic and hyperbolic motions. We show that in all cases the semiclassical wave function is determined by our identities and so our identities can be seen as defining a semiclassical Kepler/Coulomb problem. The results are interpreted in terms of two dynamical systems: one a complex valued solution to the classical mechanics for a Coulomb potential and the other the drift field for a semiclassical Nelson diffusion.

The stochastic Burgers equation with vorticity: Semiclassical asymptotic series solutions with applications

We consider a stochastic Burgers type equation which incorporates a vector potential. The solution of this equation is not of gradient form and so this equation can be described as a stochastic Burgers equation with vorticity. Building on previous work on the standard stochastic Burgers equation, we discuss the related Hamilton-Jacobi theory in detail and show how to find semiclassical series expansions for our stochastic Burgers equation with vorticity. We examine the behaviour of the solution in the inviscid limit and discuss the geometric structure of the resulting singularities. We illustrate these results with an example of a Burgers type fluid in a rotating bucket under a harmonic oscillator potential. We conclude with a discussion of the relationship between the correspondence limit of Nelsons stochastic mechanics and stationary state solutions for Burgers equations and illustrate an example of the equations studied in this paper arising from considerations of the Coulomb potential.

A stochastic Burgers-Zeldovich model for the formation of planetary ring systems and the satellites of Jupiter and Saturn

We consider a proto-ring nebula of a gas giant such as Neptune as a cloud of gas/dust particles whose behaviour is governed by the stochastic mechanics associated to the Kepler problem. This leads to a stochastic Burgers-Zeldovich type model for the formation of planetesimals involving a stochastic Burgers equation with vorticity which could help to explain the turbulent behaviour observed in ring systems. The Burgers fluid density and the distribution of the mass M(T) of a spherical planetesimal of radius δ are computed for times T. For circular orbits, sufficient conditions on certain time averages of δ2 are given ensuring that VarM(T) ∼ 0 as T ∼ ∞. Some applications are given to the satellites of Jupiter and Saturn, in particular giving a possible explanation of the equal mass families of satellites.

Semiclassical stochastic mechanics for the Coulomb potential with applications to modelling dark matter

Little is known about dark matter particles save that their most important interactions with ordinary matter are gravitational and that, if they exist, they are stable, slow moving and relatively massive. Based on these assumptions, a semiclassical approximation to the Schrodinger equation under the action of a Coulomb potential should be relevant for modelling their behaviour. We investigate the semiclassical limit of the Schrodinger equation for a particle of mass M under a Coulomb potential in the context of Nelson’s stochastic mechanics. This is done using a Freidlin-Wentzell asymptotic series expansion in the parameter ϵ=ħ/M for the Nelson diffusion. It is shown that for wave functions ψ ∼ exp((R + iS)/ϵ2) where R and S are real valued, the ϵ = 0 behaviour is governed by a constrained Hamiltonian system with Hamiltonian Hr and constraint Hi = 0 where the superscripts r and i denote the real and imaginary parts of the Bohr correspondence limit of the quantum mechanical Hamiltonian, independent of Nels...