Nelson's Stochastic Mechanics Research Articles

Particle Spin Described by Quantum Hamilton Equations

AbstractThe anomalous Zeeman effect made it clear that charged particles like the electron possess a magnetic dipole moment. Classically, this could be understood if the charged particle possesses an eigenrotation, that is, spin. Within Nelson's stochastic mechanics, it was shown that the model of a rotating charged ball is able to reproduce the well‐known spin expectation values. This classically motivated model of intrinsic rotation described in terms of a stochastic process is revisited here within the formalism of stochastic optimal control theory. Quantum Hamilton equations (QHE) for spinning particles are derived, which reduce to their classical counterpart in the zero quantum noise limit. These equations enable the calculation of the common spin expectation values without the use of the wave function. They also offer information on the orientation dynamics of the magnetic moment of charged particles beyond the behavior of the spin averages.

An interpretation of the Schrödinger equation in quantum mechanics from the control-theoretic point of view

In this paper, for a given (time-dependent) Schrödinger equation in quantum mechanics, an interpretation of it is investigated from the perspective of stochastic control theory with the help of Nelson stochastic mechanics.

Derivation and application of quantum Hamilton equations of motion

Non‐relativistic quantum systems are analyzed theoretically or by numerical approaches using the Schrödinger equation. Compared to the options available to treat classical mechanical systems this is limited, both in methods and in scope. However, based on Nelson's stochastic mechanics, the mathematical structure of quantum mechanics has in some aspects been developed into a form analogous to classical analytical mechanics. We show here that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton's principle of least action, allows to derive two things: i) the Schrödinger equation as the Hamilton‐Jacobi‐Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton's equations of motion to the quantum world. We derive their general form for the non‐stationary and the stationary case. For the harmonic oscillator, the stationary equations lead to the coherent states, and we establish a numerical procedure to solve for the ground state properties without using the Schrödinger equation. image

Quantum-thermal self-diffusion as a hydrodynamic mechanism for the fluctuations’ relaxation

We propose a generalization of quantum mechanical equations in the hydrodynamic form by introducing, into the Lagrangian density, terms taking into account the diffusion velocity at zero and finite temperatures and the diffusion pressure energy of the warm vacuum. Based on this, for the model of one-dimensional hydrodynamics, we construct a system of equations that are analogous to the Euler equations, but with the inclusion of quantum and thermal effects. They are a generalization of the equations of the Nelson stochastic mechanics. The numerical analysis of the system’s solutions’ behavior determined that this system can be used to describe the process of quantum-thermal fluctuation relaxation.

The geometric phase and the geometrodynamics of relativistic electron vortex beams

We have studied here the geometrodynamics of relativistic electron vortex beams from the perspective of the geometric phase associated with the scalar electron encircling the vortex line. It is pointed out that the electron vortex beam carrying orbital angular momentum is a natural consequence of the skyrmion model of a fermion. This follows from the quantization procedure of a fermion in the framework of Nelson's stochastic mechanics when a direction vector (vortex line) is introduced to depict the spin degrees of freedom. In this formalism, a fermion is depicted as a scalar particle encircling a vortex line. It is here shown that when the Berry phase acquired by the scalar electron encircling the vortex line involves quantized Dirac monopole, we have paraxial (non-paraxial) beam when the vortex line is parallel (orthogonal) to the wavefront propagation direction. Non-paraxial beams incorporate spin–orbit interaction. When the vortex line is tilted with respect to the propagation direction, the Berry phase involves non-quantized monopole. The temporal variation of the direction of the tilted vortices is studied here taking into account the renormalization group flow of the monopole charge and it is predicted that this gives rise to the spin Hall effect.

Harmonically confined Tonks-Girardeau gas: A simulation study based on Nelson's stochastic mechanics

Harmonically confined Tonks-Girardeau gas: A simulation study based on Nelson's stochastic mechanics

Non-equilibrium in Stochastic Mechanics

The notion of non-equilibrium, in the sense of a particle distribution other than ρ = |ψ|2, is imported into Nelson's stochastic mechanics, and described in terms of effective wavefunctions obeying non-linear equations. These techniques are applied to the discussion of non-locality in non-linear Schrodinger equations.

Markov processes and generalized Schrödinger equations

Starting from the forward and backward infinitesimal generators of bilateral, time-homogeneous Markov processes, the self-adjoint Hamiltonians of the generalized Schrödinger equations are first introduced by means of suitable Doob transformations. Then, by broadening with the aid of the Dirichlet forms, the results of the Nelson stochastic mechanics, we prove that it is possible to associate bilateral, and time-homogeneous Markov processes to the wave functions stationary solutions of our generalized Schrödinger equations. Particular attention is then paid to the special case of the Lévy-Schrödinger (LS) equations and to their associated Lévy-type Markov processes, and to a few examples of Cauchy background noise.

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 Nelson's 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.

The theory of stochastic space-time. 2. Quantum theory of relativity

The Nelson stochastic mechanics is derived as a consequence of the basic physical principles such as the principle of relativity of observations and the invariance of the action quantum. The unitary group of quantum mechanics is represented as the transformations of the systems of perturbing devices. It is argued that the physical spacetime has a stochastic nature, and that quantum mechanics in Nelson's formulation correctly describes this stochasticity.

The Theory of Stochastic Space-Time. 1. Gravitation as a Quantum Diffusion

The Nelson stochastic mechanics of inhomogeneous quantum diffusion in flat spacetime with a tensor of diffusion can be described as a homogeneous one in a Riemannian manifold where this tensor of diffusion plays the role of a metric tensor. It is shown that the such diffusion accelerates both a sample particle and a local frame such that their mean accelerations do not depend on their masses. This fact, explaining the principle of equivalence, allows one to represent the curvature and gravitation as consequences of the quantum fluctuations. In this diffusional treatment of gravitation it can be naturally explained the fact that the energy density of the instantaneous Newtonian interaction is negative defined.

Time of arrival of electrons in the double-slit experiment

Using Nelson's stochastic mechanics, quantum motion of electrons in the double-slit experiment is studied numerically. It is found that not only the distribution of arrival positions but also that of arrival times at the screen forms an interference pattern. It is demonstrated that the presence time and arrival time are well interpreted in terms of stochastic mechanics.

Dissipative stochastic mechanics for capturing neuronal dynamics under the influence of ion channel noise: Formalism using a special membrane

Based on the idea conveyed in the author's prior study [Fluct. Noise Lett. 6, L147 (2006)], a physical approach for the description of neuronal dynamics under the influence of ion channel noise is developed in the realm of Nelson's stochastic mechanics when open to dissipative environments. The formalism therein is scrutinized using a special membrane with some tailored properties giving the Rose-Hindmarsh dynamics in the deterministic limit. Led by the presence of multiple number of gates in an ion channel, a dual viewpoint of channel noise is established. Then, stochastic mechanics is adopted to model those channel fluctuations emerging from the uncertainty in accessing the permissible topological states of open gates. A mutual interaction between the above fluctuations and the noise, emerging from the stochasticity in the movement of gating particles between the inner and the outer faces of the membrane, is portrayed within a system plus reservoir strategy. Induced by the interaction, renormalizations of the membrane capacitance and of a membrane voltage dependent potential are found to arise. Consequently, the equations of motion, for the expectation values of the variables and the pair correlation functions, are obtained in the collective membrane voltage space.

Time of arrival in the Aharonov–Bohm effect

The quantum motion of electrons in the double-slit experiment with a confined magnetic field is numerically studied by means of Nelson's stochastic mechanics. It is found that not only the arrival positions but also the arrival times of particles form an interference pattern.

Probabilités et fluctuations quantiques

This Note is sketching a simple and natural mathematical construction for explaining the probabilistic nature of quantum mechanics. It employs nonstandard analysis and is based on Feynman's interpretation of the Heisenberg uncertainty principle, i.e., of the quantum fluctuations, which was brought to the forefront in some fractal approaches. It results, as in Nelson's stochastic mechanics, in stochastic differential equations which are deduced from infinitesimal random walks. To cite this article: M. Fliess, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Non-conventional approach in dynamics of high-intensity beams

In beam dynamics a detailed formalism based on Nelson's stochastic mechanics is presented for the description of betatron and synchrotron oscillations in both undamped and damped cases. The propagator formulation is used. The requirement of a comparison with more conventional (classical) approaches is emphasized.

Quantum Mechanics and Stochastic Mechanics for Compatible Observables at Different Times

Bohm mechanics and Nelson stochastic mechanics are confronted with quantum mechanics in the presence of noninteracting subsystems. In both cases, it is shown that correlations at different times of compatible position observables on stationary states agree with quantum mechanics only in the case of product wave functions. By appropriate Bell-like inequalities it is shown that no classical theory, in particular no stochastic process, can reproduce the quantum mechanical correlations of position variables of noninteracting systems at different times.

Orbital angular momentum in Nelson's stochastic mechanics

We provide numerical examples of integer-valued functionals of Nelson's stochastic processes. More precisely, we consider stochastic motion, according to Nelson's form of Newton's second law of dynamics, in a magnetic field having an axis z of cylindrical symmetry and a gradient in the direction of this axis. We show that there are two sets of functionals of the stochastic process having the same law as the component of quantum mechanical angular momentum along z. The functionals of the first set involve the z coordinate of the process and correctly model the behaviour of the `needle of the measuring apparatus'. The functionals of the second set involve the coordinates in a plane orthogonal to z and strongly suggest the possibility of a stochastic model of the collapse of the `system' toward the state indicated by the `needle'.

NELSON'S STOCHASTIC MECHANICS BY MEANS OF FEYNMAN PATH INTEGRALS

We show that Nelson's stochastic mechanics suitably formulated as a Hamilton–Jacobi first-order equation leads straightforwardly to the Feynman path integral formulation of quantum mechanics.

Calogero Conjecture and Hubble Law

Acceptance of the Calogero hypothesis on the “cosmic origin of quantization”in the framework of Nelson stochastic mechanics would imply that the age ofuniverse is larger than the standard quantum mechanical interpretation of theredshift measurement implies. This is due to variation of h with the radius of theuniverse and thus with time.