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Stochastic mechanics and neural networks

Kai-Hendrik Henk

Published Jan 1, 2025• 0 citations
Publisher

Abstract

Quantum mechanics is one of the most successful theories of modern physics. Nelson’s stochastic mechanics description offers an alternative approach to non-relativistic quantum mechanics based on Newtonian mechanics. In this thesis, we derive the stochastic mechanical equivalent to the quantum-mechanical Rayleigh-Ritz principle. This principle is then used to build a genetic algorithm that calculates the osmotic velocity together with the ground state energy by first minimizing the derived energy functional and then solving a Riccati equation. This new and efficient algorithm is then used to solve the ground state of two tweezer potentials used in levitodynamics. These are then compared to each other and with the harmonic oscillator by using methods from time series analysis. Additionally, we use the Itô-formula to derive two coupled stochastic differential equations, which yields a phase space description of quantum mechanics without violating the Heisenberg uncertainty principle.

DOI
10.25673/122265

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Quantum mechanics is one of the most successful theories of modern physics. Nelson’s stochastic mechanics description offers an alternative approach to non-relativistic quantum mechanics based on Newtonian mechanics. In this thesis, we derive the stochastic mechanical equivalent to the quantum-mechanical Rayleigh-Ritz principle. This principle is then used to build a genetic algorithm that calculates the osmotic velocity together with the ground state energy by first minimizing the derived energy functional and then solving a Riccati equation. This new and efficient algorithm is then used to solve the ground state of two tweezer potentials used in levitodynamics. These are then compared to each other and with the harmonic oscillator by using methods from time series analysis. Additionally, we use the Itô-formula to derive two coupled stochastic differential equations, which yields a phase space description of quantum mechanics without violating the Heisenberg uncertainty principle.
The Rayleigh-Ritz variation principle is a proven way to find ground states and energies for bound quantum systems in the Schr\"odinger picture. Advances in machine learning and neural networks make it possible to extend it from an analytical search from a subspace of the complete Hilbert space to the a numerical search in the almost complete Hilbert space. In this paper, we extend the Rayleigh-Ritz principle to Nelson's stochastic mechanics formulation of nonrelativistic quantum mechanics and propose an algorithm to find the osmotic velocities u(x)u(x)u(x), which contain the information of a quantum systems in this picture. As a proof of concept, we calculated u(x)u(x)u(x) for one-dimensional systems, the harmonic oscillator, the double well and the P\"oschl-Teller potential. To obtain exited states, we calculate ground states of supersymmetrical partner Hamiltonians for each of these potentials. We will show that this method is more efficient than the stochastic optimal control algorithm that was the usual method to obtain osmotic velocities without going back to the Schr\"odinger equation.
The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. The Schroedinger equation and the so-called Nelson equations are used to determine the diffusion tensor and the drift vectors characteristic of such a process. It is then possible to write down the forward and backward Kolmogorov equations that are used to determine the conditional probability density, as well as the Fokker-Planck equation that is used to determine the normal probability density. This method is applied to several simple cases and the results obtained are compared with those of the orthodox theory. Among the most important differences let us mention that any system evolving freely from any original state returns spontaneously to its ground state, that the definitions of the particles velocities and momenta are impossible because the sample functions of a diffusion Markov process nowhere possess a derivative with respect to time and that the so-called collapse of the wave function is a mere mirage explained by the updating choice between new and older initial conditions in the resolution of partial differential equations. We finally attempt to extend the proposed theory to the domain of the relativistic quantum mechanics. It is promising but is clearly unfinished because it has not been possible up to now to solve the Nelson and Kolmogorov equations, except in the very simple case of a free particle.
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
In this paper we establish three variational principles that provide new foundations for Nelson’s stochastic mechanics in the case of nonrelativistic particles without spin. The resulting variational picture is much richer and of a different nature with respect to the one previously considered in the literature. We first develop two stochastic variational principles whose Hamilton–Jacobi-like equations are precisely the two coupled partial differential equations that are obtained from the Schrödinger equation (Madelung equations). The two problems are zero-sum, noncooperative, stochastic differential games that are familiar in the control theory literature. They are solved here by means of a new, absolutely elementary method based on Lagrange functionals. For both games the saddle-point equilibrium solution is given by the Nelson’s process and the optimal controls for the two competing players are precisely Nelson’s current velocity v and osmotic velocity u, respectively. The first variational principle includes as special cases both the Guerra–Morato variational principle [Phys. Rev. D 27, 1774 (1983)] and Schrödinger original variational derivation of the time-independent equation. It also reduces to the classical least action principle when the intensity of the underlying noise tends to zero. It appears as a saddle-point action principle. In the second variational principle the action is simply the difference between the initial and final configurational entropy. It is therefore a saddle-point entropy production principle. From the variational principles it follows, in particular, that both v(x,t) and u(x,t) are gradients of appropriate principal functions. In the variational principles, the role of the background noise has the intuitive meaning of attempting to contrast the more classical mechanical features of the system by trying to maximize the action in the first principle and by trying to increase the entropy in the second. Combining the two variational principles, we get the quantum Hamilton principle, i.e., a variational characterization of the logarithm of the wave function ψ. The Lagrangian is the Lagrangian of classical mechanics with the complex-valued velocity v−iu replacing the classical velocity. The dynamics is given by a stochastic differential equation for real-valued diffusions with complex-valued drift and driving noise processes. From the variational principle we derive a Newton-type law. We finally define the momentum process and show that its mean and variance coincide with those of the quantum momentum operator.
The stochastic differential equation of Nelson's stochastic mechanics is integrated numerically for several simple quantum systems. The calculations are performed with use of Helfand and Greenside's method and pseudorandom numbers. The resulting trajectories are analyzed both individually and collectively to yield insight into momentum, uncertainty principles, interference, tunneling, quantum chaos, and common models of diatomic molecules from the stochastic quantization point of view. In addition to confirming Shucker's momentum theorem, these simulations illustrate, within the context of stochastic mechanics, the position-momentum and time-energy uncertainty relations, the two-slit diffraction pattern, exponential decay of an unstable system, and the greater degree of anticorrelation in a valence-bond model as compared with a molecular-orbital model of H2{\mathrm{H}}_{2}H2​. The attempt to find exponential divergence of initially nearby trajectories, potentially useful as a criterion for quantum chaos, in a periodically forced oscillator is inconclusive. A way of computing excited energies from the ground-state motion is presented. In all of these studies the use of particle trajectories allows a more insightful interpretation of physical phenomena than is possible within traditional wave mechanics.
On the basis of Nelson’s stochastic mechanics derivation of the Schrödinger equation, a formal mathematical structure of non-relativistic quantum mechanics equivalent to the one in classical analytical mechanics has been established in the literature. We recently were able to augment this structure by deriving quantum Hamilton equations of motion by finding the Nash equilibrium of a stochastic optimal control problem, which is the generalization of Hamilton’s principle of classical mechanics to quantum systems. We showed that these equations allow a description and numerical determination of the ground state of quantum problems without using the Schrödinger equation. We extend this approach here to deliver the complete discrete energy spectrum and related eigenfunctions for bound states of one-dimensional stationary quantum systems. We exemplify this analytically for the one-dimensional harmonic oscillator and numerically by analyzing a quartic double-well potential, a model of broad importance in many areas of physics. We furthermore point out a relation between the tunnel splitting of such models and mean first passage time concepts applied to Nelson’s diffusion paths in the ground state.