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Fortschritte der Physik

Schrödinger's Equation and Classical Brownian Motion

Garnet N. Ord

Published Nov 1, 1998• 5 citations
Publisher

Abstract

In the context of quantum mechanics, Schrödinger's equation has many possible interpretations, none of which appear to be entirely satisfactory. One reason for this is that there is no known microscopic model which yields both Schrödinger's equation and an accompanying ‘sensible’ interpretation. In contrast to this the diffusion equation is well known as a phenomenological equation obtainable from the microscopic model of Brownian motion. In this case the microscopic model provides an unambiguous interpretation of the partial differential equation. In this article we review work which shows that Schrödinger's equation by itself is also easily obtained from a lattice random walk version of the Brownian motion model. In this system, Schrödinger's equation arises by projection and the interpretation of wave functions is as direct and unambiguous as the interpretation of density in the diffusion case. However the direct interpretation is compatible with quantum mechanics only in a statistical sense in the context of ensembles of particles. This shows that measurement theory is the aspect of quantum theory which harbours the problems of interpretation, and we suggest the study of this and related systems to clearly illuminate the boundary between dynamics and measurement.

DOI
10.1002/(sici)1521-3978(199811)46:6/8<889::aid-prop889>3.0.co;2-z

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In the context of quantum mechanics, Schrödinger's equation has many possible interpretations, none of which appear to be entirely satisfactory. One reason for this is that there is no known microscopic model which yields both Schrödinger's equation and an accompanying ‘sensible’ interpretation. In contrast to this the diffusion equation is well known as a phenomenological equation obtainable from the microscopic model of Brownian motion. In this case the microscopic model provides an unambiguous interpretation of the partial differential equation. In this article we review work which shows that Schrödinger's equation by itself is also easily obtained from a lattice random walk version of the Brownian motion model. In this system, Schrödinger's equation arises by projection and the interpretation of wave functions is as direct and unambiguous as the interpretation of density in the diffusion case. However the direct interpretation is compatible with quantum mechanics only in a statistical sense in the context of ensembles of particles. This shows that measurement theory is the aspect of quantum theory which harbours the problems of interpretation, and we suggest the study of this and related systems to clearly illuminate the boundary between dynamics and measurement.
Introduction. There exists a very comprehensive theory concerning the time development of probabilistic systems. This is the Theory of Stochastic processes (1). Quantum mechanics as described by the Schrödinger equation rests very heavily on probabilistic interpretation, and yet, after forty years of remarkably rapid advances and successes, it is not clear how it is related to stochastic theory. The formal similarity of certain aspects of, and equations used in these two theories are well known. The diffusion equation corresponds to the Schrödinger equation, a correspondence only marred by a square root of minus unity appearing with the time derivative in the latter equation. The Kolmogoroff–Chapman relation for probabilities in Brownian motion has an analogue (2) in quantum mechanics, not for probabilities though, but for the complex probability amplitudes. The uncertainty relation also has a stochastic analogue (3). Some of these analogous characteristics are actually symptomatic of the gulf which separates these two theories. There are solutions to the diffusion equation which have a very clear physical probabilistic meaning. This is not the case with the corresponding and formally similar solutions to the Schrödinger equation and is partly due to the √ − 1 mentioned earlier. The probabilistic interpretation of the solutions to the diffusion equation is direct and immediate. On the other hand, the probabilistic interpretation of the solutions, ψ of the Schrödinger equation involve the introduction of the probability density, p = ψ*ψ. One great difference between these two theories lies in their each having distinct principles of linear superposition. In Stochastic Theory probabilities are superposed and in quantum mechanics amplitudes or wave functions are superposed.
The connection between the Schrödinger equation and Einstein's diffusion theory based on the Brownian motion of independent particles is well known. However, in contrast to diffusion theory, quantum mechanics theory has suffered controversial interpretations due to the counterintuitive concept of wavefunction. Here, while we confirm there is no difference in the mathematical form of these two equations, we derive the imaginary version of displacement. Using the diffusion theory of particles in a medium, as simple as it is, we describe that quantum mechanics is just an elegant and subtle equation to describe the probability of all the trajectories that a particle can take to propagate in time by a predictive wavefunction. Therefore, information on the position of particles through time in quantum theory is embedded in the wavefunction, which predicts the evolution of an ensemble of individual Brownian particles.
The motion of a ball through an appropriate lattice of round obstacles models the behavior of a Brownian particle and can be used to describe measurement on a macro system. On another hand, such motion is chaotic and a known conjecture asserts that the Hamiltonian of the corresponding quantum system must follow the random matrix statistics of an appropriate ensemble. We use the Hamiltonian represented by a random matrix in the Gaussian unitary ensemble to study the Schrödinger evolution of non-stationary states. For Gaussian states representing a classical system, the Brownian motion that describes the behavior of the system under measurement is obtained. For general quantum states, the Born rule for the probability of transition between states is derived. It is then shown that the Schrödinger evolution with such a Hamiltonian models measurement on macroscopic and microscopic systems, provides an explanation for the classical behavior of macroscopic bodies and for irreversibility of a measurement, and identifies the boundary between micro and macro worlds.
Abstract The motion of a ball through an appropriate lattice of round obstacles models the behavior of a Brownian particle and can be used to describe measurement on a macro system. On another hand, such motion is chaotic and a known conjecture asserts that the Hamiltonian of the corresponding quantum system must follow the random matrix statistics of an appropriate ensemble. We use the Hamiltonian represented by a random matrix in the Gaussian unitary ensemble to study the Schrödinger evolution of non-stationary states. For Gaussian states representing a classical system, the Brownian motion that describes the behavior of the system under measurement is obtained. For general quantum states, the Born rule for the probability of transition between states is derived. It is then shown that the Schrödinger evolution with such a Hamiltonian models measurement on macroscopic and microscopic systems, provides an explanation for the classical behavior of macroscopic bodies and for irreversibility of a measurement, and identifies the boundary between micro and macro worlds.
The Schrödinger free particle equation in 1+1 dimension describes second-order effects in ensembles of lattice random walks, in addition to its role in quantum mechanics, and its solutions represent the continuous limit of a property of ensembles of Brownian particles. In the present paper, the classical Schrödinger and Dirac equations have been derived from the Brownian motions of a particle, and it has been shown that the classical Schrödinger equation can be transformed into the usual Schrödinger quantum equation on applying the Heisenberg uncertainty principle between position and momentum, while the Dirac quantum equation follows from its classical counterpart on applying the Heisenberg uncertainty principle between energy and time, without applying any analytical continuation.
The Schrödinger and diffusion equations are normally related only through a formal analytic continuation. There are apparently no intermediary partial differential equations with physical interpretations that can form a conceptual bridge between the two. However, if one starts off with a symmetric binary random walk on a lattice then it is possible to show that both equations occur as approximate descriptions of different aspects of the same classical probabilistic system. This suggests that lattice calculations may prove to be a useful intermediary between classical and quantum physics.