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Oxford University Press eBooks

The Path Not Taken

Michael G. Munowitz

Published Jan 12, 2006• 0 citations
Publisher

Abstract

The quantum mechanical superposition state — the wave function — takes shape, and the rules for its deterministic evolution and probabilistic interpretation are developed. A clockwork macrocosm in which the Moon never wavers from its course gives way to a guesswork microcosm in which an electron follows no predetermined path at all. This leads to a quantum mechanical world in which the sharp values of position, velocity, momentum, energy, and all the other quantities of classical mechanics blur into the probability distributions of quantum mechanics.

DOI
10.1093/acprof:oso/9780195167375.003.0008

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The quantum mechanical superposition state — the wave function — takes shape, and the rules for its deterministic evolution and probabilistic interpretation are developed. A clockwork macrocosm in which the Moon never wavers from its course gives way to a guesswork microcosm in which an electron follows no predetermined path at all. This leads to a quantum mechanical world in which the sharp values of position, velocity, momentum, energy, and all the other quantities of classical mechanics blur into the probability distributions of quantum mechanics.
A type of mechanics will be presented that possesses some distinctive properties. On the one hand, its physical description & rules of operation are readily comprehensible & intuitively clear. On the other, it fully satisfies all observable predictions of non-relativistic quantum mechanics. Within it, particles exist at points in space, follow continuous, piecewise differentiable paths, and their linear momentum is equal to their mass times their velocity along their path. Yet the probabilities for position and momentum, conditioned on the state of the particle's environment, follow the rules of quantum theory. Indeed, all observable consequences of quantum theory are satisfied; particles can be entangled, have intrinsic spin, this spin is not local to the particle, particle identity can effect probabilities, and so forth. All the rules of quantum mechanics are obeyed, and all arise in a straightforward fashion. After this is established, connections will be drawn out between this type of mechanics and other types of quantum worlds; those that obey Bohmian mechanics, stochastic mechanics, the many worlds interpretation, and physical collapse. In the final section, a relativistic version of the mechanics will be presented.
Classical transport equations with probabilistic initial conditions can be viewed as quantum systems. In a discrete version they are probabilistic automata. The time-local probabilistic information is encoded in a classical wave function. Its unitary evolution obeys a Schrödinger equation. Statistical observables measure properties of the classical probability distribution. In the quantum formalism they are represented by operators which do not commute with the ones associated to classical observables. Momentum, quantum energy and angular momentum or charges yield conserved quantities which constrain the evolution of the classical probability distribution. The characteristic features of quantum mechanics, as the superposition of wave functions, interference, the importance of phases, non-commuting operators or a unitary time evolution, are realized by probabilistic classical transport equations. Stochastic transport equations are described by a quantum system with a stochastic Hamiltonian.
Quantum mechanics has been enormously successful in explaining the behavior of microsystems such as molecules, atoms, and nuclei. We have seen that quantum mechanics describes a quantum system by the wave function ψ, which is the solution of the Schro¨dinger equation. The wave function determines the probability amplitudes (whose modulus squared gives the probability) for all possible outcomes of a measurement corresponding to an observable on the system. In general, the wave function does not uniquely predict the result of a measurement; it only provides a probability distribution for all possible results of the measurement on a quantum system. Only if the system happens to be in an eigenstate corresponding to the observable being measured, is the result of a measurement of this observable completely predictable. On the other hand, the results of measurements corresponding to a conjugate or non-commuting observable, on the same system in the same state, are indeterminate. Quantum mechanical description of physical systems raises profound questions: How are the wave function and the probabilities for different outcomes of a measurement to be interpreted? Does the wave function represent a single system or an ensemble of systems? Does the physical system actually have the attribute corresponding to the measurement in question or does the measurement itself create the attribute? These questions led to different interpretations of quantum mechanics, which were vigorously debated during the years of its discovery as also in the following decades. The new idea of indeterminacy was especially difficult to reconcile with the long-held view that a physical theory should aim at predicting accurately, and without ambiguity, the result of any measurement on any system, big or small. These debates not only gave us a deeper understanding of quantum mechanics but also changed profoundly the way we view the micro-world.
This paper develops a neural architecture capable of reproducing several structural aspects of quantum mechanics. The proposed model, termed the Born Perceptron, extends the classical perceptron by structuring outputs as value–index pairs and interpreting normalized activations as probability distributions over indexed outcomes. By allowing indices to inhabit geometric manifolds rather than discrete labels, the perceptron naturally represents probabilistic sampling over state spaces. When stacked sequentially, such perceptrons generate stochastic trajectories reminiscent of quantum state evolution under repeated measurement. Introducing complex-valued activations yields probabilities determined by squared magnitudes, reproducing the Born rule. The framework is further extended to treat measurements distributed in time, particularly position samplings. In this regime, the neural system produces sequences of sampled positions whose joint probabilities resemble those derived in path-integral formulations of quantum mechanics. The resulting architecture offers a classical computational model that captures several conceptual features of quantum theory including measurement, collapse, trajectory evolution, and amplitude-based probability rules.
Probabilistic cellular automata with deterministic updating are discrete quantum systems. They are characterized by a probability distribution over initial configurations. We employ the quantum formalism for an investigation of random probabilistic cellular automata for which the properties of the deterministic updating are randomly distributed over space and time. We are interested in a possible continuum limit for a very large number of cells or space points. We consider bits with two colors, moving to the left or right on a linear chain. At randomly distributed scattering points they change direction and color. A numerical simulation reveals the typical features of quantum systems. We find particular initial probability distributions which reemerge periodically after a certain number of time steps, as produced by the periodic evolution of energy eigenstates in quantum mechanics. Using a description in terms of wave functions allows us to introduce statistical observables for momentum and energy. They characterize the probabilistic information without taking definite values for a given bit configuration, with a conceptual status similar to temperature in classical statistical thermal equilibrium. Conservation of energy and momentum are essential ingredients for the understanding of the evolution of the stochastic probabilistic automata. This evolution resembles in some aspects a single Dirac fermion in two dimensions with a generalized random potential.