Proof That Probability Density Approaches|ψ|2in Causal Interpretation of the Quantum Theory

Abstract

In two previous papers a causal interpretation of the quantum theory was developed which involved the hypothesis that a quantum-mechanical system contains a precisely defined particle variable x but that, at present, we are restricted to calculating the probability density $P(\mathrm{x}, t)$ that the particle is at the position x. It was shown that the assumption that $P(\mathrm{x}, t)={\left|\psi \right(\mathrm{x}, t\left)\right|}^2$ is consistent, in the sense that if it holds initially, the equations of motion of the particles will cause this relation to be maintained for all time. In this paper, we extend the theory by showing that as a result of random collisions, an arbitrary probability density will ultimately decay into one with a density of ${\left|\psi \right(\mathrm{x}, t\left)\right|}^2$. Since all quantum-mechanical experiments to date have been concerned with statistical ensembles of systems that have been colliding with other systems for a very long time, it is therefore inevitable that as we draw samples from such ensembles, the probability density of systems with particles at the point x will be equal to ${\left|\psi \right(\mathrm{x}, t\left)\right|}^2$.