Proof That Probability Density Approaches ${\left|\psi \right|}^2$ in Causal Interpretation of the Quantum Theory

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 ${\left|\psi \right(\mathrm{x}, t\left)\right|}^2$.

In the previous papers we also pointed out that, within the conceptual framework of the causal interpretation, it was possible to suggest mathematical theories more general than are permitted by the usual interpretation and that these more general theories might be needed in the domain of ${10}^{-13\mathrm{}}$ cm, where present theories seem to fail. However, if these more general theories should apply at the level of ${10}^{-13\mathrm{}}$ cm, then there would be a tendency to create discrepancies between $P$ and ${\left|\psi \right|}^2$, a tendency whose cumulative effects should be felt even at the atomic level, where the more general theory ought to approach the usual theory. However, because those discrepancies have been shown to die out as a result of collisions, we can expect that under normal conditions the difference between $P$ and ${\left|\psi \right|}^2$ would be negligible. Conditions are suggested, however, in which this difference might be appreciable, and experiments are indicated which might be able to test for the existence of such discrepancies.