Uniqueness of Bohmian Mechanics, and Solutions From Probability Conservation

TL;DR

This paper proves that in one-dimensional Bohmian mechanics, the only trajectories conserving total probability on one side are the Bohm trajectories, highlighting their uniqueness and extending the concept to separable higher-dimensional cases.

Contribution

It demonstrates the uniqueness of Bohmian trajectories based on probability conservation and clarifies the role of gauge freedom in the probability current.

Findings

Bohm trajectories uniquely conserve total side probability in 1D.

The approach extends to higher dimensions with separable wave functions.

Examples include the two-slit experiment illustrating probability conservation.

Abstract

We show that one-dimensional Bohmian mechanics is unique, in that, the Bohm trajectories are the only solutions that conserve total left (or right) probability. In Brandt et al., Phys. Lett. A, 249 (1998) 265--270, they define quantile motion--unique trajectories are solved by assuming that the total probability on each side of the particle is conserved. They argue that the quantile trajectories are identical to the Bohm trajectories. Their argument, however, fails to notice the gauge freedom in the definition of the quantum probability current. Our paper sidesteps this under-determinedness of the probability current. The one-dimensional probability conservation can be used for higher dimensional problems if the wave function is separable. Several examples are given using total left probability conservation, most notably, the two-slit experiment.