Probability and complex quantum trajectories

TL;DR

This paper extends the probability interpretation of quantum mechanics into the complex plane using complex trajectories, deriving a conserved probability density consistent with Born's rule and providing computational methods for its determination.

Contribution

It introduces a new complex-plane probability density framework based on complex trajectories, extending Born's rule beyond the real axis and deriving it from the complex Schrödinger equation.

Findings

A conserved probability density can be derived from complex trajectories.

The probability density matches Born's rule along the real line.

An integral method simplifies computing the probability density.

Abstract

It is shown that in the complex trajectory representation of quantum mechanics, the Born's Psi^{\star}\Psi probability density can be obtained from the imaginary part of the velocity field of particles on the real axis. Extending this probability axiom to the complex plane, we first attempt to find a probability density by solving an appropriate conservation equation. The characteristic curves of this conservation equation are found to be the same as the complex paths of particles in the new representation. The boundary condition in this case is that the extended probability density should agree with the quantum probability rule along the real line. For the simple, time-independent, one-dimensional problems worked out here, we find that a conserved probability density can be derived from the velocity field of particles, except in regions where the trajectories were previously suspected to be nonviable. An alternative method to find this probability density in terms of a trajectory integral, which is easier to implement on a computer and useful for single particle solutions, is also presented. Most importantly, we show, by using the complex extension of Schrodinger equation, that the desired conservation equation can be derived from this definition of probability density.