Quantum Trajectories based on the Weak Value

TL;DR

This paper investigates the physical meaning of weak trajectories derived from weak values in quantum mechanics, demonstrating their relation to classical paths and interference effects in experiments like double slit and Lloyd's mirror.

Contribution

It provides a novel interpretation of weak trajectories as averages over classical paths and explores their universal applicability in quantum interference scenarios.

Findings

Weak trajectories relate to classical path averages.

Imaginary parts of weak trajectories connect to interference variations.

Universal applicability of the average interpretation in quantum interference.

Abstract

The notion of trajectory of an individual particle is strictly inhibited in quantum mechanics because of the uncertainty principle. Nonetheless, the weak value, which has been proposed as a novel and measurable quantity definable to any quantum observable, can offer a possible description of trajectory on account of its statistical nature of the value. In this paper, we explore the physical significance provided by this weak trajectory by considering various situations where interference takes place simultaneously with the observation of particles, that is, in prototypical quantum situations for which no classical treatment is available. These include the double slit experiment and Lloyd's mirror, where in the former case it is argued that the real part of the weak trajectory describes an average over the possible classical trajectories involved in the process, and that the imaginary part is related to the variation of interference. It is shown that this average interpretation of the weak trajectory holds universally under the complex probability defined from the given transition process. These features remain essentially unaltered in the case of Lloyd's mirror where interference occurs with a single slit.