# de Broglie's Exact Trajectories

## TL;DR

This paper explores de Broglie's approach to quantum particles, proposing that particles follow virtual trajectories derived from time-independent Schrödinger or Klein-Gordon equations, akin to classical dynamics.

## Contribution

It introduces a framework where particles are associated with virtual trajectories determined by initial conditions and quantum equations, extending de Broglie's wave-like interpretation.

## Key findings

- Particles follow virtual trajectories independent of their presence.
- Trajectory determination depends on initial conditions and quantum equations.
- Framework applies to time-independent external fields.

## Abstract

De Broglie's quest for a wave-like approach capable of representing the position of a moving particle, is satisfied, in the case of time-independent external fields, by assuming that each particle runs along the virtual trajectories associated, under assigned starting conditions, with a time-independent Schr\"odinger (or Klein-Gordon) equation. Just like in classical Dynamics, indeed, the starting conditions determine, from the beginning, a set of virtual particle trajectories independent from the very presence of the particles.

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Source: https://tomesphere.com/paper/1903.01870