Primary Assumptions and Guidance Laws in Wave Mechanics

In an article written by Louis de Broglie in 1959 (30 years after the Nobel prize rewarding his foundation of Wave Mechanics), the most challenging problem raised by the Bohr, Heisenberg and Born Standard Quantum Mechanics (SQM) was pointed out in the renunciation to describe \u201ca permanent localization in space, and therefore a well-defined trajectory\u201d for any moving particle. This challenge is taken up in the present paper, showing that de Broglie\u2019s Primary Assumption p = \u127k , predicting the wave-particle duality, does also allow to obtain from the energy-dependent form of the Schr\uf6dinger and/or Klein-Gordon equations the Guidance Laws piloting particles along well-defined trajectories. The energy-independent equations, on the other hand, may only give rise - both in SQM and in the Bohmian approach - to probabilistic descriptions, overshadowing the role of de Broglie\u2019s matter waves in physical space