Abstract
The basic equations of the non-relativistic quantum mechanics with trajectories and quantum hydrodynamics are extended to the relativistic domain. This is achieved by using a Schr?dinger-like equation, which describes a particle with mass and spin-0 and with the correct relativistic relation between its linear momentum and kinetic energy. Some simple but instructive free particle examples are discussed.
Highlights
Other general approaches have been reported [10] [11], but we explore in this work an alternative methodology for extending, to the relativistic domain, the known non-relativistic quantum hydrodynamics and quantum theories with trajectories
Inserting Equation (6) in Equation (5) and following step by step Ref. [6], we can obtain the following equations, which extend to the relativistic domain the basic equations of the Madelung-de Broglie-Bohm quantum mechanics [6]:
Γ v ≈ 1 when v2 c2 ; as it should be expected when the particle moves at low speeds, Equations (7) and (8) coincide to the well-known equations of the Madelung-de Broglie-Bohm quantum mechanics [6]
Summary
The Euler equation is a particular case of the Navier-Stokes equation [3] Such hydrodynamic interpretation is considered a forebear of the de Broglie-Bohm Pilot Wave. Most of the work related to the Madelung-de-Broglie-Bohm reformulation of quantum mechanics and quantum hydrodynamics applies to particles moving slowly respect to the speed of light. In a nutshell, solving Equation (1) requires simultaneously finding the wavefunction ψ and the square of the particle v2, which determines the value of γv in Equation (2). This may look at first as an unmanageable problem; this is not the case in at least several interesting cases [12]-[17].
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