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arXiv:quant-ph/0304133 · Il Nuovo Cimento B

A de Broglie-Bohm Like Model for Klein-Gordon Equation

O. Chavoya-Aceves

Published Apr 21, 2003• 0 citations
arXivHTMLPDF

Abstract

A de Broglie-Bohm like model of Klein-Gordon equation, that leads to the correct Schrodinger equation in the low-speed limit, is presented. Under this theoretical framework, that affords an interpretation of the quantum potential, the main assumption of the de Broglie-Bohm interpretation--that the local momentum of particles is given by the gradient of the phase of the wave function--is not but approximately correct. Also, the number of particles is not locally conserved. Furthermore, the representation of physical systems through wave functions wont be complete.

DOI
10.48550/arxiv.quant-ph/0304133

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A de Broglie-Bohm like model of Klein-Gordon equation, that leads to the correct Schrodinger equation in the low-speed limit, is presented. Under this theoretical framework, that affords an interpretation of the quantum potential, the main assumption of the de Broglie-Bohm interpretation--that the local momentum of particles is given by the gradient of the phase of the wave function--is not but approximately correct. Also, the number of particles is not locally conserved. Furthermore, the representation of physical systems through wave functions wont be complete.
A de Broglie-Bohm like model of Dirac equation, that leads to the correct Pauli equations for electrons and positrons in the low-speed limit, is presented. Under this theoretical framework, that affords an interpretation of the quantum potential, the main assumption of the de Broglie-Bohm theory--that the local momentum of particles is given by the gradient of the phase of the wave function--wont be accurate. Also, the number of particles wont be locally conserved. Furthermore, the representation of physical systems through wave functions wont be complete.
The Klein-Gordon equation is interpreted in the de Broglie-Bohm manner as a single-particle relativistic quantum mechanical equation that defines unique time-like particle trajectories. The particle trajectories are determined by the conserved flow of the intrinsic energy density which can be derived from the specification of the Klein-Gordon energy-momentum tensor in an Einstein-Riemann space. The approach is illustrated by application to the simple single-particle phenomena associated with square potentials.
The Klein-Gordon equation is interpreted in the de Broglie-Bohm manner as a single-particle relativistic quantum mechanical equation that defines unique time-like particle trajectories. The particle trajectories are determined by the conserved flow of the intrinsic energy density, which can be derived from the specification of the Klein-Gordon energy-momentum tensor in an Einstein-Riemann space. The approach is illustrated by application to the simple single-particle phenomena associated with square potentials.
In the stochastic electrodynamics, various quantum effects on particles are supposed to emerge from the interactions of the particles with the fluctuating background field. We have adopted the same supposition and further supposed that the quantum potential arises from the velocity fluctuations of particles around their local averages. Based on these suppositions we have derived the expression of the quantum potential in the framework of de Broglie–Bohm theory, which is a reformulation of Schrödinger’s. Thus we are allowed to use the wave functions and Schrödinger operators explicitly so that we have been able to derive the quantum potential in simpler and more compact manner as compared with the stochastic electrodynamics’s. A new hydrodynamical version of Schrödinger equation is also presented, which has been found in the course of our study.
We define the de Broglie-Bohm (dBB) weak interpretation as the dBB interpretation restricted to particles in unbound states whose wave function is defined in the three-dimensional physical space, and the dBB strong interpretation as the usual dBB interpretation applied to all wave functions, in particular to particles in bound states whose wave function is defined in a 3N-dimensional configuration space in which N is the number of particules. We show that the current criticisms of the dBB interpretation do not apply to this weak interpretation and that, furthermore, there are theoritical and experimental reasons to justify the weak dBB interpretation. Theoretically, the main reason concern the continuity existing for such particles between quantum mechanics and classical mechanics: we demonstrate in fact that the density and the phase of the wave function of a single-particle (or a set of identical particles without interaction), when the Planck constant tends to 0, converges to the density and the action of a set of unrecognizable prepared classical particles that satisfy the statistical Hamilton-Jacobi equations. As the Hamilton-Jacobi action pilots the particle in classical mechanics, this continuity naturally concurs with the weak dBB interpretation. Experimentally, we show that the measurement results of the main quantum experiments (Young's slits experiment, Stern and Gerlach, EPR-B) are compatible with the de Broglie-Bohm weak interpretation and everything takes place as if these unbounded particles had trajectories. In addition, we propose two potential solutions to complete the dBB weak interpretation.
Following de Broglie, Bohm, and others, it is assumed that quantum mechanics may be interpreted causally and that the $\ensuremath{\psi}$ function plays the role of a generating function for particle trajectories. By arguing that the $\ensuremath{\psi}$ function should not be interpreted as a probability amplitude, a new method for generating particle trajectories is postulated. The four-momentum of a scalar particle is assumed to be given as the gradient of an unspecified function $F(\ensuremath{\psi})$, where $\ensuremath{\psi}$ is a pure real solution of the Klein-Gordon equation. Since the location of a particle is determined solely by its trajectory, the probability distribution differs from $\ensuremath{\psi}{\ensuremath{\psi}}^{*}$; and therefore, ordinary experimental results differing from the traditional theory may, in principle, be predicted. Particle motion and trajectories are discussed for three examples: a free particle, a particle in a box, and the double slit.