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Journal of the Physical Society of Japan

A Compact Derivation of Quantum Potential in de Broglie–Bohm Theory

Osamu Matsuoka

Published Jan 13, 2023• 1 citation
Publisher

Abstract

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.

DOI
10.7566/jpsj.92.024002

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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.
Non-relativistic de Broglie–Bohm theory describes particles moving under the guidance of the wave function. In de Broglieʼs original formulation, the particle dynamics is given by a first-order differential equation. In Bohmʼs reformulation, it is given by Newtonʼs law of motion with an extra potential that depends on the wave function—the quantum potential—together with a constraint on the possible velocities. It was recently argued, mainly by numerical simulations, that relaxing this velocity constraint leads to a physically untenable theory. We provide further evidence for this by showing that for various wave functions the particles tend to escape the wave packet. In particular, we show that for a central classical potential and bound energy eigenstates the particle motion is often unbounded. This work seems particularly relevant for ways of simulating wave function evolution based on Bohmʼs formulation of the de Broglie–Bohm theory. Namely, the simulations may become unstable due to deviations from the velocity constraint.
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.
In the de Broglie–Bohm approach to quantum mechanics for a charged particle in a time-dependent electromagnetic field the time derivative of the energy is equal to the classical power plus a quantum power. We show that the average of the quantum power is zero. The de Broglie–Bohm energy is obtained from the quantum mechanical energy operator, which is the Hamiltonian with the gauge-dependent scalar potential subtracted. The time derivative of the average de Broglie–Bohm energy is shown to be equal to Ehrenfest's theorem for the quantum energy operator.
In this paper we suggested a natural interpretation of the de Broglie-Bohm quantum potential, as the energy due to the oscillating electromagnetic field (virtual photon) coupled with moving charged particle. Generalization of the Schrödinger equation is obtained. The wave function is shown to be the eigenfunction of the Sturm-Liouville problem in which we expand virtual photon to include it implicitly into consideration. It is shown that the non-locality of quantum mechanics is related only with virtual photon. As an example, the zero-energy of harmonic oscillator is obtained from classical equations.
The de Broglie–Bohm theory allows us to have got a satisfactory geometrodynamic interpretation of quantum mechanics. The fundamental element, which creates a geometrodynamic picture of the quantum world in the non-relativistic domain, a relativistic curved space-time background, and the quantum gravity domain, is the quantum potential. It is shown that, in the non-relativistic domain, the geometrodynamic nature of the quantum potential followsfrom the fact that it is an information potential containing a space-like active information on the environment; the geometric properties of the space expressed by the quantum potential determine non-local correlations between subatomic particles. Moreover, in the de Broglie–Bohm theory in a curved space-time, it is shown that the quantum, as well as the gravitational, effects of matter have geometric nature and are highly related: the quantum potential can be interpreted as the conformal degree of freedom of the space-time metric, and its presence is equivalent to the curved space-time. It is shown on the basis of some recent research that, in quantum gravity, we have a generalized geometric unification of gravitational and quantum effects of matter; Bohm's interpretation shows that the form of a quantum potential and its relation to the conformal degree of freedom of the space-time metric can be derived from the equations of motion.