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arXiv:1311.6497 · Annales de la fondation Louis de Broglie/Annales de la Fondation Louis de Broglie

A Quasi-Newtonian Approach to Bohmian Mechanics I: Quantum Potential

Mahdi Atiq, Mozafar Karamian, Mehdi Golshani

Published Nov 22, 2013• 7 citations
arXivHTMLPDF

Abstract

In this article, we investigate Bohm's view of quantum theory, especially Bohm's quantum potential, from a new perspective. We develop a quasi-Newtonian approach to Bohmian mechanics. We show that to arrive at Bohmian formulation of quantum mechanics, there is no necessity to start from the Schrödinger equation. We also obtain an equation that restricts the possible forms of quantum potential and determines the functional form of it without appealing to the wave function and the Schrödinger equation. Finally, we discuss about the significance of quantum potential in the conceptual structure of quantum theory.

DOI
10.48550/arxiv.1311.6497

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In this article, we investigate Bohm's view of quantum theory, especially Bohm's quantum potential, from a new perspective. We develop a quasi-Newtonian approach to Bohmian mechanics. We show that to arrive at Bohmian formulation of quantum mechanics, there is no necessity to start from the Schrödinger equation. We also obtain an equation that restricts the possible forms of quantum potential and determines the functional form of it without appealing to the wave function and the Schrödinger equation. Finally, we discuss about the significance of quantum potential in the conceptual structure of quantum theory.
In a previous paper, we obtained the functional form of quantum potential by a quasi-Newtonian approach and without appealing to the wave function. We also described briefly the characteristics of this approach to the Bohmian mechanics. In this article, we consider the quantization problem and we show that the 'eigenvalue postulate' is a natural consequence of continuity condition and there is no need for postulating that the spectrum of energy and angular momentum are eigenvalues of their relevant operators. In other words, the Bohmian mechanics predicts the 'eigenvalue postulate'.
In this paper we generalize the ideas of de Broglie and Bohm to the relativistic case which is based on the relativistic Schrödinger equation. In this regard, the relativistic forms of the guidance equation and quantum potential are derived. In our formulation of Relativistic Bohmian Mechanic, the quantum equilibrium hypothesis (ρ=∣ψ∣2)(ρ=|ψ|^{2})(ρ=∣ψ∣2) and the probabilistic interpretation of the wave function are not an intrinsic feature of the theory as well as expected from the theoretical structure of Bohmian mechanics but still we can extract the statistical predictions of this theory. In fact with assuming the quantum equilibrium hypothesis in non-relativistic case and then acceleration to particles by an external field we go to the relativistic regime. So in this case the quantum equilibrium would not be established and theory will have testable predictions that can be experienced, or this will be compared to the result of relativistic quantum mechanics. In some simple cases such calculations have been done.
In this paper we express the space of rotation as a Riemannian space and try to generalize the classical equations of motion of a homogeneous spherical solid body in the domain of quantum mechanics. This is done within Bohm's view of quantum mechanics, but we do not use the Schrödinger equation. Instead, we assume that in addition to the classical potential there is an extra potential and try to obtain it. In doing this, we start from a classical picture based on Hamilton-Jacobi formalism and statistical mechanics but we use an interpretation which is different from the classical one. Then, we introduce a proper action and extremize it. This procedure gives us a mathematical identity for the extra potential that limits its form. The classical mechanics is a trivial solution of this method. In the simplest cases where the extra potential is not a constant, a mathematical identity determines it uniquely. In fact the first nontrivial potential, apart from some constant coefficients which are determined by experiment, is the usual Bohmian quantum potential.
Bohmian mechanics is a deterministic theory of quantum mechanics that is based on a set of n velocity functions for n particles, where these functions depend on the wavefunction from the n-body time-dependent Schroedinger equation. It is well know that Bohmian mechanics is not applicable to stationary states, since the velocity field for stationary states is the zero function. Recently, an alternative to Bohmian mechanics has been formulated, based on a conservation of energy equation, where the velocity fields are not the zero function, but this formalism is only applicable to stationary states with real valued wavefunctions. In this paper, Bohmian mechanics is merged with the alternative to Bohmian mechanics. This is accomplished by introducing an interpretation of the Bohm quantum potential. The final formalism gives dynamic particles for all states, including stationary states. The final main working equation contains two kinetic energy terms and a term that contains a factor that can be interpreted as a pressure. The derivation is a simple n-body generalization of the recent generalization, or refinement, of the Madelung equations.
Quantum mechanics is the most successful theory to describe microscopic phenomena. It was derived in different ways over the past 100 years by Heisenberg, Schrödinger, and Feynman. At the same time, other interpretations have been suggested, including the Bohm-De Broglie interpretation and the so-called Bohmian mechanics. Here, we show that Bohmian mechanics, which utilizes the concept of the Bohm quantum potential, can also serve as a starting point for quantizing classical non-relativistic systems. By incorporating the Bohm quantum potential into the Vlasov framework, we obtain a mean-field theory that captures the corpuscular nature of matter, in agreement with quantum mechanics within the Random Phase Approximation (RPA).
Bohmian mechanics is the most naively obvious embedding imaginable of Schrödinger’s equation into a completely coherent physical theory. It describes a world in which particles move in a highly non-Newtonian sort of way, one which may at first appear to have little to do with the spectrum of predictions of quantum mechanics. It turns out, however, that as a consequence of the defining dynamical equations of Bohmian mechanics, when a system has wave function ψ its configuration is typically random, with probability density ρ given by |ψ|², the quantum equilibrium distribution. It also turns out that the entire quantum formalism, operators as observables and all the rest, naturally emerges in Bohmian mechanics from the analysis of “measurements.” This analysis reveals the status of operators as observables in the description of quantum phenomena, and facilitates a clear view of the range of applicability of the usual quantum mechanical formulas.