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IIIB2. The Bohm Model of Particles
Underlying Quantum Mechanics.
1. Introduction.
The Bohm model (references 1-3) of a particle theory underlying quantum mechanics is a scheme that apparently gives exact agreement with low energy Schrödinger equation quantum mechanics, and at the same time explains the probability law. If this model were valid, it would constitute evidence for the existence of particles—theoretical evidence, but still substantial evidence. We will point out, however, that the Bohm model has two flaws that disqualifies it as a valid theory.
2. Perception of Only One Version?
The first, explained more fully in section IIIB3, is that all branches of the wave function continue forever, but there is no rationale given for explaining why we perceive only the branch that the particles travel on. Why is the "particled" version of the observer's brain the conscious version?
3. No Source Equation in the Bohm Model.
The second flaw has to do with the way in which particles are introduced into the model. The mathematics of the model is concerned only with the derivation of particlelike trajectories from (1) the usual Schrödinger equation (actually, from the quantum mechanical equation of continuity) and (2) the requirement that the |ψ(x)|2 probability law holds. But there is nothing in the mathematics which guarantees that there actually are particles that follow the trajectories, with one and only one particle per single-particle wave function. That is, there is no source equation—such as the electrostatic equation L@E=ρ/ε0, where the charge density ρ is the source that forces the existence of the electric field—that forces the existence of a wave function when there is a particle present. So rather than being a consequence of the mathematics, the most essential fact of all—the one-to-one correspondence between particles and wave functions—is put in by assumption in the Bohm model. And that is not acceptable in a satisfactory underlying theory of particles.
Bohm himself refers to this problem (reference 1, sections 4 and 9) and suggests possible solutions. But the suggestions are never developed, even in the much later reference 3. The reason, I believe, is that it is extremely difficult, and perhaps even impossible, to construct an underlying theory of particles with a source term (1) that does not perceptibly alter, say, the spectrum of the atoms, (2) that has reasonable initial conditions on the particle positions and velocities in relation to the wave functions, (3) that preserves, to a high degree of accuracy, the |ψ(x)|2 density of trajectories, and (4) that preserves the separability (product wave function) of isolated systems.
References.
[1] David Bohm, Phys. Rev. 85 166 (1952).
[2] David Bohm, Phys. Rev. 85 180 (1952).
[3] D. Bohm and B. J. Hiley, The Undivided Universe, (Routledge, New York, 1993).
© 2007 Casey Blood, Ph.D. All rights reserved.
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