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arXiv:quant-ph/9804073

Bohm`s Interpretation of Quantum Mechanics and the Reconstruction of the Probability Distribution

H. Rupertsberger

Published Apr 30, 1998• 0 citations
arXivHTMLPDF

Abstract

Within Bohm`s interpretation of quantum mechanics particles follow classical trajectories that are determined by the full solution of the time dependent Schroedinger equation. If this interpretation is consistent it must be possible to determine the probability distribution at time t from the probability distribution at time t=0 by using these trajectories. In this paper it is shown that this is the case indeed.

DOI
10.48550/arxiv.quant-ph/9804073

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Within Bohm`s interpretation of quantum mechanics particles follow classical trajectories that are determined by the full solution of the time dependent Schroedinger equation. If this interpretation is consistent it must be possible to determine the probability distribution at time t from the probability distribution at time t=0 by using these trajectories. In this paper it is shown that this is the case indeed.
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.
This paper presents a deterministic temporal interpretation of quantum mechanics. It proposes that a particle has multiple possible trajectories, but it does not move along them simultaneously. Instead, the particle switches between trajectories over time, spending more time on the trajectory where the probability of detection is higher according to the Schrödinger equation. The particle does not have a fixed position until it interacts with a detector. This approach explains the statistical patterns observed in quantum experiments, such as the double-slit experiment, while maintaining determinism and remaining fully consistent with quantum mechanics.
A brief account of the world view of classical physics is given first. We then recapitulate as to why the Copenhagen interpretation of the quantum mechanics had to renounce most of the attractive features of the clasical world view such as a causal description, locality, scientific realism and introduce a fundamental distinction between system and apparatus. The crucial role is played in this by the Bohr's insistence on the wavefunction providing the most complete description possible for an even individual system. The alternative of introducing extra dynamical variables, called hidden variables, in addition to the wavefunction of the system so as to be able to retain at least some of the desirable features of classical physics, is then explored. The first such successful attempt was that of Bohm in 1952 who showed that a realistic interpretation of the quantum mechanics can be given which maintains a causal description as well as does not treat systems and measuring appeartus differently. We begin with the construction of the Bohm's theory. He introduces particle positions as the hidden variables. The particle positions play a special role in Bohm theory. The particle trajectories are guided by the wavefunction. The Bohm theory is deterministic. The probability enters through a special assumption, ``quantum equilibrium'' hypothesis, for the initial conditions on the ensemble of particle trajectories. The ``wave or particle'' dilemma is resolved by a ``wave and particle'' resolution. The measurements in Bohm theory can be described without mysticism. Bohm's theory is however nonlocal.
In Bohm's causal or trajectory interpretation of quantum mechanics, it is straightforward to determine, from a sufficiently large number of calculated particle trajectories, probability distributions for time delays caused by a potential barrier, or for transmission times through a barrier. We show that these distributions can be calculated directly and more efficiently from probability currents, without the calculation of Bohm trajectories as an intermediate step. The ideas are illustrated for Gaussian wave packets incident on a square potential barrier and used to explain why average causal delay times differ from the average delay times calculated in other approaches.
Born’s interpretation of the squared modulus of a wave function (|ψ|2) as the probability density of finding a particle within a specified region of space is a key ingredient of the standard framework of quantum mechanics, thereby implying that the standard interpretation of quantum mechanics is inherently epistemological. On the other hand, the possibility of an alternative interpretation of quantum mechanics by interpreting |ψ|2 as the probability density of a particle being actually present within a specified region was first suggested by de Broglie [1]. Later, Bohm [2-4] developed the details of such an ontological model of quantum mechanics by using the notion of an observer-independent spacetime trajectory of an individual particle T&F Cat # K11224, Chapter 8, Page 114, 15-9-2010 that is determined by its wave function through an equation of motion which is formulated in a way consistent with the Schrödinger time evolution. Bohm’s model, thus, explicitly refuted the counterarguments (such as those put forward by Pauli [5] and von Neumann [6]) that claimed to have ruled out the formulation of such a model. Subsequently, much work has been done on various aspects of the Bohmian model [7-10]. That such a model is not unique has been elaborately discussed [11,12] while different versions of the ontological model of quantum mechanics have been proposed [13-21].