Non-linear Wave Mechanics, A Causal Interpretation;


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Since the uncertainty relation can be derived from the wavefunction in other interpretations of quantum mechanics, it can be likewise derived in the epistemic sense mentioned above on the de Broglie—Bohm theory. To put the statement differently, the particles' positions are only known statistically. As in classical mechanics , successive observations of the particles' positions refine the experimenter's knowledge of the particles' initial conditions.

Thus, with succeeding observations, the initial conditions become more and more restricted. For the derivation of the uncertainty relation, see Heisenberg uncertainty principle , noting that this article describes the principle from the viewpoint of the Copenhagen interpretation.

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De Broglie—Bohm theory highlighted the issue of nonlocality : it inspired John Stewart Bell to prove his now-famous theorem , [58] which in turn led to the Bell test experiments. In the Einstein—Podolsky—Rosen paradox , the authors describe a thought experiment that one could perform on a pair of particles that have interacted, the results of which they interpreted as indicating that quantum mechanics is an incomplete theory.

Decades later John Bell proved Bell's theorem see p. In particular, Bell proved that any local theory with unique results must make empirical predictions satisfying a statistical constraint called "Bell's inequality". Aspect's results show experimentally that Bell's inequality is in fact violated, meaning that the relevant quantum-mechanical predictions are correct.

In these Bell test experiments, entangled pairs of particles are created; the particles are separated, traveling to remote measuring apparatus. The orientation of the measuring apparatus can be changed while the particles are in flight, demonstrating the apparent nonlocality of the effect. The de Broglie—Bohm theory makes the same empirically correct predictions for the Bell test experiments as ordinary quantum mechanics.

It is able to do this because it is manifestly nonlocal. It is often criticized or rejected based on this; Bell's attitude was: "It is a merit of the de Broglie—Bohm version to bring this [nonlocality] out so explicitly that it cannot be ignored. The de Broglie—Bohm theory describes the physics in the Bell test experiments as follows: to understand the evolution of the particles, we need to set up a wave equation for both particles; the orientation of the apparatus affects the wavefunction.

The particles in the experiment follow the guidance of the wavefunction. It is the wavefunction that carries the faster-than-light effect of changing the orientation of the apparatus. An analysis of exactly what kind of nonlocality is present and how it is compatible with relativity can be found in Maudlin. Bohm's formulation of de Broglie—Bohm theory in terms of a classically looking version has the merits that the emergence of classical behavior seems to follow immediately for any situation in which the quantum potential is negligible, as noted by Bohm in Modern methods of decoherence are relevant to an analysis of this limit.

See Allori et al. Work by Robert E. Wyatt in the early s attempted to use the Bohm "particles" as an adaptive mesh that follows the actual trajectory of a quantum state in time and space. In the "quantum trajectory" method, one samples the quantum wavefunction with a mesh of quadrature points. One then evolves the quadrature points in time according to the Bohm equations of motion.

At each time step, one then re-synthesizes the wavefunction from the points, recomputes the quantum forces, and continues the calculation. This approach has been adapted, extended, and used by a number of researchers in the chemical physics community as a way to compute semi-classical and quasi-classical molecular dynamics. Wyatt and his work on "computational Bohmian dynamics". Eric R. Bittner 's group at the University of Houston has advanced a statistical variant of this approach that uses Bayesian sampling technique to sample the quantum density and compute the quantum potential on a structureless mesh of points.

There remain difficulties using the Bohmian approach, mostly associated with the formation of singularities in the quantum potential due to nodes in the quantum wavefunction. Various schemes have been developed to overcome this; however, no general solution has yet emerged. These methods, as does Bohm's Hamilton—Jacobi formulation, do not apply to situations in which the full dynamics of spin need to be taken into account. The properties of trajectories in the de Broglie — Bohm theory differ significantly from the Moyal quantum trajectories. The second issue with Bohmian mechanics may, at first sight, appear rather harmless, but which on a closer look develops considerable destructive power: the issue of empty branches.

These are the components of the post-measurement state that do not guide any particles because they do not have the actual configuration q in their support. At first sight, the empty branches do not appear problematic but on the contrary very helpful as they enable the theory to explain unique outcomes of measurements.

On a closer view, though, one must admit that these empty branches do not actually disappear. Now, if the Everettian theory may be accused of ontological extravagance, then Bohmian mechanics could be accused of ontological wastefulness. On top of the ontology of empty branches comes the additional ontology of particle positions that are, on account of the quantum equilibrium hypothesis, forever unknown to the observer.

Yet, the actual configuration is never needed for the calculation of the statistical predictions in experimental reality, for these can be obtained by mere wavefunction algebra. From this perspective, Bohmian mechanics may appear as a wasteful and redundant theory. I think it is considerations like these that are the biggest obstacle in the way of a general acceptance of Bohmian mechanics. Many authors have expressed critical views of the de Broglie—Bohm theory by comparing it to Everett's many-worlds approach.

Many but not all proponents of the de Broglie—Bohm theory such as Bohm and Bell interpret the universal wavefunction as physically real. According to some supporters of Everett's theory, if the never collapsing wavefunction is taken to be physically real, then it is natural to interpret the theory as having the same many worlds as Everett's theory.

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In the Everettian view the role of the Bohmian particle is to act as a "pointer", tagging, or selecting, just one branch of the universal wavefunction the assumption that this branch indicates which wave packet determines the observed result of a given experiment is called the "result assumption" [64] ; the other branches are designated "empty" and implicitly assumed by Bohm to be devoid of conscious observers.

Dieter Zeh comments on these "empty" branches: [65]. David Deutsch has expressed the same point more "acerbically": [64] [66]. Everett's many-worlds interpretation is an attempt to demonstrate that the wavefunction alone is sufficient to account for all our observations.


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When we see the particle detectors flash or hear the click of a Geiger counter , then Everett's theory interprets this as our wavefunction responding to changes in the detector's wavefunction , which is responding in turn to the passage of another wavefunction which we think of as a "particle", but is actually just another wave packet. For this reason Everett sometimes referred to his own many-worlds approach as the "pure wave theory".

Talking of Bohm's approach, Everett says: [67]. In the Everettian view, then, the Bohm particles are superfluous entities, similar to, and equally as unnecessary as, for example, the luminiferous ether , which was found to be unnecessary in special relativity. This argument of Everett is sometimes called the "redundancy argument", since the superfluous particles are redundant in the sense of Occam's razor. These authors claim [64] that the "result assumption" see above is inconsistent with the view that there is no measurement problem in the predictable outcome i.

De Broglie’s wave theory and the two-wave description of matter | SpringerLink

These authors also claim [64] that a standard tacit assumption of the de Broglie—Bohm theory that an observer becomes aware of configurations of particles of ordinary objects by means of correlations between such configurations and the configuration of the particles in the observer's brain is unreasonable. This conclusion has been challenged by Valentini , [69] who argues that the entirety of such objections arises from a failure to interpret de Broglie—Bohm theory on its own terms. According to Peter R. Holland , in a wider Hamiltonian framework, theories can be formulated in which particles do act back on the wave function.

De Broglie—Bohm theory has been derived many times and in many ways. Below are six derivations, all of which are very different and lead to different ways of understanding and extending this theory. De Broglie—Bohm theory has a history of different formulations and names. In this section, each stage is given a name and a main reference. At the end of the presentation, Wolfgang Pauli pointed out that it was not compatible with a semi-classical technique Fermi had previously adopted in the case of inelastic scattering.

Contrary to a popular legend, de Broglie actually gave the correct rebuttal that the particular technique could not be generalized for Pauli's purpose, although the audience might have been lost in the technical details and de Broglie's mild manner left the impression that Pauli's objection was valid. He was eventually persuaded to abandon this theory nonetheless because he was "discouraged by criticisms which [it] roused". An analysis of de Broglie's presentation is given in Bacciagaluppi et al.

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This sealed the fate of de Broglie's theory for the next two decades. Peter R.

The Resonant Nonlinear Schrodinger Equation in the Causal Interpretation of Quantum Theory

Holland has pointed out that, earlier in , Einstein had actually submitted a preprint with a similar proposal but, not convinced, had withdrawn it before publication. After publishing a popular textbook on Quantum Mechanics that adhered entirely to the Copenhagen orthodoxy, Bohm was persuaded by Einstein to take a critical look at von Neumann's theorem. It was an independent origination of the pilot wave theory, and extended it to incorporate a consistent theory of measurement, and to address a criticism of Pauli that de Broglie did not properly respond to; it is taken to be deterministic though Bohm hinted in the original papers that there should be disturbances to this, in the way Brownian motion disturbs Newtonian mechanics.

The de Broglie—Bohm theory is an example of a hidden-variables theory. However, Bell's theorem complicates this hope, as it demonstrates that there can be no local hidden-variable theory that is compatible with the predictions of quantum mechanics. The Bohmian interpretation is causal but not local. Bohm's paper was largely ignored or panned by other physicists.

Albert Einstein , who had suggested that Bohm search for a realist alternative to the prevailing Copenhagen approach , did not consider Bohm's interpretation to be a satisfactory answer to the quantum nonlocality question, calling it "too cheap", [82] while Werner Heisenberg considered it a "superfluous 'ideological superstructure' ". I just received your long letter of 20th November, and I also have studied more thoroughly the details of your paper. I do not see any longer the possibility of any logical contradiction as long as your results agree completely with those of the usual wave mechanics and as long as no means is given to measure the values of your hidden parameters both in the measuring apparatus and in the observe [sic] system.

He subsequently described Bohm's theory as "artificial metaphysics". According to physicist Max Dresden , when Bohm's theory was presented at the Institute for Advanced Study in Princeton, many of the objections were ad hominem , focusing on Bohm's sympathy with communists as exemplified by his refusal to give testimony to the House Un-American Activities Committee.

In , Chris Philippidis, Chris Dewdney and Basil Hiley were the first to perform numeric computations on the basis of the quantum potential to deduce ensembles of particle trajectories. Eventually John Bell began to defend the theory. In "Speakable and Unspeakable in Quantum Mechanics" [Bell ], several of the papers refer to hidden-variables theories which include Bohm's.

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The trajectories of the Bohm model that would result for particular experimental arrangements were termed "surreal" by some. Bohmian mechanics is the same theory, but with an emphasis on the notion of current flow, which is determined on the basis of the quantum equilibrium hypothesis that the probability follows the Born rule. The term "Bohmian mechanics" is also often used to include most of the further extensions past the spin-less version of Bohm.

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