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Measurement in the De Broglie-Bohm Interpretation: Double-Slit, Stern-Gerlach, and EPR-B

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Measurement in the De Broglie-Bohm Interpretation: Double-Slit, Stern-Gerlach, and EPR-B Hindawi Publishing Corporation Physics Research International Volume 2014, Article ID 605908, 16 pages http://dx.doi.org/10.1155/2014/605908 Research Article Measurement in the de Broglie-Bohm Interpretation: Double-Slit, Stern-Gerlach, and EPR-B Michel Gondran1 and Alexandre Gondran2 1 University Paris Dauphine, Lamsade, 75 016 Paris, France 2 Ecole´ Nationale de l’Aviation Civile, 31000 Toulouse, France Correspondence should be addressed to Alexandre Gondran; [email protected] Received 24 February 2014; Accepted 7 July 2014; Published 10 August 2014 Academic Editor: Anand Pathak Copyright © 2014 M. Gondran and A. Gondran. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We propose a pedagogical presentation of measurement in the de Broglie-Bohm interpretation. In this heterodox interpretation, the position of a quantum particle exists and is piloted by the phase of the wave function. We show how this position explains determinism and realism in the three most important experiments of quantum measurement: double-slit, Stern-Gerlach, and EPR- B. First, we demonstrate the conditions in which the de Broglie-Bohm interpretation can be assumed to be valid through continuity with classical mechanics. Second, we present a numerical simulation of the double-slit experiment performed by Jonsson¨ in 1961 with electrons. It demonstrates the continuity between classical mechanics and quantum mechanics. Third, we present an analytic expression of the wave function in the Stern-Gerlach experiment. This explicit solution requires the calculation of a Pauli spinor with a spatial extension. This solution enables us to demonstrate the decoherence of the wave function and the three postulates of quantum measurement. Finally, we study the Bohm version of the Einstein-Podolsky-Rosen experiment. Its theoretical resolution in space and time shows that a causal interpretation exists where each atom has a position and a spin. 1. Introduction expressed the same idea in many books, both popular and scholarly), Bohm’s theory cannot exist. Yet it does exist, and “Isawtheimpossibledone”[1]. This is how John Bell describes is even older than Bohm’s papers themselves. In fact, the basic his inexpressible surprise in 1952 upon the publication of idea behind it was formulated in 1927 by Louis de Broglie in a an article by Bohm [2]. The impossibility came from a model he called “pilot wave theory”. Since this theory provides theorembyJohnvonNeumannoutlinedin1932inhisbook explanations of what, in “high circles”, is declared inexplicable, The Mathematical Foundations of Quantum Mechanics [3], it is worth consideration, even by physicists [...]whodonot which seemed to show the impossibility of adding “hidden think it gives us the final answer to the question how reality variables” to quantum mechanics. This impossibility, with really is” [4]. its physical interpretation, became almost a postulate of And in 1987, Bell wonders about his teachers’ silence quantum mechanics, based on von Neumann’s indisputable concerning the Broglie-Bohm pilot wave: authority as a mathematician. Bernard d’Espagnat notes in “ButwhythenhadBornnottoldmeofthis“pilotwave”? 1979 the following: If only to point out what was wrong with it? Why did “At the university, Bell had, like all of us, received from his von Neumann not consider it? More extraordinarily, why teachers a message which, later still, Feynman would brilliantly did people go on producing “impossibility” proofs after 1952, state as follows: “No one can explain more than we have and as recently as 1978? [sic] While even Pauli, Rosenfeld, explained here [...]. We do not have the slightest idea of a and Heisenberg could produce no more devastating criticism more fundamental mechanism from which the former results ofBohm’sversionthantobranditas“metaphysical”and (the interference fringes) could follow”. If indeed we are to “ideological”? Why is the pilot-wave picture ignored in text believe Feynman (and Banesh Hoffman, and many others, who books? Should it not be taught, not as the only way, but 2 Physics Research International as an antidote to the prevailing complacency? To show that solution to the Schrodinger¨ equation: vagueness, subjectivity and indeterminism are not forced on Ψ (x,) ℏ2 us by experimental facts, but through a deliberate theoretical ℏ =− ΔΨ (x,) +(x) Ψ (x,) , (1) choice?” [5]. 2 More than thirty years after John Bell’s questions, the Ψ (x,0) =Ψ0 (x) . (2) interpretation of the de Broglie-Bohm pilot wave is still ignored by both the international community and the text- Ψ(x,) = √ℏ(x,) ((ℏ(x,)/ books. With the variable change exp ℏ)) What is this pilot wave theory? For de Broglie, a quantum ,theSchrodinger¨ equation can be decomposed into particle is not only defined by its wave function. He assumes Madelung equations [8](1926): that the quantum particle also has a position which is piloted ℏ 2 ℏ (x,) 1 2 ℏ Δ√ (x,) by the wave function [6]. However, only the probability + (∇ℏ (x,)) +(x) − =0, density of this position is known. The position exists in 2 2 √ℏ (x,) itself (ontologically) but is unknown to the observer. It only becomes known during the measurement. (3) The goal of the present paper is to present the Broglie- ℏ (x,) ∇ℏ (x,) Bohm pilot wave through the study of the three most impor- + (ℏ (x,) )=0 div (4) tant experiments of quantum measurement: the double-slit experiment which is the crucial experiment of the wave- with initial conditions: particle duality, the Stern and Gerlach experiment with the measurement of the spin, and the EPR-B experiment with the ℏ ℏ ℏ ℏ (x,0) =0 (x) ,(x,0) =0 (x) . (5) problem of nonlocality. The paper is organized as follows. In Section 2,we Madelung equations correspond to a set of noninteracting ℏ demonstrate the conditions in which the de Broglie-Bohm quantum particles all prepared in the same way (same 0 (x) ℏ interpretationcanbeassumedtobevalidthroughcontinuity and 0(x)). with classical mechanics. This involves the de Broglie-Bohm A quantum particle is said to be statistically prepared if interpretation for a set of particles prepared in the same ℏ ℏ its initial probability density 0 (x) and its initial action 0(x) way. In Section 3, we present a numerical simulation of converge, when ℏ→0, to nonsingular functions 0(x) and the double-slit experiment performed by Jonsson¨ in 1961 0(x). It is the case of an electronic or 60 beam in the double- with electrons [7]. The method of Feynman path integrals slit experiment or an atomic beam in the Stern and Gerlach allows us to calculate the time-dependent wave function. experiment. We will see that it is also the case of a beam of The evolution of the probability density just outside the entangled particles in the EPR-B experiment. Then, we have slits leads one to consider the dualism of the wave-particle the following theorem [9, 10]. interpretation. And the de Broglie-Bohm trajectories provide an explanation for the impact positions of the particles. Theorem 1. For statistically prepared quantum particles, the ℏ ℏ Finally, we show the continuity between classical and quan- probability density (x,)and the action (x,),solutionsto tum trajectories with the convergence of these trajectories the Madelung equations (3), (4),and(5), converge, when ℏ→ ℎ 0 to classical trajectories when tends to .InSection 4, 0, to the classical density (x,)and the classical action (x,), we present an analytic expression of the wave function in solutions to the statistical Hamilton-Jacobi equations: the Stern-Gerlach experiment. This explicit solution requires the calculation of a Pauli spinor with a spatial extension. (x,) 1 2 + (∇ (x,)) +(x,) =0, (6) This solution enables us to demonstrate the decoherence 2 of the wave function and the three postulates of quantum (x,0) = (x) , measurement: quantization, Born interpretation, and wave 0 (7) function reduction. The spinor spatial extension also enables (x,) ∇ (x,) + ( (x,) )=0, (8) the introduction of the de Broglie-Bohm trajectories which div gives a very simple explanation of the particles’ impact and of the measurement process. In Section 5,westudytheEPR- (x,0) =0 (x) . (9) B experiment, the Bohm version of the Einstein-Podolsky- Rosen experiment. Its theoretical resolution in space and time We give some indications on the demonstration of this shows that a causal interpretation exists where each atom theorem when the wave function Ψ(x,) is written as a has a position and a spin. Finally, we recall that a physical function of the initial wave function Ψ0(x) by the Feynman explanation of nonlocal influences is possible. paths integral [11]: Ψ (x,) = ∫ (,) ℏ ( (x,;x )) Ψ (x )x , 2. The de Broglie-Bohm Interpretation exp ℏ 0 0 0 0 (10) The de Broglie-Bohm interpretation is based on the following where (, ℏ) is an independent function of x and of x0.Fora demonstration. Let us consider a wave function Ψ(x,) statistically prepared quantum particle, the wave function is Physics Research International 3 √ ℏ ℏ written: Ψ(x,)=(,ℏ)∫ 0 (x0) exp((/ℏ)(0(x0)+(x,; But the three postulates of measurement are not necessary: the postulate of quantization, the Born postulate of proba- x0)))0. The theorem of the stationary phase shows that if ℏ Ψ(x,) ∼ ((/ℏ) ( (x )+ bilistic interpretation of the wave function, and the postulate tends towards 0, we have exp minx0 0 0 (x,;x ))) ℎ(x,) of the reduction of the wave function. We see that these 0 ;thatistosay,thequantumaction postulates of measurement can be explained on each example converges to the function: as we will show in the following. We replace these three postulates by a single one, the (x,) = min (0 (x0)+ (x,;x0)) “quantum equilibrium hypothesis,” [13–15] that describes the x0 (11) interaction between the initial wave function Ψ0(x) and the initial particle position X(0): for a set of identically prepared which is the solution to the Hamilton-Jacobi equation (6) particles having =0wave function Ψ0(x), it is assumed that with the initial condition (7).
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