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

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Measurement in the De Broglie-Bohm Interpretation: Double-Slit, Stern-Gerlach and EPR-B Michel Gondran, Alexandre Gondran Measurement in the de Broglie-Bohm interpretation: Double-slit, Stern-Gerlach and EPR-B Michel Gondran, Alexandre Gondran To cite this version: Michel Gondran, Alexandre Gondran. Measurement in the de Broglie-Bohm interpretation: Double- slit, Stern-Gerlach and EPR-B. Physics Research International, Hindawi, 2014, 2014. hal-00862895v3 HAL Id: hal-00862895 https://hal.archives-ouvertes.fr/hal-00862895v3 Submitted on 24 Jan 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Measurement in the de Broglie-Bohm interpretation: Double-slit, Stern-Gerlach and EPR-B Michel Gondran University Paris Dauphine, Lamsade, 75 016 Paris, France∗ Alexandre Gondran École Nationale de l’Aviation Civile, 31000 Toulouse, Francey 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 Jönsson in 1961 with electrons. It demonstrates the continuity between classical mechanics and quantum mechanics: evolution of the probability density at various distances and convergence of the quantum trajectories to the classical trajectories when h tends to 0. 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 to demonstrate the decoherence of the wave function and the three postulates of quantum measurement: quantization, the Born interpretation and wave function reduction. The spinor spatial extension also enables the introduction of the de Broglie-Bohm trajectories, which gives a very simple explanation of the particles’ impact and of the measurement process. Finally, we study the EPR-B experiment, the Bohm version of the Einstein-Podolsky-Rosen ex- periment. Its theoretical resolution in space and time shows that a causal interpretation exists where each atom has a position and a spin. This interpretation avoids the flaw of the previous causal interpretation. We recall that a physical explanation of non-local influences is possible. I. INTRODUCTION question "how reality really is."4 And in 1987, Bell wonders about his teachers’ silence "I saw the impossible done".1 This is how John Bell concerning the Broglie-Bohm pilot-wave: describes his inexpressible surprise in 1952 upon the pub- "But why then had Born not told me of this ’pilot lication of an article by David Bohm2. The impossibility wave’? If only to point out what was wrong with it? Why came from a theorem by John von Neumann outlined in did von Neumann not consider it? More extraordinarily, 1932 in his book The Mathematical Foundations of Quan- why did people go on producing "impossibility" proofs af- tum Mechanics,3 which seemed to show the impossibil- ter 1952, and as recently as 1978? While even Pauli, ity of adding "hidden variables" to quantum mechanics. Rosenfeld, and Heisenberg could produce no more devas- This impossibility, with its physical interpretation, be- tating criticism of Bohm’s version than to brand it as came almost a postulate of quantum mechanics, based "metaphysical" and "ideological"? Why is the pilot-wave on von Neumann’s indisputable authority as a mathe- picture ignored in text books? Should it not be taught, matician. As Bernard d’Espagnat notes in 1979: not as the only way, but as an antidote to the prevailing "At the university, Bell had, like all of us, received complacency? To show that vagueness, subjectivity and indeterminism are not forced on us by experimental facts, from his teachers a message which, later still, Feynman 5 would brilliantly state as follows: "No one can explain but through a deliberate theoretical choice?" more than we have explained here [...]. We don’t have More than thirty years after John Bell’s questions, the the slightest idea of a more fundamental mechanism from interpretation of the de Broglie-Bohm pilot wave is still which the former results (the interference fringes) could ignored by both the international community and the follow". If indeed we are to believe Feynman (and Banesh textbooks. Hoffman, and many others, who expressed the same idea What is this pilot wave theory? For de Broglie, a quan- in many books, both popular and scholarly), Bohm’s the- tum particle is not only defined by its wave function. He ory cannot exist. Yet it does exist, and is even older than assumes that the quantum particle also has a position Bohm’s papers themselves. In fact, the basic idea behind which is piloted by the wave function.6 However only the it was formulated in 1927 by Louis de Broglie in a model probability density of this position is known. The posi- he called "pilot wave theory". Since this theory provides tion exists in itself (ontologically) but is unknown to the explanations of what, in "high circles", is declared in- observer. It only becomes known during the measure- explicable, it is worth consideration, even by physicists ment. [...] who do not think it gives us the final answer to the The goal of the present paper is to present the Broglie- 2 Bohm pilot-wave through the study of the three most @ρ~(x; t) rS~(x; t) + div ρ~(x; t) = 0 (4) important experiments of quantum measurement: the @t m double-slit experiment which is the crucial experiment of the wave-particle duality, the Stern and Gerlach experi- with initial conditions: ment with the measurement of the spin, and the EPR-B ~ ~ ~ ~ experiment with the problem of non-locality. ρ (x; 0) = ρ0 (x) and S (x; 0) = S0 (x): (5) The paper is organized as follows. In section II, we demonstrate the conditions in which the de Broglie-Bohm Madelung equations correspond to a set of non- interpretation can be assumed to be valid through con- interacting quantum particles all prepared in the same ~ ~ tinuity with classical mechanics. This involves the de way (same ρ0 (x) and S0 (x)). Broglie-Bohm interpretation for a set of particles pre- A quantum particle is said to be statistically prepared ~ pared in the same way. In section III, we present a if its initial probability density ρ0 (x) and its initial action ~ numerical simulation of the double-slit experiment per- S0 (x) converge, when ~ ! 0, to non-singular functions formed by Jönsson in 1961 with electrons15. The method ρ0(x) and S0(x). It is the case of an electronic or C60 of Feynman path integrals allows to calculate the time- beam in the double slit experiment or an atomic beam in dependent wave function. The evolution of the proba- the Stern and Gerlach experiment. We will seen that it bility density just outside the slits leads one to consider is also the case of a beam of entrangled particles in the EPR-B experiment. Then, we have the following theo- the dualism of the wave-particle interpretation. And the 8,9 de Broglie-Bohm trajectories provide an explanation for rem: the impact positions of the particles. Finally, we show For statistically prepared quantum particles, the proba- the continuity between classical and quantum trajecto- bility density ρ~(x; t) and the action S~(x; t), solutions to ries with the convergence of these trajectories to classi- the Madelung equations (3)(4)(5), converge, when ~ ! 0, cal trajectories when h tends to 0. In section IV, we to the classical density ρ(x; t) and the classical action present an analytic expression of the wave function in S(x; t), solutions to the statistical Hamilton-Jacobi equa- the Stern-Gerlach experiment. This explicit solution re- tions: quires the calculation of a Pauli spinor with a spatial @S (x; t) 1 + (rS(x; t))2 + V (x; t) = 0 (6) extension. This solution enables to demonstrate the de- @t 2m coherence of the wave function and the three postulates S(x; 0) = S0(x) (7) of quantum measurement: quantization, Born interpre- @ρ (x; t) rS (x; t) tation and wave function reduction. The spinor spatial + div ρ (x; t) = 0 (8) extension also enables the introduction of the de Broglie- @t m Bohm trajectories which gives a very simple explanation ρ(x; 0) = ρ0(x): (9) of the particles’ impact and of the measurement pro- cess. In section V, we study the EPR-B experiment, We give some indications on the demonstration of this the Bohm version of the Einstein-Podolsky-Rosen exper- theorem when the wave function Ψ(x; t) is written as a iment. Its theoretical resolution in space and time shows function of the initial wave function Ψ0(x) by the Feyn- 10 that a causal interpretation exists where each atom has man paths integral : a position and a spin. Finally, we recall that a physical Z i explanation of non-local influences is possible. Ψ(x; t) = F (t; ~) exp Scl(x; t; x0 Ψ0(x0)dx0 ~ (10) II. THE DE BROGLIE-BOHM where F (t; ~) is an independent function of x INTERPRETATION and of x0.
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