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 particle exists and is piloted by the phase of the . 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 . 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 . 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 .

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 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,𝑡) ℏ2 us by experimental facts, but through a deliberate theoretical 𝑖ℏ =− ΔΨ (x,𝑡) +𝑉(x) Ψ (x,𝑡) , (1) choice?” [5]. 𝜕𝑡 2𝑚 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,𝑡) = √𝜌ℏ(x,𝑡) (𝑖(𝑆ℏ(x,𝑡)/ 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 𝜕𝑆 (x,𝑡) 1 2 ℏ Δ√𝜌 (x,𝑡) by the wave function [6]. However, only the probability + (∇𝑆ℏ (x,𝑡)) +𝑉(x) − =0, density of this position is known. The position exists in 𝜕𝑡 2𝑚 2𝑚 √𝜌ℏ (x,𝑡) 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- 𝜕𝜌ℏ (x,𝑡) ∇𝑆ℏ (x,𝑡) Bohm pilot wave through the study of the three most impor- + (𝜌ℏ (x,𝑡) )=0 𝜕𝑡 div 𝑚 (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 𝜌 (x,0) =𝜌0 (x) ,𝑆(x,0) =𝑆0 (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 𝜌0 (x) ℏ interpretationcanbeassumedtobevalidthroughcontinuity and 𝑆0(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 𝜌0 (x) and its initial action 𝑆0(x) way. In Section 3, we present a numerical simulation of converge, when ℏ→0, to nonsingular functions 𝜌0(x) and the double-slit experiment performed by Jonsson¨ in 1961 𝑆0(x). It is the case of an electronic or 𝐶60 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 𝜌 (x,𝑡)and the action 𝑆 (x,𝑡),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 𝜌(x,𝑡)and the classical action 𝑆(x,𝑡), 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. 𝜕𝑆 (x,𝑡) 1 2 + (∇𝑆 (x,𝑡)) +𝑉(x,𝑡) =0, (6) This solution enables us to demonstrate the decoherence 𝜕𝑡 2𝑚 of the wave function and the three postulates of quantum 𝑆 (x,0) =𝑆 (x) , measurement: quantization, Born interpretation, and wave 0 (7) function reduction. The spinor spatial extension also enables 𝜕𝜌 (x,𝑡) ∇𝑆 (x,𝑡) + (𝜌 (x,𝑡) )=0, (8) the introduction of the de Broglie-Bohm trajectories which 𝜕𝑡 div 𝑚 gives a very simple explanation of the particles’ impact and of the measurement process. In Section 5,westudytheEPR- 𝜌 (x,0) =𝜌0 (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,𝑡) 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]: 𝑖 Ψ (x,𝑡) = ∫ 𝐹 (𝑡, ℏ) ( 𝑆 (x,𝑡;x )) Ψ (x )𝑑x , 2. The de Broglie-Bohm Interpretation exp ℏ 𝑐𝑙 0 0 0 0 (10)

The de Broglie-Bohm interpretation is based on the following where 𝐹(𝑡, ℏ) is an independent function of x and of x0.Fora demonstration. Let us consider a wave function Ψ(x,𝑡) statistically prepared quantum particle, the wave function is Physics Research International 3

√ ℏ ℏ written: Ψ(x,𝑡)=𝐹(𝑡,ℏ)∫ 𝜌0 (x0) exp((𝑖/ℏ)(𝑆0(x0)+𝑆𝑐𝑙(x,𝑡; But the three postulates of measurement are not necessary: the postulate of quantization, the Born postulate of proba- x0)))𝑑𝑥0. The theorem of the stationary phase shows that if ℏ Ψ(x,𝑡) ∼ ((𝑖/ℏ) (𝑆 (x )+ bilistic interpretation of the wave function, and the postulate tends towards 0, we have exp minx0 0 0 𝑆 (x,𝑡;x ))) 𝑆ℎ(x,𝑡) of the reduction of the wave function. We see that these 𝑐𝑙 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 𝑆 (x,𝑡) = min (𝑆0 (x0)+𝑆𝑐𝑙 (x,𝑡;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 𝑡=0wave function Ψ0(x), it is assumed that with the initial condition (7). Moreover, as the quantum den- X(0) ℎ the initial particle positions are distributed according to sity 𝜌 (x,𝑡)satisfies the continuity equation4 ( ), we deduce, ℎ ℎ since 𝑆 (x,𝑡) tends towards 𝑆(x,𝑡),that𝜌 (x,𝑡) converges 󵄨 󵄨2 ℎ to the classical density 𝜌(x,𝑡), which satisfies the continuity 𝑃 [X (0) = x] ≡𝑃(x,0) = 󵄨Ψ0 (x)󵄨 =𝜌0 (x) . (14) equation (8). We obtain both announced convergences. These statistical Hamilton-Jacobi equations6 ( ), (7), (8), ItistheBornruleattheinitialtime. and (9) correspond to a set of classical particles prepared Then, the 𝑃( ( x,𝑡)≡𝑃[X(𝑡) = x]) inthesameway(thesame𝜌0(x) and 𝑆0(x)). These classical ℎ for a set of particles moving with the velocity field v (x,𝑡)= particles are trajectories obtained in Eulerian representation ∇𝑆ℎ(x, 𝑡)/𝑚 with the velocity field k(x,𝑡) = ∇𝑆(x, 𝑡)/𝑚, but the density satisfies the property of the “equivariance” of the |Ψ(x,𝑡)|2 and the action are not sufficient to describe it completely. To probability distribution [13]: know its position at time 𝑡,itisnecessarytoknowitsinitial position. Because the Madelung equations converge to the 2 ℎ 𝑃 [X (𝑡) = x] ≡𝑃(x,𝑡) = |Ψ (x,𝑡)| =𝜌 (x,𝑡) . (15) statistical Hamilton-Jacobi equations, it is logical to do the same in quantum mechanics. We conclude that a statistically prepared quantum particle is not completely described by its ItistheBornruleattime𝑡. wave function. It is necessary to add this initial position and Then, the de Broglie-Bohm interpretation is based on a an equation to define the evolution of this position in the continuity between classical and quantum mechanics where time. It is the de Brogglie-Bohm interpretation where the the quantum particles are statistically prepared with an initial position is called the “hidden variable.” probability density that satisfies the “quantum equilibrium The two first postulates of quantum mechanics, describ- hypothesis” (14). It is the case of the three studied experi- ing the and its evolution [12], must be com- ments. pleted in this heterodox interpretation. At initial time 𝑡=0, We will revisit these three measurement experiments the state of the particle is given by the initial wave function through mathematical calculations and numerical simula- Ψ0(x) (a wave packet) and its initial position X(0);itisthenew tions. For each one, we present the statistical interpretation first postulate. The new second postulate gives the evolution that is common to the Copenhagen interpretation and the de onthewavefunctionandontheposition.Forasinglespinless Broglie-Bohm pilot wave and then the trajectories specific to particle in a potential 𝑉(x), the evolution of the wave function the de Broglie-Bohm interpretation. We show that the precise is given by the usual Schrodinger¨ equations (1), (2)andthe definition of the initial conditions, that is, the preparation of evolution of the particle position is given by the particles, plays a fundamental methodological role. 󵄨 󵄨 𝑑X (𝑡) Jℎ(x,𝑡)󵄨 ∇𝑆ℎ(x,𝑡)󵄨 = 󵄨 = 󵄨 , 3. Double-Slit Experiment with Electrons ℎ 󵄨 󵄨 (12) 𝑑𝑡 𝜌 (x,𝑡)󵄨x=X(𝑡) 𝑚 󵄨𝑥=X(𝑡) Young’s double-slit experiment [16] has long been the cru- cial experiment for the interpretation of the wave-particle where duality. They have been realized with massive objects, such ℏ as electrons [7, 17–19], neutrons [20, 21], cold neutrons [22], Jℎ (x,𝑡) = (Ψ∗ (x,𝑡) ∇Ψ (x,𝑡) −Ψ(x,𝑡) ∇Ψ∗ (x,𝑡)) and atoms [23], and, more recently, with coherent ensembles 2𝑚𝑖 of ultracold atoms [24, 25] and even with mesoscopic single (13) quantum objects such as 𝐶60 and 𝐶70 [26, 27]. For Feynman, this experiment addresses “the basic element of the mysterious is the usual quantum current. behavior [of electrons] in its most strange form. [It is] a In the case of a particle with spin, as in the Stern phenomenon which is impossible, absolutely impossible to and Gerlach experiment, the Schrodinger¨ equation must be explaininanyclassicalwayandwhichhasinittheheartof replaced by the Pauli or Dirac equations. quantum mechanics. In reality, it contains the only mystery” The third quantum mechanics postulate which describes [28]. The de Broglie-Bohm interpretation and the numerical the measurement (the ) can be conserved. simulation help us here to revisit the double-slit experiment 4 Physics Research International

y

0.2 𝜇m x 0.2 𝜇m 10 𝜇m

S 0.8 𝜇m 10 𝜇m

z 70 cm 35 cm 35 cm Figure 2: General view of the evolution of the probability density fromthesourcetothescreenintheJonsson¨ experiment. A lighter Figure 1: Diagram of the Jonnson’s¨ double-slit experiment per- shade means that the density is higher; that is, the probability of formed with electrons. presence is high.

with electrons performed by Jonsson¨ in 1961 and to provide an answer to Feynman’s mystery. These simulations [29]follow those conducted in 1979 by Philippidis et al. [30]whichare today classics. However, these simulations [30]havesome limitations because they did not consider realistic slits. The slits, which can be clearly represented by a function 𝐺(𝑦) with 𝐺(𝑦) =1 for −𝛽≤𝑦≤𝛽and 𝐺(𝑦) =0 for |𝑦| >,ifthey 𝛽 3𝜇m are 2𝛽 in width, were modeled by a Gaussian function 𝐺(𝑦) = −𝑦2/2𝛽2 𝑒 . Interference was found, but the calculation could not account for diffraction at the edge of the slits. Consequently, these simulations could not be used to defend the de Broglie- Bohm interpretation. Figure 1 shows a diagram of the double-slit experiment 3.5 cm by Jonsson.¨ An electron gun emits electrons one by one in the horizontal plane, through a hole of a few micrometers, Figure 3: Close-up of the evolution of the probability density in the 8 at a velocity V = 1.8 × 10 m/s along the horizontal 𝑥-axis. first 3 cm after the slits in the Jonsson¨ experiment. After traveling for 𝑑1 =35cm, they encounter a plate pierced with two horizontal slits 𝐴 and 𝐵, each 0.2 𝜇mwideand spaced 1 𝜇m from each other. A screen located at 𝑑2 =35cm after the slits collects these electrons. The impact of each with Ψ𝐴(𝑦, 𝑡) =∫ 𝐾(𝑦, 𝑡,𝑎 𝑦 ,𝑡1)Ψ(𝑦𝑎,𝑡1)𝑑𝑦𝑎, Ψ𝐵(𝑦, 𝑡) = electron appears on the screen as the experiment unfolds. 𝐴 ∫ 𝐾(𝑦, 𝑡, 𝑦 ,𝑡 )Ψ(𝑦 ,𝑡 )𝑑𝑦 𝐾(𝑦, 𝑡, 𝑦 ,𝑡 ) = (𝑚/2𝑖ℏ(𝑡 − After thousands of impacts, we find that the distribution of 𝐵 𝑏 1 𝑏 1 𝑏,and 𝛼 1 2 1/2 𝑖𝑚(𝑦−𝑦𝛼) /2ℏ(𝑡−𝑡1) electrons on the screen shows interference fringes. 𝑡1)) 𝑒 . The slits are very long along the 𝑧-axis, so there is no effect Figure 3 shows a close-up of the evolution of the prob- of diffraction along this axis. In the simulation, we therefore ability density just after the slits. We note that interference only consider the wave function along the 𝑦-axis; the variable will only occur a few centimeters after the slits. Thus, if the 𝑥 will be treated classically with 𝑥=V𝑡. Electrons emerging detection screen is 1 cm from the slits, there is no interference from an electron gun are represented by the same initial wave and one can determine by which slit each electron has passed. function Ψ0(𝑦). In this experiment, the measurement is performed by the detection screen, which only reveals the existence or absence 3.1. Probability Density. Figure 2 gives a general view of the of the fringes. evolutionoftheprobabilitydensityfromthesourcetothe The calculation method enables us to compare the detection screen (a lighter shade means that the density is evolution of the cross-section of the probability density at higher; i.e., the probability of presence is high). The calcula- various distances after the slits (0.35 mm, 3.5 mm, 3.5 cm, and tionsweremadeusingthemethodofFeynmanpathintegrals 35 cm) where the two slits 𝐴 and 𝐵 are open simultaneously −11 |Ψ +Ψ|2 [29]. The wave function after the slits (𝑡1 =𝑑1/V ≃ 2.10 s (interference: 𝐴 𝐵 ) with the evolution of the sum of −11 𝐴 𝐵 <𝑡<𝑡1 +𝑑2/V ≃ 4.10 s) is deduced from the values of the the probability densities where the slits and are open 2 2 wave function at slits 𝐴 and 𝐵: Ψ(𝑦, 𝑡)𝐴 =Ψ (𝑦, 𝑡)𝐵 +Ψ (𝑦, 𝑡) independently (the sum of two diffractions: |Ψ𝐴| +|Ψ𝐵| ). Physics Research International 5

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 (𝜇m) (𝜇m)

(a) 0.35 mm (b) 3.5 mm

−2 −1 012−10 −5 0510 (𝜇m) (𝜇m)

(c) 3.5 cm (d) 35 cm

2 2 2 Figure 4: Comparison of the probability density |Ψ𝐴 +Ψ𝐵| (full line) and |Ψ𝐴| +|Ψ𝐵| (dotted line) at various distances after the slits: (a) 0.35 mm, (b): 3.5 mm, (c): 3.5 cm, and (d): 35 cm.

Figure 4 shows that the difference between these two phe- electron when it is stopped by the first screen. Figure 6 shows nomenaappearsonlyafewcentimetersaftertheslits. a close-up of these trajectories just after they leave their slits. The different trajectories explain both the impact of electrons on the detection screen and the interference fringes. 3.2. Impacts on Screen and de Broglie-Bohm Trajectories. The This is the simplest and most natural interpretation to explain interference fringes are observed after a certain period of time the impact positions: “the position of an impact is simply the when the impacts of the electrons on the detection screen position of the particle at the time of impact.” This was the become sufficiently numerous. Classical quantum theory view defended by Einstein at the Solvay Congress of 1927.The only explains the impact of individual particles statistically. position is the only measured variable of the experiment. However, in the de Broglie-Bohm interpretation, a parti- InthedeBroglie-Bohminterpretation,theimpactsonthe cle has an initial position and follows a path whose velocity at screen are the real positions of the electron as in classical each instant is given by (12). On the basis of this assumption mechanics and the three postulates of the measurement of we conduct a simulation experiment by drawing random quantum mechanics can be trivially explained: the position initial positions of the electrons in the initial wave packet is an eigenvalue of the position operator because the position (quantum equilibrium hypothesis). variable is identical to its operator (XΨ=xΨ), the Born Figure 5 shows, after its initial starting position, 100 postulate is satisfied with the “equivariance” property, and the possible quantum trajectories of an electron passing through reduction of the wave packet is not necessary to explain the one of the two slits: we have not represented the paths of the impacts. 6 Physics Research International

4 quantum mechanics and classical mechanics. All particles have quantum properties, but specific quantum behavior only 3 appears in certain experimental conditions: here when the ℎ𝑡/𝑚 2 ratio is sufficiently large. Interferences only appear gradually and the quantum particle behaves at any time as 1 both a wave and a particle.

m) 0 𝜇 ( −1 4. The Stern-Gerlach Experiment −2 In 1922, by studying the deflection of a beam of silver atoms −3 in a strongly inhomogeneous magnetic field (cf. Figure 8) Gerlach and Stern [31, 32] obtained an experimental result −4 that contradicts the common sense prediction: the beam, −35 −30 −20 −10 0 10203035 instead of expanding, splits into two separate beams giving 𝑁+ 𝑁− (cm) two spots of equal intensity and on a detector, at equal distances from the axis of the original beam. Figure 5: 100 electron trajectories for the Jonsson¨ experiment. Historically, this is the experiment which helped establish spin quantization. Theoretically, it is the seminal experiment posing the problem of measurement in quantum mechanics. Today it is the theory of decoherence with the diagonalization of the that is put forward to explain the first 1 part of the measurement process [33–38]. However, although 0.8 these authors consider the Stern-Gerlach experiment as fundamental, they do not propose a calculation of the spin 0.6 decoherence time. 0.4 We present an analytical solution to this decoherence 0.2 time and the diagonalization of the density matrix. This solution requires the calculation of the Pauli spinor with a

m) 0 spatial extension as the equation: 𝜇 ( −0.2

−0.4 𝜃0 −𝑖(𝜑 /2) cos 𝑒 0 −1/2 2 2 2 −0.6 0 2 −𝑧 /4𝜎0 Ψ (𝑧) = (2𝜋𝜎0 ) 𝑒 ( ). (16) 𝜃0 𝑖(𝜑 /2) −0.8 𝑒 0 sin 2 −1 012345678910 (cm) Quantum mechanics textbooks [12, 28, 39, 40]donottake Figure 6: Close-up on the 100 trajectories of the electrons just after into account the spatial extension of the spinor (16)and the slits. simply use the simplified spinor without spatial extension:

𝜃0 −𝑖(𝜑 /2) 𝑒 0 cos 2 Ψ0 =( ). Through numerical simulations, we will demonstrate (17) 𝜃0 𝑖(𝜑 /2) ℎ 𝑒 0 how, when the Planck constant tends to 0, the quantum sin 2 trajectories converge to the classical trajectories. In reality a constant is not able to tend to 0 by definition. The convergence to classical trajectories is obtained if the term However, as we shall see, the different evolutions of the spatial ℎ𝑡/𝑚,so →0 ℎ→0is equivalent to 𝑚→+∞(i.e., the mass extension between the two spinor components will have a oftheparticlegrows)or𝑡→0(i.e., the distance splits-screen key role in the explanation of the measurement process. 𝑑2 →0). Figure 7 shows the 100 trajectories that start at This spatial extension enables us, in following the precursory the same 100 initial points when Planck’s constant is divided, works of Takabayasi [41, 42], Bohm et al. [43, 44], Dewdney respectively, into 10, 100, 1000, and 10000 (equivalent to et al. [45], and Holland [46], to revisit the Stern and Gerlach multiplying the mass by 10, 100, 1000, and 10000). We obtain experiment, to explain the decoherence, and to demonstrate quantum trajectories converging to the classical trajectories, the three postulates of the measure: quantization, Born when ℎ tends to 0. statistical interpretation, and wave function reduction. The study of the slits clearly shows that, in the de Broglie- Silver atoms contained in the oven 𝐸 (Figure 8)areheated Bohm interpretation, there is no physical separation between to a high temperature and escape through a narrow opening. Physics Research International 7

1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2

m) 0 h/10m) 0 h/100 𝜇 𝜇 ( ( −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −1 −30 −20 −10 0 102030 −30 −20 −10 0102030 (cm) (cm)

1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2

m) 0 h/1000m) 0 h/10000 𝜇 𝜇 ( ( −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −1 −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 (cm) (cm)

Figure 7: Convergence of 100 electron trajectories when ℎ is divided by 10, 100, 1000, and 10000.

z assume that at the entrance to the electromagnet, 𝐴1 and at the initial time 𝑡=0, each atom can be approximately P1 y T described by a Gaussian spinor in 𝑧 given by (16) correspond- N+ x A1 ing to a pure state. The variable 𝑦 will be treated classically −4 E y= t with 𝑦=V𝑡. 𝜎0 =10 m corresponds to the size of the slot 𝑇 along the 𝑧-axis. The approximation by a Gaussian N− T = 1000∘K initial spinor will allow explicit calculations. Because the slot ( = 500 m/s) is much wider along the 𝑥-axis, the variable 𝑥 will be also Δl D treated classically. To obtain an explicit solution of the Stern- Figure 8: Schematic configuration of the Stern-Gerlach experiment. Gerlach experiment, we take the numerical values used in the Cohen-Tannoudji textbook [12]. For the silver atom, we −25 have 𝑚 = 1.8 × 10 kg, V0 = 500 m/s (corresponding to the ∘ temperature of 𝑇 = 1000 K). In (16)andinFigure 9, 𝜃0 and Asecondaperture,𝑇, selects those atoms whose velocity, v0, 𝜑0 are the polar angles characterizing the initial orientation is parallel to the 𝑦-axis. The atomic beam crosses the gap of of the magnetic moment and 𝜃0 corresponds to the angle the electromagnet 𝐴1 before condensing on the detector, 𝑃1. with the 𝑧-axis. The experiment is a statistical mixture of pure Before crossing the electromagnet, the magnetic moment of states where the 𝜃0 and the 𝜑0 are randomly chosen: 𝜃0 is each silver atom is oriented randomly (isotropically). In the drawn in a uniform way from [0, 𝜋] and 𝜑0 is drawn in a beam,werepresenteachatombyitswavefunction;onecan uniform way from [0, 2𝜋]. 8 Physics Research International

𝜓 z Ψ=( + ) The evolution of the spinor 𝜓− in a magnetic field B |−⟩ is then given by the :

𝜕𝜓+ 𝜕𝑡 ℏ2 𝜓 𝜓 𝜃0 𝑖ℏ ( )=− Δ( +)+𝜇 B𝜎( +), 𝜓 𝐵 𝜓 (18) 𝜕𝜓− 2𝑚 − − 𝜕𝑡 y

where 𝜇𝐵 = 𝑒ℏ/2𝑚𝑒 is the Bohr magneton and where 𝜑0 𝜎=(𝜎𝑥,𝜎𝑦,𝜎𝑧) corresponds to the three Pauli matrices. The particle first enters an electromagnetic field B directed along 󸀠 󸀠 x the 𝑧-axis, 𝐵𝑥 =𝐵0𝑥, 𝐵𝑦 =0,and𝐵𝑧 =𝐵0 −𝐵0𝑧,with 󸀠 3 𝐵0 =5Tesla, 𝐵0 =|𝜕𝐵/𝜕𝑧|=10Tesla/m over a length Δ𝑙 = 1 cm. On exiting the magnetic field, the particle is 𝑃 𝐷=20 |+⟩ free until it reaches the detector 1 placed at a cm distance. Figure 9: Orientation of the magnetic moment. 𝜃0 and 𝜑0 are the The particle stays within the magnetic field for a time Δ𝑡 = −5 polar angles characterizing the spin vector in the de Broglie-Bohm Δ𝑙/V =2×10 s. During this time [0, Δ𝑡],thespinoris[47] interpretation. (see the Appendix)

2 𝜃0 2 −1/2 −(𝑧−(𝜇 𝐵󸀠 /2𝑚)𝑡2) /4𝜎2 𝑖((𝜇 𝐵󸀠 𝑡𝑧−(𝜇2 𝐵󸀠2/6𝑚)𝑡3+𝜇 𝐵 𝑡+(ℏ𝜑 /2))/ℏ) (2𝜋𝜎 ) 𝑒 𝐵 0 0 𝑒 𝐵 0 𝐵 0 𝐵 0 0 cos 2 0 Ψ (𝑧,) 𝑡 ≃( ). (19) 2 𝜃0 2 −1/2 −(𝑧+(𝜇 𝐵󸀠 /2𝑚)𝑡2) /4𝜎2 𝑖((−𝜇 𝐵󸀠 𝑡𝑧−(𝜇2 𝐵󸀠2/6𝑚)𝑡3−𝜇 𝐵 𝑡−(ℏ𝜑 /2))/ℏ) 𝑖 (2𝜋𝜎 ) 𝑒 𝐵 0 0 𝑒 𝐵 0 𝐵 0 𝐵 0 0 sin 2 0

After the magnetic field, at time 𝑡 + Δ𝑡 (𝑡 ≥0) in the free electromagnet: space, the spinor becomes [44–48](seetheAppendix) −1/2 𝜃 2 2 2 2 0 −(𝑧−𝑧Δ−𝑢𝑡) /2𝜎0 𝜌𝜃 (𝑧, 𝑡 )+ Δ𝑡 ≃ (2𝜋𝜎 ) (cos 𝑒 0 0 2 Ψ (𝑧, 𝑡 )+ Δ𝑡 (22) 2 𝜃0 −(𝑧+𝑧 +𝑢𝑡)2/2𝜎2 + 𝑒 Δ 0 ). 𝜃0 2 −1/2 −(𝑧−𝑧 −𝑢𝑡)2/4𝜎2 𝑖((𝑚𝑢𝑧+ℏ𝜑 )/ℏ) sin (2𝜋𝜎 ) 𝑒 Δ 0 𝑒 + 2 cos 2 0 ≃( ), 𝜃0 2 −1/2 −(𝑧+𝑧 +𝑢𝑡)2/4𝜎2 𝑖((−𝑚𝑢𝑧+ℏ𝜑 )/ℏ) Figure 10 shows the probability density of a pure state (with sin (2𝜋𝜎 ) 𝑒 Δ 0 𝑒 − 2 0 𝜃0 =𝜋/3) as a function of 𝑧 at several values of 𝑡 (the plots (20) are labeled 𝑦=V𝑡). The beam separation does not appear at the end of the magnetic field (1 cm) but 16 cm further along. It is the moment of the decoherence. + − where The decoherence time, where the two spots 𝑁 and 𝑁 are separated, is then given by 𝜇 𝐵󸀠 (Δ𝑡)2 𝜇 𝐵󸀠 (Δ𝑡) 𝐵 0 −5 𝐵 0 3𝜎 −𝑧 𝑧Δ = =10 m,𝑢= =1m/s. 𝑡 ≃ 0 Δ =3×10−4 . 2𝑚 𝑚 𝐷 𝑢 s (23) (21) This decoherence time is usually the time required to diagonalize the marginal density matrix of spin variables Equation (20) takes into account the spatial extension of the associated with a pure state [49]: spinorandwenotethatthetwospinorcomponentshave very different 𝑧 values. All interpretations are based on this 󵄨 󵄨2 ∗ equation. ∫ 󵄨𝜓+ (𝑧,) 𝑡 󵄨 𝑑𝑧 ∫ 𝜓+ (𝑧,) 𝑡 𝜓− (𝑧,) 𝑡 𝑑𝑧 𝜌𝑆 (𝑡) =( ). 󵄨 󵄨2 ∫ 𝜓 (𝑧,) 𝑡 𝜓∗ (𝑧,) 𝑡 𝑑𝑧 ∫ 󵄨𝜓 (𝑧,) 𝑡 󵄨 𝑑𝑧 4.1. The Decoherence Time. We deduce from (20)theprob- − + 󵄨 − 󵄨 abilitydensityofapurestateinthefreespaceafterthe (24) Physics Research International 9

0 cm 6 cm 16 cm 21 cm

−0.6 0 0.6−0.6 0 0.6−0.6 0 0.6−0.6 0 0.6 (mm) (mm) (mm) (mm)

Figure 10: Evolution of the probability density of a pure state with 𝜃0 =𝜋/3.

+ By drilling a hole in the detector 𝑃1 to the location of N + 1 the spot 𝑁 (Figure 8), we select all the atoms that are in |+⟩ = ( 1 ) 0 thespinstate 0 . The new spinor of these atoms (mm) is obtained by making the component Ψ− of the spinor Ψ z − −1 N identically zero (and not only numerically equal to zero) at the time when the atom crosses the detector 𝑃1;atthis −5 −4 −3 −2 −1 012345 time the component Ψ− is indeed stopped by detector 𝑃1. x (mm) The future trajectory of the silver atom after crossing the 𝑃 𝑃 detector 1 will be guided by this new (normalized) spinor. Figure 11: 1000 silver atom impacts on the detector 1. The wave function reduction is therefore not linked to the electromagnet but to the detector 𝑃1 causing an irreversible elimination of the spinor component Ψ−.

𝑡≥𝑡 𝜓 (𝑧, 𝑡 + Δ𝑡)𝜓 (𝑧, 𝑡 + Δ𝑡) For 𝐷,theproduct + − is null 4.3. Impacts and Quantization Explained by de Broglie-Bohm and the density matrix is diagonal: the probability density of Trajectories. Finally, it remains to provide an explanation of the initial pure state (20)isdiagonal: the individual impacts of silver atoms. The spatial extension of the spinor (16)allowsustotakeintoaccounttheparticle’s 𝑧 2 𝜃0 initial position 0 andtointroducetheBroglie-Bohmtrajec- cos 0 𝑆 2 −1 2 tories [2, 6, 45, 46, 50] which is the natural assumption to 𝜌 (𝑡+Δ𝑡) = (2𝜋𝜎0 ) ( 𝜃 ). (25) explain the individual impacts. 0 2 0 sin 2 Figure 12 presents, for a silver atom with the initial spinor orientation (𝜃0 =𝜋/3, 𝜑0 =0),aplotinthe(𝑂𝑦𝑧) plane of a set of 10 trajectories whose initial position 𝑧0 has been 4.2. Proof of the Postulates of Quantum Measurement. We randomly chosen from a Gaussian distribution with standard then obtain atoms with a spin oriented only along the 𝑧-axis deviation 𝜎0.Thespinorientations𝜃(𝑧, 𝑡) are represented by (positively or negatively). Let us consider the spinor Ψ(𝑧, 𝑡 + arrows. Δ𝑡) given by (20). Experimentally, we do not measure the The final orientation, obtained after the decoherence time spin directly but the 𝑧̃ position of the particle impact on 𝑃1 𝑡𝐷, depends on the initial particle position 𝑧0 in the spinor (Figure 11). with a spatial extension and on the initial angle 𝜃0 of the spin + 𝑧∈𝑁̃ 𝜓 𝜃0 If ,theterm − of (20)isnumericallyequal with the 𝑧-axis. We obtain +𝜋/2 if 𝑧0 >𝑧 and −𝜋/2 if 𝑧0 < 1 Ψ ( ) 𝜃0 to zero and the spinor is proportional to 0 ,oneofthe 𝑧 with eigenvectors of the spin operator 𝑆𝑧 =(ℏ/2)𝜎𝑧: Ψ(𝑧,̃ 𝑡 + 2 −1/4 −(𝑧̃ −𝑧 −𝑢𝑡)2/4𝜎2 𝑖((𝑚𝑢𝑧̃ +ℏ𝜑 )/ℏ) Δ𝑡) ≃ (2𝜋𝜎 ) (𝜃 /2)𝑒 1 Δ 0 𝑒 1 + ( 1 ) 𝜃 0 cos 0 0 . 𝜃0 −1 2 0 𝑧 =𝜎0𝐹 (sin ), (26) Then, we have 𝑆𝑧Ψ = (ℏ/2)𝜎𝑧Ψ=+(ℏ/2)Ψ. 2 − If 𝑧∈𝑁̃ ,theterm𝜓+ of (20)isnumerically Ψ ( 0 ) equaltozeroandthespinor is proportional to 1 ,the where 𝐹 is the repartition function of the normal centered- other eigenvector of the spin operator 𝑆𝑧: Ψ(𝑧,̃ 𝑡 + Δ𝑡) ≃ 2 2 reduced law. If we ignore the position of the atom in its wave 2 −1/4 −(𝑧̃2+𝑧Δ+𝑢𝑡) /4𝜎0 𝑖((−𝑚𝑢𝑧̃2+ℏ𝜑−)/ℏ) 0 (2𝜋𝜎0 ) sin(𝜃0/2)𝑒 𝑒 ( 1 ).Then, function, we lose the determinism given by (26). we have 𝑆𝑧Ψ = (ℏ/2)𝜎𝑧Ψ=−(ℏ/2)Ψ. Therefore, the In the de Broglie-Bohm interpretation with a realistic measurement of the spin corresponds to an eigenvalue of the interpretation of the spin, the “measured” value is not spinoperator.Itisaproofofthepostulateofquantization. 2 independent of the context of the measure and is contextual. Equation (25)givestheprobabilitycos(𝜃0/2) (resp., It conforms to the Kochen and Specker theorem [51]: realism 2 sin (𝜃0/2)) to measure the particle in the spin state +ℏ/2 and noncontextuality are inconsistent with certain quantum (resp., −ℏ/2); this proves the Born probabilistic postulate. mechanics predictions. 10 Physics Research International

0.8 0.6

0.6 0.4

0.4 0.2

0.2 0 (mm) (mm) z z 0 −0.2

−0.2 −0.4

−0.4 −0.6 0 5101520 0 5 10 15 20 y (cm) y (cm)

Figure 12: Ten silver atom trajectories with initial spin orientation Figure 13: Ten silver atom trajectories where the initial orientation (𝜃0 =𝜋/3)and initial position 𝑧0;arrowsrepresentthespin (𝜃0,𝜑0) has been randomly chosen; arrows represent the spin orientation 𝜃(𝑧, 𝑡) along the trajectories. orientation 𝜃(𝑧, 𝑡) along the trajectories.

Now let us consider a mixture of pure states where the foreachatom,andtheimpossibilityofinteractionfasterthan initial orientation (𝜃0,𝜑0)fromthespinorhasbeenrandomly the speed of light. chosen. These are the conditions of the initial Stern and Here, we show that there exists a de Broglie-Bohm Gerlach experiment. Figure 13 represents a simulation of 10 interpretation which answers these two questions positively. quantum trajectories of silver atoms from which the initial To demonstrate this nonlocal realism, two methodological positions 𝑧0 arealsorandomlychosen. conditions are necessary. The first condition is the same as in Finally, the de Broglie-Bohm trajectories propose a clear the Stern-Gerlach experiment: the solution to the entangled interpretation of the spin measurement in quantum mechan- state is obtained by resolving the Pauli equation from an ics. There is interaction with the measuring apparatus as initial singlet wave function with a spatial extension as is generally stated, and there is indeed a minimum time 1 󵄨 󵄨 󵄨 󵄨 required for measurement. However this measurement and Ψ0 (r𝐴, r𝐵)= 𝑓(r𝐴)𝑓(r𝐵)(󵄨+𝐴⟩ 󵄨−𝐵⟩−󵄨−𝐴⟩ 󵄨+𝐵⟩) this time do not have the signification that is usually applied √2 to them. The result of the Stern-Gerlach experiment is not (27) the measure of the spin projection along the 𝑧-axis, but the orientation of the spin either in the direction of the magnetic and not from a simplified wave function without spatial field gradient or in the opposite direction. It depends on extension: the position of the particle in the wave function. We have 1 󵄨 󵄨 󵄨 󵄨 Ψ0 (r𝐴, r𝐵) = (󵄨+𝐴⟩ 󵄨−𝐵⟩ − 󵄨−𝐴⟩ 󵄨+𝐵⟩). therefore a simple explanation for the noncompatibility of √2 (28) spin measurements along different axes. The measurement duration is then the time necessary for the particle to point 𝑓 function and |±⟩ vectors are presented later. its spin in the final direction. The resolution in space of the Pauli equation is essential: it enables the spin measurement by spatial quantization and 5. EPR-B Experiment explains the determinism and the disentangling process. To explain the interaction and the evolution between the spin of Nonseparability is one of the most puzzling aspects of the two particles, we consider a two-step version of the EPR- quantum mechanics. For over thirty years, the EPR-B, the B experiment. It is our second methodological condition. A spin version of the Einstein-Podolsky-Rosen experiment [52] first causal interpretation of EPR-B experiment was proposed proposed by Bohm and Aharanov [53, 54], the Bell theorem in 1987 by Dewdney et al. [66, 67] using these two conditions. [55], and the BCHSH inequalities [5, 55, 56]havebeenatthe However, this interpretation had a flaw46 [ ,page418]:the heart of the debate on hidden variables and nonlocality. Many spin module of each particle depends directly on the singlet experiments since Bell’s paper have demonstrated violations wave function, and thus the spin module of each particle of these inequalities and have vindicated quantum theory varied during the experiment from 0 to ℏ/2.Wepresentade [57–63].Now,EPRpairsofmassiveatomsarealsoconsidered Broglie-Bohm interpretation that avoids this flaw68 [ ]. [64, 65]. The usual conclusion of these experiments is to reject Figure 14 presents the Einstein-Podolsky-Rosen-Bohm the nonlocal realism for two reasons: the impossibility of experiment. A source 𝑆 creates, in 𝑂, pairs of identical atoms decomposing a pair of entangled atoms into two states, one 𝐴 and 𝐵, but with opposite spins. The atoms 𝐴 and 𝐵 Physics Research International 11

z z󳰀

z z󳰀 z x󳰀 y

EB x EA

Atom B O Atom A y

y(Δt + tD) y(t0 +Δt+tD) yt0 yΔt ytD

z z 𝛿 z󳰀

x x

Figure 14: Schematic configuration of the EPR-B experiment. split following the 𝑦-axisinoppositedirectionsandhead on a pair of particles 𝐴 and 𝐵 in a singlet state. This is the towards two identical Stern-Gerlach apparatus E𝐴 and E𝐵. experiment first proposed in 1987 by Dewdney et al.66 [ , 67]. The electromagnet E𝐴 “measures” the spin of 𝐴 along the 𝑧- Consider that at time 𝑡0 the particle 𝐴 arrives at the axis and the electromagnet E𝐵 “measures” the spin of 𝐵 along entrance of electromagnet E𝐴. After this exit of the magnetic 󸀠 the 𝑧 -axis, which is obtained after a rotation of an angle 𝛿 field E𝐴,attime𝑡0 +Δ𝑡+𝑡,thewavefunction(27)becomes around the 𝑦-axis. The initial wave function of the entangled [68] state is the singlet state (27), where r =(𝑥,𝑧), 𝑓(r)= 2 2 2 2 −1/2 −(𝑥 +𝑧 )/4𝜎0 Ψ(r𝐴, r𝐵,𝑡0 +Δ𝑡+𝑡) (2𝜋𝜎0 ) 𝑒 , |±𝐴⟩,and|±𝐵⟩ are the eigenvectors of the operators 𝜎𝑧 and 𝜎𝑧 : 𝜎𝑧 |±𝐴⟩=±|±𝐴⟩, 𝜎𝑧 |±𝐵⟩= 1 ±|± ⟩ 𝐴 𝐵 𝐴 𝑦 𝐵 = 𝑓(r ) 𝐵 . We treat the dependence with classically: speed √2 𝐵 (31) −V𝑦 for 𝐴 and V𝑦 for 𝐵.ThewavefunctionΨ(r𝐴, r𝐵,𝑡) of 𝐴 𝐵 + 󵄨 󵄨 − 󵄨 󵄨 the two identical particles and ,electricallyneutraland ×(𝑓 (r𝐴,𝑡)󵄨+𝐴⟩ 󵄨−𝐵⟩−𝑓 (r𝐴,𝑡)󵄨−𝐴⟩ 󵄨+𝐵⟩) with magnetic moments 𝜇0,subjecttomagneticfieldsE𝐴 and E𝐵,admitsonthebasisof|±𝐴⟩ and |±𝐵⟩ four components with 𝑎,𝑏 Ψ (r𝐴, r𝐵,𝑡) and satisfies the two-body Pauli equation [46, ± 𝑖((±𝑚𝑢𝑧/ℏ)+𝜑±(𝑡)) page 417]: 𝑓 (r,𝑡) ≃𝑓(𝑥, 𝑧Δ ∓𝑧 ∓𝑢𝑡) 𝑒 , (32) 𝑎,𝑏 2 2 𝜕Ψ ℏ ℏ E 𝑎 𝑧 𝑢 𝑖ℏ =(− Δ − Δ )Ψ𝑎,𝑏 +𝜇𝐵 𝐴 (𝜎 ) Ψ𝑐,𝑏 where Δ and are given by (21). 𝐴 𝐵 𝑗 𝑗 𝑐 𝜕𝑡 2𝑚 2𝑚 The atomic density 𝜌(𝑧𝐴,𝑧𝐵,𝑡0 +Δ𝑡+𝑡)is found by (29) ∗ integrating Ψ (r𝐴, r𝐵,𝑡0 +Δ𝑡+𝑡)Ψ(r𝐴, r𝐵,𝑡0 +Δ𝑡+𝑡)on 𝑥𝐴 E 𝑏 𝑎,𝑑 +𝜇𝐵 𝐵 (𝜎 ) Ψ and 𝑥𝐵: 𝑗 𝑗 𝑑 with the initial conditions: 𝜌(𝑧𝐴,𝑧𝐵,𝑡0 +Δ𝑡+𝑡) 𝑎,𝑏 𝑎,𝑏 Ψ (r , r ,0)=Ψ (r , r ), −1/2 2 2 𝐴 𝐵 0 𝐴 𝐵 (30) 2 −(𝑧𝐵) /2𝜎0 = ((2𝜋𝜎0 ) 𝑒 ) Ψ𝑎,𝑏(r , r ) where 0 𝐴 𝐵 corresponds to the singlet state (27). (33) To obtain an explicit solution of the EPR-B experiment, 2 −1/2 × ((2𝜋𝜎0 ) we take the numerical values of the Stern-Gerlach experi- ment. 1 −(𝑧 −𝑧 −𝑢𝑡)2/2𝜎2 −(𝑧 +𝑧 +𝑢𝑡)2/2𝜎2 × (𝑒 𝐴 Δ 0 +𝑒 𝐴 Δ 0 )) . One of the difficulties of the interpretation of the EPR- 2 B experiment is the existence of two simultaneous measure- ments. By doing these measurements one after the other, the We deduce that the beam of particle 𝐴 is divided into two, interpretation of the experiment will be facilitated. That is while the beam of particle 𝐵 stays undivided: the purpose of the two-step version of the experiment EPR-B studied below. (i) the density of 𝐴 is the same, whether particle 𝐴 is entangled with 𝐵 or not; 5.1.FirstStepEPR-B:SpinMeasurementof𝐴. In the first step (ii) the density of 𝐵 is not affected by the “measurement” we make a Stern and Gerlach “measurement” for atom 𝐴, of 𝐴. 12 Physics Research International

Our first conclusion is that the position of 𝐵 does not Step 1 (spin measurement of 𝐴). In (31)particle𝐴 can be depend on the measurement of 𝐴; only the spins are involved. considered independent of 𝐵. We can therefore give it the We conclude from (31)thatthespinsof𝐴 and 𝐵 remain wave function: opposite throughout the experiment. These are the two Ψ𝐴 (r ,𝑡 +Δ𝑡+𝑡) properties used in the causal interpretation. 𝐴 0 𝐴 𝐴 𝜃0 + 󵄨 𝜃0 𝑖𝜑𝐴 − 󵄨 𝐵 = 𝑓 (r ,𝑡)󵄨+ ⟩+ 𝑒 0 𝑓 (r ,𝑡)󵄨− ⟩ 5.2. Second Step EPR-B: Spin Measurement of . The second cos 2 𝐴 󵄨 𝐴 sin 2 𝐴 󵄨 𝐴 step is a continuation of the first and corresponds to the (36) EPR-B experiment broken down into two steps. On a pair of particles 𝐴 and 𝐵 in a singlet state, first we made a Stern and which is the wave function of a free particle in a Stern-Gerlach 𝐴 𝑡 𝑡 +Δ𝑡+ 𝐴 𝐴 Gerlach measurement on the atom between 0 and 0 apparatus and whose initial spin is given by (𝜃0 ,𝜑0 ). For 𝑡𝐷; secondly, we make a Stern and Gerlach measurement on 𝐴 𝐴 𝐴 an initial polarization (𝜃0 ,𝜑0 ) and an initial position (𝑧0 ), the 𝐵 atom with an electromagnet E𝐵 forming an angle 𝛿 with E 𝑡 +Δ𝑡+𝑡 𝑡 + 2(Δ𝑡 + 𝑡 ) we obtain, in the de Broglie-Bohm interpretation [44]ofthe 𝐴 during 0 𝐷 and 0 𝐷 . Stern and Gerlach experiment, an evolution of the position At the exit of magnetic field E𝐴,attime𝑡0 +Δ𝑡+𝑡𝐷, 𝐴 (𝑧𝐴(𝑡)) and of the spin orientation of 𝐴 (𝜃 (𝑧𝐴(𝑡), 𝑡))[48]. the wave function is given by (31). Immediately after the The case of particle 𝐵 is different. 𝐵 follows a rectilinear measurement of 𝐴, still at time 𝑡0 +Δ𝑡+𝑡𝐷,thewavefunction 𝐵 𝐵 trajectory with 𝑦𝐵(𝑡) = V𝑦𝑡, 𝑧𝐵(𝑡) = 𝑧 ,and𝑥𝐵(𝑡) = 𝑥 .By of 𝐵 depends on the measurement ± of 𝐴: 0 0 contrast, the orientation of its spin moves with the orientation 󵄨 𝐵 𝐴 𝐵 Ψ𝐵/±𝐴 (r𝐵,𝑡0 +Δ𝑡+𝑡1)=𝑓(r𝐵) 󵄨∓𝐵⟩. (34) of the spin of 𝐴: 𝜃 (𝑡) = 𝜋 −𝜃 (𝑧𝐴(𝑡), 𝑡) and 𝜑 (𝑡) = 𝜑(𝑧 (𝑡), 𝑡) −𝜋 𝐵 𝑡 + 2(Δ𝑡 + 𝑡 ) 𝐴 .Wecanassociatethefollowingwavefunction Then, the measurement of at time 0 𝐷 yields, with the particle 𝐵: in this two-step version of the EPR-B experiment, the same 𝐵 results for spatial quantization and correlations of spins as in Ψ (r𝐵,𝑡0 +Δ𝑡+𝑡) the EPR-B experiment. 𝐵 𝐵 𝜃 (𝑡) 󵄨 𝜃 (𝑡) 𝑖𝜑𝐵(𝑡) 󵄨 =𝑓(r𝐵)(cos 󵄨+𝐵⟩+sin 𝑒 󵄨−𝐵⟩). 5.3. Causal Interpretation of the EPR-B Experiment. We as- 2 󵄨 2 󵄨 𝐴 sume, at the creation of the two entangled particles and (37) 𝐵,thateachofthetwoparticles𝐴 and 𝐵 has an initial 𝐴 𝐴 𝐴 wave function with opposite spins: Ψ0 (r𝐴,𝜃0 ,𝜑0 )=𝑓(r𝐴) This wave function is specific, because it depends upon initial 𝐴 𝐴 𝐴 𝐴 𝑖𝜑0 𝐵 𝐵 𝐵 conditions of (position and spin). The orientation of spin (cos(𝜃0 /2)|+𝐴⟩+sin(𝜃0 /2)𝑒 |−𝐴⟩) and Ψ0 (r𝐵,𝜃0 ,𝜑0 )= 𝐵 of the particle 𝐵 is driven by the particle 𝐴 through the singlet 𝐵 𝐵 𝑖𝜑0 𝐵 𝐴 𝑓(r𝐵)(cos(𝜃0 /2)|+𝐵⟩+sin(𝜃0 /2)𝑒 |−𝐵⟩) with 𝜃0 =𝜋−𝜃0 𝐵 𝐴 wave function. Thus, the singlet wave function is the nonlocal and 𝜑0 =𝜑0 −𝜋. The two particles 𝐴 and 𝐵 are statistically variable. prepared as in the Stern and Gerlach experiment. Then the Pauliprincipletellsusthatthetwo-bodywavefunctionmust Step 2 (spin measurement of 𝐵). At the time 𝑡0 +Δ𝑡+𝑡𝐷, 𝐵 be antisymmetric; after calculation, we find the same singlet immediately after the measurement of 𝐴, 𝜃 (𝑡0 +Δ𝑡+𝑡𝐷)=𝜋 𝐴 state (27): or 0 in accordance with the value of 𝜃 (𝑧𝐴(𝑡), 𝑡) and the 󸀠 󸀠 𝐴 𝐴 𝐵 𝐵 wave function of 𝐵 is given by (34). The frame (𝑂𝑥 𝑦𝑧 ) Ψ0 (r𝐴,𝜃 ,𝜑 , r𝐵,𝜃 ,𝜑 ) corresponds to the frame (𝑂𝑥𝑦𝑧) after a rotation of an angle 𝐵 𝑖𝜑𝐴 󵄨 󵄨 󵄨 󵄨 𝛿 around the 𝑦-axis. 𝜃 corresponds to the 𝐵-spin angle with =−𝑒 𝑓(r𝐴)𝑓(r𝐵)×(󵄨+𝐴⟩ 󵄨−𝐵⟩−󵄨−𝐴⟩ 󵄨+𝐵⟩). 󸀠𝐵 󸀠 the 𝑧-axis and 𝜃 to the 𝐵-spin angle with the 𝑧 -axis; then (35) 󸀠𝐵 𝜃 (𝑡0 +Δ𝑡+𝑡𝐷)=𝜋+𝛿or 𝛿.Inthissecondstep,we Thus, we can consider that the singlet wave function is the are exactly in the case of a particle in a simple Stern and wave function of a family of two fermions 𝐴 and 𝐵 with Gerlach experiment (with magnet E𝐵)withaspecificinitial opposite spins: the direction of initial spins 𝐴 and 𝐵 exists but polarization equal to 𝜋+𝛿or 𝛿 and not random like in Step 1. is not known. It is a local hidden variable which is therefore Then, the measurement of 𝐵,attime𝑡0 + 2(Δ𝑡 +𝐷 𝑡 ),gives, necessarytoaddintheinitialconditionsofthemodel. in this interpretation of the two-step version of the EPR-B ThisisnottheinterpretationfollowedbytheBohmschool experiment, the same results as in the EPR-B experiment. [44–46, 66, 67] in the interpretation of the singlet wave function; they do not assume the existence of wave functions 5.4. Physical Explanation of Nonlocal Influences. From the Ψ𝐴(r ,𝜃𝐴,𝜑𝐴) Ψ𝐵(r ,𝜃𝐵,𝜑𝐵) 0 𝐴 0 0 and 0 𝐵 0 0 for each particle but only wave function of two entangled particles, we find spins, 𝐴 𝐴 𝐵 𝐵 the singlet state Ψ0(r𝐴,𝜃 ,𝜑 , r𝐵,𝜃 ,𝜑 ).Inconsequence, trajectories, and also a wave function for each of the two they suppose a zero spin for each particle at the initial particles. In this interpretation, the quantum particle has time and a spin module of each particle varied during the a local position like a classical particle, but it has also a experiment from 0 to ℏ/2 [46,page418]. nonlocal behavior through the wave function. So, it is the Here, we assume that at the initial time we know the spin wave function that creates the nonclassical properties. We of each particle (given by each initial wave function) and the can keep a view of a local realist world for the particle, but initial position of each particle. we should add a nonlocal vision through the wave function. Physics Research International 13

As we saw in Step 1, the nonlocal influences in the EPR-B appears only some centimeters after the slits and shows the experiment only concern the spin orientation not the motion continuity with classical mechanics; in the Stern-Gerlach of the particles themselves. Indeed only spins are entangled experiment, the spin-up/down measurement appears also in the wave function (27) not positions and motions like in after a given time, called decoherence time; in the EPR- the initial EPR experiment. This is a key point in the search B experiment, only the spin of 𝐵 is affected by the spin for a physical explanation of nonlocal influences. measurement of 𝐴 notitsdensity.Moreover,thedeBroglie- The simplest explanation of this nonlocal influence isto Bohm trajectories propose a clear explanation of the spin reintroduce the concept of ether (or the preferred frame) but a measurement in quantum mechanics. new format given by Lorentz-Poincare´ and by Einstein in 1920 However, we have seen two very different cases in the [69]: “Recapitulating, we may say that according to the general measurement process. In the first case (double-slit exper- theory of relativity space is endowed with physical qualities; iment), there is no influence of the measuring apparatus in this sense, therefore, there exists an ether. According to the (the screen) on the quantum particle. In the second case general theory of relativity space without ether is unthinkable; (Stern-Gerlach experiment, EPR-B), there is an interaction [sic] for in such space there not only would be no propagation with the measuring apparatus (the magnetic field) and the of light, but also no possibility of existence for standards of quantum particle. The result of the measurement depends space and time (measuring-rods and clocks), nor therefore any on the position of the particle in the wave function. The space-time intervals in the physical sense. But this ether may measurement duration is then the time necessary for the not be thought of as endowed with the quality characteristic of stabilisation of the result. ponderable media, as consisting of parts which may be tracked This heterodox interpretation clearly explains experi- through time. The idea of motion may not be applied toit.” ments with a set of quantum particles that are statistically Taking into account the new experiments, especially prepared. These particles verify the “quantum equilibrium Aspect’s experiments, Popper [70, page XVIII] defends a hypothesis” and the de Broglie-Bohm interpretation estab- similar view in 1982: lishes continuity with classical mechanics. However, there “I feel not quite convinced that the experiments are correctly is no reason that the de Broglie-Bohm interpretation can interpreted; but if they are, we just have to accept action at a be extended to quantum particles that are not statistically distance. I think (with J.P. Vigier) that this would of course be prepared. This situation occurs when the wave packet cor- veryimportant,butIdonotforamomentthinkthatitwould responds to a quasiclassical coherent state, introduced in shake, or even touch, realism. Newton and Lorentz were realists 1926 by Schrodinger¨ [71]. The field quantum theory and the and accepted action at a distance; and Aspect’s experiments second quantification are built on these coherent states72 [ ]. would be the first crucial experiment between Lorentz’s and It is also the case, for the hydrogen atom, of localized wave Einstein’s interpretation of the Lorentz transformations.” packets whose motions are on the classical trajectory (an Finally, in the de Broglie-Bohm interpretation, the EPR-B old dream of Schrodinger’s).¨ Their existence was predicted experiments of nonlocality have therefore a great importance in 1994 by Bialynicki-Birula et al. [73–75] and discovered not to eliminate realism and determinism but as Popper said recently by Maeda and Gallagher [76]onRydbergatoms. to rehabilitate the existence of a certain type of ether, like For these nonstatistically prepared quantum particles, we Lorentz’s ether and like Einstein’s ether in 1920. have shown [9, 10] that the natural interpretation is the Schrodinger¨ interpretation proposed at the Solvay congress in 1927. Everything happens as if the quantum mechanics 6. Conclusion interpretation depended on the preparation of the particles In the three experiments presented in this paper, the variable (statistically or not statistically prepared). It is perhaps a that is measured in fine is the position of the particle given response to the “theory of the double solution” that Louis de by this impact on a screen. In the double-slit, the set of these Broglie was seeking since 1927: “Iintroducedasthe“double positions gives the interferences; in the Stern-Gerlach and the solution theory” the idea that it was necessary to distinguish two different solutions that are both linked to the wave equation, EPR-B experiments, it is the position of the particle impact 𝑢 that defines the spin value. one that I called wave ,whichwasarealphysicalwave representedbyasingularityasitwasnotnormalizabledue It is this position that the de Broglie-Bohm interpretation to a local anomaly defining the particle, the other one as adds to the wave function to define a complete state of the Schrodinger’s¨ Ψ wave, which is a probability representation as quantum particle. The de Broglie-Bohm interpretation is then 0 it is normalizable without singularities”[77]. based only on the initial conditions Ψ (x) and X(0) and the evolution equations (1)and(12). If we add as initial condition the “quantum equilibrium hypothesis” (14), we have seen that we can deduce, for these three examples, the three postulates Appendix of measurement. These three postulates are not necessary if we solve the time-dependent Schrodinger¨ equation (double- Calculating the Spinor Evolution in the slit experiment) or the Pauli equation with spatial extension Stern-Gerlach Experiment (Stern-Gerlach and EPR experiments). However, these sim- ulations enable us to better understand those experiments: In the magnetic field 𝐵=(𝐵𝑥,0,𝐵𝑧),thePauliequation in the double-slit experiment, the interference phenomenon (18) gives coupled Schrodinger¨ equations for each spinor 14 Physics Research International

component: The experimental conditions give ℏΔ𝑡/2𝑚𝜎0 =4× −11 −4 10 m ≪𝜎0 =10 m. We deduce the approximations 𝜕𝜓± 𝑖ℏ (𝑥, 𝑧,) 𝑡 𝜎𝑡 ≃𝜎0 and 𝜕𝑡 ℏ2 =− ∇2𝜓 (𝑥, 𝑧,) 𝑡 ±𝜇 (𝐵 −𝐵󸀠 𝑧) 𝜓 (𝑥, 𝑧,) 𝑡 (A.1) 𝜓 (𝑧,) 𝑡 2𝑚 ± 𝐵 0 0 ± 𝐾

2 2 3 󸀠 2 −1/4 −(𝑧+𝐾𝑡2/2𝑚) /4𝜎2 𝑖 𝐾 𝑡 ∓𝑖𝜇 𝐵 𝑥𝜓 (𝑥, 𝑧,) 𝑡 . ≃ (2𝜋𝜎 ) 𝑒 0 [−𝐾𝑡𝑧 − ]. 𝐵 0 ∓ 0 exp ℏ 6𝑚 If one affects the transformation [47] (A.7)

𝑖𝜇𝐵𝐵0𝑡 𝜓± (𝑥, 𝑧,) 𝑡 = exp (± ) 𝜓± (𝑥, 𝑧,) 𝑡 (A.2) ℏ At the end of the magnetic field, at time Δ𝑡,thespinoris (A.1)becomes equal to 𝜕𝜓 𝑖ℏ ± (𝑥, 𝑧,) 𝑡 𝜕𝑡 𝜓+ (𝑧, )Δ𝑡 Ψ (𝑧, )Δ𝑡 =( ) (A.8) 𝜓− (𝑧, )Δ𝑡 ℏ2 =− ∇2𝜓 (𝑥, 𝑧,) 𝑡 ∓𝜇 𝐵󸀠 𝑧𝜓 (𝑥, 𝑧,) 𝑡 (A.3) 2𝑚 ± 𝐵 0 ± 2𝜇 𝐵 𝑡 with ∓𝑖𝜇 𝐵󸀠 𝑥𝜓 (𝑥, 𝑧,) 𝑡 (±𝑖 𝐵 0 ). 𝐵 0 ∓ exp ℏ 2 −1/4 −((𝑧−𝑧 )2/4𝜎2)+(𝑖/ℏ)𝑚𝑢𝑧 𝜃0 𝑖𝜑 𝜓 𝑧, Δ𝑡 = (2𝜋𝜎 ) 𝑒 Δ 0 𝑒 + , The coupling term oscillates rapidly with the Larmor fre- + ( ) 0 cos 11 −1 󸀠 2 quency 𝜔𝐿 =2𝜇𝐵𝐵0/ℏ = 1, 4 × 10 s .Since|𝐵0|≫|𝐵0𝑧| 󸀠 2 −1/4 −((𝑧+𝑧 )2/4𝜎2)−(𝑖/ℏ)𝑚𝑢𝑧 𝜃0 𝑖𝜑 and |𝐵0|≫|𝐵𝑥|, the period of oscillation is short compared Δ 0 − 0 𝜓− (𝑧, )Δ𝑡 =(2𝜋𝜎0 ) 𝑒 𝑖 sin 𝑒 , to the motion of the wave function. Averaging over a period 2 that is long compared to the oscillation period, the coupling 𝜇 𝐵󸀠 (Δ𝑡)2 𝜇 𝐵󸀠 (Δ𝑡) term vanishes, which entails [47] 𝑧 = 𝐵 0 ,𝑢=0 0 , Δ 2𝑚 𝑚 2 𝜕𝜓± ℏ 2 󸀠 2 3 𝑖ℏ (𝑥, 𝑧,) 𝑡 =− ∇ 𝜓± (𝑥, 𝑧,) 𝑡 ∓𝜇𝐵𝐵0𝑧𝜓± (𝑥, 𝑧,) 𝑡 . 𝜑0 𝜇𝐵𝐵0Δ𝑡 𝐾 (Δ𝑡) 𝜕𝑡 2𝑚 𝜑+ = − − ; (A.4) 2 ℏ 6𝑚ℏ 𝜑 𝜇 𝐵 Δ𝑡 𝐾2(Δ𝑡)3 Since the variable 𝑥 isnotinvolvedinthisequationand 𝜑 =− 0 + 0 0 − . 0 − 2 ℏ 6𝑚ℏ 𝜓±(𝑥, 𝑧) does not depend on 𝑥, 𝜓±(𝑥, 𝑧, 𝑡) does not depend (A.9) on 𝑥: 𝜓±(𝑥, 𝑧, 𝑡) ≡ 𝜓±(𝑧, 𝑡). Then we can explicitly compute the preceding equations for all 𝑡 in [0, Δ𝑡] with Δ𝑡 = Δ𝑙/V = 5 2×10 s. We remark that the passage through the magnetic field We obtain gives the equivalent of a velocity +𝑢 in the direction 0𝑧 𝜃 𝜓 −𝑢 𝜓 0 𝑖(𝜑0/2) 󸀠 to the function + and a velocity to the function −. 𝜓 (𝑧,) 𝑡 =𝜓𝐾 (𝑧,) 𝑡 cos 𝑒 ,𝐾=−𝜇𝐵𝐵 , + 2 0 Thenwehaveafreeparticlewiththeinitialwavefunction (A.5) 𝜓 (𝑥, 𝑧, 𝑡) = 𝜃 (A.8). The Pauli equation resolution again yields ± 0 −𝑖(𝜑0/2) 󸀠 𝜓 (𝑥, 𝑡)𝜓 (𝑧, 𝑡) 𝜓− (𝑧,) 𝑡 =𝜓𝐾 (𝑧,) 𝑡 𝑖 sin 𝑒 ,𝐾=+𝜇𝐵𝐵0 𝑥 ± and with the experimental conditions we 2 2 2 2 −1/4 −𝑥 /4𝜎0 have 𝜓𝑥(𝑥, 𝑡) ≃ 0(2𝜋𝜎 ) 𝑒 and 2 2 2 𝜎𝑡 =𝜎0 + (ℏ𝑡/2𝑚𝜎0) and

−1/4 2 2 2 2 −(𝑧+𝐾𝑡 /2𝑚) /4𝜎𝑡 𝜓+ (𝑧, 𝑡 )+ Δ𝑡 𝜓𝐾 (𝑧,) 𝑡 =(2𝜋𝜎𝑡 ) 𝑒 −1/4 𝜃 ≃(2𝜋𝜎2) 0 𝑖 ℏ ℏ𝑡 𝐾2𝑡3 0 cos 2 × [ − −1 ( )−𝐾𝑡𝑧− exp tan 2 ℏ 2 2𝑚𝜎0 6𝑚 −(𝑧−𝑧 −𝑢𝑡)2/4𝜎2+(𝑖/ℏ)(𝑚𝑢𝑧−(1/2)𝑚𝑢2𝑡+ℏ𝜑 ) [ × exp Δ 0 + , (A.10) 2 2 2 2 (𝑧 + 𝐾𝑡 /2𝑚) ℏ 𝑡 𝜓− (𝑧, 𝑡 )+ Δ𝑡 + ] , 2 2 8𝑚𝜎 𝜎 −1/4 𝜃 0 𝑡 ] 2 0 ≃(2𝜋𝜎0 ) 𝑖 sin (A.6) 2 −(𝑧+𝑧 +𝑢𝑡)2/4𝜎2+(𝑖/ℏ)(−𝑚𝑢𝑧−(1/2)𝑚𝑢2𝑡+ℏ𝜑 ) where (A.6)isaclassicalresult[11]. × exp Δ 0 − . Physics Research International 15

Conflict of Interests [18] P. G. Merlin, G. F. Missiroli, and G. Pozzi, “On the statistical aspect of electron interference phenomena,” American Journal The authors declare that there is no conflict of interests of Physics,vol.44,no.3,pp.306–307,1976. regarding the publication of this paper. [19] A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, and H. Ezawa, “Demonstration of single-electron buildup of an interference References pattern,” American Journal of Physics,vol.57,pp.117–120,1989. [20] H. V. Halbon Jr. and P. Preiswerk, “Preuve experimentale´ de [1] J. S. Bell, “On the impossible pilot wave,” in Speakable and Un- la diffraction des neutrons,” Comptes Rendus de l’Academie´ des speakable in Quantum Mechanics, Cambridge University Press, Sciences,vol.203,pp.73–75,1936. 1987. [21] H. Rauch and A. Werner, Neutron Interferometry: Lessons in [2] D. Bohm, “Asuggested interpretation of the quantum theory in Experimental Quantum Mechanics, Oxford University Press, terms of hidden variables, I and II,” Physical Review,vol.85,pp. London, UK, 2000. 166–193, 1952. [22] A. Zeilinger, R. Gahler,C.G.Shull,W.Treimer,andW.Mampe,¨ [3] J. von Neumann, Mathematical Foundations of Quantum Me- “Single- and double-slit diffraction of neutrons,” Reviews of chanics, Princeton Landmarks in Mathematics, Princeton Uni- Modern Physics, vol. 60, no. 4, pp. 1067–1073, 1988. versity Press, Princeton, NJ, USA, 1996. [23] I. Estermann and O. Stern, “Beugung von Molekularstrahlen,” [4] B. d’Espagnat, A la Recherche du Ree´ , Gauthiers-Villard, Paris, Zeitschriftur f¨ Physik,vol.61,no.1-2,pp.95–125,1930. France, 1979. [24] F. Shimizu, K. Shimizu, and H. Takuma, “Double-slit interfer- [5]J.S.Bell,Speakable and Un Speakable in Quantum Mechanics, ence with ultracold metastable neon atoms,” Physical Review A, Cambridge University Press, 2nd edition, 2004. vol.46,no.1,pp.R17–R20,1992. [6]L.deBroglie,“Lamecani´ que ondulatoire et la structure [25] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, atomique de la matiere` et du rayonnement,” Journal de Physique and E. A. Cornell, “Observation of Bose-Einstein condensation et Le Radium,vol.8,no.5,pp.225–241,1927,AnEnglish in a dilute atomic vapor,” Science,vol.269,no.5221,pp.198–201, translation can be found in G. Bacciagalluppi and A. Valentini, 1995. Quantum Theory of the Crossroads (Cambridge University [26] M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, Press, Cambridge, UK, 2009). and A. Zellinger, “Wave-particle duality of C60 molecules,” [7] C. Jonsson,¨ “Elektroneninte rferenzen an mehreren kunstlich¨ Nature,vol.401,no.6754,pp.680–682,1999. hergestellten Feinspalten,” Zeitschriftur f¨ Physik,vol.161,pp. [27] O. Nairz, M. Arndt, and A. Zeilinger, “Experimental challenges 454–474, 1961. in fullerene interferometry,” JournalofModernOptics,vol.47, [8] E. Madelung, “Quantentheorie in hydrodynamischer form,” no. 14-15, pp. 2811–2821, 2000. Zeitschriftur f¨ Physik,vol.40,no.3-4,pp.322–326,1927. [28] R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman [9] M. Gondran and A. Gondran, “Discerned and non-discerned Lectures on Physics. Vol. 3: Quantum Mechanics, Addison- particles in classical mechanics and convergence of quantum Wesley, New York, NY, USA, 1965. mechanics to classical mechanics,” Annales de la Fondation [29] M. Gondran and A. Gondran, “Numerical simulation of the Louis de Broglie, vol. 36, no. 1, pp. 117–135, 2011. double slit interference with ultracold atoms,” The American [10] M. Gondran and A. Gondran, “The two limits of the Journal of Physics,vol.73,no.6,pp.507–515,2005. Schrodinger¨ equation in the semi-classical approximation: [30] C. Philippidis, C. Dewdney, and B. J. Hiley, “Quantum interfer- discerned and non-discerned particles in classical mechanics,” ence and the ,” Il Nuovo Cimento B: Serie 11, in Foundations of Probability and Physics-6,vol.1424ofAIP vol.52,no.1,pp.15–28,1979. Conference Proceedings, pp. 111–115, 2012. [31] W. Gerlach and O. Stern, “Der experimentelle Nachweis des [11] R. Feynman and A. Hibbs, Quantum Mechanics and Paths magnetischen moments des silberatoms,” Zeitschriftur f¨ Physik, Integrals, McGraw-Hill, 1965. vol. 8, no. 1, pp. 110–111, 1921. [12] C. Cohen-Tannoudji, B. Diu, and F. Laloe,¨ Quantum Mechanics, [32] W. Gerlach and O. Stern, “Der experimentelle Nachweis der JohnWiley&Sons,NewYork,NY,USA,1977. Richtungsquantelung im Magnetfeld,” Zeitschriftur f¨ Physik,vol. [13] D. Durr,¨ S. Goldstein, and N. Zanghi, “Quantum equilibrium 9, no. 1, pp. 349–352, 1922. and the origin of absolute uncertainty,” Journal of Statistical [33] H. D. Zeh, “On the interpretation of measurement in quantum Physics,vol.67,no.5-6,pp.843–907,1992. theory,” Foundations of Physics,vol.1,pp.69–76,1970,Reprinted [14] A. S. Sanz and S. Miret-Artes,´ “Quantum phase analysis with in Wheeler and Zurek, pp. 342–349, 1983. quantum trajectories: a step towards the creation of a Bohmian [34] W. H. Zurek, “Environment-induced superselection rules,” thinking,” The American Journal of Physics,vol.80,no.6,pp. Physical Review D: Particles and Fields,vol.26,no.8,pp.1862– 525–533, 2012. 1880, 1982. [15] T. Norsen, “The pilot-wave perspective on quantum scattering [35] J. A. Wheeler and W. H. Zurek, Quantum Theory and Measure- and tunneling,” American Journal of Physics,vol.81,no.4,pp. ment, Princeton Series in Physics, Princeton University Press, 258–266, 2013. Princeton, NJ, USA, 1983. [16] T. Young, “On the theory of light and colors,” Philosophical [36] W. H. Zurek, “Decoherence, einselection, and the quantum transactions of the Royal Society, vol. 92, pp. 12–48, 1802. origins of the classical,” ReviewsofModernPhysics,vol.75,no. [17]C.DavissonandL.H.Germer,“Thescatteringofelectronsbya 3,pp.715–775,2003. single crystal of nickel,” Nature,vol.119,no.2998,pp.558–560, [37] R. Omnes,` “Consistent interpretations of quantum mechanics,” 1927. Reviews of Modern Physics,vol.64,no.2,pp.339–382,1992. 16 Physics Research International

[38] M. Schlosshauer, Decoherence and the Quantumto—Classical [59] A. Aspect, J. Dalibard, and G. Roger, “Experimental test of Bell’s Transition, Springer, 2007. inequalities using time-varying analyzers,” Physical Review [39] J. J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, Letters,vol.49,no.25,pp.1804–1807,1982. 1985. [60] W.Tittel,J.Brendel,H.Zbinden,andN.Gisin,“Violationofbell [40] M. Le Bellac, Quantum Physics, Cambridge University Press, inequalities by photons more than 10 km apart,” Physical Review Cambridge, UK, 2006. Letters,vol.81,no.17,pp.3563–3566,1998. [61] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. [41] T. Takabayasi, “On the formulation of quantum mechanics asso- Zeilinger, “Violation of Bell’s inequality under strict Einstein ciated with classical pictures,” Progress of Theoretical Physics,vol. locality conditions,” Physical Review Letters,vol.81,no.23,pp. 8, no. 2, pp. 143–182, 1952. 5039–5043, 1998. [42] T. Takabayasi, “The formulation of quantum mechanics in [62] R. A. Bertlmann and A. Zeilinger, Eds., Quantum [Un]speaka- terms of ensemble in phase space,” Progress of Theoretical bles: From Bell to Quantum Information, Springer, 2002. Physics,vol.11,no.4-5,pp.341–373,1954. [63] M. Genovese, “Research on hidden variable theories: a review of [43] D. Bohm, R. Schiller, and J. Tiomno, “A causal interpretation of recent progresses,” Physics Reports,vol.413,no.6,pp.319–396, the pauli equation,” Il Nuovo Cimento Series, supplement 1, pp. 2005. 48–66, 1955. [64] A. Beige, W. J. Munro, and P. L. Knight, “Bell’s inequality test [44] D. Bohm and B. J. Hiley, The Undivided Universe,Routledge, with entangled atoms,” Physical Review A,vol.62,no.5,Article London, UK, 1993. ID 052102, 2000. [45] C. Dewdney, P. R. Holland, and A. Kyprianidis, “What happens [65] M. A. Rowe, D. Kielpinski, V. Meyer et al., “Experimental in a spin measurement?” Physics Letters A,vol.119,no.6,pp. violation of a Bell’s inequality with efficient detection,” Nature, 259–267, 1986. vol. 409, no. 6822, pp. 791–794, 2001. [46] P. R. Holland, The Quantum Theory of Motion, Cambridge [66] C. Dewdney, P. R. Holland, and A. Kyprianidis, “A causal University Press, Cambridge, Mass, USA, 1993. account of nonlocal Einstein-Podolsky-Rosen spin correla- [47] D. E. Platt, “A modern analysis of the Stern-Gerlach experi- tions,” Journal of Physics A: Mathematical and General,vol.20, ment,” American Journal of Physics,vol.60,no.4,pp.306–308, no.14,pp.4717–4732,1987. 1992. [67] C. Dewdney, P.R. Holland, A. Kyprianidis, and J. P.Vigier, “Spin [48] M. Gondran, A. Gondran, and A. Kenoufi, “Decoherence time and non-locality in quantum mechanics,” Nature,vol.336,no. and spin measurement in the Stern-Gerlach experiment,” AIP 6199, pp. 536–544, 1988. Conference Proceedings,vol.1424,pp.116–120,2012. [68] M. Gondran and A. Gondran, “A new causal interpretation [49]G.B.Roston,M.Casas,A.Plastino,andA.R.Plastino,“Quan- of EPR-B experiment,” in Proceedings of the International tum entanglement, spin-1/2 and the Stern-Gerlach experiment,” Conference Quantum Theory: Reconsideration of Foundations-6 European Journal of Physics, vol. 26, no. 4, pp. 657–672, 2005. (QTRF '12), pp. 370–375, Vaxjo, Sweden, June 2012. [50] A. Challinor, A. Lasenby, S. Gull, and C. Doran, “A relativistic [69] A. Einstein, Ether and the Theory of Relativity, University of causal account of a spin measurement,” Physics Letters A,vol. Leyden, 1920. 218, no. 3–6, pp. 128–138, 1996. [70] K. Popper, Quantum Theory and the Schism in Physics: From The Postscript to the Logic of Scientific Discovery, Hutchinsons, [51] S. Kochen and E. P. Specker, “The problem of hidden variables London, UK, edited by W. W. Bartley III, 1982. in quantum mechanics,” Journal of Applied Mathematics and Mechanics,vol.17,pp.59–87,1967. [71] E. Schrodinger,¨ “Der stetige Ubergang¨ von der Mikro- zur Makromechanik,” Die Naturwissenschaften,vol.14,no.28,pp. [52] A. Einstein, B. Podolsky, and N. Rosen, “Can quantum- 664–666, 1926. mechanical description of physical reality be considered com- plete?” Physical Review,vol.47,no.10,pp.777–780,1935. [72] R. J. Glauber, C. deWitt, A. Blandin, and C. Cohen-Tanoudji, in Quantum Optics and Electronics, Les Houches Lectures 1964,p. [53] D. Bohm, Quantum Theory, Prentice-Hall, New York, NY, USA, 63,GordonandBreach,NewYork,NY,USA,1965. 1951. [73] I. Bialynicki-Birula, M. Kalinski,andJ.H.Eberly,“Lagrange´ [54] D. Bohm and Y. Aharanov, “Discussion of experimental proof equilibrium points in celestial mechanics and nonspreading for the paradox of Einstein, Rosen, and Podolsky,” vol. 108, pp. wave packets for strongly driven rydberg electrons,” Physical 1070–1076, 1957. Review Letters, vol. 73, no. 13, pp. 1777–1780, 1994. [55] J. S. Bell, “On the Einstein Podolsky rosen paradox,” Physics,vol. [74] A. Buchleitner and D. Delande, “Nondispersive electronic wave 1, no. 3, pp. 195–200, 1964. packets in multiphoton processes,” Physical Review Letters,vol. [56] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, 75,no.8,pp.1487–1490,1995. “Proposed experiment to test local hidden-variable theories,” [75] A. Buchleitner, D. Delande, and J. Zakrzewski, “Non-dispersive Physical Review Letters,vol.23,no.15,pp.880–884,1969. wave packets in periodically driven quantum systems,” Physics [57] S. J. Freedman and J. F. Clauser, “Experimental test of local Reports,vol.368,no.5,pp.409–547,2002. hidden-variable theories,” Physical Review Letters,vol.28,no. [76] H. Maeda and T. F. Gallagher, “Nondispersing wave packets,” 14, pp. 938–941, 1972. Physical Review Letters,vol.92,ArticleID133004,2004. [58] A. Aspect, P. Grangier, and G. Roger, “Experimental realiza- [77] L. de Broglie and J. L. Andrade e Silva, La Reinterpr´ etation´ de la tion of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: Mecanique´ Ondulatoire, Gauthier-Villars, 1971. a new violation of Bell’s inequalities,” Physical Review Letters, vol.49,no.2,pp.91–94,1982. Journal of Journal of The Scientific Journal of Advances in Gravity Photonics World Journal Soft Matter Condensed Matter Physics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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