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

Authors: Michel Gondran; Alexandre Gondran
Journal: Physics Research International
 (2014)

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## Abstract

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. 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.


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## Body

## 1. Introduction

“I saw the impossible done” [1]. This is how John Bell describes his inexpressible surprise in 1952 upon the publication of an article by Bohm [2]. The impossibility came from a theorem by John von Neumann outlined in 1932 in his bookThe Mathematical Foundations of Quantum Mechanics [3], which seemed to show the impossibility of adding “hidden variables” to quantum mechanics. This impossibility, with its physical interpretation, became almost a postulate of quantum mechanics, based on von Neumann’s indisputable authority as a mathematician. Bernard d’Espagnat notes in 1979 the following:“At the university, Bell had, like all of us, received from his teachers a message which, later still, Feynman would brilliantly state as follows: “No one can explain more than we have explained here […]. We do not have the slightest idea of a more fundamental mechanism from which the former results (the interference fringes) could follow”. If indeed we are to believe Feynman (and Banesh Hoffman, and many others, who expressed the same idea in many books, both popular and scholarly), Bohm’s theory cannot exist. Yet it does exist, and is even older than Bohm’s papers themselves. In fact, the basic idea behind it was formulated in 1927 by Louis de Broglie in a model he called “pilot wave theory”. Since this theory provides explanations of what, in “high circles”, is declared inexplicable, it is worth consideration, even by physicists […] who do not think it gives us the final answer to the question how reality really is” [4].And in 1987, Bell wonders about his teachers’ silence concerning the Broglie-Bohm pilot wave:“But why then had Born not told me of this “pilot wave”? If only to point out what was wrong with it? Why did von Neumann not consider it? More extraordinarily, why did people go on producing “impossibility” proofs after 1952, and as recently as 1978?[sic]While even Pauli, Rosenfeld, and Heisenberg could produce no more devastating criticism of Bohm’s version than to brand it as “metaphysical” and “ideological”? Why is the pilot-wave picture ignored in text books? Should it not be taught, not as the only way, but as an antidote to the prevailing complacency? To show that vagueness, subjectivity and indeterminism are not forced on us by experimental facts, but through a deliberate theoretical choice?” [5].More than thirty years after John Bell’s questions, the interpretation of the de Broglie-Bohm pilot wave is still ignored by both the international community and the textbooks.What is this pilot wave theory? For de Broglie, a quantum particle is not only defined by its wave function. He assumes that the quantum particle also has a position which is piloted by the wave function [6]. However, only the probability density of this position is known. The position exists in itself (ontologically) but is unknown to the observer. It only becomes known during the measurement.The goal of the present paper is to present the Broglie-Bohm pilot wave through the study of the three most important experiments of quantum measurement: the double-slit experiment which is the crucial experiment of the wave-particle duality, the Stern and Gerlach experiment with the measurement of the spin, and the EPR-B experiment with the problem of nonlocality.The paper is organized as follows. In Section2, we demonstrate the conditions in which the de Broglie-Bohm interpretation can be assumed to be valid through continuity with classical mechanics. This involves the de Broglie-Bohm interpretation for a set of particles prepared in the same way. In Section 3, we present a numerical simulation of the double-slit experiment performed by Jönsson in 1961 with electrons [7]. The method of Feynman path integrals allows us to calculate the time-dependent wave function. The evolution of the probability density just outside the slits leads one to consider the dualism of the wave-particle interpretation. And the de Broglie-Bohm trajectories provide an explanation for the impact positions of the particles. Finally, we show the continuity between classical and quantum trajectories with the convergence of these trajectories to classical trajectories when h tends to 0. In Section 4, 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: quantization, 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. In Section 5, we study the EPR-B experiment, 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. Finally, we recall that a physical explanation of nonlocal influences is possible.


## 2. The de Broglie-Bohm Interpretation

The de Broglie-Bohm interpretation is based on the following demonstration. Let us consider a wave functionΨ(x,t) solution to the Schrödinger equation:
(1)iħ∂Ψ(x,t)∂t=-ħ22mΔΨ(x,t)+V(x)Ψ(x,t),(2)Ψ(x,0)=Ψ0(x).
With the variable change Ψ(x,t)=ρħ(x,t)exp⁡(i(Sħ(x,t)/ħ)), the Schrödinger equation can be decomposed into Madelung equations [8] (1926):
(3)∂Sħ(x,t)∂t+12m(∇Sħ(x,t))2+V(x)-ħ22mΔρħ(x,t)ρħ(x,t)=0,(4)∂ρħ(x,t)∂t+div⁡(ρħ(x,t)∇Sħ(x,t)m)=0
with initial conditions:
(5)ρħ(x,0)=ρ0ħ(x),Sħ(x,0)=S0ħ(x).Madelung equations correspond to a set of noninteracting quantum particles all prepared in the same way (sameρ0ħ(x) and S0ħ(x)).A quantum particle is said to bestatistically prepared if its initial probability density ρ0ħ(x) and its initial action S0ħ(x) converge, when ħ→0, to nonsingular functions ρ0(x) and S0(x). It is the case of an electronic or C60 beam in the double-slit experiment or an atomic beam in the Stern and Gerlach experiment. We will see that it is also the case of a beam of entangled particles in the EPR-B experiment. Then, we have the following theorem [9, 10].Theorem 1.
For statistically prepared quantum particles, the probability densityρħ(x,t) and the action Sħ(x,t), solutions to the Madelung equations (3), (4), and (5), converge, when ħ→0, to the classical density ρ(x,t) and the classical action S(x,t), solutions to the statistical Hamilton-Jacobi equations:
(6)∂S(x,t)∂t+12m(∇S(x,t))2+V(x,t)=0,(7)S(x,0)=S0(x),(8)∂ρ(x,t)∂t+div⁡(ρ(x,t)∇S(x,t)m)=0,(9)ρ(x,0)=ρ0(x).We give some indications on the demonstration of this theorem when the wave functionΨ(x,t) is written as a function of the initial wave function Ψ0(x) by the Feynman paths integral [11]:
(10)Ψ(x,t)=∫F(t,ħ)exp⁡(iħScl(x,t;x0))Ψ0(x0)dx0,
where F(t,ħ) is an independent function of x and of x0. For a statistically prepared quantum particle, the wave function is written: Ψ(x,t)=F(t,ħ)∫‍ρ0ħ(x0)exp⁡((i/ħ)(S0ħ(x0)+Scl(x,t;x0)))dx0. The theorem of the stationary phase shows that if ħ tends towards 0, we have Ψ(x,t)~exp⁡((i/ħ)min⁡x0(S0(x0)+Scl(x,t;x0))); that is to say, the quantum action Sh(x,t) converges to the function:
(11)S(x,t)=min⁡x0⁡(S0(x0)+Scl(x,t;x0))
which is the solution to the Hamilton-Jacobi equation (6) with the initial condition (7). Moreover, as the quantum density ρh(x,t) satisfies the continuity equation (4), we deduce, since Sh(x,t) tends towards S(x,t), that ρh(x,t) converges to the classical density ρ(x,t), which satisfies the continuity equation (8). We obtain both announced convergences.These statistical Hamilton-Jacobi equations (6), (7), (8), and (9) correspond to a set of classical particles prepared in the same way (the same ρ0(x) and S0(x)). These classical particles are trajectories obtained in Eulerian representation with the velocity field v(x,t)=∇S(x,t)/m, but the density and the action are not sufficient to describe it completely. To know its position at time t, it is necessary to know its initial position. Because the Madelung equations converge to the statistical Hamilton-Jacobi equations, it is logical to do the same in quantum mechanics. We conclude that astatistically prepared quantum particle is not completely described by its wave function. It is necessary to add this initial position and an equation to define the evolution of this position in the time. It is the de Brogglie-Bohm interpretation where the position is called the “hidden variable.”The two first postulates of quantum mechanics, describing the quantum state and its evolution [12], must be completed in this heterodox interpretation. At initial time t=0, the state of the particle is given by the initial wave function Ψ0(x) (a wave packet) and its initial position X(0); it is the new first postulate. The new second postulate gives the evolution on the wave function and on the position. For a single spinless particle in a potential V(x), the evolution of the wave function is given by the usual Schrödinger equations (1), (2) and the evolution of the particle position is given by
(12)dX(t)dt=Jh(x,t)ρh(x,t)|x=X(t)=∇Sh(x,t)m|x=X(t),
where
(13)Jh(x,t)=ħ2mi(Ψ*(x,t)∇Ψ(x,t)-Ψ(x,t)∇Ψ*(x,t))
is the usual quantum current.In the case of a particle with spin, as in the Stern and Gerlach experiment, the Schrödinger equation must be replaced by the Pauli or Dirac equations.The third quantum mechanics postulate which describes the measurement operator (the observable) can be conserved. But the three postulates of measurement are not necessary: the postulate of quantization, the Born postulate of probabilistic interpretation of the wave function, and the postulate of the reduction of the wave function. We see that these postulates of measurement can be explained on each example as we will show in the following.We replace these three postulates by a single one, the “quantum equilibrium hypothesis,” [13–15] that describes the interaction between the initial wave function Ψ0(x) and the initial particle position X(0): for a set of identically prepared particles having t=0 wave function Ψ0(x), it is assumed that the initial particle positions X(0) are distributed according to
(14)P[X(0)=x]≡P(x,0)=|Ψ0(x)|2=ρ0h(x).
It is the Born rule at the initial time.Then, the probability distribution (P(x,t)≡P[X(t)=x]) for a set of particles moving with the velocity field vh(x,t)=∇Sh(x,t)/m satisfies the property of the “equivariance” of the |Ψ(x,t)|2 probability distribution [13]:
(15)P[X(t)=x]≡P(x,t)=|Ψ(x,t)|2=ρh(x,t).
It is the Born rule at time t.Then, the de Broglie-Bohm interpretation is based on a continuity between classical and quantum mechanics where the quantum particles are statistically prepared with an initial probability density that satisfies the “quantum equilibrium hypothesis” (14). It is the case of the three studied experiments.We will revisit these three measurement experiments through mathematical calculations and numerical simulations. For each one, we present the statistical interpretation that is common to the Copenhagen interpretation and the de Broglie-Bohm pilot wave and then the trajectories specific to the de Broglie-Bohm interpretation. We show that the precise definition of the initial conditions, that is, the preparation of the particles, plays a fundamental methodological role.


## 3. Double-Slit Experiment with Electrons

Young’s double-slit experiment [16] has long been the crucial experiment for the interpretation of the wave-particle duality. They have been realized with massive objects, such as electrons [7, 17–19], neutrons [20, 21], cold neutrons [22], and atoms [23], and, more recently, with coherent ensembles of ultracold atoms [24, 25] and even with mesoscopic single quantum objects such as C60 and C70 [26, 27]. For Feynman, this experiment addresses “the basic element of the mysterious behavior [of electrons] in its most strange form. [It is] a phenomenon which is impossible, absolutely impossible to explain in any classical way and which has in it the heart of quantum mechanics. In reality, it contains the only mystery” [28]. The de Broglie-Bohm interpretation and the numerical simulation help us here to revisit the double-slit experiment with electrons performed by Jönsson in 1961 and to provide an answer to Feynman’s mystery. These simulations [29] follow those conducted in 1979 by Philippidis et al. [30] which are today classics. However, these simulations [30] have some 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≤β and G(y)=0 for |y|>β, if they are 2β in width, were modeled by a Gaussian function G(y)=e-y2/2β2. 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.Figure1 shows a diagram of the double-slit experiment by Jönsson. An electron gun emits electrons one by one in the horizontal plane, through a hole of a few micrometers, at a velocity v=1.8×108 m/s along the horizontal x-axis. After traveling for d1=35 cm, they encounter a plate pierced with two horizontal slits A and B, each 0.2 μm wide and spaced 1 μm from each other. A screen located at d2=35 cm after the slits collects these electrons. The impact of each electron appears on the screen as the experiment unfolds. After thousands of impacts, we find that the distribution of electrons on the screen shows interference fringes.Figure 1
Diagram of the Jönnson’s double-slit experiment performed with electrons.The slits are very long along thez-axis, so there is no effect of diffraction along this axis. In the simulation, we therefore only consider the wave function along the y-axis; the variable x will be treated classically with x=vt. Electrons emerging from an electron gun are represented by the same initial wave function Ψ0(y).


### 3.1. Probability Density

Figure2 gives a general view of the evolution of the probability density from the source to the detection screen (a lighter shade means that the density is higher; i.e., the probability of presence is high). The calculations were made using the method of Feynman path integrals [29]. The wave function after the slits (t1=d1/v≃2.10-11 s <t<t1+d2/v≃4.10-11 s) is deduced from the values of the wave function at slits A and B: Ψ(y,t)=ΨA(y,t)+ΨB(y,t) with ΨA(y,t)=∫A‍K(y,t,ya,t1)Ψ(ya,t1)dya, ΨB(y,t)=∫B‍K(y,t,yb,t1)Ψ(yb,t1)dyb, and K(y,t,yα,t1)=(m/2iħ(t-t1))1/2eim(y-yα)2/2ħ(t-t1).Figure 2
General view of the evolution of the probability density from the source to the screen in the Jönsson experiment. A lighter shade means that the density is higher; that is, the probability of presence is high.Figure3 shows a close-up of the evolution of the probability density just after the slits. We note that interference will only occur a few centimeters after the slits. Thus, if the detection screen is 1 cm from the slits, there is no interference and one can determine by which slit each electron has passed. In this experiment, the measurement is performed by the detection screen, which only reveals the existence or absence of the fringes.Figure 3
Close-up of the evolution of the probability density in the first 3 cm after the slits in the Jönsson experiment.The calculation method enables us to compare the evolution of the cross-section of the probability density at various distances after the slits (0.35 mm, 3.5 mm, 3.5 cm, and 35 cm) where the two slitsA and B are open simultaneously (interference: |ΨA+ΨB|2) with the evolution of the sum of the probability densities where the slits A and B are open independently (the sum of two diffractions: |ΨA|2+|ΨB|2). Figure 4 shows that the difference between these two phenomena appears only a few centimeters after the slits.Comparison of the probability density|ΨA+ΨB|2 (full line) and |ΨA|2+|ΨB|2 (dotted line) at various distances after the slits: (a) 0.35 mm, (b): 3.5 mm, (c): 3.5 cm, and (d): 35 cm.
(a)
0.35 mm
(b)
3.5 mm
(c)
3.5 cm
(d)
35 cm


### 3.2. Impacts on Screen and de Broglie-Bohm Trajectories

The interference fringes are observed after a certain period of time when the impacts of the electrons on the detection screen become sufficiently numerous. Classical quantum theory only explains the impact of individual particles statistically.However, in the de Broglie-Bohm interpretation, a particle has an initial position and follows a path whose velocity at each instant is given by (12). On the basis of this assumption we conduct a simulation experiment by drawing random initial positions of the electrons in the initial wave packet (quantum equilibrium hypothesis).Figure5 shows, after its initial starting position, 100 possible quantum trajectories of an electron passing through one of the two slits: we have not represented the paths of the electron when it is stopped by the first screen. Figure 6 shows a close-up of these trajectories just after they leave their slits.Figure 5
100 electron trajectories for the Jönsson experiment.Figure 6
Close-up on the 100 trajectories of the electrons just after the slits.The different trajectories explain both the impact of electrons on the detection screen and the interference fringes. This is the simplest and most natural interpretation to explain the impact positions: “the position of an impact is simply the position of the particle at the time of impact.” This was the view defended by Einstein at the Solvay Congress of 1927. The position is the only measured variable of the experiment.In the de Broglie-Bohm interpretation, the impacts on the screen are the real positions of the electron as in classical mechanics and the three postulates of the measurement of quantum mechanics can be trivially explained: the position is an eigenvalue of the position operator because the position variable is identical to its operator (XΨ=xΨ), the Born postulate is satisfied with the “equivariance” property, and the reduction of the wave packet is not necessary to explain the impacts.Through numerical simulations, we will demonstrate how, when the Planck constanth tends to 0, the quantum 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 ht/m→0, so h→0 is equivalent to m→+∞ (i.e., the mass of the particle grows) or t→0 (i.e., the distance splits-screen d2→0). Figure 7 shows the 100 trajectories that start at the same 100 initial points when Planck’s constant is divided, respectively, into 10, 100, 1000, and 10000 (equivalent to multiplying the mass by 10, 100, 1000, and 10000). We obtain quantum trajectories converging to the classical trajectories, when h tends to 0.Figure 7
Convergence of 100 electron trajectories whenh is divided by 10, 100, 1000, and 10000.The study of the slits clearly shows that, in the de Broglie-Bohm interpretation, there is no physical separation between quantum mechanics and classical mechanics. All particles have quantum properties, but specific quantum behavior only appears in certain experimental conditions: here when the ratioht/m is sufficiently large. Interferences only appear gradually and the quantum particle behaves at any time as both a wave and a particle.


## 3.1. Probability Density

Figure2 gives a general view of the evolution of the probability density from the source to the detection screen (a lighter shade means that the density is higher; i.e., the probability of presence is high). The calculations were made using the method of Feynman path integrals [29]. The wave function after the slits (t1=d1/v≃2.10-11 s <t<t1+d2/v≃4.10-11 s) is deduced from the values of the wave function at slits A and B: Ψ(y,t)=ΨA(y,t)+ΨB(y,t) with ΨA(y,t)=∫A‍K(y,t,ya,t1)Ψ(ya,t1)dya, ΨB(y,t)=∫B‍K(y,t,yb,t1)Ψ(yb,t1)dyb, and K(y,t,yα,t1)=(m/2iħ(t-t1))1/2eim(y-yα)2/2ħ(t-t1).Figure 2
General view of the evolution of the probability density from the source to the screen in the Jönsson experiment. A lighter shade means that the density is higher; that is, the probability of presence is high.Figure3 shows a close-up of the evolution of the probability density just after the slits. We note that interference will only occur a few centimeters after the slits. Thus, if the detection screen is 1 cm from the slits, there is no interference and one can determine by which slit each electron has passed. In this experiment, the measurement is performed by the detection screen, which only reveals the existence or absence of the fringes.Figure 3
Close-up of the evolution of the probability density in the first 3 cm after the slits in the Jönsson experiment.The calculation method enables us to compare the evolution of the cross-section of the probability density at various distances after the slits (0.35 mm, 3.5 mm, 3.5 cm, and 35 cm) where the two slitsA and B are open simultaneously (interference: |ΨA+ΨB|2) with the evolution of the sum of the probability densities where the slits A and B are open independently (the sum of two diffractions: |ΨA|2+|ΨB|2). Figure 4 shows that the difference between these two phenomena appears only a few centimeters after the slits.Comparison of the probability density|ΨA+ΨB|2 (full line) and |ΨA|2+|ΨB|2 (dotted line) at various distances after the slits: (a) 0.35 mm, (b): 3.5 mm, (c): 3.5 cm, and (d): 35 cm.
(a)
0.35 mm
(b)
3.5 mm
(c)
3.5 cm
(d)
35 cm


## 3.2. Impacts on Screen and de Broglie-Bohm Trajectories

The interference fringes are observed after a certain period of time when the impacts of the electrons on the detection screen become sufficiently numerous. Classical quantum theory only explains the impact of individual particles statistically.However, in the de Broglie-Bohm interpretation, a particle has an initial position and follows a path whose velocity at each instant is given by (12). On the basis of this assumption we conduct a simulation experiment by drawing random initial positions of the electrons in the initial wave packet (quantum equilibrium hypothesis).Figure5 shows, after its initial starting position, 100 possible quantum trajectories of an electron passing through one of the two slits: we have not represented the paths of the electron when it is stopped by the first screen. Figure 6 shows a close-up of these trajectories just after they leave their slits.Figure 5
100 electron trajectories for the Jönsson experiment.Figure 6
Close-up on the 100 trajectories of the electrons just after the slits.The different trajectories explain both the impact of electrons on the detection screen and the interference fringes. This is the simplest and most natural interpretation to explain the impact positions: “the position of an impact is simply the position of the particle at the time of impact.” This was the view defended by Einstein at the Solvay Congress of 1927. The position is the only measured variable of the experiment.In the de Broglie-Bohm interpretation, the impacts on the screen are the real positions of the electron as in classical mechanics and the three postulates of the measurement of quantum mechanics can be trivially explained: the position is an eigenvalue of the position operator because the position variable is identical to its operator (XΨ=xΨ), the Born postulate is satisfied with the “equivariance” property, and the reduction of the wave packet is not necessary to explain the impacts.Through numerical simulations, we will demonstrate how, when the Planck constanth tends to 0, the quantum 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 ht/m→0, so h→0 is equivalent to m→+∞ (i.e., the mass of the particle grows) or t→0 (i.e., the distance splits-screen d2→0). Figure 7 shows the 100 trajectories that start at the same 100 initial points when Planck’s constant is divided, respectively, into 10, 100, 1000, and 10000 (equivalent to multiplying the mass by 10, 100, 1000, and 10000). We obtain quantum trajectories converging to the classical trajectories, when h tends to 0.Figure 7
Convergence of 100 electron trajectories whenh is divided by 10, 100, 1000, and 10000.The study of the slits clearly shows that, in the de Broglie-Bohm interpretation, there is no physical separation between quantum mechanics and classical mechanics. All particles have quantum properties, but specific quantum behavior only appears in certain experimental conditions: here when the ratioht/m is sufficiently large. Interferences only appear gradually and the quantum particle behaves at any time as both a wave and a particle.


## 4. The Stern-Gerlach Experiment

In 1922, by studying the deflection of a beam of silver atoms in a strongly inhomogeneous magnetic field (cf. Figure8) Gerlach and Stern [31, 32] obtained an experimental result that contradicts the common sense prediction: the beam, instead of expanding, splits into two separate beams giving two spots of equal intensity N+ and N- on a detector, at equal distances from the axis of the original beam.Figure 8
Schematic configuration of the Stern-Gerlach 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 density matrix that is put forward to explain the first part of the measurement process [33–38]. However, although these authors consider the Stern-Gerlach experiment as fundamental, they do not propose a calculation of the spin decoherence time.We present an analytical solution to this decoherence time and the diagonalization of the density matrix. This solution requires the calculation of the Pauli spinor with a spatial extension as the equation:(16)Ψ0(z)=(2πσ02)-1/2e-z2/4σ02(cos⁡θ02e-i(φ0/2)sinθ02ei(φ0/2)).
Quantum mechanics textbooks [12, 28, 39, 40] do not take into account the spatial extension of the spinor (16) and simply use the simplified spinor without spatial extension:
(17)Ψ0=(cos⁡θ02e-i(φ0/2)sinθ02ei(φ0/2)).
However, as we shall see, the different evolutions of the spatial extension between the two spinor components will have a key role in the explanation of the measurement process. This spatial extension enables us, in following the precursory works of Takabayasi [41, 42], Bohm et al. [43, 44], Dewdney et al. [45], and Holland [46], to revisit the Stern and Gerlach experiment, to explain the decoherence, and to demonstrate the three postulates of the measure: quantization, Born statistical interpretation, and wave function reduction.Silver atoms contained in the ovenE (Figure 8) are heated to a high temperature and escape through a narrow opening. A second aperture, T, selects those atoms whose velocity, v0, is parallel to the y-axis. The atomic beam crosses the gap of the electromagnet A1 before condensing on the detector, P1. Before crossing the electromagnet, the magnetic moment of each silver atom is oriented randomly (isotropically). In the beam, we represent each atom by its wave function; one can assume that at the entrance to the electromagnet, A1 and at the initial time t=0, each atom can be approximately described by a Gaussian spinor in z given by (16) corresponding to a pure state. The variable y will be treated classically with y=vt. σ0=10-4 m corresponds to the size of the slot T along the z-axis. The approximation by a Gaussian initial spinor will allow explicit calculations. Because the slot is much wider along the x-axis, the variable x will be also treated classically. To obtain an explicit solution of the Stern-Gerlach experiment, we take the numerical values used in the Cohen-Tannoudji textbook [12]. For the silver atom, we have m=1.8×10-25 kg, v0=500 m/s (corresponding to the temperature of T=1000°K). In (16) and in Figure 9, θ0 and φ0 are the polar angles characterizing the initial orientation of the magnetic moment and θ0 corresponds to the angle with the z-axis. The experiment is a statistical mixture of pure states where the θ0 and the φ0 are randomly chosen: θ0 is drawn in a uniform way from [0,π] and φ0 is drawn in a uniform way from [0,2π].Figure 9
Orientation of the magnetic moment.θ0 and φ0 are the polar angles characterizing the spin vector in the de Broglie-Bohm interpretation.The evolution of the spinorΨ=(ψ+ψ-) in a magnetic field B is then given by the Pauli equation:
(18)iħ(∂ψ+∂t∂ψ-∂t)=-ħ22mΔ(ψ+ψ-)+μBBσ(ψ+ψ-),
where μB=eħ/2me is the Bohr magneton and where σ=(σx,σy,σz) corresponds to the three Pauli matrices. The particle first enters an electromagnetic field B directed along the z-axis, Bx=B0′x, By=0, and Bz=B0-B0′z, with B0=5 Tesla, B0′=|∂B/∂z|=103 Tesla/m over a length Δl=1 cm. On exiting the magnetic field, the particle is free until it reaches the detector P1 placed at a D=20 cm distance.The particle stays within the magnetic field for a timeΔt=Δl/v=2×10-5 s. During this time [0,Δt], the spinor is [47] (see the Appendix)(19)Ψ(z,t)≃(cos⁡θ02(2πσ02)-1/2e-(z-(μBB0′/2m)t2)2/4σ02ei((μBB0′tz-(μB2B0′2/6m)t3+μBB0t+(ħφ0/2))/ħ)isinθ02(2πσ02)-1/2e-(z+(μBB0′/2m)t2)2/4σ02ei((-μBB0′tz-(μB2B0′2/6m)t3-μBB0t-(ħφ0/2))/ħ)).After the magnetic field, at timet+Δt(t≥0) in the free space, the spinor becomes [44–48] (see the Appendix)
(20)Ψ(z,t+Δt)≃(cos⁡θ02(2πσ02)-1/2e-(z-zΔ-ut)2/4σ02ei((muz+ħφ+)/ħ)sinθ02(2πσ02)-1/2e-(z+zΔ+ut)2/4σ02ei((-muz+ħφ-)/ħ)),
where
(21)zΔ=μBB0′(Δt)22m=10-5m,u=μBB0′(Δt)m=1m/s.
Equation (20) takes into account the spatial extension of the spinor and we note that the two spinor components have very different z values. All interpretations are based on this equation.


### 4.1. The Decoherence Time

We deduce from (20) the probability density of a pure state in the free space after the electromagnet:
(22)ρθ0(z,t+Δt)≃(2πσ02)-1/2(cos⁡2θ02e-(z-zΔ-ut)2/2σ0200000000000000000000000+sin2θ02e-(z+zΔ+ut)2/2σ02).
Figure 10 shows the probability density of a pure state (with θ0=π/3) as a function of z at several values of t (the plots 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.Figure 10
Evolution of the probability density of a pure state withθ0=π/3.The decoherence time, where the two spotsN+ and N- are separated, is then given by
(23)tD≃3σ0-zΔu=3×10-4s.This decoherence time is usually the time required to diagonalize the marginal density matrix of spin variables associated with a pure state [49]:
(24)ρS(t)=(∫|ψ+(z,t)|2dz∫ψ+(z,t)ψ-*(z,t)dz∫ψ-(z,t)ψ+*(z,t)dz∫|ψ-(z,t)|2dz).Fort≥tD, the product ψ+(z,t+Δt)ψ-(z,t+Δt) is null and the density matrix is diagonal: the probability density of the initial pure state (20) is diagonal:
(25)ρS(t+Δt)=(2πσ02)-1(cos⁡2θ0200sin2θ02).


### 4.2. Proof of the Postulates of Quantum Measurement

We then obtain atoms with a spin oriented only along thez-axis (positively or negatively). Let us consider the spinor Ψ(z,t+Δt) given by (20). Experimentally, we do not measure the spin directly but the z~ position of the particle impact on P1 (Figure 11).Figure 11
1000 silver atom impacts on the detectorP1.Ifz~∈N+, the term ψ- of (20) is numerically equal to zero and the spinor Ψ is proportional to (10), one of the eigenvectors of the spin operator Sz=(ħ/2)σz: Ψ(z~,t+Δt)≃(2πσ02)-1/4cos⁡⁡(θ0/2)e-(z~1-zΔ-ut)2/4σ02ei((muz~1+ħφ+)/ħ)(10). Then, we have SzΨ=(ħ/2)σzΨ=+(ħ/2)Ψ.Ifz~∈N-, the term ψ+ of (20) is numerically equal to zero and the spinor Ψ is proportional to (01), the other eigenvector of the spin operator Sz: Ψ(z~,t+Δt)≃(2πσ02)-1/4sin⁡(θ0/2)e-(z~2+zΔ+ut)2/4σ02ei((-muz~2+ħφ-)/ħ)(01). Then, we have SzΨ=(ħ/2)σzΨ=-(ħ/2)Ψ. Therefore, the measurement of the spin corresponds to an eigenvalue of the spin operator. It is a proof of the postulate of quantization.Equation (25) gives the probability cos⁡2(θ0/2) (resp., sin2(θ0/2)) to measure the particle in the spin state +ħ/2 (resp., -ħ/2); this proves the Born probabilistic postulate.By drilling a hole in the detectorP1 to the location of the spot N+ (Figure 8), we select all the atoms that are in the spin state |+〉=(10). The new spinor of these atoms is obtained by making the component Ψ- of the spinor Ψ identically zero (and not only numerically equal to zero) at the time when the atom crosses the detector P1; at this time the component Ψ- is indeed stopped by detector P1. The future trajectory of the silver atom after crossing the detector P1 will be guided by this new (normalized) spinor. 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-.


### 4.3. Impacts and Quantization Explained by de Broglie-Bohm Trajectories

Finally, it remains to provide an explanation of the individual impacts of silver atoms. The spatial extension of the spinor (16) allows us to take into account the particle’s initial position z0 and to introduce the Broglie-Bohm trajectories [2, 6, 45, 46, 50] which is the natural assumption to explain the individual impacts.Figure12 presents, for a silver atom with the initial spinor orientation (θ0=π/3, φ0=0), a plot in the (Oyz) plane of a set of 10 trajectories whose initial position z0 has been randomly chosen from a Gaussian distribution with standard deviation σ0. The spin orientations θ(z,t) are represented by arrows.Figure 12
Ten silver atom trajectories with initial spin orientation(θ0=π/3) and initial position z0; arrows represent the spin orientation θ(z,t) along the trajectories.The final orientation, obtained after the decoherence timetD, depends on the initial particle position z0 in the spinor with a spatial extension and on the initial angle θ0 of the spin with the z-axis. We obtain +π/2 if z0>zθ0 and -π/2 if z0<zθ0 with
(26)zθ0=σ0F-1(sin2θ02),
where F is the repartition function of the normal centered-reduced law. If we ignore the position of the atom in its wave function, we lose the determinism given by (26).In the de Broglie-Bohm interpretation with a realistic interpretation of the spin, the “measured” value is not independent of the context of the measure and is contextual. It conforms to the Kochen and Specker theorem [51]: realism and noncontextuality are inconsistent with certain quantum mechanics predictions.Now let us consider a mixture of pure states where the initial orientation (θ0,φ0) from the spinor has been randomly chosen. These are the conditions of the initial Stern and Gerlach experiment. Figure 13 represents a simulation of 10 quantum trajectories of silver atoms from which the initial positions z0 are also randomly chosen.Figure 13
Ten silver atom trajectories where the initial orientation (θ0,φ0) has been randomly chosen; arrows represent the spin orientation θ(z,t) along the trajectories.Finally, the de Broglie-Bohm trajectories propose a clear interpretation of the spin measurement in quantum mechanics. There is interaction with the measuring apparatus as is generally stated, and there is indeed a minimum time required for measurement. However this measurement and this time do not have the signification that is usually applied to them. The result of the Stern-Gerlach experiment is not the measure of the spin projection along thez-axis, but the orientation of the spin either in the direction of the magnetic field gradient or in the opposite direction. It depends on the position of the particle in the wave function. We have therefore a simple explanation for the noncompatibility of spin measurements along different axes. The measurement duration is then the time necessary for the particle to point its spin in the final direction.


## 4.1. The Decoherence Time

We deduce from (20) the probability density of a pure state in the free space after the electromagnet:
(22)ρθ0(z,t+Δt)≃(2πσ02)-1/2(cos⁡2θ02e-(z-zΔ-ut)2/2σ0200000000000000000000000+sin2θ02e-(z+zΔ+ut)2/2σ02).
Figure 10 shows the probability density of a pure state (with θ0=π/3) as a function of z at several values of t (the plots 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.Figure 10
Evolution of the probability density of a pure state withθ0=π/3.The decoherence time, where the two spotsN+ and N- are separated, is then given by
(23)tD≃3σ0-zΔu=3×10-4s.This decoherence time is usually the time required to diagonalize the marginal density matrix of spin variables associated with a pure state [49]:
(24)ρS(t)=(∫|ψ+(z,t)|2dz∫ψ+(z,t)ψ-*(z,t)dz∫ψ-(z,t)ψ+*(z,t)dz∫|ψ-(z,t)|2dz).Fort≥tD, the product ψ+(z,t+Δt)ψ-(z,t+Δt) is null and the density matrix is diagonal: the probability density of the initial pure state (20) is diagonal:
(25)ρS(t+Δt)=(2πσ02)-1(cos⁡2θ0200sin2θ02).


## 4.2. Proof of the Postulates of Quantum Measurement

We then obtain atoms with a spin oriented only along thez-axis (positively or negatively). Let us consider the spinor Ψ(z,t+Δt) given by (20). Experimentally, we do not measure the spin directly but the z~ position of the particle impact on P1 (Figure 11).Figure 11
1000 silver atom impacts on the detectorP1.Ifz~∈N+, the term ψ- of (20) is numerically equal to zero and the spinor Ψ is proportional to (10), one of the eigenvectors of the spin operator Sz=(ħ/2)σz: Ψ(z~,t+Δt)≃(2πσ02)-1/4cos⁡⁡(θ0/2)e-(z~1-zΔ-ut)2/4σ02ei((muz~1+ħφ+)/ħ)(10). Then, we have SzΨ=(ħ/2)σzΨ=+(ħ/2)Ψ.Ifz~∈N-, the term ψ+ of (20) is numerically equal to zero and the spinor Ψ is proportional to (01), the other eigenvector of the spin operator Sz: Ψ(z~,t+Δt)≃(2πσ02)-1/4sin⁡(θ0/2)e-(z~2+zΔ+ut)2/4σ02ei((-muz~2+ħφ-)/ħ)(01). Then, we have SzΨ=(ħ/2)σzΨ=-(ħ/2)Ψ. Therefore, the measurement of the spin corresponds to an eigenvalue of the spin operator. It is a proof of the postulate of quantization.Equation (25) gives the probability cos⁡2(θ0/2) (resp., sin2(θ0/2)) to measure the particle in the spin state +ħ/2 (resp., -ħ/2); this proves the Born probabilistic postulate.By drilling a hole in the detectorP1 to the location of the spot N+ (Figure 8), we select all the atoms that are in the spin state |+〉=(10). The new spinor of these atoms is obtained by making the component Ψ- of the spinor Ψ identically zero (and not only numerically equal to zero) at the time when the atom crosses the detector P1; at this time the component Ψ- is indeed stopped by detector P1. The future trajectory of the silver atom after crossing the detector P1 will be guided by this new (normalized) spinor. 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-.


## 4.3. Impacts and Quantization Explained by de Broglie-Bohm Trajectories

Finally, it remains to provide an explanation of the individual impacts of silver atoms. The spatial extension of the spinor (16) allows us to take into account the particle’s initial position z0 and to introduce the Broglie-Bohm trajectories [2, 6, 45, 46, 50] which is the natural assumption to explain the individual impacts.Figure12 presents, for a silver atom with the initial spinor orientation (θ0=π/3, φ0=0), a plot in the (Oyz) plane of a set of 10 trajectories whose initial position z0 has been randomly chosen from a Gaussian distribution with standard deviation σ0. The spin orientations θ(z,t) are represented by arrows.Figure 12
Ten silver atom trajectories with initial spin orientation(θ0=π/3) and initial position z0; arrows represent the spin orientation θ(z,t) along the trajectories.The final orientation, obtained after the decoherence timetD, depends on the initial particle position z0 in the spinor with a spatial extension and on the initial angle θ0 of the spin with the z-axis. We obtain +π/2 if z0>zθ0 and -π/2 if z0<zθ0 with
(26)zθ0=σ0F-1(sin2θ02),
where F is the repartition function of the normal centered-reduced law. If we ignore the position of the atom in its wave function, we lose the determinism given by (26).In the de Broglie-Bohm interpretation with a realistic interpretation of the spin, the “measured” value is not independent of the context of the measure and is contextual. It conforms to the Kochen and Specker theorem [51]: realism and noncontextuality are inconsistent with certain quantum mechanics predictions.Now let us consider a mixture of pure states where the initial orientation (θ0,φ0) from the spinor has been randomly chosen. These are the conditions of the initial Stern and Gerlach experiment. Figure 13 represents a simulation of 10 quantum trajectories of silver atoms from which the initial positions z0 are also randomly chosen.Figure 13
Ten silver atom trajectories where the initial orientation (θ0,φ0) has been randomly chosen; arrows represent the spin orientation θ(z,t) along the trajectories.Finally, the de Broglie-Bohm trajectories propose a clear interpretation of the spin measurement in quantum mechanics. There is interaction with the measuring apparatus as is generally stated, and there is indeed a minimum time required for measurement. However this measurement and this time do not have the signification that is usually applied to them. The result of the Stern-Gerlach experiment is not the measure of the spin projection along thez-axis, but the orientation of the spin either in the direction of the magnetic field gradient or in the opposite direction. It depends on the position of the particle in the wave function. We have therefore a simple explanation for the noncompatibility of spin measurements along different axes. The measurement duration is then the time necessary for the particle to point its spin in the final direction.


## 5. EPR-B Experiment

Nonseparability is one of the most puzzling aspects of quantum mechanics. For over thirty years, the EPR-B, the spin version of the Einstein-Podolsky-Rosen experiment [52] proposed by Bohm and Aharanov [53, 54], the Bell theorem [55], and the BCHSH inequalities [5, 55, 56] have been at the heart of the debate on hidden variables and nonlocality. Many experiments since Bell’s paper have demonstrated violations of these inequalities and have vindicated quantum theory [57–63]. Now, EPR pairs of massive atoms are also considered [64, 65]. The usual conclusion of these experiments is to reject the nonlocal realism for two reasons: the impossibility of decomposing a pair of entangled atoms into two states, one for each atom, and the impossibility of interaction faster than the speed of light.Here, we show that there exists a de Broglie-Bohm interpretation which answers these two questions positively. To demonstrate this nonlocal realism, two methodological conditions are necessary. The first condition is the same as in the Stern-Gerlach experiment: the solution to the entangled state is obtained by resolving the Pauli equation from an initial singlet wave function with a spatial extension as(27)Ψ0(rA,rB)=12f(rA)f(rB)(|+A〉|-B〉-|-A〉|+B〉)
and not from a simplified wave function without spatial extension:
(28)Ψ0(rA,rB)=12(|+A〉|-B〉-|-A〉|+B〉).f function and |±〉 vectors are presented later.The resolution in space of the Pauli equation is essential: it enables the spin measurement by spatial quantization and explains the determinism and the disentangling process. To explain the interaction and the evolution between the spin of the two particles, we consider a two-step version of the EPR-B experiment. It is our second methodological condition. A first causal interpretation of EPR-B experiment was proposed in 1987 by Dewdney et al. [66, 67] using these two conditions. However, this interpretation had a flaw [46, page 418]: the spin module of each particle depends directly on the singlet wave function, and thus the spin module of each particle varied during the experiment from 0 to ħ/2. We present a de Broglie-Bohm interpretation that avoids this flaw [68].Figure14 presents the Einstein-Podolsky-Rosen-Bohm experiment. A source S creates, in O, pairs of identical atoms A and B, but with opposite spins. The atoms A and B split following the y-axis in opposite directions and head towards two identical Stern-Gerlach apparatus EA and EB. The electromagnet EA “measures” the spin of A along the z-axis and the electromagnet EB “measures” the spin of B along the z′-axis, which is obtained after a rotation of an angle δ around the y-axis. The initial wave function of the entangled state is the singlet state (27), where r=(x,z), f(r)=(2πσ02)-1/2e-(x2+z2)/4σ02, |±A〉, and |±B〉 are the eigenvectors of the operators σzA and σzB: σzA|±A〉=±|±A〉, σzB|±B〉=±|±B〉. We treat the dependence with y classically: speed -vy for A and vy for B. The wave function Ψ(rA,rB,t) of the two identical particles A and B, electrically neutral and with magnetic moments μ0, subject to magnetic fields EA and EB, admits on the basis of |±A〉 and |±B〉 four components Ψa,b(rA,rB,t) and satisfies the two-body Pauli equation [46, page 417]:
(29)iħ∂Ψa,b∂t=(-ħ22mΔA-ħ22mΔB)Ψa,b+μBjEA(σj)caΨc,b+μBjEB(σj)dbΨa,d
with the initial conditions:
(30)Ψa,b(rA,rB,0)=Ψ0a,b(rA,rB),
where Ψ0a,b(rA,rB) corresponds to the singlet state (27).Figure 14
Schematic configuration of the EPR-B experiment.To obtain an explicit solution of the EPR-B experiment, we take the numerical values of the Stern-Gerlach experiment.One of the difficulties of the interpretation of the EPR-B experiment is the existence of two simultaneous measurements. By doing these measurements one after the other, the interpretation of the experiment will be facilitated. That is the purpose of the two-step version of the experiment EPR-B studied below.


### 5.1. First Step EPR-B: Spin Measurement ofA

In the first step we make a Stern and Gerlach “measurement” for atomA, on a pair of particles A and B in a singlet state. This is the experiment first proposed in 1987 by Dewdney et al. [66, 67].Consider that at timet0 the particle A arrives at the entrance of electromagnet EA. After this exit of the magnetic field EA, at time t0+Δt+t, the wave function (27) becomes [68]
(31)Ψ(rA,rB,t0+Δt+t)=12f(rB)×(f+(rA,t)|+A〉|-B〉-f-(rA,t)|-A〉|+B〉)
with
(32)f±(r,t)≃f(x,z∓zΔ∓ut)ei((±muz/ħ)+φ±(t)),
where zΔ and u are given by (21).The atomic densityρ(zA,zB,t0+Δt+t) is found by integrating Ψ*(rA,rB,t0+Δt+t)Ψ(rA,rB,t0+Δt+t) on xA and xB:
(33)ρ(zA,zB,t0+Δt+t)=((2πσ02)-1/2e-(zB)2/2σ02)×((2πσ02)-1/200000000×12(e-(zA-zΔ-ut)2/2σ02+e-(zA+zΔ+ut)2/2σ02)).
We deduce that the beam of particle A is divided into two, while the beam of particle B stays undivided: (i)
the density ofA is the same, whether particle A is entangled with B or not;(ii)
the density ofB is not affected by the “measurement” of A.Our first conclusion is that the position ofB does not depend on the measurement of A; only the spins are involved. We conclude from (31) that the spins of A and B remain opposite throughout the experiment. These are the two properties used in the causal interpretation.


### 5.2. Second Step EPR-B: Spin Measurement ofB

The second step is a continuation of the first and corresponds to the EPR-B experiment broken down into two steps. On a pair of particlesA and B in a singlet state, first we made a Stern and Gerlach measurement on the A atom between t0 and t0+Δt+tD; secondly, we make a Stern and Gerlach measurement on the B atom with an electromagnet EB forming an angle δ with EA during t0+Δt+tD and t0+2(Δt+tD).At the exit of magnetic fieldEA, at time t0+Δt+tD, the wave function is given by (31). Immediately after the measurement of A, still at time t0+Δt+tD, the wave function of B depends on the measurement ± of A:
(34)ΨB/±A(rB,t0+Δt+t1)=f(rB)|∓B〉.
Then, the measurement of B at time t0+2(Δt+tD) yields, in this two-step version of the EPR-B experiment, the same results for spatial quantization and correlations of spins as in the EPR-B experiment.


### 5.3. Causal Interpretation of the EPR-B Experiment

We assume, at the creation of the two entangled particlesA and B, that each of the two particles A and B has an initial wave function with opposite spins: Ψ0A(rA,θ0A,φ0A)=f(rA)(cos⁡⁡(θ0A/2)|+A〉+sin(θ0A/2)eiφ0A|-A〉) and Ψ0B(rB,θ0B,φ0B)=f(rB)(cos⁡⁡(θ0B/2)|+B〉+sin(θ0B/2)eiφ0B|-B〉) with θ0B=π-θ0A and φ0B=φ0A-π. The two particles A and B are statistically prepared as in the Stern and Gerlach experiment. Then the Pauli principle tells us that the two-body wave function must be antisymmetric; after calculation, we find the same singlet state (27):
(35)Ψ0(rA,θA,φA,rB,θB,φB)=-eiφAf(rA)f(rB)×(|+A〉|-B〉-|-A〉|+B〉).
Thus, we can consider that the singlet wave function is the wave function of a family of two fermions A and B with opposite spins: the direction of initial spins A and B exists but is notknown. It is a local hidden variable which is therefore necessary to add in the initial conditions of the model.This is not the interpretation followed by the Bohm school [44–46, 66, 67] in the interpretation of the singlet wave function; they do not assume the existence of wave functions Ψ0A(rA,θ0A,φ0A) and Ψ0B(rB,θ0B,φ0B) for each particle but only the singlet state Ψ0(rA,θA,φA,rB,θB,φB). In consequence, they suppose a zero spin for each particle at the initial time and a spin module of each particle varied during the experiment from 0 to ħ/2 [46, page 418].Here, we assume that at the initial time we know the spin of each particle (given by each initial wave function) and the initial position of each particle.Step 1 (spin measurement ofA).
In (31) particle A can be considered independent of B. We can therefore give it the wave function:
(36)ΨA(rA,t0+Δt+t)=cos⁡θ0A2f+(rA,t)|+A〉+sinθ0A2eiφ0Af-(rA,t)|-A〉
which is the wave function of a free particle in a Stern-Gerlach apparatus and whose initial spin is given by (θ0A,φ0A). For an initial polarization (θ0A,φ0A) and an initial position (z0A), we obtain, in the de Broglie-Bohm interpretation [44] of the Stern and Gerlach experiment, an evolution of the position (zA(t)) and of the spin orientation of A (θA(zA(t),t)) [48].
The case of particleB is different. B follows a rectilinear trajectory with yB(t)=vyt, zB(t)=z0B, and xB(t)=x0B. By contrast, the orientation of its spin moves with the orientation of the spin of A: θB(t)=π-θA(zA(t),t) and φB(t)=φ(zA(t),t)-π. We can associate the following wave function with the particle B:
(37)ΨB(rB,t0+Δt+t)=f(rB)(cos⁡θB(t)2|+B〉+sinθB(t)2eiφB(t)|-B〉).
This wave function is specific, because it depends upon initial conditions of A (position and spin). The orientation of spin of the particle B is driven by the particle Athrough the singlet wave function. Thus, the singlet wave function is the nonlocal variable.Step 2 (spin measurement ofB).
At the timet0+Δt+tD, immediately after the measurement of A, θB(t0+Δt+tD)=π or 0 in accordance with the value of θA(zA(t),t) and the wave function of B is given by (34). The frame (Ox′yz′) corresponds to the frame (Oxyz) after a rotation of an angle δ around the y-axis. θB corresponds to the B-spin angle with the z-axis and θ′B to the B-spin angle with the z′-axis; then θ′B(t0+Δt+tD)=π+δ or δ. In this second step, we are exactly in the case of a particle in a simple Stern and Gerlach experiment (with magnet EB) with a specific initial polarization equal to π+δ or δ and not random like in Step 1. Then, the measurement of B, at time t0+2(Δt+tD), gives, in this interpretation of the two-step version of the EPR-B experiment, the same results as in the EPR-B experiment.


### 5.4. Physical Explanation of Nonlocal Influences

From the wave function of two entangled particles, we find spins, trajectories, and also a wave function for each of the two particles. In this interpretation, the quantum particle has a local position like a classical particle, but it has also a nonlocal behavior through the wave function. So, it is the wave function that creates the nonclassical properties. We can keep a view of a local realist world for the particle, but we should add a nonlocal vision through the wave function. As we saw in Step1, the nonlocal influences in the EPR-B experiment only concern the spin orientation not the motion of the particles themselves. Indeed only spins are entangled in the wave function (27) not positions and motions like in the initial EPR experiment. This is a key point in the search for a physical explanation of nonlocal influences.The simplest explanation of this nonlocal influence is to reintroduce the concept of ether (or the preferred frame) but a new format given by Lorentz-Poincaré and by Einstein in 1920 [69]: “Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. According to the general theory of relativity space without ether is unthinkable;[sic]for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.”Taking into account the new experiments, especially Aspect’s experiments, Popper [70, page XVIII] defends a similar view in 1982:“I feel not quite convinced that the experiments are correctly interpreted; but if they are, we just have to accept action at a distance. I think (with J.P. Vigier) that this would of course be very important, but I do not for a moment think that it would shake, or even touch, realism. Newton and Lorentz were realists and accepted action at a distance; and Aspect’s experiments would be the first crucial experiment between Lorentz’s and Einstein’s interpretation of the Lorentz transformations.”Finally, in the de Broglie-Bohm interpretation, the EPR-B experiments of nonlocality have therefore a great importance not to eliminate realism and determinism but as Popper said to rehabilitate the existence of a certain type of ether, like Lorentz’s ether and like Einstein’s ether in 1920.


## 5.1. First Step EPR-B: Spin Measurement ofA

In the first step we make a Stern and Gerlach “measurement” for atomA, on a pair of particles A and B in a singlet state. This is the experiment first proposed in 1987 by Dewdney et al. [66, 67].Consider that at timet0 the particle A arrives at the entrance of electromagnet EA. After this exit of the magnetic field EA, at time t0+Δt+t, the wave function (27) becomes [68]
(31)Ψ(rA,rB,t0+Δt+t)=12f(rB)×(f+(rA,t)|+A〉|-B〉-f-(rA,t)|-A〉|+B〉)
with
(32)f±(r,t)≃f(x,z∓zΔ∓ut)ei((±muz/ħ)+φ±(t)),
where zΔ and u are given by (21).The atomic densityρ(zA,zB,t0+Δt+t) is found by integrating Ψ*(rA,rB,t0+Δt+t)Ψ(rA,rB,t0+Δt+t) on xA and xB:
(33)ρ(zA,zB,t0+Δt+t)=((2πσ02)-1/2e-(zB)2/2σ02)×((2πσ02)-1/200000000×12(e-(zA-zΔ-ut)2/2σ02+e-(zA+zΔ+ut)2/2σ02)).
We deduce that the beam of particle A is divided into two, while the beam of particle B stays undivided: (i)
the density ofA is the same, whether particle A is entangled with B or not;(ii)
the density ofB is not affected by the “measurement” of A.Our first conclusion is that the position ofB does not depend on the measurement of A; only the spins are involved. We conclude from (31) that the spins of A and B remain opposite throughout the experiment. These are the two properties used in the causal interpretation.


## 5.2. Second Step EPR-B: Spin Measurement ofB

The second step is a continuation of the first and corresponds to the EPR-B experiment broken down into two steps. On a pair of particlesA and B in a singlet state, first we made a Stern and Gerlach measurement on the A atom between t0 and t0+Δt+tD; secondly, we make a Stern and Gerlach measurement on the B atom with an electromagnet EB forming an angle δ with EA during t0+Δt+tD and t0+2(Δt+tD).At the exit of magnetic fieldEA, at time t0+Δt+tD, the wave function is given by (31). Immediately after the measurement of A, still at time t0+Δt+tD, the wave function of B depends on the measurement ± of A:
(34)ΨB/±A(rB,t0+Δt+t1)=f(rB)|∓B〉.
Then, the measurement of B at time t0+2(Δt+tD) yields, in this two-step version of the EPR-B experiment, the same results for spatial quantization and correlations of spins as in the EPR-B experiment.


## 5.3. Causal Interpretation of the EPR-B Experiment

We assume, at the creation of the two entangled particlesA and B, that each of the two particles A and B has an initial wave function with opposite spins: Ψ0A(rA,θ0A,φ0A)=f(rA)(cos⁡⁡(θ0A/2)|+A〉+sin(θ0A/2)eiφ0A|-A〉) and Ψ0B(rB,θ0B,φ0B)=f(rB)(cos⁡⁡(θ0B/2)|+B〉+sin(θ0B/2)eiφ0B|-B〉) with θ0B=π-θ0A and φ0B=φ0A-π. The two particles A and B are statistically prepared as in the Stern and Gerlach experiment. Then the Pauli principle tells us that the two-body wave function must be antisymmetric; after calculation, we find the same singlet state (27):
(35)Ψ0(rA,θA,φA,rB,θB,φB)=-eiφAf(rA)f(rB)×(|+A〉|-B〉-|-A〉|+B〉).
Thus, we can consider that the singlet wave function is the wave function of a family of two fermions A and B with opposite spins: the direction of initial spins A and B exists but is notknown. It is a local hidden variable which is therefore necessary to add in the initial conditions of the model.This is not the interpretation followed by the Bohm school [44–46, 66, 67] in the interpretation of the singlet wave function; they do not assume the existence of wave functions Ψ0A(rA,θ0A,φ0A) and Ψ0B(rB,θ0B,φ0B) for each particle but only the singlet state Ψ0(rA,θA,φA,rB,θB,φB). In consequence, they suppose a zero spin for each particle at the initial time and a spin module of each particle varied during the experiment from 0 to ħ/2 [46, page 418].Here, we assume that at the initial time we know the spin of each particle (given by each initial wave function) and the initial position of each particle.Step 1 (spin measurement ofA).
In (31) particle A can be considered independent of B. We can therefore give it the wave function:
(36)ΨA(rA,t0+Δt+t)=cos⁡θ0A2f+(rA,t)|+A〉+sinθ0A2eiφ0Af-(rA,t)|-A〉
which is the wave function of a free particle in a Stern-Gerlach apparatus and whose initial spin is given by (θ0A,φ0A). For an initial polarization (θ0A,φ0A) and an initial position (z0A), we obtain, in the de Broglie-Bohm interpretation [44] of the Stern and Gerlach experiment, an evolution of the position (zA(t)) and of the spin orientation of A (θA(zA(t),t)) [48].
The case of particleB is different. B follows a rectilinear trajectory with yB(t)=vyt, zB(t)=z0B, and xB(t)=x0B. By contrast, the orientation of its spin moves with the orientation of the spin of A: θB(t)=π-θA(zA(t),t) and φB(t)=φ(zA(t),t)-π. We can associate the following wave function with the particle B:
(37)ΨB(rB,t0+Δt+t)=f(rB)(cos⁡θB(t)2|+B〉+sinθB(t)2eiφB(t)|-B〉).
This wave function is specific, because it depends upon initial conditions of A (position and spin). The orientation of spin of the particle B is driven by the particle Athrough the singlet wave function. Thus, the singlet wave function is the nonlocal variable.Step 2 (spin measurement ofB).
At the timet0+Δt+tD, immediately after the measurement of A, θB(t0+Δt+tD)=π or 0 in accordance with the value of θA(zA(t),t) and the wave function of B is given by (34). The frame (Ox′yz′) corresponds to the frame (Oxyz) after a rotation of an angle δ around the y-axis. θB corresponds to the B-spin angle with the z-axis and θ′B to the B-spin angle with the z′-axis; then θ′B(t0+Δt+tD)=π+δ or δ. In this second step, we are exactly in the case of a particle in a simple Stern and Gerlach experiment (with magnet EB) with a specific initial polarization equal to π+δ or δ and not random like in Step 1. Then, the measurement of B, at time t0+2(Δt+tD), gives, in this interpretation of the two-step version of the EPR-B experiment, the same results as in the EPR-B experiment.


## 5.4. Physical Explanation of Nonlocal Influences

From the wave function of two entangled particles, we find spins, trajectories, and also a wave function for each of the two particles. In this interpretation, the quantum particle has a local position like a classical particle, but it has also a nonlocal behavior through the wave function. So, it is the wave function that creates the nonclassical properties. We can keep a view of a local realist world for the particle, but we should add a nonlocal vision through the wave function. As we saw in Step1, the nonlocal influences in the EPR-B experiment only concern the spin orientation not the motion of the particles themselves. Indeed only spins are entangled in the wave function (27) not positions and motions like in the initial EPR experiment. This is a key point in the search for a physical explanation of nonlocal influences.The simplest explanation of this nonlocal influence is to reintroduce the concept of ether (or the preferred frame) but a new format given by Lorentz-Poincaré and by Einstein in 1920 [69]: “Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. According to the general theory of relativity space without ether is unthinkable;[sic]for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.”Taking into account the new experiments, especially Aspect’s experiments, Popper [70, page XVIII] defends a similar view in 1982:“I feel not quite convinced that the experiments are correctly interpreted; but if they are, we just have to accept action at a distance. I think (with J.P. Vigier) that this would of course be very important, but I do not for a moment think that it would shake, or even touch, realism. Newton and Lorentz were realists and accepted action at a distance; and Aspect’s experiments would be the first crucial experiment between Lorentz’s and Einstein’s interpretation of the Lorentz transformations.”Finally, in the de Broglie-Bohm interpretation, the EPR-B experiments of nonlocality have therefore a great importance not to eliminate realism and determinism but as Popper said to rehabilitate the existence of a certain type of ether, like Lorentz’s ether and like Einstein’s ether in 1920.


## 6. Conclusion

In the three experiments presented in this paper, the variable that is measured in fine is the position of the particle given by this impact on a screen. In the double-slit, the set of these positions gives the interferences; in the Stern-Gerlach and the EPR-B experiments, it is the position of the particle impact that defines the spin value.It is this position that the de Broglie-Bohm interpretation adds to the wave function to define a complete state of the quantum particle. The de Broglie-Bohm interpretation is then based only on the initial conditionsΨ0(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 of measurement. These three postulates are not necessary if we solve the time-dependent Schrödinger equation (double-slit experiment) or the Pauli equation with spatial extension (Stern-Gerlach and EPR experiments). However, these simulations enable us to better understand those experiments: in the double-slit experiment, the interference phenomenon appears only some centimeters after the slits and shows the continuity with classical mechanics; in the Stern-Gerlach experiment, the spin-up/down measurement appears also after a given time, called decoherence time; in the EPR-B experiment, only the spin of B is affected by the spin measurement of A not its density. Moreover, the de Broglie-Bohm trajectories propose a clear explanation of the spin measurement in quantum mechanics.However, we have seen two very different cases in the measurement process. In the first case (double-slit experiment), there is no influence of the measuring apparatus (the screen) on the quantum particle. In the second case (Stern-Gerlach experiment, EPR-B), there is an interaction with the measuring apparatus (the magnetic field) and the quantum particle. The result of the measurement depends on the position of the particle in the wave function. The measurement duration is then the time necessary for the stabilisation of the result.This heterodox interpretation clearly explains experiments with a set of quantum particles that are statistically prepared. These particles verify the “quantum equilibrium hypothesis” and the de Broglie-Bohm interpretation establishes continuity with classical mechanics. However, there is no reason that the de Broglie-Bohm interpretation can be extended to quantum particles that are not statistically prepared. This situation occurs when the wave packet corresponds to a quasiclassical coherent state, introduced in 1926 by Schrödinger [71]. The field quantum theory and the second quantification are built on these coherent states [72]. It is also the case, for the hydrogen atom, of localized wave packets whose motions are on the classical trajectory (an old dream of Schrödinger’s). Their existence was predicted in 1994 by Bialynicki-Birula et al. [73–75] and discovered recently by Maeda and Gallagher [76] on Rydberg atoms. For these nonstatistically prepared quantum particles, we have shown [9, 10] that the natural interpretation is the Schrödinger interpretation proposed at the Solvay congress in 1927. Everything happens as if the quantum mechanics interpretation depended on the preparation of the particles (statistically or not statistically prepared). It is perhaps a response to the “theory of the double solution” that Louis de Broglie was seeking since 1927: “I introduced as the “double solution theory” the idea that it was necessary to distinguish two different solutions that are both linked to the wave equation, one that I called waveu, which was a real physical wave represented by a singularity as it was not normalizable due to a local anomaly defining the particle, the other one as Schrödinger’sΨwave, which is a probability representation as it is normalizable without singularities” [77].



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