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Research ArticleMeasurement 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, France2 École 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 theCreative CommonsAttributionLicense, which permits unrestricted use, distribution, and reproduction in anymedium, provided the originalwork is properly cited.

We propose a pedagogical presentation of measurement in the de Broglie-Bohm interpretation. In this heterodox interpretation,the position of a quantum particle exists and is piloted by the phase of the wave function. We show how this position explainsdeterminism and realism in the three most important experiments of quantummeasurement: 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 continuitywith classical mechanics. Second, we present a numerical simulation of the double-slit experiment performed by Jönsson in 1961with electrons. It demonstrates the continuity between classical mechanics and quantum mechanics. Third, we present an analyticexpression of the wave function in the Stern-Gerlach experiment. This explicit solution requires the calculation of a Pauli spinorwith a spatial extension. This solution enables us to demonstrate the decoherence of the wave function and the three postulates ofquantum measurement. Finally, we study the Bohm version of the Einstein-Podolsky-Rosen experiment. Its theoretical resolutionin space and time shows that a causal interpretation exists where each atom has a position and a spin.

1. Introduction

“I saw the impossible done” [1].This is how John Bell describeshis inexpressible surprise in 1952 upon the publication ofan article by Bohm [2]. The impossibility came from atheorem by John von Neumann outlined in 1932 in his bookThe Mathematical Foundations of Quantum Mechanics [3],which seemed to show the impossibility of adding “hiddenvariables” to quantum mechanics. This impossibility, withits physical interpretation, became almost a postulate ofquantum mechanics, based on von Neumann’s indisputableauthority as a mathematician. Bernard d’Espagnat notes in1979 the following:

“At the university, Bell had, like all of us, received from histeachers a message which, later still, Feynman would brilliantlystate as follows: “No one can explain more than we haveexplained here [. . .]. We do not have the slightest idea of amore fundamental mechanism from which the former results(the interference fringes) could follow”. If indeed we are tobelieve Feynman (and Banesh Hoffman, andmany others, who

expressed the same idea in many books, both popular andscholarly), Bohm’s theory cannot exist. Yet it does exist, andis even older than Bohm’s papers themselves. In fact, the basicidea behind it was formulated in 1927 by Louis de Broglie in amodel he called “pilot wave theory”. Since this theory providesexplanations of what, in “high circles”, is declared inexplicable,it is worth consideration, even by physicists [. . .] who do notthink it gives us the final answer to the question how realityreally is” [4].

And in 1987, Bell wonders about his teachers’ silenceconcerning 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 didvon Neumann not consider it? More extraordinarily, whydid 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 criticismof Bohm’s version than to brand it as “metaphysical” and“ideological”? Why is the pilot-wave picture ignored in textbooks? Should it not be taught, not as the only way, but

Hindawi Publishing CorporationPhysics Research InternationalVolume 2014, Article ID 605908, 16 pageshttp://dx.doi.org/10.1155/2014/605908

2 Physics Research International

as an antidote to the prevailing complacency? To show thatvagueness, subjectivity and indeterminism are not forced onus by experimental facts, but through a deliberate theoreticalchoice?” [5].

More than thirty years after John Bell’s questions, theinterpretation of the de Broglie-Bohm pilot wave is stillignored by both the international community and the text-books.

What is this pilot wave theory? For de Broglie, a quantumparticle is not only defined by its wave function. He assumesthat the quantum particle also has a position which is pilotedby the wave function [6]. However, only the probabilitydensity of this position is known. The position exists initself (ontologically) but is unknown to the observer. It onlybecomes 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 impor-tant experiments of quantum measurement: the double-slitexperiment which is the crucial experiment of the wave-particle duality, the Stern and Gerlach experiment with themeasurement of the spin, and the EPR-B experiment with theproblem of nonlocality.

The paper is organized as follows. In Section 2, wedemonstrate the conditions in which the de Broglie-Bohminterpretation can be assumed to be valid through continuitywith classical mechanics. This involves the de Broglie-Bohminterpretation for a set of particles prepared in the sameway. In Section 3, we present a numerical simulation ofthe double-slit experiment performed by Jönsson in 1961with electrons [7]. The method of Feynman path integralsallows us to calculate the time-dependent wave function.The evolution of the probability density just outside theslits leads one to consider the dualism of the wave-particleinterpretation. And the de Broglie-Bohm trajectories providean explanation for the impact positions of the particles.Finally, we show the continuity between classical and quan-tum trajectories with the convergence of these trajectoriesto classical trajectories when ℎ tends to 0. In Section 4,we present an analytic expression of the wave function inthe Stern-Gerlach experiment. This explicit solution requiresthe calculation of a Pauli spinor with a spatial extension.This solution enables us to demonstrate the decoherenceof the wave function and the three postulates of quantummeasurement: quantization, Born interpretation, and wavefunction reduction. The spinor spatial extension also enablesthe introduction of the de Broglie-Bohm trajectories whichgives a very simple explanation of the particles’ impact andof 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 timeshows that a causal interpretation exists where each atomhas a position and a spin. Finally, we recall that a physicalexplanation of nonlocal influences is possible.

2. The de Broglie-Bohm Interpretation

Thede Broglie-Bohm interpretation is based on the followingdemonstration. Let us consider a wave function Ψ(x, 𝑡)

solution to the Schrödinger equation:

𝑖ℏ

𝜕Ψ (x, 𝑡)𝜕𝑡

= −

ℏ

2

2𝑚

ΔΨ (x, 𝑡) + 𝑉 (x) Ψ (x, 𝑡) , (1)

Ψ (x, 0) = Ψ0(x) . (2)

With the variable change Ψ(x, 𝑡) = √𝜌ℏ(x, 𝑡) exp (𝑖(𝑆ℏ(x, 𝑡)/ℏ)), the Schrödinger equation can be decomposed intoMadelung equations [8] (1926):

𝜕𝑆

ℏ(x, 𝑡)𝜕𝑡

+

1

2𝑚

(∇𝑆

ℏ(x, 𝑡))

2

+ 𝑉 (x) − ℏ2

2𝑚

Δ√𝜌

ℏ(x, 𝑡)

√𝜌

ℏ(x, 𝑡)

= 0,

(3)

𝜕𝜌

ℏ(x, 𝑡)𝜕𝑡

+ div(𝜌ℏ (x, 𝑡) ∇𝑆ℏ(x, 𝑡)𝑚

) = 0 (4)

with initial conditions:

𝜌

ℏ(x, 0) = 𝜌ℏ

0(x) , 𝑆ℏ (x, 0) = 𝑆ℏ

0(x) . (5)

Madelung equations correspond to a set of noninteractingquantum particles all prepared in the same way (same 𝜌ℏ

0(x)

and 𝑆ℏ0(x)).

A quantum particle is said to be statistically prepared ifits initial probability density 𝜌ℏ

0(x) and its initial action 𝑆ℏ

0(x)

converge, when ℏ → 0, to nonsingular functions 𝜌0(x) and

𝑆

0(x). It is the case of an electronic or𝐶

60beam in the double-

slit experiment or an atomic beam in the Stern and Gerlachexperiment. We will see that it is also the case of a beam ofentangled particles in the EPR-B experiment. Then, we havethe following theorem [9, 10].

Theorem 1. For statistically prepared quantum particles, theprobability density 𝜌ℏ(x, 𝑡) and the action 𝑆ℏ(x, 𝑡), solutions tothe 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:

𝜕𝑆 (x, 𝑡)𝜕𝑡

+

1

2𝑚

(∇𝑆 (x, 𝑡))2 + 𝑉 (x, 𝑡) = 0, (6)

𝑆 (x, 0) = 𝑆0(x) , (7)

𝜕𝜌 (x, 𝑡)𝜕𝑡

+ div (𝜌 (x, 𝑡) ∇𝑆 (x, 𝑡)𝑚

) = 0,(8)

𝜌 (x, 0) = 𝜌0(x) . (9)

We give some indications on the demonstration of thistheorem when the wave function Ψ(x, 𝑡) is written as afunction of the initial wave function Ψ

0(x) by the Feynman

paths integral [11]:

Ψ (x, 𝑡) = ∫𝐹 (𝑡, ℏ) exp ( 𝑖ℏ

𝑆

𝑐𝑙(x, 𝑡; x

0))Ψ

0(x

0) 𝑑x

0, (10)

where 𝐹(𝑡, ℏ) is an independent function of x and of x0. For a

statistically prepared quantum particle, the wave function is

Physics Research International 3

written:Ψ(x, 𝑡) = 𝐹(𝑡, ℏ) ∫√𝜌ℏ0(x

0) exp((𝑖/ℏ)(𝑆ℏ

0(x

0)+𝑆

𝑐𝑙(x, 𝑡;

x0)))𝑑𝑥

0. The theorem of the stationary phase shows that if ℏ

tends towards 0, we have Ψ(x, 𝑡) ∼ exp((𝑖/ℏ)minx0(𝑆0(x0) +𝑆

𝑐𝑙(x, 𝑡; x

0))); that is to say, the quantum action 𝑆ℎ(x, 𝑡)

converges to the function:

𝑆 (x, 𝑡) = minx0 (𝑆0 (x0) + 𝑆𝑐𝑙 (x, 𝑡; x0)) (11)

which is the solution to the Hamilton-Jacobi equation (6)with the initial condition (7). Moreover, as the quantum den-sity 𝜌ℎ(x, 𝑡) satisfies the continuity equation (4), we deduce,since 𝑆ℎ(x, 𝑡) tends towards 𝑆(x, 𝑡), that 𝜌ℎ(x, 𝑡) convergesto the classical density 𝜌(x, 𝑡), which satisfies the continuityequation (8). We obtain both announced convergences.

These statistical Hamilton-Jacobi equations (6), (7), (8),and (9) correspond to a set of classical particles preparedin the same way (the same 𝜌

0(x) and 𝑆

0(x)). These classical

particles are trajectories obtained in Eulerian representationwith the velocity field k(x, 𝑡) = ∇𝑆(x, 𝑡)/𝑚, but the densityand the action are not sufficient to describe it completely. Toknow its position at time 𝑡, it is necessary to know its initialposition. Because the Madelung equations converge to thestatistical Hamilton-Jacobi equations, it is logical to do thesame in quantum mechanics. We conclude that a statisticallyprepared quantum particle is not completely described by itswave function. It is necessary to add this initial position andan equation to define the evolution of this position in thetime. It is the de Brogglie-Bohm interpretation where theposition is called the “hidden variable.”

The two first postulates of quantum mechanics, describ-ing the quantum state and its evolution [12], must be com-pleted in this heterodox interpretation. At initial time 𝑡 = 0,the state of the particle is given by the initial wave functionΨ

0(x) (awave packet) and its initial positionX(0); it is the new

first postulate. The new second postulate gives the evolutionon thewave function and on the position. For a single spinlessparticle in a potential𝑉(x), the evolution of thewave functionis given by the usual Schrödinger equations (1), (2) and theevolution of the particle position is given by

𝑑X (𝑡)𝑑𝑡

=

Jℎ(x, 𝑡)𝜌

ℎ(x, 𝑡)

x=X(𝑡)=

∇𝑆

ℎ(x, 𝑡)𝑚

𝑥=X(𝑡), (12)

where

Jℎ (x, 𝑡) = ℏ2𝑚𝑖

(Ψ

∗(x, 𝑡) ∇Ψ (x, 𝑡) − Ψ (x, 𝑡) ∇Ψ∗ (x, 𝑡))

(13)

is the usual quantum current.In the case of a particle with spin, as in the Stern

and Gerlach experiment, the Schrödinger equation must bereplaced by the Pauli or Dirac equations.

The third quantum mechanics postulate which describesthemeasurement operator (the observable) can be conserved.

But the three postulates of measurement are not necessary:the postulate of quantization, the Born postulate of proba-bilistic interpretation of the wave function, and the postulateof the reduction of the wave function. We see that thesepostulates of measurement can be explained on each exampleas we will show in the following.

We replace these three postulates by a single one, the“quantum equilibrium hypothesis,” [13–15] that describes theinteraction between the initial wave function Ψ

0(x) and the

initial particle position X(0): for a set of identically preparedparticles having 𝑡 = 0 wave functionΨ

0(x), it is assumed that

the initial particle positionsX(0) are distributed according to

𝑃 [X (0) = x] ≡ 𝑃 (x, 0) =

Ψ

0(x)

2

= 𝜌

ℎ

0(x) . (14)

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) =∇𝑆

ℎ(x, 𝑡)/𝑚 satisfies the property of the “equivariance” of the

|Ψ(x, 𝑡)|2 probability distribution [13]:

𝑃 [X (𝑡) = x] ≡ 𝑃 (x, 𝑡) = |Ψ (x, 𝑡)|2 = 𝜌ℎ (x, 𝑡) . (15)

It is the Born rule at time 𝑡.Then, the de Broglie-Bohm interpretation is based on a

continuity between classical and quantum mechanics wherethe quantum particles are statistically prepared with an initialprobability density that satisfies the “quantum equilibriumhypothesis” (14). It is the case of the three studied experi-ments.

We will revisit these three measurement experimentsthrough mathematical calculations and numerical simula-tions. For each one, we present the statistical interpretationthat is common to the Copenhagen interpretation and the deBroglie-Bohm pilot wave and then the trajectories specific tothe de Broglie-Bohm interpretation.We show that the precisedefinition of the initial conditions, that is, the preparation ofthe particles, plays a fundamental methodological role.

3. Double-Slit Experiment with Electrons

Young’s double-slit experiment [16] has long been the cru-cial experiment for the interpretation of the wave-particleduality. They have been realized with massive objects, suchas electrons [7, 17–19], neutrons [20, 21], cold neutrons [22],and atoms [23], and, more recently, with coherent ensemblesof ultracold atoms [24, 25] and even with mesoscopic singlequantum objects such as 𝐶

60and 𝐶

70[26, 27]. For Feynman,

this experiment addresses “the basic element of the mysteriousbehavior [of electrons] in its most strange form. [It is] aphenomenon which is impossible, absolutely impossible toexplain in any classical way and which has in it the heart ofquantum mechanics. In reality, it contains the only mystery”[28]. The de Broglie-Bohm interpretation and the numericalsimulation help us here to revisit the double-slit experiment


0.2 𝜇m0.2 𝜇m

0.8 𝜇m10𝜇m

35 cm35 cm

y

z

x

S

Figure 1: Diagram of the Jönnson’s double-slit experiment per-formed with electrons.

with electrons performedby Jönsson in 1961 and to provide ananswer to Feynman’s mystery. These simulations [29] followthose conducted in 1979 by Philippidis et al. [30] which aretoday classics. However, these simulations [30] have somelimitations because they did not consider realistic slits. Theslits, which can be clearly represented by a function𝐺(𝑦)with𝐺(𝑦) = 1 for −𝛽 ≤ 𝑦 ≤ 𝛽 and 𝐺(𝑦) = 0 for |𝑦| > 𝛽, if theyare 2𝛽 in width, weremodeled by a Gaussian function𝐺(𝑦) =𝑒

−𝑦2/2𝛽2

. Interference was found, but the calculation could notaccount 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 experimentby Jönsson. An electron gun emits electrons one by one inthe 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 𝑑

1= 35 cm, they encounter a plate pierced

with two horizontal slits 𝐴 and 𝐵, each 0.2 𝜇m wide andspaced 1𝜇m from each other. A screen located at 𝑑

2= 35 cm

after the slits collects these electrons. The impact of eachelectron appears on the screen as the experiment unfolds.After thousands of impacts, we find that the distribution ofelectrons on the screen shows interference fringes.

The slits are very long along the 𝑧-axis, so there is no effectof diffraction along this axis. In the simulation, we thereforeonly consider the wave function along the 𝑦-axis; the variable𝑥 will be treated classically with 𝑥 = V𝑡. Electrons emergingfrom an electron gun are represented by the same initial wavefunction Ψ

0(𝑦).

3.1. Probability Density. Figure 2 gives a general view of theevolution of the probability density from the source to thedetection screen (a lighter shade means that the density ishigher; i.e., the probability of presence is high). The calcula-tions were made using themethod of Feynman path integrals[29]. The wave function after the slits (𝑡

1= 𝑑

1/V ≃ 2.10−11 s

< 𝑡 < 𝑡

1+ 𝑑

2/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) + Ψ

𝐵(𝑦, 𝑡)

70 cm

10𝜇m

Figure 2: General view of the evolution of the probability densityfrom the source to the screen in the Jönsson experiment. A lightershade means that the density is higher; that is, the probability ofpresence is high.

3.5 cm

3𝜇m

Figure 3: Close-up of the evolution of the probability density in thefirst 3 cm after the slits in the Jönsson experiment.

with Ψ𝐴(𝑦, 𝑡) = ∫

𝐴𝐾(𝑦, 𝑡, 𝑦

𝑎, 𝑡

1)Ψ(𝑦

𝑎, 𝑡

1)𝑑𝑦

𝑎, Ψ

𝐵(𝑦, 𝑡) =

∫

𝐵𝐾(𝑦, 𝑡, 𝑦

𝑏, 𝑡

1)Ψ(𝑦

𝑏, 𝑡

1)𝑑𝑦

𝑏, and 𝐾(𝑦, 𝑡, 𝑦

𝛼, 𝑡

1) = (𝑚/2𝑖ℏ(𝑡 −

𝑡

1))

1/2𝑒

𝑖𝑚(𝑦−𝑦𝛼)2/2ℏ(𝑡−𝑡1).

Figure 3 shows a close-up of the evolution of the prob-ability density just after the slits. We note that interferencewill only occur a few centimeters after the slits. Thus, if thedetection screen is 1 cm from the slits, there is no interferenceand one can determine by which slit each electron has passed.In this experiment, the measurement is performed by thedetection screen, which only reveals the existence or absenceof the fringes.

The calculation method enables us to compare theevolution of the cross-section of the probability density atvarious distances after the slits (0.35mm, 3.5mm, 3.5 cm, and35 cm) where the two slits 𝐴 and 𝐵 are open simultaneously(interference: |Ψ

𝐴+ Ψ

𝐵|

2) with the evolution of the sum ofthe probability densities where the slits 𝐴 and 𝐵 are openindependently (the sum of two diffractions: |Ψ

𝐴|

2+ |Ψ

𝐵|

2).


−1 −0.5

(𝜇m)0 0.5 1

(a) 0.35mm

−1 −0.5

(𝜇m)0 0.5 1

(b) 3.5mm

−2 −1

(𝜇m)0 1 2

(c) 3.5 cm

−10 −5

(𝜇m)0 5 10

(d) 35 cm

Figure 4: Comparison of the probability density |Ψ𝐴+ Ψ

𝐵|

2 (full line) and |Ψ𝐴|

2+ |Ψ

𝐵|

2 (dotted line) at various distances after the slits: (a)0.35mm, (b): 3.5mm, (c): 3.5 cm, and (d): 35 cm.

Figure 4 shows that the difference between these two phe-nomena appears only a few centimeters after the slits.

3.2. Impacts on Screen and de Broglie-Bohm Trajectories. Theinterference fringes are observed after a certain period of timewhen the impacts of the electrons on the detection screenbecome sufficiently numerous. Classical quantum theoryonly explains the impact of individual particles statistically.

However, in the de Broglie-Bohm interpretation, a parti-cle has an initial position and follows a path whose velocity ateach instant is given by (12). On the basis of this assumptionwe conduct a simulation experiment by drawing randominitial positions of the electrons in the initial wave packet(quantum equilibrium hypothesis).

Figure 5 shows, after its initial starting position, 100possible quantum trajectories of an electron passing throughone of the two slits: we have not represented the paths of the

electron when it is stopped by the first screen. Figure 6 showsa close-up of these trajectories just after they leave their slits.

The different trajectories explain both the impact ofelectrons on the detection screen and the interference fringes.This is the simplest andmost natural interpretation to explainthe impact positions: “the position of an impact is simply theposition of the particle at the time of impact.” This was theview defended by Einstein at the Solvay Congress of 1927.Theposition is the only measured variable of the experiment.

In the de Broglie-Bohm interpretation, the impacts on thescreen are the real positions of the electron as in classicalmechanics and the three postulates of the measurement ofquantum mechanics can be trivially explained: the positionis an eigenvalue of the position operator because the positionvariable is identical to its operator (XΨ = xΨ), the Bornpostulate is satisfied with the “equivariance” property, and thereduction of the wave packet is not necessary to explain theimpacts.


−35 −30 −20

−4

−3

−10

−2

−1

(𝜇m)

(cm)0 10 20 30 35

0

1

2

3

4

Figure 5: 100 electron trajectories for the Jönsson experiment.

−0.8

−0.6

−0.4

−0.2

−1

(𝜇m)

(cm)0 1 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

1

Figure 6: Close-up on the 100 trajectories of the electrons just afterthe slits.

Through numerical simulations, we will demonstratehow, when the Planck constant ℎ tends to 0, the quantumtrajectories converge to the classical trajectories. In realitya constant is not able to tend to 0 by definition. Theconvergence to classical trajectories is obtained if the termℎ𝑡/𝑚 → 0, so ℎ → 0 is equivalent to𝑚 → +∞ (i.e., the massof the particle grows) or 𝑡 → 0 (i.e., the distance splits-screen𝑑

2→ 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 tomultiplying the mass by 10, 100, 1000, and 10000). We obtainquantum trajectories converging to the classical trajectories,when ℎ tends to 0.

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 particleshave quantumproperties, but specific quantumbehavior onlyappears in certain experimental conditions: here when theratio ℎ𝑡/𝑚 is sufficiently large. Interferences only appeargradually and the quantum particle behaves at any time asboth a wave and a particle.

4. The Stern-Gerlach Experiment

In 1922, by studying the deflection of a beam of silver atomsin a strongly inhomogeneous magnetic field (cf. Figure 8)Gerlach and Stern [31, 32] obtained an experimental resultthat contradicts the common sense prediction: the beam,instead of expanding, splits into two separate beams givingtwo spots of equal intensity𝑁+ and𝑁− on a detector, at equaldistances from the axis of the original beam.

Historically, this is the experiment which helped establishspin quantization. Theoretically, it is the seminal experimentposing the problem of measurement in quantum mechanics.Today it is the theory of decoherence with the diagonalizationof the density matrix that is put forward to explain the firstpart of the measurement process [33–38]. However, althoughthese authors consider the Stern-Gerlach experiment asfundamental, they do not propose a calculation of the spindecoherence time.

We present an analytical solution to this decoherencetime and the diagonalization of the density matrix. Thissolution requires the calculation of the Pauli spinor with aspatial extension as the equation:

Ψ

0(𝑧) = (2𝜋𝜎

2

0)

−1/2

𝑒

−𝑧2/4𝜎2

0(

cos𝜃

0

2

𝑒

−𝑖(𝜑0/2)

sin𝜃

0

2

𝑒

𝑖(𝜑0/2)

). (16)

Quantum mechanics textbooks [12, 28, 39, 40] do not takeinto account the spatial extension of the spinor (16) andsimply use the simplified spinor without spatial extension:

Ψ

0= (

cos𝜃

0

2

𝑒

−𝑖(𝜑0/2)

sin𝜃

0

2

𝑒

𝑖(𝜑0/2)

). (17)

However, as we shall see, the different evolutions of the spatialextension between the two spinor components will have akey role in the explanation of the measurement process.This spatial extension enables us, in following the precursoryworks of Takabayasi [41, 42], Bohm et al. [43, 44], Dewdneyet al. [45], and Holland [46], to revisit the Stern and Gerlachexperiment, to explain the decoherence, and to demonstratethe three postulates of the measure: quantization, Bornstatistical interpretation, and wave function reduction.

Silver atoms contained in the oven𝐸 (Figure 8) are heatedto a high temperature and escape through a narrow opening.


−0.8

−0.6

−0.4

−0.2

−1

(𝜇m)

(cm)−30 −20 −10

h/10 h/100

h/1000 h/10000

0 10 20 30

0

0.2

0.4

0.6

0.8

1

−0.8

−0.6

−0.4

−0.2

−1

(𝜇m)

(cm)−30 −20 −10 0 10 20 30

0

0.2

0.4

0.6

0.8

1

−0.8

−0.6

−0.4

−0.2

−1

(𝜇m)

(cm)−30 −20 −10 0 10 20 30

0

0.2

0.4

0.6

0.8

1

−0.8

−0.6

−0.4

−0.2

−1

(𝜇m)

(cm)−30 −20 −10 0 10 20 30

0

0.2

0.4

0.6

0.8

1

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

x

y

z

y = �t

D

A1

P1

Δl(� = 500m/s)T = 1000∘K

E

TN+

N−

Figure 8: Schematic configuration of the Stern-Gerlach experiment.

A second aperture, 𝑇, selects those atoms whose velocity, v0,

is parallel to the 𝑦-axis. The atomic beam crosses the gap ofthe electromagnet 𝐴

1before condensing on the detector, 𝑃

1.

Before crossing the electromagnet, the magnetic moment ofeach silver atom is oriented randomly (isotropically). In thebeam, we represent each atom by its wave function; one can

assume that at the entrance to the electromagnet, 𝐴1and

at the initial time 𝑡 = 0, each atom can be approximatelydescribed by a Gaussian spinor in 𝑧 given by (16) correspond-ing to a pure state. The variable 𝑦 will be treated classicallywith 𝑦 = V𝑡. 𝜎

0= 10

−4 m corresponds to the size of theslot 𝑇 along the 𝑧-axis. The approximation by a Gaussianinitial spinor will allow explicit calculations. Because the slotis much wider along the 𝑥-axis, the variable 𝑥 will be alsotreated classically. To obtain an explicit solution of the Stern-Gerlach experiment, we take the numerical values used inthe Cohen-Tannoudji textbook [12]. For the silver atom, wehave𝑚 = 1.8 × 10−25 kg, V

0= 500m/s (corresponding to the

temperature of 𝑇 = 1000∘K). In (16) and in Figure 9, 𝜃0and

𝜑

0are the polar angles characterizing the initial orientation

of the magnetic moment and 𝜃0corresponds to the angle

with the 𝑧-axis.The experiment is a statistical mixture of purestates where the 𝜃

0and the 𝜑

0are randomly chosen: 𝜃

0is

drawn in a uniform way from [0, 𝜋] and 𝜑0is drawn in a

uniform way from [0, 2𝜋].


x

y

z

𝜑0

𝜃0

|−⟩

|+⟩

Figure 9: Orientation of the magnetic moment. 𝜃0and 𝜑

0are the

polar angles characterizing the spin vector in the de Broglie-Bohminterpretation.

The evolution of the spinor Ψ = ( 𝜓+𝜓− ) in a magnetic fieldB is then given by the Pauli equation:

𝑖ℏ(

𝜕𝜓

+

𝜕𝑡

𝜕𝜓

−

𝜕𝑡

) = −

ℏ

2

2𝑚

Δ(

𝜓

+

𝜓

−

) + 𝜇

𝐵B𝜎(𝜓+

𝜓

−

) , (18)

where 𝜇𝐵

= 𝑒ℏ/2𝑚

𝑒is the Bohr magneton and where

𝜎 = (𝜎

𝑥, 𝜎

𝑦, 𝜎

𝑧) corresponds to the three Pauli matrices. The

particle first enters an electromagnetic field B directed alongthe 𝑧-axis, 𝐵

𝑥= 𝐵

0𝑥, 𝐵

𝑦= 0, and 𝐵

𝑧= 𝐵

0− 𝐵

0𝑧, with

𝐵

0= 5 Tesla, 𝐵

0= |𝜕𝐵/𝜕𝑧| = 10

3 Tesla/m over a lengthΔ𝑙 = 1 cm. On exiting the magnetic field, the particle isfree until it reaches the detector 𝑃

1placed at a 𝐷 = 20 cm

distance.Theparticle stayswithin themagnetic field for a timeΔ𝑡 =

Δ𝑙/V = 2 × 10−5 s. During this time [0, Δ𝑡], the spinor is [47](see the Appendix)

Ψ (𝑧, 𝑡) ≃ (

cos𝜃

0

2

(2𝜋𝜎

2

0)

−1/2

𝑒

−(𝑧−(𝜇𝐵𝐵

0/2𝑚)𝑡

2)2

/4𝜎2

0𝑒

𝑖((𝜇𝐵𝐵

0𝑡𝑧−(𝜇

2

𝐵𝐵2

0/6𝑚)𝑡

3+𝜇𝐵𝐵0𝑡+(ℏ𝜑0/2))/ℏ)

𝑖 sin𝜃

0

2

(2𝜋𝜎

2

0)

−1/2

𝑒

−(𝑧+(𝜇𝐵𝐵

0/2𝑚)𝑡

2)2

/4𝜎2

0𝑒

𝑖((−𝜇𝐵𝐵

0𝑡𝑧−(𝜇

2

𝐵𝐵2

0/6𝑚)𝑡

3−𝜇𝐵𝐵0𝑡−(ℏ𝜑0/2))/ℏ)

). (19)

After the magnetic field, at time 𝑡 + Δ𝑡 (𝑡 ≥ 0) in the freespace, the spinor becomes [44–48] (see the Appendix)

Ψ (𝑧, 𝑡 + Δ𝑡)

≃ (

cos𝜃

0

2

(2𝜋𝜎

2

0)

−1/2

𝑒

−(𝑧−𝑧Δ−𝑢𝑡)2/4𝜎2

0𝑒

𝑖((𝑚𝑢𝑧+ℏ𝜑+)/ℏ)

sin𝜃

0

2

(2𝜋𝜎

2

0)

−1/2

𝑒

−(𝑧+𝑧Δ+𝑢𝑡)2/4𝜎2

0𝑒

𝑖((−𝑚𝑢𝑧+ℏ𝜑−)/ℏ)

),

(20)

where

𝑧

Δ=

𝜇

𝐵𝐵

0(Δ𝑡)

2

2𝑚

= 10

−5 m, 𝑢 =𝜇

𝐵𝐵

0(Δ𝑡)

𝑚

= 1m/s.(21)

Equation (20) takes into account the spatial extension of thespinor and we note that the two spinor components havevery different 𝑧 values. All interpretations are based on thisequation.

4.1. The Decoherence Time. We deduce from (20) the prob-ability density of a pure state in the free space after the

electromagnet:

𝜌

𝜃0(𝑧, 𝑡 + Δ𝑡) ≃ (2𝜋𝜎

2

0)

−1/2

(cos2𝜃

0

2

𝑒

−(𝑧−𝑧Δ−𝑢𝑡)2/2𝜎2

0

+sin2𝜃

0

2

𝑒

−(𝑧+𝑧Δ+𝑢𝑡)2/2𝜎2

0) .

(22)

Figure 10 shows the probability density of a pure state (with𝜃

0= 𝜋/3) as a function of 𝑧 at several values of 𝑡 (the plots

are labeled 𝑦 = V𝑡). The beam separation does not appear atthe end of the magnetic field (1 cm) but 16 cm further along.It is the moment of the decoherence.

The decoherence time, where the two spots 𝑁+ and 𝑁−are separated, is then given by

𝑡

𝐷≃

3𝜎

0− 𝑧

Δ

𝑢

= 3 × 10

−4 s. (23)

This decoherence time is usually the time required todiagonalize the marginal density matrix of spin variablesassociated with a pure state [49]:

𝜌

𝑆(𝑡) = (

∫

𝜓

+(𝑧, 𝑡)

2

𝑑𝑧 ∫𝜓

+(𝑧, 𝑡) 𝜓

∗

−(𝑧, 𝑡) 𝑑𝑧

∫𝜓

−(𝑧, 𝑡) 𝜓

∗

+(𝑧, 𝑡) 𝑑𝑧 ∫

𝜓

−(𝑧, 𝑡)

2

𝑑𝑧

) .

(24)


−0.6

(mm)−0.6

(mm)−0.6

(mm)−0.6

(mm)

0 cm 6 cm 16 cm 21 cm

0 0.6 0 0.6 0 0.6 0 0.6

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

−5 −4 −3 −2 −1

−1

z(m

m)

x (mm)0 1 2 3 4 5

0

1N+

N−

Figure 11: 1000 silver atom impacts on the detector 𝑃1.

For 𝑡 ≥ 𝑡𝐷, the product 𝜓

+(𝑧, 𝑡 + Δ𝑡)𝜓

−(𝑧, 𝑡 + Δ𝑡) is null

and the density matrix is diagonal: the probability density ofthe initial pure state (20) is diagonal:

𝜌

𝑆(𝑡 + Δ𝑡) = (2𝜋𝜎

2

0)

−1

(

cos2𝜃

0

2

0

0 sin2𝜃

0

2

) . (25)

4.2. Proof of the Postulates of Quantum Measurement. Wethen obtain atoms with a spin oriented only along the 𝑧-axis(positively or negatively). Let us consider the spinor Ψ(𝑧, 𝑡 +Δ𝑡) given by (20). Experimentally, we do not measure thespin directly but the �̃� position of the particle impact on 𝑃

1

(Figure 11).If �̃� ∈ 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)𝜎

𝑧: Ψ(�̃�, 𝑡 +

Δ𝑡) ≃ (2𝜋𝜎

2

0)

−1/4 cos(𝜃0/2)𝑒

−(�̃�1−𝑧Δ−𝑢𝑡)2/4𝜎2

0𝑒

𝑖((𝑚𝑢�̃�1+ℏ𝜑+)/ℏ)(

1

0).

Then, we have 𝑆𝑧Ψ = (ℏ/2)𝜎

𝑧Ψ = +(ℏ/2)Ψ.

If �̃� ∈ 𝑁−, the term 𝜓+

of (20) is numericallyequal to zero and the spinor Ψ is proportional to ( 0

1), the

other eigenvector of the spin operator 𝑆𝑧: Ψ(�̃�, 𝑡 + Δ𝑡) ≃

(2𝜋𝜎

2

0)

−1/4 sin(𝜃0/2)𝑒

−(�̃�2+𝑧Δ+𝑢𝑡)2/4𝜎2

0𝑒

𝑖((−𝑚𝑢�̃�2+ℏ𝜑−)/ℏ)(

0

1). Then,

we have 𝑆𝑧Ψ = (ℏ/2)𝜎

𝑧Ψ = −(ℏ/2)Ψ. Therefore, the

measurement of the spin corresponds to an eigenvalue of thespin operator. It is a proof of the postulate of quantization.

Equation (25) gives the probability cos2(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 detector 𝑃1to the location of

the spot 𝑁+ (Figure 8), we select all the atoms that are inthe spin state |+⟩ = ( 1

0). 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 𝑃

1; at this

time the component Ψ−is indeed stopped by detector 𝑃

1.

The future trajectory of the silver atom after crossing thedetector 𝑃

1will be guided by this new (normalized) spinor.

The wave function reduction is therefore not linked to theelectromagnet but to the detector 𝑃

1causing an irreversible

elimination of the spinor component Ψ−.

4.3. Impacts and Quantization Explained by de Broglie-BohmTrajectories. Finally, it remains to provide an explanation ofthe individual impacts of silver atoms. The spatial extensionof the spinor (16) allows us to take into account the particle’sinitial position 𝑧

0and to introduce the Broglie-Bohm trajec-

tories [2, 6, 45, 46, 50] which is the natural assumption toexplain the individual impacts.

Figure 12 presents, for a silver atomwith the initial spinororientation (𝜃

0= 𝜋/3, 𝜑

0= 0), a plot in the (𝑂𝑦𝑧) plane

of a set of 10 trajectories whose initial position 𝑧0has been

randomly chosen from aGaussian distribution with standarddeviation 𝜎

0. The spin orientations 𝜃(𝑧, 𝑡) are represented by

arrows.The final orientation, obtained after the decoherence time

𝑡

𝐷, depends on the initial particle position 𝑧

0in the spinor

with a spatial extension and on the initial angle 𝜃0of the spin

with the 𝑧-axis. We obtain +𝜋/2 if 𝑧0> 𝑧

𝜃0 and −𝜋/2 if 𝑧0<

𝑧

𝜃0 with

𝑧

𝜃0= 𝜎

0𝐹

−1(sin2

𝜃

0

2

) , (26)

where 𝐹 is the repartition function of the normal centered-reduced law. If we ignore the position of the atom in its wavefunction, we lose the determinism given by (26).

In the de Broglie-Bohm interpretation with a realisticinterpretation of the spin, the “measured” value is notindependent of the context of the measure and is contextual.It conforms to the Kochen and Specker theorem [51]: realismand noncontextuality are inconsistent with certain quantummechanics predictions.


0 5 10 15 20

0

0.2

0.4

0.6

0.8

−0.4

−0.2

y (cm)

z(m

m)

Figure 12: Ten silver atom trajectories with initial spin orientation(𝜃

0= 𝜋/3) and initial position 𝑧

0; arrows represent the spin

orientation 𝜃(𝑧, 𝑡) along the trajectories.

Now let us consider a mixture of pure states where theinitial orientation (𝜃

0, 𝜑

0) from the spinor has been randomly

chosen. These are the conditions of the initial Stern andGerlach experiment. Figure 13 represents a simulation of 10quantum trajectories of silver atoms from which the initialpositions 𝑧

0are also randomly chosen.

Finally, the de Broglie-Bohm trajectories propose a clearinterpretation of the spin measurement in quantummechan-ics. There is interaction with the measuring apparatus asis generally stated, and there is indeed a minimum timerequired for measurement. However this measurement andthis time do not have the signification that is usually appliedto them. The result of the Stern-Gerlach experiment is notthe measure of the spin projection along the 𝑧-axis, but theorientation of the spin either in the direction of the magneticfield gradient or in the opposite direction. It depends onthe position of the particle in the wave function. We havetherefore a simple explanation for the noncompatibility ofspin measurements along different axes. The measurementduration is then the time necessary for the particle to pointits spin in the final direction.

5. EPR-B Experiment

Nonseparability is one of the most puzzling aspects ofquantum mechanics. For over thirty years, the EPR-B, thespin 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 theheart of the debate on hidden variables and nonlocality.Manyexperiments since Bell’s paper have demonstrated violationsof these inequalities and have vindicated quantum theory[57–63]. Now, EPR pairs ofmassive atoms are also considered[64, 65].Theusual conclusion of these experiments is to rejectthe nonlocal realism for two reasons: the impossibility ofdecomposing a pair of entangled atoms into two states, one

0 5 10 15 20

0

0.2

0.4

0.6

−0.6

−0.4

−0.2

y (cm)

z(m

m)

Figure 13: Ten silver atom trajectories where the initial orientation(𝜃

0, 𝜑

0) has been randomly chosen; arrows represent the spin

orientation 𝜃(𝑧, 𝑡) along the trajectories.

for each atom, and the impossibility of interaction faster thanthe speed of light.

Here, we show that there exists a de Broglie-Bohminterpretation which answers these two questions positively.To demonstrate this nonlocal realism, two methodologicalconditions are necessary. The first condition is the same as inthe Stern-Gerlach experiment: the solution to the entangledstate is obtained by resolving the Pauli equation from aninitial singlet wave function with a spatial extension as

Ψ

0(r

𝐴, r

𝐵) =

1

√2

𝑓 (r𝐴) 𝑓 (r

𝐵) (

+

𝐴⟩

−

𝐵⟩ −

−

𝐴⟩

+

𝐵⟩)

(27)

and not from a simplified wave function without spatialextension:

Ψ

0(r

𝐴, r

𝐵) =

1

√2

(

+

𝐴⟩

−

𝐵⟩ −

−

𝐴⟩

+

𝐵⟩) . (28)

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 andexplains the determinism and the disentangling process. Toexplain the interaction and the evolution between the spin ofthe two particles, we consider a two-step version of the EPR-B experiment. It is our second methodological condition. Afirst causal interpretation of EPR-B experiment was proposedin 1987 by Dewdney et al. [66, 67] using these two conditions.However, this interpretation had a flaw [46, page 418]: thespin module of each particle depends directly on the singletwave function, and thus the spin module of each particlevaried during the experiment from 0 to ℏ/2. We present a deBroglie-Bohm interpretation that avoids this flaw [68].

Figure 14 presents the Einstein-Podolsky-Rosen-Bohmexperiment. A source 𝑆 creates, in 𝑂, pairs of identical atoms𝐴 and 𝐵, but with opposite spins. The atoms 𝐴 and 𝐵


x

y

z

z

z

z

x

z z

x

yz

xz

𝛿

�y(Δt + tD) �y(t0 + Δt + tD) �yt0 �yΔt �ytD

O Atom AAtom B

EAEB

Figure 14: Schematic configuration of the EPR-B experiment.

split following the 𝑦-axis in opposite directions and headtowards two identical Stern-Gerlach apparatus E

𝐴and E

𝐵.

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 𝛿around the 𝑦-axis. The initial wave function of the entangledstate is the singlet state (27), where r = (𝑥, 𝑧), 𝑓(r) =(2𝜋𝜎

2

0)

−1/2𝑒

−(𝑥2+𝑧2)/4𝜎2

0 , |±𝐴⟩, and |±

𝐵⟩ are the eigenvectors

of the operators 𝜎𝑧𝐴

and 𝜎𝑧𝐵: 𝜎

𝑧𝐴|±

𝐴⟩ = ±|±

𝐴⟩, 𝜎

𝑧𝐵|±

𝐵⟩ =

±|±

𝐵⟩. We treat the dependence with 𝑦 classically: speed

−V𝑦for 𝐴 and V

𝑦for 𝐵. The wave function Ψ(r

𝐴, r

𝐵, 𝑡) of

the two identical particles 𝐴 and 𝐵, electrically neutral andwithmagnetic moments 𝜇

0, subject to magnetic fields E

𝐴and

E𝐵, admits on the basis of |±

𝐴⟩ and |±

𝐵⟩ four components

Ψ

𝑎,𝑏(r

𝐴, r

𝐵, 𝑡) and satisfies the two-body Pauli equation [46,

page 417]:

𝑖ℏ

𝜕Ψ

𝑎,𝑏

𝜕𝑡

= (−

ℏ

2

2𝑚

Δ

𝐴−

ℏ

2

2𝑚

Δ

𝐵)Ψ

𝑎,𝑏+ 𝜇𝐵

E𝐴𝑗(𝜎

𝑗)

𝑎

𝑐Ψ

𝑐,𝑏

+ 𝜇𝐵

E𝐵𝑗(𝜎

𝑗)

𝑏

𝑑Ψ

𝑎,𝑑

(29)

with the initial conditions:

Ψ

𝑎,𝑏(r

𝐴, r

𝐵, 0) = Ψ

𝑎,𝑏

0(r

𝐴, r

𝐵) ,

(30)

where Ψ𝑎,𝑏0(r

𝐴, r

𝐵) corresponds to the singlet state (27).

To obtain an explicit solution of the EPR-B experiment,we take the numerical values of the Stern-Gerlach experi-ment.

One of the difficulties of the interpretation of the EPR-B experiment is the existence of two simultaneous measure-ments. By doing these measurements one after the other, theinterpretation of the experiment will be facilitated. That isthe purpose of the two-step version of the experiment EPR-Bstudied below.

5.1. First Step EPR-B: Spin Measurement of𝐴. In the first stepwe make a Stern and Gerlach “measurement” for atom 𝐴,

on a pair of particles 𝐴 and 𝐵 in a singlet state. This is theexperiment first proposed in 1987 by Dewdney et al. [66, 67].

Consider that at time 𝑡0the particle 𝐴 arrives at the

entrance of electromagnet E𝐴. After this exit of the magnetic

field E𝐴, at time 𝑡

0+ Δ𝑡 + 𝑡, the wave function (27) becomes

[68]

Ψ (r𝐴, r

𝐵, 𝑡

0+ Δ𝑡 + 𝑡)

=

1

√2

𝑓 (r𝐵)

× (𝑓

+(r

𝐴, 𝑡)

+

𝐴⟩

−

𝐵⟩ − 𝑓

−(r

𝐴, 𝑡)

−

𝐴⟩

+

𝐵⟩)

(31)

with

𝑓

±(r, 𝑡) ≃ 𝑓 (𝑥, 𝑧 ∓ 𝑧

Δ∓ 𝑢𝑡) 𝑒

𝑖((±𝑚𝑢𝑧/ℏ)+𝜑±(𝑡)),

(32)

where 𝑧Δand 𝑢 are given by (21).

The atomic density 𝜌(𝑧𝐴, 𝑧

𝐵, 𝑡

0+ Δ𝑡 + 𝑡) is found by

integrating Ψ∗(r𝐴, r

𝐵, 𝑡

0+ Δ𝑡 + 𝑡)Ψ(r

𝐴, r

𝐵, 𝑡

0+ Δ𝑡 + 𝑡) on 𝑥

𝐴

and 𝑥𝐵:

𝜌 (𝑧

𝐴, 𝑧

𝐵, 𝑡

0+ Δ𝑡 + 𝑡)

= ((2𝜋𝜎

2

0)

−1/2

𝑒

−(𝑧𝐵)2/2𝜎2

0)

× ((2𝜋𝜎

2

0)

−1/2

×

1

2

(𝑒

−(𝑧𝐴−𝑧Δ−𝑢𝑡)2/2𝜎2

0+ 𝑒

−(𝑧𝐴+𝑧Δ+𝑢𝑡)2/2𝜎2

0)) .

(33)

We deduce that the beam of particle 𝐴 is divided into two,while the beam of particle 𝐵 stays undivided:

(i) the density of 𝐴 is the same, whether particle 𝐴 isentangled with 𝐵 or not;

(ii) the density of 𝐵 is not affected by the “measurement”of 𝐴.


Our first conclusion is that the position of 𝐵 does notdepend on themeasurement of𝐴; only the spins are involved.We conclude from (31) that the spins of 𝐴 and 𝐵 remainopposite throughout the experiment. These are the twoproperties used in the causal interpretation.

5.2. Second Step EPR-B: Spin Measurement of 𝐵. The secondstep is a continuation of the first and corresponds to theEPR-B experiment broken down into two steps. On a pair ofparticles 𝐴 and 𝐵 in a singlet state, first we made a Stern andGerlachmeasurement on the𝐴 atom between 𝑡

0and 𝑡

0+Δ𝑡+

𝑡

𝐷; secondly, we make a Stern and Gerlach measurement on

the 𝐵 atomwith an electromagnet E𝐵forming an angle 𝛿with

E𝐴during 𝑡

0+ Δ𝑡 + 𝑡

𝐷and 𝑡

0+ 2(Δ𝑡 + 𝑡

𝐷).

At the exit of magnetic field E𝐴, at time 𝑡

0+ Δ𝑡 + 𝑡

𝐷,

the wave function is given by (31). Immediately after themeasurement of𝐴, still at time 𝑡

0+Δ𝑡+𝑡

𝐷, the wave function

of 𝐵 depends on the measurement ± of 𝐴:

Ψ

𝐵/±𝐴(r

𝐵, 𝑡

0+ Δ𝑡 + 𝑡

1) = 𝑓 (r

𝐵)

∓

𝐵⟩ . (34)

Then, the measurement of 𝐵 at time 𝑡0+ 2(Δ𝑡 + 𝑡

𝐷) yields,

in this two-step version of the EPR-B experiment, the sameresults for spatial quantization and correlations of spins as inthe EPR-B experiment.

5.3. Causal Interpretation of the EPR-B Experiment. We as-sume, at the creation of the two entangled particles 𝐴 and𝐵, that each of the two particles 𝐴 and 𝐵 has an initialwave function with opposite spins: Ψ𝐴

0(r

𝐴, 𝜃

𝐴

0, 𝜑

𝐴

0) = 𝑓(r

𝐴)

(cos(𝜃𝐴0/2)|+

𝐴⟩ + sin(𝜃𝐴

0/2)𝑒

𝑖𝜑𝐴

0|−

𝐴⟩) and Ψ𝐵

0(r

𝐵, 𝜃

𝐵

0, 𝜑

𝐵

0) =

𝑓(r𝐵)(cos(𝜃𝐵

0/2)|+

𝐵⟩ + sin(𝜃𝐵

0/2)𝑒

𝑖𝜑𝐵

0|−

𝐵⟩) with 𝜃𝐵

0= 𝜋 − 𝜃

𝐴

0

and 𝜑𝐵0= 𝜑

𝐴

0− 𝜋. The two particles 𝐴 and 𝐵 are statistically

prepared as in the Stern and Gerlach experiment. Then thePauli principle tells us that the two-body wave function mustbe antisymmetric; after calculation, we find the same singletstate (27):

Ψ

0(r

𝐴, 𝜃

𝐴, 𝜑

𝐴, r

𝐵, 𝜃

𝐵, 𝜑

𝐵)

= −𝑒

𝑖𝜑𝐴

𝑓 (r𝐴) 𝑓 (r

𝐵) × (

+

𝐴⟩

−

𝐵⟩ −

−

𝐴⟩

+

𝐵⟩) .

(35)

Thus, we can consider that the singlet wave function is thewave function of a family of two fermions 𝐴 and 𝐵 withopposite spins: the direction of initial spins𝐴 and 𝐵 exists butis not known. It is a local hidden variable which is thereforenecessary to add in the initial conditions of the model.

This is not the interpretation followed by theBohmschool[44–46, 66, 67] in the interpretation of the singlet wavefunction; they do not assume the existence of wave functionsΨ

𝐴

0(r

𝐴, 𝜃

𝐴

0, 𝜑

𝐴

0) and Ψ𝐵

0(r

𝐵, 𝜃

𝐵

0, 𝜑

𝐵

0) for each particle but only

the singlet state Ψ0(r

𝐴, 𝜃

𝐴, 𝜑

𝐴, r

𝐵, 𝜃

𝐵, 𝜑

𝐵). In consequence,

they suppose a zero spin for each particle at the initialtime and a spin module of each particle varied during theexperiment from 0 to ℏ/2 [46, page 418].

Here, we assume that at the initial time we know the spinof each particle (given by each initial wave function) and theinitial position of each particle.

Step 1 (spin measurement of 𝐴). In (31) particle 𝐴 can beconsidered independent of 𝐵. We can therefore give it thewave function:

Ψ

𝐴(r

𝐴, 𝑡

0+ Δ𝑡 + 𝑡)

= cos𝜃

𝐴

0

2

𝑓

+(r

𝐴, 𝑡)

+

𝐴⟩ + sin

𝜃

𝐴

0

2

𝑒

𝑖𝜑𝐴

0𝑓

−(r

𝐴, 𝑡)

−

𝐴⟩

(36)

which is thewave function of a free particle in a Stern-Gerlachapparatus and whose initial spin is given by (𝜃𝐴

0, 𝜑

𝐴

0). For

an initial polarization (𝜃𝐴0, 𝜑

𝐴

0) and an initial position (𝑧𝐴

0),

we obtain, in the de Broglie-Bohm interpretation [44] of theStern and Gerlach experiment, an evolution of the position(𝑧

𝐴(𝑡)) and of the spin orientation of 𝐴 (𝜃𝐴(𝑧

𝐴(𝑡), 𝑡)) [48].

The case of particle 𝐵 is different. 𝐵 follows a rectilineartrajectory with 𝑦

𝐵(𝑡) = V

𝑦𝑡, 𝑧

𝐵(𝑡) = 𝑧

𝐵

0, and 𝑥

𝐵(𝑡) = 𝑥

𝐵

0. By

contrast, the orientation of its spinmoveswith the orientationof the spin of 𝐴: 𝜃𝐵(𝑡) = 𝜋 − 𝜃𝐴(𝑧

𝐴(𝑡), 𝑡) and 𝜑𝐵(𝑡) =

𝜑(𝑧

𝐴(𝑡), 𝑡) − 𝜋. We can associate the following wave function

with the particle 𝐵:

Ψ

𝐵(r

𝐵, 𝑡

0+ Δ𝑡 + 𝑡)

= 𝑓 (r𝐵) (cos 𝜃

𝐵(𝑡)

2

+

𝐵⟩ + sin 𝜃

𝐵(𝑡)

2

𝑒

𝑖𝜑𝐵(𝑡)

−

𝐵⟩) .

(37)

This wave function is specific, because it depends upon initialconditions of 𝐴 (position and spin). The orientation of spinof the particle 𝐵 is driven by the particle𝐴 through the singletwave function.Thus, the singlet wave function is the nonlocalvariable.

Step 2 (spin measurement of 𝐵). At the time 𝑡0+ Δ𝑡 + 𝑡

𝐷,

immediately after the measurement of𝐴, 𝜃𝐵(𝑡0+Δ𝑡+𝑡

𝐷) = 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 (𝑂𝑥𝑦𝑧)corresponds to the frame (𝑂𝑥𝑦𝑧) after a rotation of an angle𝛿 around the 𝑦-axis. 𝜃𝐵 corresponds to the 𝐵-spin angle withthe 𝑧-axis and 𝜃𝐵 to the 𝐵-spin angle with the 𝑧-axis; then𝜃

𝐵(𝑡

0+ Δ𝑡 + 𝑡

𝐷) = 𝜋 + 𝛿 or 𝛿. In this second step, we

are exactly in the case of a particle in a simple Stern andGerlach experiment (with magnet E

𝐵) 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 𝑡

0+ 2(Δ𝑡 + 𝑡

𝐷), gives,

in this interpretation of the two-step version of the EPR-Bexperiment, the same results as in the EPR-B experiment.

5.4. Physical Explanation of Nonlocal Influences. From thewave function of two entangled particles, we find spins,trajectories, and also a wave function for each of the twoparticles. In this interpretation, the quantum particle hasa local position like a classical particle, but it has also anonlocal behavior through the wave function. So, it is thewave function that creates the nonclassical properties. Wecan keep a view of a local realist world for the particle, butwe should add a nonlocal vision through the wave function.


As we saw in Step 1, the nonlocal influences in the EPR-Bexperiment only concern the spin orientation not the motionof the particles themselves. Indeed only spins are entangledin the wave function (27) not positions and motions like inthe initial EPR experiment. This is a key point in the searchfor a physical explanation of nonlocal influences.

The simplest explanation of this nonlocal influence is toreintroduce the concept of ether (or the preferred frame) but anew format given by Lorentz-Poincaré and byEinstein in 1920[69]: “Recapitulating, we may say that according to the generaltheory of relativity space is endowed with physical qualities;in this sense, therefore, there exists an ether. According to thegeneral theory of relativity space without ether is unthinkable;[sic] for in such space there not only would be no propagationof light, but also no possibility of existence for standards ofspace and time (measuring-rods and clocks), nor therefore anyspace-time intervals in the physical sense. But this ether maynot be thought of as endowed with the quality characteristic ofponderable media, as consisting of parts which may be trackedthrough time. The idea of motion may not be applied to it.”

Taking into account the new experiments, especiallyAspect’s experiments, Popper [70, page XVIII] defends asimilar view in 1982:

“I feel not quite convinced that the experiments are correctlyinterpreted; but if they are, we just have to accept action at adistance. I think (with J.P. Vigier) that this would of course bevery important, but I do not for a moment think that it wouldshake, or even touch, realism. Newton and Lorentz were realistsand accepted action at a distance; and Aspect’s experimentswould be the first crucial experiment between Lorentz’s andEinstein’s interpretation of the Lorentz transformations.”

Finally, in the de Broglie-Bohm interpretation, the EPR-Bexperiments of nonlocality have therefore a great importancenot to eliminate realism and determinism but as Popper saidto rehabilitate the existence of a certain type of ether, likeLorentz’s ether and like Einstein’s ether in 1920.

6. Conclusion

In the three experiments presented in this paper, the variablethat is measured in fine is the position of the particle givenby this impact on a screen. In the double-slit, the set of thesepositions gives the interferences; in the Stern-Gerlach and theEPR-B experiments, it is the position of the particle impactthat defines the spin value.

It is this position that the de Broglie-Bohm interpretationadds to the wave function to define a complete state of thequantumparticle.ThedeBroglie-Bohm interpretation is thenbased only on the initial conditions Ψ0(x) and X(0) and theevolution equations (1) and (12). If we add as initial conditionthe “quantum equilibrium hypothesis” (14), we have seen thatwe can deduce, for these three examples, the three postulatesof measurement. These three postulates are not necessary ifwe solve the time-dependent Schrödinger equation (double-slit experiment) or the Pauli equation with spatial extension(Stern-Gerlach and EPR experiments). However, these sim-ulations enable us to better understand those experiments:in the double-slit experiment, the interference phenomenon

appears only some centimeters after the slits and shows thecontinuity with classical mechanics; in the Stern-Gerlachexperiment, the spin-up/down measurement appears alsoafter a given time, called decoherence time; in the EPR-B experiment, only the spin of 𝐵 is affected by the spinmeasurement of 𝐴 not its density. Moreover, the de Broglie-Bohm trajectories propose a clear explanation of the spinmeasurement in quantum mechanics.

However, we have seen two very different cases in themeasurement process. In the first case (double-slit exper-iment), 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 interactionwith the measuring apparatus (the magnetic field) and thequantum particle. The result of the measurement dependson the position of the particle in the wave function. Themeasurement duration is then the time necessary for thestabilisation of the result.

This heterodox interpretation clearly explains experi-ments with a set of quantum particles that are statisticallyprepared. These particles verify the “quantum equilibriumhypothesis” and the de Broglie-Bohm interpretation estab-lishes continuity with classical mechanics. However, thereis no reason that the de Broglie-Bohm interpretation canbe extended to quantum particles that are not statisticallyprepared. This situation occurs when the wave packet cor-responds to a quasiclassical coherent state, introduced in1926 by Schrödinger [71]. The field quantum theory and thesecond quantification are built on these coherent states [72].It is also the case, for the hydrogen atom, of localized wavepackets whose motions are on the classical trajectory (anold dream of Schrödinger’s). Their existence was predictedin 1994 by Bialynicki-Birula et al. [73–75] and discoveredrecently by Maeda and Gallagher [76] on Rydberg atoms.For these nonstatistically prepared quantum particles, wehave shown [9, 10] that the natural interpretation is theSchrödinger interpretation proposed at the Solvay congressin 1927. Everything happens as if the quantum mechanicsinterpretation depended on the preparation of the particles(statistically or not statistically prepared). It is perhaps aresponse to the “theory of the double solution” that Louis deBroglie was seeking since 1927: “I introduced as the “doublesolution theory” the idea that it was necessary to distinguish twodifferent solutions that are both linked to the wave equation,one that I called wave 𝑢, which was a real physical waverepresented by a singularity as it was not normalizable dueto a local anomaly defining the particle, the other one asSchrödinger’s Ψ wave, which is a probability representation asit is normalizable without singularities” [77].

Appendix

Calculating the Spinor Evolution in theStern-Gerlach Experiment

In the magnetic field 𝐵 = (𝐵𝑥, 0, 𝐵

𝑧), the Pauli equation

(18) gives coupled Schrödinger equations for each spinor


component:

𝑖ℏ

𝜕𝜓

±

𝜕𝑡

(𝑥, 𝑧, 𝑡)

= −

ℏ

2

2𝑚

∇

2𝜓

±(𝑥, 𝑧, 𝑡) ± 𝜇

𝐵(𝐵

0− 𝐵

0𝑧)𝜓

±(𝑥, 𝑧, 𝑡)

∓ 𝑖𝜇

𝐵𝐵

0𝑥𝜓

∓(𝑥, 𝑧, 𝑡) .

(A.1)

If one affects the transformation [47]

𝜓

±(𝑥, 𝑧, 𝑡) = exp(±

𝑖𝜇

𝐵𝐵

0𝑡

ℏ

)𝜓

±(𝑥, 𝑧, 𝑡)

(A.2)

(A.1) becomes

𝑖ℏ

𝜕𝜓

±

𝜕𝑡

(𝑥, 𝑧, 𝑡)

= −

ℏ

2

2𝑚

∇

2𝜓

±(𝑥, 𝑧, 𝑡) ∓ 𝜇

𝐵𝐵

0𝑧𝜓

±(𝑥, 𝑧, 𝑡)

∓ 𝑖𝜇

𝐵𝐵

0𝑥𝜓

∓(𝑥, 𝑧, 𝑡) exp(±𝑖

2𝜇

𝐵𝐵

0𝑡

ℏ

) .

(A.3)

The coupling term oscillates rapidly with the Larmor fre-quency 𝜔

𝐿= 2𝜇

𝐵𝐵

0/ℏ = 1, 4 × 10

11 s−1. Since |𝐵0| ≫ |𝐵

0𝑧|

and |𝐵0| ≫ |𝐵

0𝑥|, the period of oscillation is short compared

to the motion of the wave function. Averaging over a periodthat is long compared to the oscillation period, the couplingterm vanishes, which entails [47]

𝑖ℏ

𝜕𝜓

±

𝜕𝑡

(𝑥, 𝑧, 𝑡) = −

ℏ

2

2𝑚

∇

2𝜓

±(𝑥, 𝑧, 𝑡) ∓ 𝜇

𝐵𝐵

0𝑧𝜓

±(𝑥, 𝑧, 𝑡) .

(A.4)

Since the variable 𝑥 is not involved in this equation and𝜓

0

±(𝑥, 𝑧) does not depend on 𝑥, 𝜓

±(𝑥, 𝑧, 𝑡) does not depend

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 =2 × 10

5 s.We obtain

𝜓

+(𝑧, 𝑡) = 𝜓

𝐾(𝑧, 𝑡) cos

𝜃

0

2

𝑒

𝑖(𝜑0/2), 𝐾 = −𝜇

𝐵𝐵

0,

𝜓

−(𝑧, 𝑡) = 𝜓

𝐾(𝑧, 𝑡) 𝑖 sin

𝜃

0

2

𝑒

−𝑖(𝜑0/2), 𝐾 = +𝜇

𝐵𝐵

0

(A.5)

𝜎

2

𝑡= 𝜎

2

0+ (ℏ𝑡/2𝑚𝜎

0)

2and

𝜓

𝐾(𝑧, 𝑡) = (2𝜋𝜎

2

𝑡)

−1/4

𝑒

−(𝑧+𝐾𝑡2/2𝑚)2

/4𝜎2

𝑡

× exp 𝑖ℏ

[

[

−

ℏ

2

tan−1 ( ℏ𝑡2𝑚𝜎

2

0

) − 𝐾𝑡𝑧 −

𝐾

2𝑡

3

6𝑚

+

(𝑧 + 𝐾𝑡

2/2𝑚)

2

ℏ

2𝑡

2

8𝑚𝜎

2

0𝜎

2

𝑡

]

]

,

(A.6)

where (A.6) is a classical result [11].

The experimental conditions give ℏΔ𝑡/2𝑚𝜎0

= 4 ×

10

−11 m ≪ 𝜎0= 10

−4 m. We deduce the approximations𝜎

𝑡≃ 𝜎

0and

𝜓

𝐾(𝑧, 𝑡)

≃ (2𝜋𝜎

2

0)

−1/4

𝑒

−(𝑧+𝐾𝑡2/2𝑚)2

/4𝜎2

0 exp 𝑖ℏ

[−𝐾𝑡𝑧 −

𝐾

2𝑡

3

6𝑚

] .

(A.7)

At the end of the magnetic field, at time Δ𝑡, the spinor isequal to

Ψ (𝑧, Δ𝑡) = (

𝜓

+(𝑧, Δ𝑡)

𝜓

−(𝑧, Δ𝑡)

) (A.8)

with

𝜓

+(𝑧, Δ𝑡) = (2𝜋𝜎

2

0)

−1/4

𝑒

−((𝑧−𝑧Δ)2/4𝜎2

0)+(𝑖/ℏ)𝑚𝑢𝑧 cos

𝜃

0

2

𝑒

𝑖𝜑+,

𝜓

−(𝑧, Δ𝑡) = (2𝜋𝜎

2

0)

−1/4

𝑒

−((𝑧+𝑧Δ)2/4𝜎2

0)−(𝑖/ℏ)𝑚𝑢𝑧

𝑖 sin𝜃

0

2

𝑒

𝑖𝜑−,

𝑧

Δ=

𝜇

𝐵𝐵

0(Δ𝑡)

2

2𝑚

, 𝑢 =

𝜇

0𝐵

0(Δ𝑡)

𝑚

,

𝜑

+=

𝜑

0

2

−

𝜇

𝐵𝐵

0Δ𝑡

ℏ

−

𝐾

2(Δ𝑡)

3

6𝑚ℏ

;

𝜑

−= −

𝜑

0

2

+

𝜇

0𝐵

0Δ𝑡

ℏ

−

𝐾

2(Δ𝑡)

3

6𝑚ℏ

.

(A.9)

We remark that the passage through the magnetic fieldgives the equivalent of a velocity +𝑢 in the direction 0𝑧to the function 𝜓

+and a velocity −𝑢 to the function 𝜓

−.

Then we have a free particle with the initial wave function(A.8). The Pauli equation resolution again yields 𝜓

±(𝑥, 𝑧, 𝑡) =

𝜓

𝑥(𝑥, 𝑡)𝜓

±(𝑧, 𝑡) and with the experimental conditions we

have 𝜓𝑥(𝑥, 𝑡) ≃ (2𝜋𝜎

2

0)

−1/4𝑒

−𝑥2/4𝜎2

0 and

𝜓

+(𝑧, 𝑡 + Δ𝑡)

≃ (2𝜋𝜎

2

0)

−1/4

cos𝜃

0

2

× exp−(𝑧−𝑧Δ−𝑢𝑡)2/4𝜎2

0+(𝑖/ℏ)(𝑚𝑢𝑧−(1/2)𝑚𝑢

2𝑡+ℏ𝜑+)

,

𝜓

−(𝑧, 𝑡 + Δ𝑡)

≃ (2𝜋𝜎

2

0)

−1/4

𝑖 sin𝜃

0

2

× exp−(𝑧+𝑧Δ+𝑢𝑡)2/4𝜎2

0+(𝑖/ℏ)(−𝑚𝑢𝑧−(1/2)𝑚𝑢

2𝑡+ℏ𝜑−)

.

(A.10)


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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