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Measurement in the de Broglie-Bohm interpretation: Double-slit, Stern-Gerlach and EPR-B Michel Gondran, Alexandre Gondran

To cite this version:

Michel Gondran, Alexandre Gondran. Measurement in the de Broglie-Bohm interpretation: Double- slit, Stern-Gerlach and EPR-B. Research International, Hindawi, 2014, 2014. ￿hal-00862895v3￿

HAL Id: hal-00862895 https://hal.archives-ouvertes.fr/hal-00862895v3 Submitted on 24 Jan 2014

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Measurement in the de Broglie-Bohm interpretation: Double-slit, Stern-Gerlach and EPR-B

Michel Gondran University Paris Dauphine, Lamsade, 75 016 Paris, France∗

Alexandre Gondran École Nationale de l’Aviation Civile, 31000 Toulouse, France†

We propose a pedagogical presentation of measurement in the de Broglie-Bohm interpretation. In this heterodox interpretation, the position of a particle exists and is piloted by the phase of the function. We show how this position explains 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 . , we present a numerical simulation of the double-slit experiment performed by Jönsson in 1961 with . It demonstrates the continuity between and : evolution of the probability density at various distances and convergence of the quantum trajectories to the classical trajectories when h tends to 0. Third, we present an analytic expression of the in the Stern-Gerlach experiment. This explicit solution requires the calculation of a Pauli with a spatial extension. This solution enables to demonstrate the decoherence of the wave function and the three postulates of quantum measurement: quantization, the Born interpretation and wave function reduction. The spinor spatial extension also enables the introduction of the de Broglie-Bohm trajectories, which gives a very simple explanation of the particles’ impact and of the measurement process. Finally, we study the EPR-B experiment, the Bohm version of the Einstein-Podolsky-Rosen ex- periment. Its theoretical resolution in space and time shows that a causal interpretation exists where each has a position and a . This interpretation avoids the flaw of the previous causal interpretation. We recall that a physical explanation of non-local influences is possible.

I. INTRODUCTION question "how reality really is."4 And in 1987, Bell wonders about his teachers’ silence "I saw the impossible done".1 This is how John Bell concerning the Broglie-Bohm pilot-wave: describes his inexpressible surprise in 1952 upon the pub- "But why then had Born not told me of this ’pilot lication of an article by David Bohm2. The impossibility wave’? If only to point out what was wrong with it? Why came from a theorem by outlined in did von Neumann not consider it? More extraordinarily, 1932 in his book The Mathematical Foundations of Quan- why did people go on producing "impossibility" proofs af- tum Mechanics,3 which seemed to show the impossibil- ter 1952, and as recently as 1978? While even Pauli, ity of adding "hidden variables" to quantum mechanics. Rosenfeld, and Heisenberg could produce no more devas- This impossibility, with its physical interpretation, be- tating criticism of Bohm’s version than to brand it as came almost a postulate of quantum mechanics, based "metaphysical" and "ideological"? Why is the pilot-wave on von Neumann’s indisputable authority as a mathe- picture ignored in text books? Should it not be taught, matician. As Bernard d’Espagnat notes in 1979: not as the only way, but as an antidote to the prevailing "At the university, Bell had, like all of us, received complacency? To show that vagueness, subjectivity and indeterminism are not forced on us by experimental facts, from his teachers a message which, later still, Feynman 5 would brilliantly state as follows: "No one can explain but through a deliberate theoretical choice?" more than we have explained here [...]. We don’t have More than thirty years after John Bell’s questions, the the slightest idea of a more fundamental mechanism from interpretation of the de Broglie-Bohm pilot wave is still which the former results (the interference fringes) could ignored by both the international community and the follow". If indeed we are to believe Feynman (and Banesh textbooks. Hoffman, and many others, who expressed the same idea What is this pilot wave theory? For de Broglie, a quan- in many books, both popular and scholarly), Bohm’s the- tum particle is not only defined by its wave function. He ory cannot exist. Yet it does exist, and is even older than assumes that the quantum particle also has a position Bohm’s papers themselves. In fact, the basic idea behind which is piloted by the wave function.6 However only the it was formulated in 1927 by in a model probability density of this position is known. The posi- he called "pilot wave theory". Since this theory provides tion exists in itself (ontologically) but is unknown to the explanations of what, in "high circles", is declared in- observer. It only becomes known during the measure- explicable, it is worth consideration, even by ment. [...] who do not think it gives us the final answer to the The goal of the present paper is to present the Broglie- 2

Bohm pilot-wave through the study of the three most ∂ρ~(x, t)  ∇S~(x, t) + div ρ~(x, t) = 0 (4) important experiments of quantum measurement: the ∂t m double-slit experiment which is the crucial experiment of the wave-particle duality, the Stern and Gerlach experi- with initial conditions: ment with the measurement of the spin, and the EPR-B ~ ~ ~ ~ experiment with the problem of non-locality. ρ (x, 0) = ρ0 (x) and S (x, 0) = S0 (x). (5) The paper is organized as follows. In section II, we demonstrate the conditions in which the de Broglie-Bohm correspond to a set of non- interpretation can be assumed to be valid through con- interacting quantum particles all prepared in the same ~ ~ tinuity with classical mechanics. This involves the de way (same ρ0 (x) and S0 (x)). Broglie-Bohm interpretation for a set of particles pre- A quantum particle is said to be statistically prepared ~ pared in the same way. In section III, we present a if its initial probability density ρ0 (x) and its initial action ~ numerical simulation of the double-slit experiment per- S0 (x) converge, when ~ → 0, to non-singular functions formed by Jönsson in 1961 with electrons15. The method ρ0(x) and S0(x). It is the case of an electronic or C60 of Feynman path integrals allows to calculate the time- beam in the double slit experiment or an atomic beam in dependent wave function. The evolution of the proba- the Stern and Gerlach experiment. We will seen that it bility density just outside the slits leads one to consider is also the case of a beam of entrangled particles in the EPR-B experiment. Then, we have the following theo- the dualism of the wave-particle interpretation. And the 8,9 de Broglie-Bohm trajectories provide an explanation for rem: the impact positions of the particles. Finally, we show For statistically prepared quantum particles, the proba- the continuity between classical and quantum trajecto- bility density ρ~(x, t) and the action S~(x, t), solutions to ries with the convergence of these trajectories to classi- the Madelung equations (3)(4)(5), converge, when ~ → 0, cal trajectories when h tends to 0. In section IV, we to the classical density ρ(x, t) and the classical action present an analytic expression of the wave function in S(x, t), solutions to the statistical Hamilton-Jacobi equa- the Stern-Gerlach experiment. This explicit solution re- tions: quires the calculation of a Pauli spinor with a spatial ∂S (x, t) 1 + (∇S(x, t))2 + V (x, t) = 0 (6) extension. This solution enables to demonstrate the de- ∂t 2m of the wave function and the three postulates S(x, 0) = S0(x) (7) of quantum measurement: quantization, Born interpre- ∂ρ (x, t)  ∇S (x, t) tation and wave function reduction. The spinor spatial + div ρ (x, t) = 0 (8) extension also enables the introduction of the de Broglie- ∂t m Bohm trajectories which gives a very simple explanation ρ(x, 0) = ρ0(x). (9) of the particles’ impact and of the measurement pro- cess. In section V, we study the EPR-B experiment, We give some indications on the demonstration of this the Bohm version of the Einstein-Podolsky-Rosen exper- theorem when the wave function Ψ(x, t) is written as a iment. Its theoretical resolution in space and time shows function of the initial wave function Ψ0(x) by the Feyn- 10 that a causal interpretation exists where each atom has man paths integral : a position and a spin. Finally, we recall that a physical Z  i  explanation of non-local influences is possible. Ψ(x, t) = F (t, ~) exp Scl(x, t; x0 Ψ0(x0)dx0 ~ (10) II. THE DE BROGLIE-BOHM where F (t, ~) is an independent function of x INTERPRETATION and of x0. For a statistically prepared quantum particle, the wave function is written Ψ(x, t) = R p i F (t, ) ρ~(x0) exp( (S~(x0) + Scl(x, t; x0))dx0. The The de Broglie-Bohm interpretation is based on the ~ 0 ~ 0 following demonstration. Let us consider a wave function theorem of the stationary phase shows that, if ~ tends i Ψ(x, t) solution to the Schrödinger equation: towards 0, we have Ψ(x, t) ∼ exp( minx0 (S0(x0) + S (x, t; x )), that is to say that the~ quantum action 2 cl 0 ∂Ψ(x, t) h i = − ~ 4Ψ(x, t) + V (x)Ψ(x, t) (1) S (x, t) converges to the function ~ ∂t 2m

Ψ(x, 0) = Ψ0(x). (2) S(x, t) = minx0 (S0(x0) + Scl(x, t; x0)) (11)

With the variable change Ψ(x, t) = which is the solution to the Hamilton-Jacobi equation (6) p S~(x,t) ρ~(x, t) exp(i ), the Schrödinger equation ~ with the initial condition (7). Moreover, as the quantum can be decomposed into Madelung equations7(1926): density ρh(x, t) satisfies the continuity equation (4), we deduce, since Sh(x, t) tends towards S(x, t), that ρh(x, t) ~ 2 p ∂S (x, t) 1 2 ~ 4 ρ~(x, t) converges to the classical density ρ(x, t), which satisfies + (∇S~(x, t)) +V (x)− p = 0 ∂t 2m 2m ρ~(x, t) the continuity equation (8). We obtain both announced (3) convergences. 3

These statistical Hamilton-Jacobi equations (6,7,8,9) Then, the (P (x, t) ≡ P [X(t) = correspond to a set of classical particles prepared in the x]) for a set of particles moving with the velocity field h same way (same ρ0(x) and S0(x)). These classical parti- h ∇S (x,t) v (x, t) = m satisfies the property of the "equiv- cles are trajectories obtained in Eulerian representation 2 12 ∇S(x,t) ariance" of the |Ψ(x, t)| probability distribution: with the velocity field v (x,t) = m , but the density and the action are not sufficient to describe it completely. P [X(t) = x] ≡ P (x, t) = |Ψ(x, t)|2 = ρh(x, t). (15) To know its position at time t, it is necessary to know its initial position. Because the Madelung equations con- It is the at time t. verge to the statistical Hamilton-Jacobi equations, it is Then, the de Broglie-Bohm interpretation is based logical to do the same in quantum mechanics. We con- on a continuity between classical and quantum mechan- clude that a statistically prepared quantum particle is ics where the quantum particles are statistical prepared not completely described by its wave function. It is nec- with an initial probability densitiy satisfies the "quantum essary to add this initial position and an equation to equilibrium hypothesis" (14). It is the case of the three define the evolution of this position in the time. It is studied experiments. the de Brogglie-Bohm interpretation where the position We will revisit these three measurement experiments is called the "hidden variable". through mathematical calculations and numerical simu- The two first postulates of quantum mechanics, de- lations. For each one, we present the statistical interpre- scribing the and its evolution,11 must be tation that is common to the Copenhagen interpretation completed in this heterodox interpretation. At initial and the de Broglie-Bohm pilot wave, then the trajecto- time t=0, the state of the particle is given by the initial ries specific to the de Broglie-Bohm interpretation. We wave function Ψ0(x) (a wave packet) and its initial posi- show that the precise definition of the initial conditions, tion X(0); it is the new first postulate. The new second i.e. the preparation of the particles, plays a fundamental postulate gives the evolution on the wave function and on methodological role. the position. For a single spin-less particle in a potential V (x), the evolution of the wave function is given by the usual Schrödinger equation (1,2) and the evolution of the III. DOUBLE-SLIT EXPERIMENT WITH particle position is given by ELECTRONS

dX(t) Jh(x, t) ∇Sh(x, t) Young’s double-slit experiment16 has long been the = h |x=X(t) = |x=X(t) (12) dt ρ (x, t) m crucial experiment for the interpretation of the wave- where particle duality. There have been realized with mas- sive objects, such as electrons15,17, neutrons18, cold neu- 19 20 Jh(x, t) = ~ (Ψ∗(x, t)∇Ψ(x, t) − Ψ(x, t)∇Ψ∗(x, t)) trons , , and more recently, with coherent en- 2mi sembles of ultra-cold atoms21, and even with mesoscopic (13) 22 single quantum objects such as C60 and C70 . For Feyn- is the usual quantum current. man, this experiment addresses "the basic element of In the case of a particle with spin, as in the Stern and the mysterious behavior [of electrons] in its most strange Gerlach experiment, the Schrödinger equation must be form. [It is] a phenomenon which is impossible, abso- replaced by the Pauli or Dirac equations. lutely impossible to explain in any classical way and which The third quantum mechanics postulate which de- has in it the heart of quantum mechanics. In reality, it scribes the measurement (the ) can be contains the only mystery."23 The de Broglie-Bohm in- conserved. But the three postulates of measurement are terpretation and the numerical simulation help us here not necessary: the postulate of quantization, the Born to revisit the double-slit experiment with electrons per- postulate of probabilistic interpretation of the wave func- formed by Jönsson in 1961 and to provide an answer tion and the postulate of the reduction of the wave func- to Feynman’mystery. These simulations24 follow those tion. We see that these postulates of measurement can conducted in 1979 by Philippidis, Dewdney and Hiley25 be explained on each example as we will shown in the which are today a classics. However, these simulations25 following. have some limitations because they did not consider re- We replace these three postulates by a single one, the alistic slits. The slits, which can be clearly represented 12–14 "quantum equilibrium hypothesis", that describes by a function G(y) with G(y) = 1 for −β ≤ y ≤ β and the interaction between the initial wave function Ψ0(x) G(y) = 0 for |y| > β, if they are 2β in width, were mod- and the initial particle position X(0): For a set of iden- 2 2 eled by a G(y) = e−y /2β . Interfer- tically prepared particles having t = 0 wave function ence was found, but the calculation could not account for Ψ (x), it is assumed that the initial particle positions 0 diffraction at the edge of the slits. Consequently, these X(0) are distributed according to: simulations could not be used to defend the de Broglie- 2 h Bohm interpretation. P [X(0) = x] ≡ P (x, 0) = |Ψ0(x)| = ρ0 (x). (14) Figure 1 shows a diagram of the double slit experiment It is the Born rule at the initial time. by Jönsson. An gun emits electrons one by one 4

FIG. 2. General view of the evolution of the probability den- sity from the source to the screen in the Jönsson experiment. A lighter shade means that the density is higher i.e. the prob- FIG. 1. Diagram of the Jönnson’s double slit experiment ability of presence is high. performed with electrons. in the horizontal plane, through a hole of a few microme- ters, at a velocity v = 1.8 × 108m/s along the horizontal x-axis. After traveling for d1 = 35cm, they encounter a plate pierced with two horizontal slits A and B, each 0.2µm wide and spaced 1µm from each other. A screen located at d2 = 35cm after the slits collects these elec- trons. 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. The slits are very long along the z-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. FIG. 3. Close-up of the evolution of the probability density Electrons emerging from an electron gun are represented in the first 3cm after the slits in the Jönsson experiment. by the same initial wave function Ψ0(y).

electron has passed. In this experiment, the measure- A. Probability density ment is performed by the detection screen, which only reveals the existence or absence of the fringes. Figure 2 gives a general view of the evolution of The calculation method enables us to compare the evo- the probability density from the source to the detec- lution of the cross-section of the probability density at tion screen (a lighter shade means that the density is various distances after the slits (0.35mm, 3.5mm, 3.5cm higher i.e. the probability of presence is high). The and 35cm) where the two slits A and B are open simul- 2 calculations were made using the method of Feynman taneously (interference: |ΨA + ΨB| ) with the evolution path integrals24. The wave function after the slits of the sum of the probability densities where the slits A −11 −11 (t1 = d1/v ' 2.10 s < t < t1 + d2/v ' 4.10 s) and B are open independently (the sum of two diffrac- 2 2 is deduced from the values of the wave function tions: |ΨA| + |ΨB| ). Figure 4 shows that the difference at slits A and B: Ψ(y, t) = ΨA(y, t) + ΨB(y, t) between these two phenomena appears only a few cen- R with ΨA(y, t) = K(y, t, ya, t1)Ψ(ya, t1)dya, timeters after the slits. R A ΨB(y, t) = B K(y, t, yb, t1)Ψ(yb, t1)dyb and 2 1/2 im(y−yα) /2 (t−t1) K(y, t, yα, t1) = (m/2i~(t − t1)) e ~ . Figure 3 shows a close-up of the evolution of the prob- B. Impacts on screen and de Broglie-Bohm ability density just after the slits. We note that inter- trajectories ference will only occur a few centimeters after the slits. Thus, if the detection screen is 1cm from the slits, there is The interference fringes are observed after a certain no interference and one can determine by which slit each period of time when the impacts of the electrons on the 5

(a) : 0,35mm (b) : 3,5mm 1

0.8

0.6

0.4

0.2 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 µm µm

m 0 µ

−0.2 (c) : 3,5cm (d) : 35cm −0.4

−0.6

−0.8

−1 0 1 2 3 4 5 6 7 8 9 10 −2 −1 0 1 2 −10 −5 0 5 10 cm µm µm

2 FIG. 6. Close-up on the 100 trajectories of the electrons just FIG. 4. Comparison of the probability density |ΨA + ΨB | 2 2 after the slits. (full line) and |ΨA| +|ΨB | (dotted line) at various distances after the slits: (a) 0.35mm, (b): 3.5mm, (c): 3.5cm and (d): 35cm. The different trajectories explain both the impact of electrons on the detection screen and the interference detection screen become sufficiently numerous. Classical fringes. This is the simplest and most natural interpre- quantum theory only explains the impact of individual tation to explain the impact positions: "The position of particles statistically. an impact is simply the position of the particle at the However, in the de Broglie-Bohm interpretation: a par- time of impact." This was the view defended by Einstein ticle has an initial position and follows a path whose ve- at the Solvay Congress of 1927. The position is the only locity at each instant is given by equation (12). On the measured variable of the experiment. of this assumption we conduct a simulation experi- In the de Broglie-Bohm interpretation, the impacts on ment by drawing random initial positions of the electrons the screen are the real positions of the electron as in clas- in the initial wave packet ("quantum equilibrium hypoth- sical mechanics and the three postulates of the measure- esis"). ment of quantum mechanics can be trivialy explained: the position is an eigenvalue of the be- cause the position variable is identical to its operator (XΨ 4 = xΨ), the Born postulate is satisfied with the "equiv- ariance" property, and the reduction of the wave packet 3 is not necessary to explain the impacts. 2 Through numerical simulations, we will demonstrate how, when the h tends to 0, the quan- 1 tum trajectories converge to the classical trajectories. In

m 0 reality a constant is not able to tend to 0 by definition. µ The convergence to classical trajectories is obtained if −1 the term ht/m → 0; so h → 0 is equivalent to m → +∞ −2 (i.e. the mass of the particle grows) or t → 0 (i.e. the distance slits-screem d2 → 0). Figure 7 shows the 100 −3 trajectories that start at the same 100 initial points when

−4 Planck’s constant is divided respectively by 10, 100, 1000 and 10000 (equivalent to multiplying the mass by 10, 100, −35 −30 −20 −10 0 10 20 30 35 cm 1000 and 10000). We obtain quantum trajectories con- verging to the classical trajectories, when h tends to 0. FIG. 5. 100 electron trajectories for the Jönsson experiment. The study of the slits clearly shows that, in the de Broglie-Bohm interpretation, there is no physical separa- Figure 5 shows, after its initial starting position, 100 tion between quantum mechanics and classical mechan- possible quantum trajectories of an electron passing ics. All particles have quantum properties, but specif- through one of the two slits: We have not represented ically quantum behavior only appears in certain exper- the paths of the electron when it is stopped by the first imental conditions: here when the ratio ht/m is suffi- screen. Figure 6 shows a close-up of these trajectories ciently large. Interferences only appear gradually and just after they leave their slits. the quantum particle behaves at any time as both a wave 6

1 1 a spatial extension as the equation: 0.8 0.8 0.6 0.6 2 ϕ0 0.4 0.4 z  θ0 −i  1 − 2 cos e 2 0.2 0.2 0 2 − 2 4σ 2 Ψ (z) = (2πσ0) e 0 θ ϕ0 . (16) m 0 h/10 m 0 h/100 0 i 2 µ µ sin e −0.2 −0.2 2 −0.4 −0.4 −0.6 −0.6 11,23,28,29 −0.8 −0.8 Quantum mechanics textbooks do not take into −1 −1 −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 account the spatial extension of the spinor (16) and sim- cm cm ply use the simplified spinor without spatial extension: 1 1 0.8 0.8 ϕ0  θ0 −i 2  0.6 0.6 0 cos 2 e 0.4 0.4 Ψ = ϕ0 . (17) θ0 i 2 0.2 0.2 sin 2 e

m 0 h/1000 m 0 h/10000 µ µ −0.2 −0.2 −0.4 −0.4 However, as we shall see, the different evolutions of the −0.6 −0.6 −0.8 −0.8 spatial extension between the two spinor components will −1 −1 −30 −20 −10 0 10 20 30 −30 −20 −10 0 10 20 30 have a key role in the explanation of the measurement cm cm process. This spatial extension enables us, in follow- ing the precursory works of Takabayasi30, Bohm31,32, FIG. 7. Convergence of 100 electron trajectories when h is 33 34 divided by 10, 100, 1 000 and 10 000. Dewdney et al. and Holland , to revisit the Stern and Gerlach experiment, to explain the decoherence and to demonstrate the three postulates of the measure: quanti- and a particle. zation, Born statistical interpretation and wave function reduction. Silver atoms contained in the oven E (Figure 8) are IV. THE STERN-GERLACH EXPERIMENT heated to a high temperature and escape through a nar- row opening. A second aperture, T, selects those atoms In 1922, by studying the deflection of a beam of silver whose velocity, v0, is parallel to the y-axis. The atomic atoms in a strongly inhomogeneous magnetic field (cf. beam crosses the gap of the electromagnet A1 before con- Figure 8) and Walter Gerlach26 obtained an densing on the detector, P1 . Before crossing the elec- experimental result that contradicts the common sense tromagnet, the of each silver atom prediction: the beam, instead of expanding, splits into is oriented randomly (isotropically). In the beam, we two separate beams giving two spots of equal intensity represent each atom by its wave function; one can as- N + and N − on a detector, at equal distances from the sume that at the entrance to the electromagnet, A1, and axis of the original beam. at the initial time t = 0, each atom can be approxi- matively described by a Gaussian spinor in z given by (16) corresponding to a pure state. The variable y will −4 be treated classically with y = vt. σ0 = 10 m corre- sponds to the size of the slot T along the z-axis. The approximation by a Gaussian initial spinor will allow ex- plicit 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 ex- periment, we take the numerical values used in the Co- henTannoudji textbook11. For the silver atom, we have −25 FIG. 8. Schematic configuration of the Stern-Gerlach experi- m = 1.8 × 10 kg, v0 = 500 m/s (corresponding to the ◦ ment. temperature of T = 1000 K). In equation (16) and in figure 9, θ0 and ϕ0 are the polar angles characterizing Historically, this is the experiment which helped es- the initial orientation of the magnetic moment, θ0 corre- tablish spin quantization. Theoretically, it is the seminal sponds to the angle with the z-axis. The experiment is experiment posing the problem of measurement in quan- a statistical mixture of pure states where the θ0 and the tum mechanics. Today it is the theory of decoherence ϕ0 are randomly chosen: θ0 is drawn in a uniform way with the diagonalization of the that is from [0, π] and that ϕ0 is drawn in a uniform way from put forward to explain the first part of the measurement [0, 2π]. 27  ψ  process . However, although these authors consider the The evolution of the spinor Ψ = + in a magnetic Stern-Gerlach experiment as fundamental, they do not ψ− propose a calculation of the spin decoherence time. field B is then given by the : We present an analytical solution to this decoherence  ∂ψ+  2     time and the diagonalization of the density matrix. This ∂t ~ ψ+ ψ+ i~ ∂ψ = − ∆ + µBBσ (18) − ψ− ψ− solution requires the calculation of the Pauli spinor with ∂t 2m 7

e where µB = ~ is the and where σ = 2me (σx, σy, σz) corresponds to the three . The particle first enters an electromagnetic field B directed 0 0 along the z-axis, Bx = B0x, By = 0, Bz = B0 − B0z, 0 ∂B 3 with B0 = 5 Tesla, B0 = ∂z = 10 T esla/m over a length ∆l = 1 cm. On exiting the magnetic field, the particle is free until it reaches the detector P1 placed at a D = 20 cm distance. FIG. 9. Orientation of the magnetic moment. θ0 and ϕ0 The particle stays within the magnetic field for a time are the polar angles characterizing the spin vector in the de ∆l −5 ∆t = v = 2×10 s. During this time [0, ∆t], the spinor Broglie-Bohm interpretation. is:37 (see Appendix A)

0  µB B 2 2 2 02  (z− 0 t ) µ B ϕ 2m µ B0 tz− B 0 t3+µ B t+ ~ 0 θ 1 − 2 B 0 6m B 0 2 0 2 − 2 4σ0 i  cos 2 (2πσ0) e e ~  Ψ(z, t) ' 0 . (19)  µB B 2 2 2 02  (z+ 0 t ) µ B ϕ  2m −µ B0 tz− B 0 t3−µ B t− ~ 0  θ 1 − 2 B 0 6m B 0 2 0 2 − 2 4σ0 i i sin 2 (2πσ0) e e ~

After the magnetic field, at time t + ∆t (t ≥ 0) in the 0 cm 6 cm 16 cm 21 cm free space, the spinor becomes:32–34,37,38 (see Appendix A)

 (z−z −ut)2  − ∆ muz+ ϕ θ 2 − 1 2 i ~ + −0.6 0 0.6 −0.6 0 0.6 −0.6 0 0.6 −0.6 0 0.6 0 2 4σ0 cos 2 (2πσ0 ) e e ~ mm mm mm mm Ψ(z, t + ∆t) '  2  (z+z∆+ut)  1 − −muz+~ϕ−  θ0 2 − 4σ2 i sin (2πσ ) 2 e 0 e ~ 2 0 FIG. 10. Evolution of the probability density of a pure state (20) with θ = π/3. where 0 µ B0 (∆t)2 µ B0 (∆t) z = B 0 = 10−5m, u = B 0 = 1m/s. ∆ 2m m This decoherence time is usually the time required to (21) diagonalize the marginal density matrix of spin variables Equation (20) takes into account the spatial extension of associated with a pure state39: the spinor and we note that the two spinor components  R 2 R ∗  have very different z values. All interpretations are based S |ψ+(z, t)| dz ψ+(z, t)ψ−(z, t)dz ρ (t) = R ∗ R 2 on this equation. ψ−(z, t)ψ+(z, t)dz |ψ−(z, t)| dz (24) For t ≥ tD, the product ψ+(z, t + ∆t)ψ−(z, t + ∆t) is A. The decoherence time null and the density matrix is diagonal: the probability density of the initial pure state (20) is diagonal: We deduce from (20) the probability density of a pure state in the free space after the electromagnet:  2 θ0  S 2 −1 cos 2 0 ρ (t + ∆t) = (2πσ ) 2 (25) 2 0 θ0 (z−z∆−ut) 0 sin 2 2 − 1 2 θ0 − 2 ρ (z, t + ∆t) ' (2πσ ) 2 cos e 2σ0 θ0 0 2

2 (z+z∆+ut) ! B. Proof of the postulates of quantum 2 θ0 − 2 + sin e 2σ0 (22) measurement 2

Figure 10 shows the probability density of a pure state We then obtain atoms with a spin oriented only along (with θ0 = π/3) as a function of z at several values of t the z-axis (positively or negatively). Let us consider the (the plots are labeled y = vt). The beam separation does spinor Ψ(z, t + ∆t) given by equation (20). Experimen- not appear at the end of the magnetic field (1 cm), but tally, we do not measure the spin directly, but the ze po- 16 cm further along. It is the moment of the decoherence. sition of the particle impact on P1 (Figure 11). The decoherence time, where the two spots N + and N − + If ze ∈ N , the term ψ− of (20) is nu- are separated, is then given by the equation: merically equal to zero and the spinor Ψ is pro- portional to 1, one of the eigenvectors of the 3σ0 − z∆ −4 0 tD ' = 3 × 10 s. (23) ~ u spin operator Sz = 2 σz: Ψ(˜z, t + ∆t) ' 8

+ N 0.8 1

0 0.6 z (mm) − −1 N

0.4 −5 −4 −3 −2 −1 0 1 2 3 4 5 x (mm)

0.2

FIG. 11. 1000 silver atom impacts on the detector P1. z (mm)

0

2 (˜z1−z∆−ut) 1 − muz˜1+~ϕ+ θ 2 1 −0.2 2 − 4 0 4σ0 i  (2πσ0) cos 2 e e ~ 0 . Then, we ~ ~ have SzΨ = 2 σzΨ = + 2 Ψ. − −0.4 If z ∈ N , the term ψ+ of (20) is numerically equal to 0 5 10 15 20 e 0 y (cm) zero and the spinor Ψ is proportional to 1 , the other eigenvector of the spin operator Sz: Ψ(˜z, t + ∆t) ' 2 FIG. 12. Ten silver atom trajectories with initial spin orien- (˜z2+z∆+ut) π 1 − −muz˜2+~ϕ− 2 − θ0 4σ2 i 0 tation (θ0 = 3 ) and initial position z0; arrows represent the (2πσ ) 4 sin e 0 e ~ . Then, we 0 2 1 spin orientation θ(z, t) along the trajectories. ~ ~ have SzΨ = 2 σzΨ = − 2 Ψ. Therefore, the measure- ment of the spin corresponds to an eigenvalue of the spin operator. It is a proof of the postulate of quantization.

2 θ0 Equation (25) gives the probability cos 2 (resp. 2 θ0 ~ sin 2 ) to measure the particle in the spin state + 2 ~ (resp. − 2 ); this proves the Born probabilistic postulate. The final orientation, obtained after the decoherence By drilling a hole in the detector P1 to the location of time tD, depends on the initial particle position z0 in the the spot N + (figure 8), we select all the atoms that are in spinor with a spatial extension and on the initial angle 1 π θ0 the spin state |+i = 0 . The new spinor of these atoms θ0 of the spin with the z-axis. We obtain + 2 if z0 > z π is obtained by making the component Ψ− of the spinor Ψ θ0 and − 2 if z0 < z with 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 de- tector P . The future trajectory of the silver atom after 1  θ  θ0 −1 2 0 crossing the detector P1 will be guided by this new (nor- z = σ0F sin (26) malized) spinor. The wave function reduction is therefore 2 not linked to the electromagnet, but to the detector P1 causing an irreversible elimination of the spinor compo- nent Ψ−. 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 equation (26). C. Impacts and quantizations explained by de Broglie-Bohm trajectories In the de Broglie-Bohm interpretation with a realistic interpretation of the spin, the "measured" value is not Finally, it remains to provide an explanation of the in- independent of the context of the measure and is contex- dividual impacts of silver atoms. The spatial extension tual. It conforms to the Kochen and Specker theorem:41 of the spinor (16) allows to take into account the parti- Realism and non-contextuality are inconsistent with cer- cle’s initial position z0 and to introduce the Broglie-Bohm tain quantum mechanics predictions. trajectories2,6,33,34,40 which is the natural assumption to explain the individual impacts. Now let us consider a mixture of pure states where Figure 12 presents, for a silver atom with the initial the initial orientation (θ0, ϕ0) from the spinor has been π spinor orientation (θ0 = 3 , ϕ0 = 0), a plot in the (Oyz) randomly chosen. These are the conditions of the ini- plane of a set of 10 trajectories whose initial position z0 tial Stern and Gerlach experiment. Figure 13 represents has been randomly chosen from a Gaussian distribution a simulation of 10 quantum trajectories of silver atoms with σ0. The spin orientations θ(z, t) from which the initial positions z0 are also randomly cho- are represented by arrows. sen. 9

0.6 methodological conditions are necessary. The first condi- tion is the same as in the Stern-Gerlach experiment: the solution to the entangled state is obtained by resolving 0.4 the Pauli equation from an initial singlet wave function with a spatial extension as: 0.2 1 Ψ0(rA, rB) = √ f(rA)f(rB)(|+Ai|−Bi − |−Ai|+Bi), 0 2

z (mm) (27) and not from a simplified wave function without spatial −0.2 extension:

−0.4 1 Ψ0(rA, rB) = √ (|+Ai|−Bi − |−Ai|+Bi). (28) 2

−0.6 0 5 10 15 20 y (cm) f function and |±i vectors are presented later. The resolution in space of the Pauli equation is essen- FIG. 13. Ten silver atom trajectories where the initial ori- tial: it enables the spin measurement by spatial quantiza- entation (θ0, ϕ0) has been randomly chosen; arrows represent tion and explains the determinism and the disentangling the spin orientation θ(z, t) along the trajectories. 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 Finally, the de Broglie-Bohm trajectories propose a methodological condition. A first causal interpretation of clear interpretation of the spin measurement in quantum EPR-B experiment was proposed in 1987 by Dewdney, mechanics. There is interaction with the measuring ap- Holland and Kyprianidis35 using these two conditions. paratus as is generally stated; and there is indeed a min- However, this interpretation had a flaw34 (p. 418): the imum time required to measure. However this measure- spin module of each particle depends directly on the sin- ment and this time do not have the signification that is glet wave function, and thus the spin module of each usually applied to them. The result of the Stern-Gerlach ~ particle varied during the experiment from 0 to 2 . We experiment is not the measure of the spin projection present a de Broglie-Bohm interpretation that avoid this along the z-axis, but the orientation of the spin either flaw.36 in the direction of the magnetic field gradient, or in the opposite direction. It depends on the position of the par- ticle in the wave function. We have therefore a simple explanation for the non-compatibility of spin measure- ments along different axes. The measurement duration is then the time necessary for the particle to point its spin in the final direction.

V. 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 exper- FIG. 14. Schematic configuration of the EPR-B experiment. iment42 proposed by Bohm43, the Bell theorem44 and 5,44,45 the BCHSH inequalities have been at the heart of Figure 14 presents the Einstein-Podolsky-Rosen-Bohm the debate on hidden variables and non-locality. Many experiment. A source S created in O pairs of identical experiments since Bell’s paper have demonstrated viola- atoms A and B, but with opposite spins. The atoms A tions of these inequalities and have vindicated quantum and B split following the y-axis in opposite directions, theory46. Now, EPR pairs of massive atoms are also con- 47 and head towards two identical Stern-Gerlach apparatus sidered . The usual conclusion of these experiments is E and E . The electromagnet E "measures" the spin to reject the non-local realism for two reasons: the im- A B A of A along the z-axis and the electromagnet EB "mea- possibility of decomposing a pair of entangled atoms into sures" the spin of B along the z0-axis, which is obtained two states, one for each atom, and the impossibility of after a of an angle δ around the y-axis. The interaction faster than the speed of . initial wave function of the entangled state is the singlet Here, we show that there exists a de Broglie-Bohm x2+z2 1 − 2 − 4σ2 interpretation which answers these two questions pos- state (27) where r = (x, z), f(r) = (2πσ0) 2 e 0 , itively. To demonstrate this non-local realism, two |±Ai and |±Bi are the eigenvectors of the operators σzA 10

and σzB : σzA |±Ai = ±|±Ai, σzB |±Bi = ±|±Bi. We • the density of A is the same, whether particle A is treat the dependence with y classically: speed −vy for A entangled with B or not, and vy for B. The wave function Ψ(rA, rB, t) of the two A and B, electrically neutral and with • the density of B is not affected by the "measure- ment" of A. magnetic moments µ0, subject to magnetic fields EA and EB, admits on the basis of |±Ai and |±Bi four com- a,b Our first conclusion is: the position of B does not de- ponents Ψ (rA, rB, t) and satisfies the two-body Pauli 34 pend on the measurement of A, only the spins are in- equation (p. 417): volved. We conclude from equation (31) that the spins ∂Ψa,b  2 2  of A and B remain opposite throughout the experiment. i = − ~ ∆ − ~ ∆ Ψa,b ~ ∂t 2m A 2m B These are the two properties used in the causal interpre- tation. EA a c,b EB b a,d + µBj (σj)c Ψ + µBj (σj)dΨ (29) with the initial conditions: B. Second step EPR-B: Spin measurement of B a,b a,b Ψ (rA, rB, 0) = Ψ0 (rA, rB) (30) The second step is a continuation of the first and corre- where Ψa,b(r , r ) corresponds to the (27). 0 A B sponds to the EPR-B experiment broken down into two To obtain an explicit solution of the EPR-B experi- steps. On a pair of particles A and B in a singlet state, ment, we take the numerical values of the Stern-Gerlach first we made a Stern and Gerlach measurement on the experiment. A atom between t and t + 4t + t , secondly, we make One of the difficulties of the interpretation of the EPR- 0 0 D a Stern and Gerlach measurement on the B atom with B experiment is the existence of two simultaneous mea- an electromagnet E forming an angle δ with E during surements. By doing these measurements one after the B A t + 4t + t and t + 2(4t + t ). other, the interpretation of the experiment will be facili- 0 D 0 D At the exit of magnetic field E , at time t + 4t + t , tated. That is the purpose of the two-step version of the A 0 D the wave function is given by (31). Immediately after the experiment EPR-B studied below. measurement of A, still at time t0 + 4t + tD, the wave function of B depends on the measurement ± of A: A. First step EPR-B: Spin measurement of A ΨB/±A(rB, t0 + 4t + t1) = f(rB)|∓Bi. (34)

In the first step we make a Stern and Gerlach "mea- Then, the measurement of B at time t0 + 2(4t + tD) surement" for atom A, on a pair of particles A and B in yields, in this two-step version of the EPR-B experiment, a singlet state. This is the experiment first proposed in the same results for spatial quantization and correlations 1987 by Dewdney, Holland and Kyprianidis.35 of spins as in the EPR-B experiment. Consider that at time t0 the particle A arrives at the entrance of electromagnet EA. After this exit of the mag- netic field EA, at time t0 +4t+t, the wave function (27) C. Causal interpretation of the EPR-B experiment becomes36:

1 + Ψ(rA, rB , t0 + 4t + t) = √ f(rB ) × ( f (rA, t)|+Ai|−B i We assume, at the creation of the two entangled 2 particles A and B, that each of the two particles A − − f (rA, t)|−Ai|+B i) and B has an initial wave function with opposite spins:  θA θA A  (31) A A A 0 0 iϕ0 Ψ0 (rA, θ0 , ϕ0 ) = f(rA) cos 2 |+Ai + sin 2 e |−Ai B B B with and Ψ0 (rB, θ0 , ϕ0 ) = B B ±muz ±  θ θ B  ± i( +ϕ (t)) 0 0 iϕ0 B A f (r, t) ' f(x, z ∓ z4 ∓ ut)e ~ (32) f(rB) cos 2 |+Bi + sin 2 e |−Bi with θ0 = π − θ0 B A where z and u are given by equation (21). and ϕ0 = ϕ0 − π. The two particles A and B are ∆ statistically prepared as in the Stern and Gerlach The atomic density ρ(zA, zB, t0 + ∆t + t) is found by ∗ experiment. Then the Pauli principle tells us that the integrating Ψ (rA, rB, t0 + 4t + t)Ψ(rA, rB, t0 + 4t + t) on x and x : two-body wave function must be antisymmetric; after A B calculation we find the same singlet state (27): 2 ! (zB ) 1 − 2 − 2 A ρ(z , z , t + ∆t + t) = (2πσ ) 2 e 2σ0 (33) A A B B iϕ A B 0 0 Ψ0(rA, θ , ϕ , rB, θ , ϕ ) = − e f(rA)f(rB) (35) × (|+ i|− i − |− i|+ i) . 2 2 !! A B A B (zA−z∆−ut) (zA+z∆+ut) 2 − 1 1 − 2 − 2 × (2πσ ) 2 e 2σ0 + e 2σ0 . 0 2 Thus, we can consider that the singlet wave function is the wave function of a family of two A and B We deduce that the beam of particle A is divided into with opposite spins: the direction of initial spin A and two, while the beam of particle B stays undivided: B exists, but is not known. It is a local hidden variable 11 which is therefore necessary to add in the initial condi- t0 + 2(4t + tD)), gives, in this interpretation of the two- tions of the model. step version of the EPR-B experiment, the same results This is not the interpretation followed by the Bohm as in the EPR-B experiment. school32–35 in the interpretation of the singlet wave func- tion; they do not assume the existance of wave func- A A A B B B tions Ψ0 (rA, θ0 , ϕ0 ) and Ψ0 (rB, θ0 , ϕ0 ) for each parti- A A B B cle, but only the singlet state Ψ0(rA, θ , ϕ , rB, θ , ϕ ). D. Physical explanation of non-local influences In consequence, they suppose a zero spin for each parti- cle at the initial time and a spin module of each particle ~ 34 From the wave function of two entangled particles, we varied during the experiment from 0 to 2 (p. 418). Here, we assume that at the initial time we know the find spins, trajectories and also a wave function for each spin of each particle (given by each initial wave function) of the two particles. In this interpretation, the quantum and the initial position of each particle. particle has a local position like a classical particle, but Step 1: spin measurement of A it has also a non-local behavior through the wave func- In the equation (31) particle A can be considered inde- tion. So, it is the wave function that creates the non pendent of B. We can therefore give it the wave function classical properties. We can keep a view of a local realist world for the particle, but we should add a non-local vi- θA sion through the wave function. As we saw in step 1, the ΨA(r , t + 4t + t) =cos 0 f +(r , t)|+ i A 0 2 A A non-local influences in the EPR-B experiment only con- A the spin orientation, not the of the particles θ0 iϕA − + sin e 0 f (rA, t)|−Ai(36) themselves. Indeed only spins are entangled in the wave 2 function (27) not positions and like in the initial which is the wave function of a in a Stern EPR experiment. This is a key point in the search for a Gerlach apparatus and whose initial spin is given by physical explanation of non-local influences. A A A A (θ0 , ϕ0 ). For an initial (θ0 , ϕ0 ) and an ini- The simplest explanation of this non-local influence A tial position (z0 ), we obtain, in the de Broglie-Bohm is to reintroduce the concept of ether (or the preferred 32 interpretation of the Stern and Gerlach experiment, an frame), but a new format given by Lorentz-Poincaré and evolution of the position (zA(t)) and of the spin orienta- by Einstein in 192048:"Recapitulating, we may say that A 38 tion of A (θ (zA(t), t)) . according to the general space is en- The case of particle B is different. B follows a rectilin- dowed with physical qualities; in this sense, therefore, B ear trajectory with yB(t) = vyt, zB(t) = z0 and xB(t) = there exists an ether. According to the general theory of B x0 . By contrast, the orientation of its spin moves with relativity space without ether is unthinkable; for in such B A the orientation of the spin of A: θ (t) = π − θ (zA(t), t) space there not only would be no propagation of light, B and ϕ (t) = ϕ(zA(t), t) − π. We can then associate the but also no possibility of existence for standards of space wave function: and time (measuring-rods and clocks), nor therefore any  θB(t) space-time intervals in the physical sense. But this ether ΨB(r , t + 4t + t) = f(r ) cos |+ i (37) may not be thought of as endowed with the quality charac- B 0 B 2 B teristic of ponderable media, as consisting of parts which B  θ (t) B may be tracked through time. The idea of motion may + sin eiϕ (t)|− i . 2 B not be applied to it." Taking into account the new experiments, especially This wave function is specific, because it depends upon Aspect’s experiments, Popper49 (p. XVIII) defends a initial conditions of A (position and spin). The orienta- similar view in 1982: tion of spin of the particle B is driven by the particle A through the singlet wave function. Thus, the singlet wave "I feel not quite convinced that the experiments are cor- function is the non-local variable. rectly interpreted; but if they are, we just have to accept Step 2: Spin measurement of B action at a distance. I think (with J.P. Vigier) that this would of course be very important, but I do not for a mo- At the time t0 + ∆t + tD, immediately after the mea- B ment think that it would shake, or even touch, realism. surement of A, θ (t0 + ∆t + tD) = π or 0 in accordance A and Lorentz were realists and accepted action at with the value of θ (zA(t), t) and the wave function of B is given by (34). The frame (Ox0yz0) corresponds to a distance; and Aspect’s experiments would be the first the frame (Oxyz) after a rotation of an angle δ around crucial experiment between Lorentz’s and Einstein’s in- the y-axis. θB corresponds to the B-spin angle with the terpretation of the Lorentz transformations." z-axis, and θ0B to the B-spin angle with the z0-axis, then Finally, in the de Broglie-Bohm interpretation, the 0B θ (t0 + ∆t + tD) = π + δ or δ. In this second step, we EPR-B experiments of non-locality have therefore a great are exactly in the case of a particle in a simple Stern and importance, not to eliminate realism and determinism, Gerlach experiment (with magnet EB) with a specific but as Popper said, to rehabilitate the existence of a cer- initial polarization equal to π + δ or δ and not random tain type of ether, like Lorentz’s ether and like Einstein’s like in step 1. Then, the measurement of B, at time ether in 1920. 12

VI. CONCLUSION lagher55 on Rydberg atoms. For these non statistically prepared quantum particles, we have shown8,9 that the In the three experiments presented in this article, the natural interpetation is the Schrödinger interpretation variable that is measured in fine is the position of the proposed at the Solvay congress in 1927. Everythings particle given by this impact on a screen. In the double- happens as if the quantum mechanics interpretation de- slit, the set of these positions gives the interferences; in pended on the preparation of the particles (statistically the Stern-Gerlach and the EPR-B experiments, it is the or not statistically prepared). It is perhaps a response to position of the particle impact that defines the spin value. the "theory of the double solution" that Louis de Broglie It is this position that the de Broglie-Bohm interpre- was seeking since 1927: "I introduced as the "double so- tation adds to the wave function to define a complete lution theory" the idea that it was necessary to distin- state of the quantum particle. The de Broglie-Bohm in- guish two different solutions that are both linked to the terpretation is then based only on the initial conditions , one that I called wave u, which was a Ψ0(x) and X(0) and the evolution equations (1) and (12). real physical wave represented by a singularity as it was If we add as initial condition the "quantum equilibrium not normalizable due to a local anomaly defining the par- hypothesis" (14), we have seen that we can deduce, for ticle, the other one as Schrödinger’s Ψ wave, which is a probability representation as it is normalizable without these three examples, the three postulates of measure- 56 ment. These three postulates are not necessary if we singularities." solve the time-dependent Schrödinger equation (double- slit experiment) or the Pauli equation with spatial exten- Appendix A: Calculating the spinor evolution in the sion (Stern-Gerlach and EPR experiments). However, Stern-Gerlach experiment these simulations enable us to better understand those experiments: In the double-slit experiment, the interfer- ence phenomena appears only some centimeters after the In the magnetic field B = (Bx, 0,Bz), the Pauli equa- slits and shows the continuity with classical mechanics; tion (18) gives coupled Schrödinger equations for each in the Stern-Gerlach experiment, the spin up/down mea- spinor component surement appears also after a given time, called decoher- 2 ∂ψ± ~ 2 ence time; in the EPR-B experiment, only the spin of i (x, z, t) = − ∇ ψ±(x, z, t) ~ ∂t 2m B is affected by the spin measurement of A, not its den- ± µ (B − B0 z)ψ (x, z, t) sity. Moreover, the de Broglie-Bohm trajectories propose B 0 0 ± 0 a clear explanation of the spin measurement in quantum ∓ iµBB0xψ∓(x, z, t). (A1) mechanics. If one effects the transformation37 However, we have seen two very different cases in the   measurement process. In the first case (double slit exper- iµBB0t ψ±(x, z, t) = exp ± ψ±(x, z, t) 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 interac- equation (A1) becomes tion with the measuring apparatus (the magnetic field) 2 and the quantum particle. The result of the measure- ∂ψ± ~ 2 i~ (x, z, t) = − ∇ ψ±(x, z, t) ment depends on the position of the particle in the wave ∂t 2m 0 function. The measurement duration is then the time ∓ µBB0zψ±(x, z, t) necessary for the stabilisation of the result.   0 2µBB0t This heterodox interpretation clearly explains exper- ∓ iµBB0xψ∓(x, z, t) exp ±i iments with a set of quantum particles that are statis- ~ tically prepared. These particles verify the "quantum The term oscillates rapidly with the Larmor fre- 2µB B0 11 −1 0 equilibrium hypothesis" and the de Broglie-Bohm inter- quency ωL = = 1, 4×10 s . Since |B0|  |B0z| 0 ~ pretation establishes continuity with classical mechanics. and |B0|  |B0x|, the period of oscillation is short com- However, there is no reason that the de Broglie-Bohm in- pared to the motion of the wave function. Averaging over terpretation can be extended to quantum particles that a period that is long compared to the oscillation period, are not statistically prepared. This situation occurs when the coupling term vanishes, which entails37 the wave packet corresponds to a quasi-classical coher- ent state, introduced in 1926 by Schrödinger50. The ∂ψ 2 i ± (x, z, t) = − ~ ∇2ψ (x, z, t) ∓ µ B0 zψ (x, z, t). field quantum theory and the second quantification are ~ ∂t 2m ± B 0 ± built on these coherent states51. It is also the case, (A2) for the atom, of localized wave packets whose Since the variable x is not involved in this equation 0 motion are on the classical trajectory (an old dream of and ψ±(x, z) does not depend on x, ψ±(x, z, t) does not Schrödinger’s). Their existence was predicted in 1994 by depend on x: ψ±(x, z, t) ≡ ψ±(z, t). Then we can explic- Bialynicki-Birula, Kalinski, Eberly, Buchleitner and De- itly compute the preceding equations for all t in [0, ∆t] 52–54 ∆l 5 lande , and discovered recently by Maeda and Gal- with ∆t = v = 2 × 10 s. 13

We obtain:

ϕ0 0 θ0 i 2 ψ+(z, t) = ψK (z, t) cos 2 e and K = −µBB0 ϕ0 0 θ0 −i 2 ψ−(z, t) = ψK (z, t)i sin 2 e and K = +µBB0 0 2 0 µBB0(∆t) µ0B0(∆t) z∆ = , u = and  2 2m m 2 2 ~t σt = σ0 + and 2 3 2mσ0 ϕ0 µBB0∆t K (∆t) ϕ+= − − ; 2 2 6m (z+ Kt )2 ~ ~ 1 − 2m    2 3 2 − 4σ2 i ~ −1 ~t ϕ µ B ∆t K (∆t) ψ (z, t) = (2πσ ) 4 e t exp − tan 0 0 0 K t 2 2mσ2 ϕ−= − + − . ~ 0 2 ~ 6m~ 2 3 Kt2 2 2 2 # K t (z + 2m ) ~ t −Ktz − + 2 2 . (A3) 6m 8mσ0 σt

where (A3) is a classical result.10 We remark that the passage through the magnetic field The experimental conditions give ~∆t = 4 × gives the equivalent of a velocity +u in the direction 0z 2mσ0 −11 −4 to the function ψ+ and a velocity −u to the function 10 m  σ0 = 10 m. We deduce the approxima- ψ−. Then we have a free particle with the initial wave tions σt ' σ0 and function (A5). The Pauli equation resolution again yields 2 (z+ Kt )2 ψ (x, z, t) = ψ (x, t)ψ (z, t) and with the experimental 2m  2 3  ± x ± 1 − 2 2 − 4σ2 i K t x 4 0 1 − ψK (z, t) ' (2πσ0) e exp −Ktz − . 2 − 4σ2 ~ 6m conditions we have ψx(x, t) ' (2πσ0) 4 e 0 and (A4) At the end of the magnetic field, at time ∆t, the spinor equals to 2 − 1 θ0 ψ (z, t + ∆t) ' (2πσ ) 4 cos + 0 2   2 ψ+(z, ∆t) (z−z∆−ut) i 1 2 Ψ(z, ∆t) = (A5) − 2 + (muz− 2 mu t+~ϕ+) ψ−(z, ∆t) × exp 4σ0 ~

2 − 1 θ0 with ψ−(z, t + ∆t) ' (2πσ ) 4 i sin 0 2 2 2 (z−z∆) i (z+z∆+ut) i 1 2 1 − + muz θ − + (−muz− mu t+ ϕ−) 2 − 4σ2 ~ 0 iϕ+ 2 2 ~ ψ (z, ∆t) = (2πσ ) 4 e 0 cos e × exp 4σ0 ~ + 0 2 2 (z+z∆) i 2 − 1 − 2 − muz θ0 iϕ ψ (z, ∆t) = (2πσ ) 4 e 4σ0 ~ i sin e − − 0 2

∗ [email protected] discerned particles in classical mechanics and convergence † [email protected] of quantum mechanics to classical mechanics", Annales de 1 J. S. Bell, "On the impossible pilot wave", 1982, reprint la Fondation Louis de Broglie, vol. 36, 117-135 (2011). in Bell 1987, in Speakable and Unspeakable in Quantum 9 M. Gondran, and A. Gondran, "The two limits of the Mechanics (Cambridge University Press, 1987). Schrödinger equation in the semi-classical approximation : 2 D. Bohm, "A suggested interpretation of the quantum the- discerned and non-discerned particles in classical mechan- ory in terms of hidden variables, I and II", Phys. Rev. 85, ics", Foundations of Probability and Physics-6, AIP Conf. 166-193 (1952). Proc. 1424,111-115 (2012). 3 J. von Neumann, Mathematical Foundations of Quantum 10 R. Feynman and A. Hibbs, Quantum Mechanics and Paths Mechanics (Princeton Univ. Press, 1996). Integrals (McGraw-Hill, Inc., 1965), pp.63. 4 B. d’Espagnat, A la recherche du réel (Gauthiers-Villard, 11 C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Me- Paris, 1979). chanics (Wiley, New York, 1977). 5 J. S. Bell, Speakable and Unspeakable in Quantum Mechan- 12 D. Dürr, S. Golstein, and N. Zanghi, "Quantum equilib- ics (Cambridge University Press, 1987), 2nd edition, 2004. rium and the origin of absolute uncertainty," J. Stat. Phys. 6 L. de Broglie, "La mécanique ondulatoire et la structure 67, 843-907 (1992). atomique de la matière et du rayonnement," J. de Phys. 13 A. S. Sanz and S. Miret-Artès, "Quantum phase analysis 8, 225-241 (1927). An English translation can be found in with quantum trajectories: A step towards the creation of G. Bacciagalluppi and A. Valentini, Quantum Theory of a Bohmian thinking", Am. J. Phys. 80, 525 (2012). the Crossroads (Cambridge University Press, Cambridge, 14 T. Norsen, "The pilot-wave perspective on quantum scat- 2009) tering and tunneling", Am. J. Phys. 81, 258-266 (2013). 7 Madelung, E.: Quantentheorie in hydrodynamischer Form. 15 C. Jönsson, “Elektroneninterferenzen an mehreren kün- Zeit. Phys. 40, 322-6 (1926). stlich hergestellten Feinspalten,” Z. Phy. 161, 454–474 8 M. Gondran and A. Gondran, "Discerned and non- (1961), English translation “Electron diffraction at mul- 14

tiple slits,” Am. J. Phys. 42, 4–11 (1974). in terms of Ensemble in ", Prog. Theor. Phys., 16 T. Young, "On the theory of light and colors,” Philos. 11 4-5, 341 (1954). Trans. RSL 92, 12–48 (1802). 31 D. Bohm, R. Schiller,and J. Tiomno, "A causal interpre- 17 C. J. Davisson and L. H. Germer, “The scattering of elec- tation of the pauli equation", Nuovo Cim. supp. 1, 48-66 trons by a single crystal of nickel,” 119, 558–560 and 67-91(1955). (1927). P. G. Merlin, C. F. Missiroli, and G. Pozzi, “On 32 D. Bohm,and B.J. Hiley, The Undivided Universe (Rout- the statistical aspect of electron interference phenomena,” ledge, London and New York, 1993). Am. J. Phys. 44, 306–307 (1976). A. Tonomura, J. Endo, 33 C. Dewdney, P.R. Holland, and A. Kypianidis, "What hap- T. Matsuda, T. Kawasaki, and H. Ezawa, “Demonstration pens in a spin measurement?", Phys. Lett. A, 119(6), 259- of single-electron buildup of an interference pattern,” Am. 267 (1986). J. Phys. 57, 117–120 (1989). 34 P.R. Holland , The Quantum Theory of Motion (Cam- 18 H. V. Halbon Jr. and P. Preiswerk, “Preuve expérimentale bridge University Press, 1993). de la diffraction des ,” C. R. Acad. Sci. Paris 203, 35 C. Dewdney, P.R. Holland, and A. Kyprianidis, "A causal 73–75 (1936). H. Rauch and A. Werner, Inter- account of non-local Einstein-Podolsky-Rosen spin corre- ferometry: Lessons in Experimental Quantum Mechanics lations," J. Phys. A, Math. Gen. 20, 4717-32 (1987). C. (Oxford Univ. Press, London, 2000). Dewdney, P.R. Holland, A. Kyprianidis, and J.P. Vigier, 19 A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. "Spin and non-locality in quantum mechanics," Nature, Mampe, “Single and double slit diffraction of neutrons,” 336, 536-44 (1988). Rev. Mod. Phys. 60, 1067–1073 (1988). 36 M. Gondran, A. Gondran, "A new causal interpretation of 20 I. Estermann and O. Stern,“Beugung von Moleku- EPR-B experiment," Quantum Theory: Reconsideration of larstrahlen,” Z. Phy. 61, 95–125 (1930). Foundations 6, AIP Conf. Proc. 1508, 370-375 (2012). 21 F. Shimizu, K. Shimizu, and H. Takuma, “Double-slit in- 37 D.E. Platt, "A modern analysis of the Stern-Gerlach ex- terference with ultracold metastable neon atoms,” Phys. periment", Am. J. Phys. 60(4), 306-308 (1992). Rev. A 46, R17–R20 (1992). M. H. Anderson, J. R. Ensher, 38 M. Gondran, A. Gondran, "A complete analysis of the M. R. Mattheus, C. E. Wieman, and E. A. Cornell, “Ob- Stern-Gerlach experiment using Pauli ", quant- servation of Bose-Einstein condensation in a dilute atomic ph/05 1276 (2005). vapor,” Science 269, 198–201 (1995). 39 G.B. Roston, M. Casas, A. Plastino and A.R. Plastino, 22 M. Arndt, O. Nairz, J. Voss-Andreae, C. Keller, G. van Eur.J.Phys. 26 (2005) 657-672. des Zouw, and A. Zeilinger, “Wave-particle duality of 40 A. Challinor, A. Lasenby, S. Gull, and Chris Doran, "A C60 molecules,” Nature 401, 680–682 (1999). O. Nairz, relativistic causal account of a spin measurement", Phys. M. Arndt, and A. Zeilinger, “Experimental challenges in Lett. A 218, 128-138 (1996). fullerene interferometry,” J. Mod. Opt. 47, 2811–2821 41 S.Kochen and E..P. Specker, "The problem of hidden vari- (2000). ables in quantum mechanics", J. Math. Mech. 17, 59-87 23 R. P. Feynman, R. B. Leighton, and M. Sands, The Feyn- (1967). man Lectures on Physics (Addison-Wesley, New York, 42 A. Einstein, B. Podolsky and N. Rosen, "Can quantum 1965). mechanical description of reality be considered complete?," 24 M. Gondran, A. Gondran, "Numerical simulation of the Phys. Rev. 47,777-780 (1935). double-slit interference with ultracold atoms", Am. J. 43 D. Bohm, Quantum Theory (New York, Prentice-Hall, Phys. 73, 6 (2005). 1951). D. Bohm, Y. Aharonov, "Discussion of experimental 25 C. Philippidis, C. Dewdney, and B. J. Hiley, "Quantum proofs for the paradox of Einstein, Rosen and Podolsky," interference and the ", Il Nuovo Cimento Phys. Rev.108, 1070 (1957). 52 B, 15-28 (1979). 44 J.S. Bell, "On the Einstein Podolsky Rosen Paradox," 26 W. Gerlach and O. Stern,"Der Experimentelle Nachweis Physics 1, 195 (1964). des Magnetischen Moments des Silberatoms", Zeit. Phys. 45 J.F. Clauser, M.A. Horne, A. Shimony and R.A. Holt, 8, 110 (1921); Zeit. Phys. 9, 349 (1922). "Proposed experiments to test local hidden-variable the- 27 H. D. Zeh, "On the Interpretation of Measurement ories," Phys. Rev. Lett. 23, 880 (1969). in Quantum Theory," Found. Phys.1, 69-76 (1970). 46 S.J. Freedman, J.F.Clauser, "Experimental test of local Reprinted in Wheeler and Zurek (1983), pp. 342-349. W. hidden-variable theories," Phys. Rev. Lett. 28, 938 (1972). H. Zurek, "Environment-Induced Superselection Rules", A. Aspect, P.Grangier and G. Roger, "Experimental re- Phys.Rev.D26 1862 (1982) ; J.A. Wheeler, and W. H. alization of Einstein-Podolsky-Rosen-Bohm GedankenEx- Zurek, Quantum Theory of Measurement (Princeton Uni- periment: a new violation of Bell’inequalities," Phys. versity Press, 1983); W. H. Zurek, "Decoherence, einselec- Rev. Lett. 49, 91 (1982). A. Aspect, J. Dalibard and G. tion and the quantum origins of the classical", Rev. Mod. Roger, "Experimental tests of Bell’inequalities using vari- Phys. 75 (2003) 715. R. Omnes, "Consistent Interpreta- able analysers," Phys. Rev. Lett. 49, 1804 (1982). W. tion of Quantum Mechanics", Rev. Mod. Phys. 64, 339 Tittel, J.Brendel, H. Zbinden and N. Gisin, "Violation (1992). M. Schlosshauer, Decoherence and the Quantum- of Bell inequalities by more than 10 km apart," to-Classical Transition (Springer-Verlag, 2007). Phys. Rev. Lett. 81, 3563 (1998). G. Weihs, T. Jennewein, 28 J.J. Sakurai, Modern Quantum Mechanics (Addison- C.Simon,H. Weinfurter and A. Zeilinger, " Violation of Wesley, 1985). Bell’inequalities under strict Einstein locality condition," 29 M. Le Bellac, Quantum Physics (Cambridge University Phys. Rev. Lett. 81, 5039 (1998). R.A. Bertlmann and Press, 2006). A. Zeilinger (eds.), Quantum [un]speakables, from Bell to 30 T.Takabayasi, "On the Formulation of Quantum Mechan- (Springer, 2002). M. Genovese, "Re- ics associated with Classical Pictures", Prog. Theor. Phys., search on hidden variables theories: a review of recent 8 2, 143 (1952); "The Formulation of Quantum Mechanics progress," Phys. Repts. 413, 319 (2005). 15

47 A. Beige, W.J. Munro and P.L. Knight, "A Bell’s inequal- C. Cohen-Tanoudji eds., Gordon and Breach, New York ity test with entangled atoms," Phys. Rev. A 62, 052102- (1965). 1-052102-9 (2000). M.A. Rowe, D. Kielpinski, V. Meyer, 52 Bialynicki-Birula, I., Kalinski, M., Eberly, J. H.: Lagrange C.A. Sackett, W.M. Itano, C. Monroe and D.J. Wineland, Equilibrium Points in Celestial Mechanics and Nonspread- "Experimental violation of a Bell’s inequality with efficient ing Wave Packets for Strongly Driven Rydberg Electrons. detection," Nature 409, 791-794 (2001). Phys. Rev. Lett. 73, 1777 (1994). 48 A. Einstein, "Ether and the Theory of Relativity", Einstein 53 Buchleitner, A., Delande, D.: Non-dispersive electronic address delivered on May 5th, 1920, in the University of wave packets in multiphoton processus. Phys. Rev. Lett. Leyden (1920). 75, 1487 (1995). 49 K. Popper, Quantum Theory and the Schism in Physics: 54 Buchleitner, A., Delande, D., Zakrzewski, J.: Non- From the Postscript to The Logic of Scientific Discovery dispersive wave packets in periodically driven quantum sys- (W. Bartley, III, Hutchinson, Londres, 1982). tems. Physics Reports 368, 409-547 (2002). 50 Schrödinger, E.: Der stetige bergang von der Mikro-zur 55 Maeda, H., Gallagher, T.F.: Non dispersing Wave Packets. Makromechanik. Naturwissenschaften 14, 664-666 (1926). Phys. Rev. Lett. 92, 133004-1 (2004). 51 Glauber, R. J.: in: and Electronics, 56 de Broglie, L., Andrade e Silva, J.L.: La Réinterprétation Les Houches Lectures 1964, C. deWitt, A. Blandin and de la mécanique ondulatoire. Gauthier-Villars (1971).