Research Article Measurement in the de Broglie-Bohm...

Research ArticleMeasurement in the de Broglie-Bohm InterpretationDouble-Slit Stern-Gerlach and EPR-B

Michel Gondran1 and Alexandre Gondran2

1 University Paris Dauphine Lamsade 75 016 Paris France2 Ecole Nationale de lrsquoAviation Civile 31000 Toulouse France

Correspondence should be addressed to Alexandre Gondran alexandregondranenacfr

Received 24 February 2014 Accepted 7 July 2014 Published 10 August 2014

Academic Editor Anand Pathak

Copyright copy 2014 M Gondran and A GondranThis 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 interpretationthe 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 Jonsson 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

ldquoI saw the impossible donerdquo [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 ldquohiddenvariablesrdquo to quantum mechanics This impossibility withits physical interpretation became almost a postulate ofquantum mechanics based on von Neumannrsquos indisputableauthority as a mathematician Bernard drsquoEspagnat notes in1979 the following

ldquoAt the university Bell had like all of us received from histeachers a message which later still Feynman would brilliantlystate as follows ldquoNo 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 followrdquo If indeed we are tobelieve Feynman (and Banesh Hoffman andmany others who

expressed the same idea in many books both popular andscholarly) Bohmrsquos theory cannot exist Yet it does exist andis even older than Bohmrsquos papers themselves In fact the basicidea behind it was formulated in 1927 by Louis de Broglie in amodel he called ldquopilot wave theoryrdquo Since this theory providesexplanations of what in ldquohigh circlesrdquo is declared inexplicableit is worth consideration even by physicists [ ] who do notthink it gives us the final answer to the question how realityreally isrdquo [4]

And in 1987 Bell wonders about his teachersrsquo silenceconcerning the Broglie-Bohm pilot wave

ldquoBut why then had Born not told me of this ldquopilot waverdquoIf only to point out what was wrong with it Why didvon Neumann not consider it More extraordinarily whydid people go on producing ldquoimpossibilityrdquo proofs after 1952and as recently as 1978 [sic] While even Pauli Rosenfeldand Heisenberg could produce no more devastating criticismof Bohmrsquos version than to brand it as ldquometaphysicalrdquo andldquoideologicalrdquo 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 pageshttpdxdoiorg1011552014605908

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 theoreticalchoicerdquo [5]

More than thirty years after John Bellrsquos 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 Jonsson in 1961with electrons [7] The method of Feynman path integralsallows us to calculate the time-dependent wave functionThe 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 particlesFinally 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 4we present an analytic expression of the wave function inthe Stern-Gerlach experiment This explicit solution requiresthe calculation of a Pauli spinor with a spatial extensionThis 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 particlesrsquo 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 119905)

solution to the Schrodinger equation

119894ℏ

120597Ψ (x 119905)120597119905

= minus

ℏ

2

2119898

ΔΨ (x 119905) + 119881 (x) Ψ (x 119905) (1)

Ψ (x 0) = Ψ0(x) (2)

With the variable change Ψ(x 119905) = radic120588

ℏ(x 119905) exp (119894(119878ℏ(x 119905)

ℏ)) the Schrodinger equation can be decomposed intoMadelung equations [8] (1926)

120597119878

ℏ(x 119905)120597119905

+

1

2119898

(nabla119878

ℏ(x 119905))

2

+ 119881 (x) minus ℏ

2

2119898

Δradic120588

ℏ(x 119905)

radic120588

ℏ(x 119905)

= 0

(3)

120597120588

ℏ(x 119905)120597119905

+ div(120588ℏ(x 119905) nabla119878

ℏ(x 119905)119898

) = 0 (4)

with initial conditions

120588

ℏ(x 0) = 120588ℏ

0(x) 119878

ℏ(x 0) = 119878ℏ

0(x) (5)

Madelung equations correspond to a set of noninteractingquantum particles all prepared in the same way (same 120588ℏ

0(x)

and 119878ℏ0(x))

A quantum particle is said to be statistically prepared ifits initial probability density 120588ℏ

0(x) and its initial action 119878ℏ

0(x)

converge when ℏ rarr 0 to nonsingular functions 1205880(x) and

119878

0(x) It is the case of an electronic or119862

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 120588ℏ

(x 119905) and the action 119878ℏ(x 119905) solutions tothe Madelung equations (3) (4) and (5) converge when ℏ rarr

0 to the classical density 120588(x 119905) and the classical action 119878(x 119905)solutions to the statistical Hamilton-Jacobi equations

120597119878 (x 119905)120597119905

+

1

2119898

(nabla119878 (x 119905))2 + 119881 (x 119905) = 0 (6)

119878 (x 0) = 1198780(x) (7)

120597120588 (x 119905)120597119905

+ div (120588 (x 119905) nabla119878 (x 119905)119898

) = 0(8)

120588 (x 0) = 1205880(x) (9)

We give some indications on the demonstration of thistheorem when the wave function Ψ(x 119905) is written as afunction of the initial wave function Ψ

0(x) by the Feynman

paths integral [11]

Ψ (x 119905) = int119865 (119905 ℏ) exp ( 119894ℏ

119878

119888119897(x 119905 x

0))Ψ

0(x

0) 119889x

0 (10)

where 119865(119905 ℏ) is an independent function of x and of x0 For a

statistically prepared quantum particle the wave function is

Physics Research International 3

writtenΨ(x 119905) = 119865(119905 ℏ) intradic120588ℏ

0(x

0) exp((119894ℏ)(119878ℏ

0(x

0)+119878

119888119897(x 119905

x0)))119889119909

0 The theorem of the stationary phase shows that if ℏ

tends towards 0 we have Ψ(x 119905) sim exp((119894ℏ)minx0(1198780(x0) +119878

119888119897(x 119905 x

0))) that is to say the quantum action 119878

ℎ(x 119905)

converges to the function

119878 (x 119905) = minx0(119878

0(x

0) + 119878

119888119897(x 119905 x

0)) (11)

which is the solution to the Hamilton-Jacobi equation (6)with the initial condition (7) Moreover as the quantum den-sity 120588ℎ

(x 119905) satisfies the continuity equation (4) we deducesince 119878ℎ(x 119905) tends towards 119878(x 119905) that 120588ℎ

(x 119905) convergesto the classical density 120588(x 119905) 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 120588

0(x) and 119878

0(x)) These classical

particles are trajectories obtained in Eulerian representationwith the velocity field k(x 119905) = nabla119878(x 119905)119898 but the densityand the action are not sufficient to describe it completely Toknow its position at time 119905 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 ldquohidden variablerdquo

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 119905 = 0the 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 potential119881(x) the evolution of thewave functionis given by the usual Schrodinger equations (1) (2) and theevolution of the particle position is given by

119889X (119905)119889119905

=

Jℎ(x 119905)120588

ℎ(x 119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816x=X(119905)

=

nabla119878

ℎ(x 119905)119898

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=X(119905)

(12)

where

Jℎ (x 119905) = ℏ

2119898119894

(Ψ

lowast(x 119905) nablaΨ (x 119905) minus Ψ (x 119905) nablaΨlowast

(x 119905))(13)

is the usual quantum currentIn the case of a particle with spin as in the Stern

and Gerlach experiment the Schrodinger 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 necessarythe 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 theldquoquantum equilibrium hypothesisrdquo [13ndash15] 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 119905 = 0 wave functionΨ

0(x) it is assumed that

the initial particle positionsX(0) are distributed according to

119875 [X (0) = x] equiv 119875 (x 0) = 1003816100381610038161003816

Ψ

0(x)1003816100381610038161003816

2

= 120588

ℎ

0(x) (14)

It is the Born rule at the initial timeThen the probability distribution (119875(x 119905) equiv 119875[X(119905) = x])

for a set of particles moving with the velocity field vℎ(x 119905) =nabla119878

ℎ(x 119905)119898 satisfies the property of the ldquoequivariancerdquo of the

|Ψ(x 119905)|2 probability distribution [13]

119875 [X (119905) = x] equiv 119875 (x 119905) = |Ψ (x 119905)|2 = 120588ℎ(x 119905) (15)

It is the Born rule at time 119905Then 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 ldquoquantum equilibriumhypothesisrdquo (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 interpretationWe 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

Youngrsquos 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 17ndash19] 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 119862

60and 119862

70[26 27] For Feynman

this experiment addresses ldquothe 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 mysteryrdquo[28] The de Broglie-Bohm interpretation and the numericalsimulation help us here to revisit the double-slit experiment


02 120583m02 120583m

08 120583m10120583m

35 cm35 cm

y

z

x

S

Figure 1 Diagram of the Jonnsonrsquos double-slit experiment per-formed with electrons

with electrons performedby Jonsson in 1961 and to provide ananswer to Feynmanrsquos 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 function119866(119910)with119866(119910) = 1 for minus120573 le 119910 le 120573 and 119866(119910) = 0 for |119910| gt 120573 if theyare 2120573 in width weremodeled by a Gaussian function119866(119910) =119890

minus119910221205732

Interference was found but the calculation could notaccount for diffraction at the edge of the slits Consequentlythese simulations could not be used to defend the de Broglie-Bohm interpretation

Figure 1 shows a diagram of the double-slit experimentby Jonsson An electron gun emits electrons one by one inthe horizontal plane through a hole of a few micrometersat a velocity V = 18 times 10

8 ms along the horizontal 119909-axisAfter traveling for 119889

1= 35 cm they encounter a plate pierced

with two horizontal slits 119860 and 119861 each 02 120583m wide andspaced 1120583m from each other A screen located at 119889

2= 35 cm

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

The slits are very long along the 119911-axis so there is no effectof diffraction along this axis In the simulation we thereforeonly consider the wave function along the 119910-axis the variable119909 will be treated classically with 119909 = V119905 Electrons emergingfrom an electron gun are represented by the same initial wavefunction Ψ

0(119910)

31 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 ie the probability of presence is high) The calcula-tions were made using themethod of Feynman path integrals[29] The wave function after the slits (119905

1= 119889

1V ≃ 210

minus11 slt 119905 lt 119905

1+ 119889

2V ≃ 410minus11 s) is deduced from the values of the

wave function at slits 119860 and 119861 Ψ(119910 119905) = Ψ

119860(119910 119905) + Ψ

119861(119910 119905)

70 cm

10120583m

Figure 2 General view of the evolution of the probability densityfrom the source to the screen in the Jonsson experiment A lightershade means that the density is higher that is the probability ofpresence is high

35 cm

3120583m

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

with Ψ

119860(119910 119905) = int

119860119870(119910 119905 119910

119886 119905

1)Ψ(119910

119886 119905

1)119889119910

119886 Ψ

119861(119910 119905) =

int

119861119870(119910 119905 119910

119887 119905

1)Ψ(119910

119887 119905

1)119889119910

119887 and 119870(119910 119905 119910

120572 119905

1) = (1198982119894ℏ(119905 minus

119905

1))

12119890

119894119898(119910minus119910120572)22ℏ(119905minus1199051)

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 passedIn 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 (035mm 35mm 35 cm and35 cm) where the two slits 119860 and 119861 are open simultaneously(interference |Ψ

119860+ Ψ

119861|

2) with the evolution of the sum ofthe probability densities where the slits 119860 and 119861 are openindependently (the sum of two diffractions |Ψ

119860|

2+ |Ψ

119861|

2)


minus1 minus05

(120583m)0 05 1

(a) 035mm

minus1 minus05

(120583m)0 05 1

(b) 35mm

minus2 minus1

(120583m)0 1 2

(c) 35 cm

minus10 minus5

(120583m)0 5 10

(d) 35 cm

Figure 4 Comparison of the probability density |Ψ119860+ Ψ

119861|

2 (full line) and |Ψ119860|

2+ |Ψ

119861|

2 (dotted line) at various distances after the slits (a)035mm (b) 35mm (c) 35 cm and (d) 35 cm

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

32 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 fringesThis is the simplest andmost natural interpretation to explainthe impact positions ldquothe position of an impact is simply theposition of the particle at the time of impactrdquo This was theview defended by Einstein at the Solvay Congress of 1927Theposition 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 ldquoequivariancerdquo property and thereduction of the wave packet is not necessary to explain theimpacts


minus35 minus30 minus20

minus4

minus3

minus10

minus2

minus1

(120583m)

(cm)0 10 20 30 35

0

1

2

3

4

Figure 5 100 electron trajectories for the Jonsson experiment

minus08

minus06

minus04

minus02

minus1

(120583m)

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

0

02

04

06

08

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 termℎ119905119898 rarr 0 so ℎ rarr 0 is equivalent to119898 rarr +infin (ie the massof the particle grows) or 119905 rarr 0 (ie the distance splits-screen119889

2rarr 0) Figure 7 shows the 100 trajectories that start at

the same 100 initial points when Planckrsquos constant is dividedrespectively into 10 100 1000 and 10000 (equivalent tomultiplying the mass by 10 100 1000 and 10000) We obtainquantum trajectories converging to the classical trajectorieswhen 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 ℎ119905119898 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 beaminstead of expanding splits into two separate beams givingtwo spots of equal intensity119873+ and119873minus 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 mechanicsToday it is the theory of decoherence with the diagonalizationof the density matrix that is put forward to explain the firstpart of the measurement process [33ndash38] 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(119911) = (2120587120590

2

0)

minus12

119890

minus119911241205902

0(

cos120579

0

2

119890

minus119894(12059302)

sin120579

0

2

119890

119894(12059302)

) (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= (

cos120579

0

2

119890

minus119894(12059302)

sin120579

0

2

119890

119894(12059302)

) (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 processThis 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 oven119864 (Figure 8) are heatedto a high temperature and escape through a narrow opening


minus08

minus06

minus04

minus02

minus1

(120583m)

(cm)minus30 minus20 minus10

h10 h100

h1000 h10000

0 10 20 30

0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)

(cm)minus30 minus20 minus10 0 10 20 30

0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

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( = 500ms)T = 1000∘K

E

TN+

Nminus

Figure 8 Schematic configuration of the Stern-Gerlach experiment

A second aperture 119879 selects those atoms whose velocity v0

is parallel to the 119910-axis The atomic beam crosses the gap ofthe electromagnet 119860

1before condensing on the detector 119875

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 1198601and

at the initial time 119905 = 0 each atom can be approximatelydescribed by a Gaussian spinor in 119911 given by (16) correspond-ing to a pure state The variable 119910 will be treated classicallywith 119910 = V119905 120590

0= 10

minus4 m corresponds to the size of theslot 119879 along the 119911-axis The approximation by a Gaussianinitial spinor will allow explicit calculations Because the slotis much wider along the 119909-axis the variable 119909 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 wehave119898 = 18 times 10

minus25 kg V0= 500ms (corresponding to the

temperature of 119879 = 1000

∘K) In (16) and in Figure 9 1205790and

120593

0are the polar angles characterizing the initial orientation

of the magnetic moment and 120579

0corresponds to the angle

with the 119911-axisThe experiment is a statistical mixture of purestates where the 120579

0and the 120593

0are randomly chosen 120579

0is

drawn in a uniform way from [0 120587] and 1205930is drawn in a

uniform way from [0 2120587]


x

y

z

1205930

1205790

|minus⟩

|+⟩

Figure 9 Orientation of the magnetic moment 1205790and 120593

0are the

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

The evolution of the spinor Ψ = (

120595+120595minus) in a magnetic field

B is then given by the Pauli equation

119894ℏ(

120597120595

+

120597119905

120597120595

minus

120597119905

) = minus

ℏ

2

2119898

Δ(

120595

+

120595

minus

) + 120583

119861B120590(120595+

120595

minus

) (18)

where 120583

119861= 119890ℏ2119898

119890is the Bohr magneton and where

120590 = (120590

119909 120590

119910 120590

119911) corresponds to the three Pauli matrices The

particle first enters an electromagnetic field B directed alongthe 119911-axis 119861

119909= 119861

1015840

0119909 119861

119910= 0 and 119861

119911= 119861

0minus 119861

1015840

0119911 with

119861

0= 5 Tesla 1198611015840

0= |120597119861120597119911| = 10

3 Teslam over a lengthΔ119897 = 1 cm On exiting the magnetic field the particle isfree until it reaches the detector 119875

1placed at a 119863 = 20 cm

distanceTheparticle stayswithin themagnetic field for a timeΔ119905 =

Δ119897V = 2 times 10minus5 s During this time [0 Δ119905] the spinor is [47](see the Appendix)

Ψ (119911 119905) ≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911minus(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3+1205831198611198610119905+(ℏ12059302))ℏ)

119894 sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((minus1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3minus1205831198611198610119905minus(ℏ12059302))ℏ)

) (19)

After the magnetic field at time 119905 + Δ119905 (119905 ge 0) in the freespace the spinor becomes [44ndash48] (see the Appendix)

Ψ (119911 119905 + Δ119905)

≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911minus119911Δminus119906119905)241205902

0119890

119894((119898119906119911+ℏ120593+)ℏ)

sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+119911Δ+119906119905)241205902

0119890

119894((minus119898119906119911+ℏ120593minus)ℏ)

)

(20)

where

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

= 10

minus5 m 119906 =

120583

119861119861

1015840

0(Δ119905)

119898

= 1ms(21)

Equation (20) takes into account the spatial extension of thespinor and we note that the two spinor components havevery different 119911 values All interpretations are based on thisequation

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

electromagnet

120588

1205790(119911 119905 + Δ119905) ≃ (2120587120590

2

0)

minus12

(cos2120579

0

2

119890


0

+sin2 1205790

2

119890

minus(119911+119911Δ+119906119905)221205902

0)

(22)

Figure 10 shows the probability density of a pure state (with120579

0= 1205873) as a function of 119911 at several values of 119905 (the plots

are labeled 119910 = V119905) The beam separation does not appear atthe end of the magnetic field (1 cm) but 16 cm further alongIt is the moment of the decoherence

The decoherence time where the two spots 119873+ and 119873minus

are separated is then given by

119905

119863≃

3120590

0minus 119911

Δ

119906

= 3 times 10

minus4 s (23)

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

120588

119878(119905) = (

int

1003816

1003816

1003816

1003816

120595

+(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911 int120595

+(119911 119905) 120595

lowast

minus(119911 119905) 119889119911

int120595

minus(119911 119905) 120595

lowast

+(119911 119905) 119889119911 int

1003816

1003816

1003816

1003816

120595

minus(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911

)

(24)


minus06

(mm)minus06

(mm)minus06

(mm)minus06

(mm)

0 cm 6 cm 16 cm 21 cm

0 06 0 06 0 06 0 06

Figure 10 Evolution of the probability density of a pure state with 1205790= 1205873

minus5 minus4 minus3 minus2 minus1

minus1

z(m

m)

x (mm)0 1 2 3 4 5

0

1N+

Nminus

Figure 11 1000 silver atom impacts on the detector 1198751

For 119905 ge 119905

119863 the product 120595

+(119911 119905 + Δ119905)120595

minus(119911 119905 + Δ119905) is null

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

120588

119878(119905 + Δ119905) = (2120587120590

2

0)

minus1

(

cos2120579

0

2

0

0 sin2120579

0

2

) (25)

42 Proof of the Postulates of Quantum Measurement Wethen obtain atoms with a spin oriented only along the 119911-axis(positively or negatively) Let us consider the spinor Ψ(119911 119905 +Δ119905) given by (20) Experimentally we do not measure thespin directly but the position of the particle impact on 119875

1

(Figure 11)If isin 119873

+ the term 120595

minusof (20) is numerically equal

to zero and the spinor Ψ is proportional to ( 1

0) one of the

eigenvectors of the spin operator 119878119911= (ℏ2)120590

119911 Ψ( 119905 +

Δ119905) ≃ (2120587120590

2

0)

minus14 cos(12057902)119890


0119890

119894((1198981199061+ℏ120593+)ℏ)(

1

0)

Then we have 119878119911Ψ = (ℏ2)120590

119911Ψ = +(ℏ2)Ψ

If isin 119873

minus the term 120595

+of (20) is numerically

equal to zero and the spinor Ψ is proportional to ( 0

1) the

other eigenvector of the spin operator 119878119911 Ψ( 119905 + Δ119905) ≃

(2120587120590

2

0)

minus14 sin(12057902)119890

minus(2+119911Δ+119906119905)241205902

0119890

119894((minus1198981199062+ℏ120593minus)ℏ)(

0

1) Then

we have 119878

119911Ψ = (ℏ2)120590

119911Ψ = minus(ℏ2)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(12057902) (resp

sin2(120579

02)) to measure the particle in the spin state +ℏ2

(resp minusℏ2) this proves the Born probabilistic postulate

By drilling a hole in the detector 1198751to the location of

the spot 119873+ (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 Ψminusof the spinor Ψ

identically zero (and not only numerically equal to zero)at the time when the atom crosses the detector 119875

1 at this

time the component Ψminusis indeed stopped by detector 119875

1

The future trajectory of the silver atom after crossing thedetector 119875

1will be guided by this new (normalized) spinor

The wave function reduction is therefore not linked to theelectromagnet but to the detector 119875

1causing an irreversible

elimination of the spinor component Ψminus

43 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 particlersquosinitial position 119911

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 (120579

0= 1205873 120593

0= 0) a plot in the (119874119910119911) plane

of a set of 10 trajectories whose initial position 1199110has been

randomly chosen from aGaussian distribution with standarddeviation 120590

0 The spin orientations 120579(119911 119905) are represented by

arrowsThe final orientation obtained after the decoherence time

119905

119863 depends on the initial particle position 119911

0in the spinor

with a spatial extension and on the initial angle 1205790of the spin

with the 119911-axis We obtain +1205872 if 1199110gt 119911

1205790 and minus1205872 if 1199110lt

119911

1205790 with

119911

1205790= 120590

0119865

minus1(sin2 1205790

2

) (26)

where 119865 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 ldquomeasuredrdquo value is notindependent of the context of the measure and is contextualIt conforms to the Kochen and Specker theorem [51] realismand noncontextuality are inconsistent with certain quantummechanics predictions


0 5 10 15 20

0

02

04

06

08

minus04

minus02

y (cm)

z(m

m)

Figure 12 Ten silver atom trajectories with initial spin orientation(120579

0= 1205873) and initial position 119911

0 arrows represent the spin

orientation 120579(119911 119905) along the trajectories

Now let us consider a mixture of pure states where theinitial orientation (120579

0 120593

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 119911

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 119911-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 nonlocalityManyexperiments since Bellrsquos paper have demonstrated violationsof these inequalities and have vindicated quantum theory[57ndash63] 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

02

04

06

minus06

minus04

minus02

y (cm)

z(m

m)

Figure 13 Ten silver atom trajectories where the initial orientation(120579

0 120593

0) has been randomly chosen arrows represent the spin


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 positivelyTo 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

119860 r

119861) =

1

radic2

119891 (r119860) 119891 (r

119861) (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(27)

and not from a simplified wave function without spatialextension

Ψ

0(r

119860 r

119861) =

1

radic2

(

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩) (28)

119891 function and |plusmn⟩ vectors are presented laterThe 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 conditionsHowever 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 ℏ2 We present a deBroglie-Bohm interpretation that avoids this flaw [68]

Figure 14 presents the Einstein-Podolsky-Rosen-Bohmexperiment A source 119878 creates in 119874 pairs of identical atoms119860 and 119861 but with opposite spins The atoms 119860 and 119861


x

y

z

z

z

z998400

x

z z998400

x

yz

x998400z998400

120575

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 119910-axis in opposite directions and headtowards two identical Stern-Gerlach apparatus E

119860and E

119861

The electromagnet E119860ldquomeasuresrdquo the spin of 119860 along the 119911-

axis and the electromagnet E119861ldquomeasuresrdquo the spin of 119861 along

the 1199111015840-axis which is obtained after a rotation of an angle 120575around the 119910-axis The initial wave function of the entangledstate is the singlet state (27) where r = (119909 119911) 119891(r) =

(2120587120590

2

0)

minus12119890

minus(1199092+1199112)41205902

0 |plusmn119860⟩ and |plusmn

119861⟩ are the eigenvectors

of the operators 120590119911119860

and 120590119911119861 120590

119911119860|plusmn

119860⟩ = plusmn|plusmn

119860⟩ 120590

119911119861|plusmn

119861⟩ =

plusmn|plusmn

119861⟩ We treat the dependence with 119910 classically speed

minusV119910for 119860 and V

119910for 119861 The wave function Ψ(r

119860 r

119861 119905) of

the two identical particles 119860 and 119861 electrically neutral andwithmagnetic moments 120583

0 subject to magnetic fields E

119860and

E119861 admits on the basis of |plusmn

119860⟩ and |plusmn

119861⟩ four components

Ψ

119886119887(r

119860 r

119861 119905) and satisfies the two-body Pauli equation [46

page 417]

119894ℏ

120597Ψ

119886119887

120597119905

= (minus

ℏ

2

2119898

Δ

119860minus

ℏ

2

2119898

Δ

119861)Ψ

119886119887+ 120583119861

E119860119895(120590

119895)

119886

119888Ψ

119888119887

+ 120583119861

E119861119895(120590

119895)

119887

119889Ψ

119886119889

(29)

with the initial conditions

Ψ

119886119887(r

119860 r

119861 0) = Ψ

119886119887

0(r

119860 r

119861)

(30)

where Ψ119886119887

0(r

119860 r

119861) corresponds to the singlet state (27)

To obtain an explicit solution of the EPR-B experimentwe 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

51 First Step EPR-B Spin Measurement of119860 In the first stepwe make a Stern and Gerlach ldquomeasurementrdquo for atom 119860

on a pair of particles 119860 and 119861 in a singlet state This is theexperiment first proposed in 1987 by Dewdney et al [66 67]

Consider that at time 1199050the particle 119860 arrives at the

entrance of electromagnet E119860 After this exit of the magnetic

field E119860 at time 119905

0+ Δ119905 + 119905 the wave function (27) becomes

[68]

Ψ (r119860 r

119861 119905

0+ Δ119905 + 119905)

=

1

radic2

119891 (r119861)

times (119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus 119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(31)

with

119891

plusmn(r 119905) ≃ 119891 (119909 119911 ∓ 119911

Δ∓ 119906119905) 119890

119894((plusmn119898119906119911ℏ)+120593plusmn(119905))

(32)

where 119911Δand 119906 are given by (21)

The atomic density 120588(119911119860 119911

119861 119905

0+ Δ119905 + 119905) is found by

integrating Ψlowast(r

119860 r

119861 119905

0+ Δ119905 + 119905)Ψ(r

119860 r

119861 119905

0+ Δ119905 + 119905) on 119909

119860

and 119909119861

120588 (119911

119860 119911

119861 119905

0+ Δ119905 + 119905)

= ((2120587120590

2

0)

minus12

119890

minus(119911119861)221205902

0)

times ((2120587120590

2

0)

minus12

times

1

2

(119890


0+ 119890

minus(119911119860+119911Δ+119906119905)221205902

0))

(33)

We deduce that the beam of particle 119860 is divided into twowhile the beam of particle 119861 stays undivided

(i) the density of 119860 is the same whether particle 119860 isentangled with 119861 or not

(ii) the density of 119861 is not affected by the ldquomeasurementrdquoof 119860


Our first conclusion is that the position of 119861 does notdepend on themeasurement of119860 only the spins are involvedWe conclude from (31) that the spins of 119860 and 119861 remainopposite throughout the experiment These are the twoproperties used in the causal interpretation

52 Second Step EPR-B Spin Measurement of 119861 The secondstep is a continuation of the first and corresponds to theEPR-B experiment broken down into two steps On a pair ofparticles 119860 and 119861 in a singlet state first we made a Stern andGerlachmeasurement on the119860 atom between 119905

0and 119905

0+Δ119905+

119905

119863 secondly we make a Stern and Gerlach measurement on

the 119861 atomwith an electromagnet E119861forming an angle 120575with

E119860during 119905

0+ Δ119905 + 119905

119863and 119905

0+ 2(Δ119905 + 119905

119863)

At the exit of magnetic field E119860 at time 119905

0+ Δ119905 + 119905

119863

the wave function is given by (31) Immediately after themeasurement of119860 still at time 119905

0+Δ119905+119905

119863 the wave function

of 119861 depends on the measurement plusmn of 119860

Ψ

119861plusmn119860(r

119861 119905

0+ Δ119905 + 119905

1) = 119891 (r

119861)

1003816

1003816

1003816

1003816

∓

119861⟩ (34)

Then the measurement of 119861 at time 1199050+ 2(Δ119905 + 119905

119863) 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

53 Causal Interpretation of the EPR-B Experiment We as-sume at the creation of the two entangled particles 119860 and119861 that each of the two particles 119860 and 119861 has an initialwave function with opposite spins Ψ119860

0(r

119860 120579

119860

0 120593

119860

0) = 119891(r

119860)

(cos(12057911986002)|+

119860⟩ + sin(120579119860

02)119890

119894120593119860

0|minus

119860⟩) and Ψ

119861

0(r

119861 120579

119861

0 120593

119861

0) =

119891(r119861)(cos(120579119861

02)|+

119861⟩ + sin(120579119861

02)119890

119894120593119861

0|minus

119861⟩) with 120579119861

0= 120587 minus 120579

119860

0

and 120593119861

0= 120593

119860

0minus 120587 The two particles 119860 and 119861 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

119860 120579

119860 120593

119860 r

119861 120579

119861 120593

119861)

= minus119890

119894120593119860

119891 (r119860) 119891 (r

119861) times (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(35)

Thus we can consider that the singlet wave function is thewave function of a family of two fermions 119860 and 119861 withopposite spins the direction of initial spins119860 and 119861 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[44ndash46 66 67] in the interpretation of the singlet wavefunction they do not assume the existence of wave functionsΨ

119860

0(r

119860 120579

119860

0 120593

119860

0) and Ψ119861

0(r

119861 120579

119861

0 120593

119861

0) for each particle but only

the singlet state Ψ0(r

119860 120579

119860 120593

119860 r

119861 120579

119861 120593

119861) 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 ℏ2 [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 119860) In (31) particle 119860 can beconsidered independent of 119861 We can therefore give it thewave function

Ψ

119860(r

119860 119905

0+ Δ119905 + 119905)

= cos120579

119860

0

2

119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩ + sin

120579

119860

0

2

119890

119894120593119860

0119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

(36)

which is thewave function of a free particle in a Stern-Gerlachapparatus and whose initial spin is given by (120579119860

0 120593

119860

0) For

an initial polarization (1205791198600 120593

119860

0) and an initial position (119911119860

0)

we obtain in the de Broglie-Bohm interpretation [44] of theStern and Gerlach experiment an evolution of the position(119911

119860(119905)) and of the spin orientation of 119860 (120579119860(119911

119860(119905) 119905)) [48]

The case of particle 119861 is different 119861 follows a rectilineartrajectory with 119910

119861(119905) = V

119910119905 119911

119861(119905) = 119911

119861

0 and 119909

119861(119905) = 119909

119861

0 By

contrast the orientation of its spinmoveswith the orientationof the spin of 119860 120579119861(119905) = 120587 minus 120579

119860(119911

119860(119905) 119905) and 120593

119861(119905) =

120593(119911

119860(119905) 119905) minus 120587 We can associate the following wave function

with the particle 119861

Ψ

119861(r

119861 119905

0+ Δ119905 + 119905)

= 119891 (r119861) (cos 120579

119861(119905)

2

1003816

1003816

1003816

1003816

+

119861⟩ + sin 120579

119861(119905)

2

119890

119894120593119861(119905) 10038161003816

1003816

1003816

minus

119861⟩)

(37)

This wave function is specific because it depends upon initialconditions of 119860 (position and spin) The orientation of spinof the particle 119861 is driven by the particle119860 through the singletwave functionThus the singlet wave function is the nonlocalvariable

Step 2 (spin measurement of 119861) At the time 1199050+ Δ119905 + 119905

119863

immediately after the measurement of119860 120579119861(1199050+Δ119905+119905

119863) = 120587

or 0 in accordance with the value of 120579119860(119911119860(119905) 119905) and the

wave function of 119861 is given by (34) The frame (1198741199091015840119910119911

1015840)

corresponds to the frame (119874119909119910119911) after a rotation of an angle120575 around the 119910-axis 120579119861 corresponds to the 119861-spin angle withthe 119911-axis and 1205791015840119861 to the 119861-spin angle with the 1199111015840-axis then120579

1015840119861(119905

0+ Δ119905 + 119905

119863) = 120587 + 120575 or 120575 In this second step we

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

119861) with a specific initial

polarization equal to 120587+120575 or 120575 and not random like in Step 1Then the measurement of 119861 at time 119905

0+ 2(Δ119905 + 119905

119863) gives

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

54 Physical Explanation of Nonlocal Influences From thewave function of two entangled particles we find spinstrajectories 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-Poincare and byEinstein in 1920[69] ldquoRecapitulating we may say that according to the generaltheory of relativity space is endowed with physical qualitiesin 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 itrdquo

Taking into account the new experiments especiallyAspectrsquos experiments Popper [70 page XVIII] defends asimilar view in 1982

ldquoI feel not quite convinced that the experiments are correctlyinterpreted but if they are we just have to accept action at adistance I think (with JP 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 Aspectrsquos experimentswould be the first crucial experiment between Lorentzrsquos andEinsteinrsquos interpretation of the Lorentz transformationsrdquo

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 likeLorentzrsquos ether and like Einsteinrsquos 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 thequantumparticleThedeBroglie-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 ldquoquantum equilibrium hypothesisrdquo (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 Schrodinger 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 experimentsin 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-updown measurement appears alsoafter a given time called decoherence time in the EPR-B experiment only the spin of 119861 is affected by the spinmeasurement of 119860 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 ldquoquantum equilibriumhypothesisrdquo 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 Schrodinger [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 Schrodingerrsquos) Their existence was predictedin 1994 by Bialynicki-Birula et al [73ndash75] and discoveredrecently by Maeda and Gallagher [76] on Rydberg atomsFor these nonstatistically prepared quantum particles wehave shown [9 10] that the natural interpretation is theSchrodinger 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 ldquotheory of the double solutionrdquo that Louis deBroglie was seeking since 1927 ldquoI introduced as the ldquodoublesolution theoryrdquo the idea that it was necessary to distinguish twodifferent solutions that are both linked to the wave equationone that I called wave 119906 which was a real physical waverepresented by a singularity as it was not normalizable dueto a local anomaly defining the particle the other one asSchrodingerrsquos Ψ wave which is a probability representation asit is normalizable without singularitiesrdquo [77]

Appendix

Calculating the Spinor Evolution in theStern-Gerlach Experiment

In the magnetic field 119861 = (119861

119909 0 119861

119911) the Pauli equation

(18) gives coupled Schrodinger equations for each spinor


component

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) plusmn 120583

119861(119861

0minus 119861

1015840

0119911)120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905)

(A1)

If one affects the transformation [47]

120595

plusmn(119909 119911 119905) = exp(plusmn

119894120583

119861119861

0119905

ℏ

)120595

plusmn(119909 119911 119905)

(A2)

(A1) becomes

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905) exp(plusmn119894

2120583

119861119861

0119905

ℏ

)

(A3)

The coupling term oscillates rapidly with the Larmor fre-quency 120596

119871= 2120583

119861119861

0ℏ = 1 4 times 10

11 sminus1 Since |1198610| ≫ |119861

1015840

0119911|

and |1198610| ≫ |119861

1015840

0119909| 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]

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905) = minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

(A4)

Since the variable 119909 is not involved in this equation and120595

0

plusmn(119909 119911) does not depend on 119909 120595

plusmn(119909 119911 119905) does not depend

on 119909 120595plusmn(119909 119911 119905) equiv 120595

plusmn(119911 119905) Then we can explicitly compute

the preceding equations for all 119905 in [0 Δ119905] with Δ119905 = Δ119897V =2 times 10

5 sWe obtain

120595

+(119911 119905) = 120595

119870(119911 119905) cos

120579

0

2

119890

119894(12059302) 119870 = minus120583

119861119861

1015840

0

120595

minus(119911 119905) = 120595

119870(119911 119905) 119894 sin

120579

0

2

119890

minus119894(12059302) 119870 = +120583

119861119861

1015840

0

(A5)

120590

2

119905= 120590

2

0+ (ℏ1199052119898120590

0)

2and

120595

119870(119911 119905) = (2120587120590

2

119905)

minus14

119890

minus(119911+11987011990522119898)2

41205902

119905

times exp 119894

ℏ

[

[

minus

ℏ

2

tanminus1(

ℏ119905

2119898120590

2

0

) minus 119870119905119911 minus

119870

2119905

3

6119898

+

(119911 + 119870119905

22119898)

2

ℏ

2119905

2

8119898120590

2

0120590

2

119905

]

]

(A6)

where (A6) is a classical result [11]

The experimental conditions give ℏΔ1199052119898120590

0= 4 times

10

minus11 m ≪ 120590

0= 10

minus4 m We deduce the approximations120590

119905≃ 120590

0and

120595

119870(119911 119905)

≃ (2120587120590

2

0)

minus14

119890

minus(119911+11987011990522119898)2

41205902

0 exp 119894

ℏ

[minus119870119905119911 minus

119870

2119905

3

6119898

]

(A7)

At the end of the magnetic field at time Δ119905 the spinor isequal to

Ψ (119911 Δ119905) = (

120595

+(119911 Δ119905)

120595

minus(119911 Δ119905)

) (A8)

with

120595

+(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911minus119911Δ)241205902

0)+(119894ℏ)119898119906119911 cos

120579

0

2

119890

119894120593+

120595

minus(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911+119911Δ)241205902

0)minus(119894ℏ)119898119906119911

119894 sin120579

0

2

119890

119894120593minus

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

119906 =

120583

0119861

1015840

0(Δ119905)

119898

120593

+=

120593

0

2

minus

120583

119861119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

120593

minus= minus

120593

0

2

+

120583

0119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

(A9)

We remark that the passage through the magnetic fieldgives the equivalent of a velocity +119906 in the direction 0119911

to the function 120595

+and a velocity minus119906 to the function 120595

minus

Then we have a free particle with the initial wave function(A8) The Pauli equation resolution again yields 120595

plusmn(119909 119911 119905) =

120595

119909(119909 119905)120595

plusmn(119911 119905) and with the experimental conditions we

have 120595119909(119909 119905) ≃ (2120587120590

2

0)

minus14119890

minus119909241205902

0 and

120595

+(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

cos120579

0

2

times expminus(119911minus119911Δminus119906119905)241205902

0+(119894ℏ)(119898119906119911minus(12)119898119906

2119905+ℏ120593+)

120595

minus(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

119894 sin120579

0

2

times expminus(119911+119911Δ+119906119905)241205902

0+(119894ℏ)(minus119898119906119911minus(12)119898119906

2119905+ℏ120593minus)

(A10)


Conflict of Interests

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

References

[1] J S Bell ldquoOn the impossible pilot waverdquo in Speakable and Un-speakable in Quantum Mechanics Cambridge University Press1987

[2] D Bohm ldquoA suggested interpretation of the quantum theory interms of hidden variables I and IIrdquo Physical Review vol 85 pp166ndash193 1952

[3] J von Neumann Mathematical Foundations of Quantum Me-chanics Princeton Landmarks in Mathematics Princeton Uni-versity Press Princeton NJ USA 1996

[4] B drsquoEspagnat A la Recherche du Ree Gauthiers-Villard ParisFrance 1979

[5] J S Bell Speakable and Un Speakable in Quantum MechanicsCambridge University Press 2nd edition 2004

[6] L de Broglie ldquoLa mecani que ondulatoire et la structureatomique de la matiere et du rayonnementrdquo Journal de Physiqueet Le Radium vol 8 no 5 pp 225ndash241 1927 An Englishtranslation can be found in G Bacciagalluppi and A ValentiniQuantum Theory of the Crossroads (Cambridge UniversityPress Cambridge UK 2009)

[7] C Jonsson ldquoElektroneninte rferenzen an mehreren kunstlichhergestellten Feinspaltenrdquo Zeitschrift fur Physik vol 161 pp454ndash474 1961

[8] E Madelung ldquoQuantentheorie in hydrodynamischer formrdquoZeitschrift fur Physik vol 40 no 3-4 pp 322ndash326 1927

[9] M Gondran and A Gondran ldquoDiscerned and non-discernedparticles in classical mechanics and convergence of quantummechanics to classical mechanicsrdquo Annales de la FondationLouis de Broglie vol 36 no 1 pp 117ndash135 2011

[10] M Gondran and A Gondran ldquoThe two limits of theSchrodinger equation in the semi-classical approximationdiscerned and non-discerned particles in classical mechanicsrdquoin Foundations of Probability and Physics-6 vol 1424 of AIPConference Proceedings pp 111ndash115 2012

[11] R Feynman and A Hibbs Quantum Mechanics and PathsIntegrals McGraw-Hill 1965

[12] C Cohen-Tannoudji B Diu and F LaloeQuantumMechanicsJohn Wiley amp Sons New York NY USA 1977

[13] D Durr S Goldstein and N Zanghi ldquoQuantum equilibriumand the origin of absolute uncertaintyrdquo Journal of StatisticalPhysics vol 67 no 5-6 pp 843ndash907 1992

[14] A S Sanz and S Miret-Artes ldquoQuantum phase analysis withquantum trajectories a step towards the creation of a Bohmianthinkingrdquo The American Journal of Physics vol 80 no 6 pp525ndash533 2012

[15] T Norsen ldquoThe pilot-wave perspective on quantum scatteringand tunnelingrdquo American Journal of Physics vol 81 no 4 pp258ndash266 2013

[16] T Young ldquoOn the theory of light and colorsrdquo Philosophicaltransactions of the Royal Society vol 92 pp 12ndash48 1802

[17] C Davisson and L H Germer ldquoThe scattering of electrons by asingle crystal of nickelrdquo Nature vol 119 no 2998 pp 558ndash5601927

[18] P G Merlin G F Missiroli and G Pozzi ldquoOn the statisticalaspect of electron interference phenomenardquo American Journalof Physics vol 44 no 3 pp 306ndash307 1976

[19] A Tonomura J Endo T Matsuda T Kawasaki and H EzawaldquoDemonstration of single-electron buildup of an interferencepatternrdquo American Journal of Physics vol 57 pp 117ndash120 1989

[20] H V Halbon Jr and P Preiswerk ldquoPreuve experimentale dela diffraction des neutronsrdquo Comptes Rendus de lrsquoAcademie desSciences vol 203 pp 73ndash75 1936

[21] H Rauch and A Werner Neutron Interferometry Lessons inExperimental Quantum Mechanics Oxford University PressLondon UK 2000

[22] A Zeilinger R Gahler C G Shull W Treimer andWMampeldquoSingle- and double-slit diffraction of neutronsrdquo Reviews ofModern Physics vol 60 no 4 pp 1067ndash1073 1988

[23] I Estermann and O Stern ldquoBeugung von MolekularstrahlenrdquoZeitschrift fur Physik vol 61 no 1-2 pp 95ndash125 1930

[24] F Shimizu K Shimizu and H Takuma ldquoDouble-slit interfer-ence with ultracold metastable neon atomsrdquo Physical Review Avol 46 no 1 pp R17ndashR20 1992

[25] M H Anderson J R Ensher M R Matthews C E Wiemanand E A Cornell ldquoObservation of Bose-Einstein condensationin a dilute atomic vaporrdquo Science vol 269 no 5221 pp 198ndash2011995

[26] M Arndt O Nairz J Vos-Andreae C Keller G van der Zouwand A Zellinger ldquoWave-particle duality of C

60moleculesrdquo

Nature vol 401 no 6754 pp 680ndash682 1999[27] O Nairz M Arndt and A Zeilinger ldquoExperimental challenges

in fullerene interferometryrdquo Journal of Modern Optics vol 47no 14-15 pp 2811ndash2821 2000

[28] R P Feynman R B Leighton and M Sands The FeynmanLectures on Physics Vol 3 Quantum Mechanics Addison-Wesley New York NY USA 1965

[29] M Gondran and A Gondran ldquoNumerical simulation of thedouble slit interference with ultracold atomsrdquo The AmericanJournal of Physics vol 73 no 6 pp 507ndash515 2005

[30] C Philippidis C Dewdney and B J Hiley ldquoQuantum interfer-ence and the quantum potentialrdquo Il Nuovo Cimento B Serie 11vol 52 no 1 pp 15ndash28 1979

[31] W Gerlach and O Stern ldquoDer experimentelle Nachweis desmagnetischen moments des silberatomsrdquo Zeitschrift fur Physikvol 8 no 1 pp 110ndash111 1921

[32] W Gerlach and O Stern ldquoDer experimentelle Nachweis derRichtungsquantelung imMagnetfeldrdquoZeitschrift fur Physik vol9 no 1 pp 349ndash352 1922

[33] H D Zeh ldquoOn the interpretation of measurement in quantumtheoryrdquo Foundations of Physics vol 1 pp 69ndash76 1970 Reprintedin Wheeler and Zurek pp 342ndash349 1983

[34] W H Zurek ldquoEnvironment-induced superselection rulesrdquoPhysical Review D Particles and Fields vol 26 no 8 pp 1862ndash1880 1982

[35] J A Wheeler and W H Zurek QuantumTheory and Measure-ment Princeton Series in Physics Princeton University PressPrinceton NJ USA 1983

[36] W H Zurek ldquoDecoherence einselection and the quantumorigins of the classicalrdquo Reviews of Modern Physics vol 75 no3 pp 715ndash775 2003

[37] R Omnes ldquoConsistent interpretations of quantum mechanicsrdquoReviews of Modern Physics vol 64 no 2 pp 339ndash382 1992


[38] M Schlosshauer Decoherence and the QuantumtomdashClassicalTransition Springer 2007

[39] J J Sakurai Modern Quantum Mechanics Addison-Wesley1985

[40] M Le Bellac Quantum Physics Cambridge University PressCambridge UK 2006

[41] T Takabayasi ldquoOn the formulation of quantummechanics asso-ciatedwith classical picturesrdquoProgress ofTheoretical Physics vol8 no 2 pp 143ndash182 1952

[42] T Takabayasi ldquoThe formulation of quantum mechanics interms of ensemble in phase spacerdquo Progress of TheoreticalPhysics vol 11 no 4-5 pp 341ndash373 1954

[43] D Bohm R Schiller and J Tiomno ldquoA causal interpretation ofthe pauli equationrdquo Il Nuovo Cimento Series supplement 1 pp48ndash66 1955

[44] D Bohm and B J Hiley The Undivided Universe RoutledgeLondon UK 1993

[45] C Dewdney P R Holland and A Kyprianidis ldquoWhat happensin a spin measurementrdquo Physics Letters A vol 119 no 6 pp259ndash267 1986

[46] P R Holland The Quantum Theory of Motion CambridgeUniversity Press Cambridge Mass USA 1993

[47] D E Platt ldquoA modern analysis of the Stern-Gerlach experi-mentrdquo American Journal of Physics vol 60 no 4 pp 306ndash3081992

[48] M Gondran A Gondran and A Kenoufi ldquoDecoherence timeand spin measurement in the Stern-Gerlach experimentrdquo AIPConference Proceedings vol 1424 pp 116ndash120 2012

[49] G B Roston M Casas A Plastino and A R Plastino ldquoQuan-tum entanglement spin-12 and the Stern-Gerlach experimentrdquoEuropean Journal of Physics vol 26 no 4 pp 657ndash672 2005

[50] A Challinor A Lasenby S Gull and C Doran ldquoA relativisticcausal account of a spin measurementrdquo Physics Letters A vol218 no 3ndash6 pp 128ndash138 1996

[51] S Kochen and E P Specker ldquoThe problem of hidden variablesin quantum mechanicsrdquo Journal of Applied Mathematics andMechanics vol 17 pp 59ndash87 1967

[52] A Einstein B Podolsky and N Rosen ldquoCan quantum-mechanical description of physical reality be considered com-pleterdquo Physical Review vol 47 no 10 pp 777ndash780 1935

[53] D BohmQuantumTheory Prentice-Hall New York NY USA1951

[54] D Bohm and Y Aharanov ldquoDiscussion of experimental prooffor the paradox of Einstein Rosen and Podolskyrdquo vol 108 pp1070ndash1076 1957

[55] J S Bell ldquoOn the Einstein Podolsky rosen paradoxrdquo Physics vol1 no 3 pp 195ndash200 1964

[56] J F Clauser M A Horne A Shimony and R A HoltldquoProposed experiment to test local hidden-variable theoriesrdquoPhysical Review Letters vol 23 no 15 pp 880ndash884 1969

[57] S J Freedman and J F Clauser ldquoExperimental test of localhidden-variable theoriesrdquo Physical Review Letters vol 28 no14 pp 938ndash941 1972

[58] A Aspect P Grangier and G Roger ldquoExperimental realiza-tion of Einstein-Podolsky-Rosen-Bohm Gedankenexperimenta new violation of Bellrsquos inequalitiesrdquo Physical Review Lettersvol 49 no 2 pp 91ndash94 1982

[59] A Aspect J Dalibard and G Roger ldquoExperimental test of Bellrsquosinequalities using time-varying analyzersrdquo Physical ReviewLetters vol 49 no 25 pp 1804ndash1807 1982

[60] W Tittel J Brendel H Zbinden andN Gisin ldquoViolation of bellinequalities by photonsmore than 10 km apartrdquo Physical ReviewLetters vol 81 no 17 pp 3563ndash3566 1998

[61] G Weihs T Jennewein C Simon H Weinfurter and AZeilinger ldquoViolation of Bellrsquos inequality under strict Einsteinlocality conditionsrdquo Physical Review Letters vol 81 no 23 pp5039ndash5043 1998

[62] R A Bertlmann and A Zeilinger Eds Quantum [Un]speaka-bles From Bell to Quantum Information Springer 2002

[63] MGenovese ldquoResearch on hidden variable theories a review ofrecent progressesrdquo Physics Reports vol 413 no 6 pp 319ndash3962005

[64] A Beige W J Munro and P L Knight ldquoBellrsquos inequality testwith entangled atomsrdquo Physical Review A vol 62 no 5 ArticleID 052102 2000

[65] M A Rowe D Kielpinski V Meyer et al ldquoExperimentalviolation of a Bellrsquos inequality with efficient detectionrdquo Naturevol 409 no 6822 pp 791ndash794 2001

[66] C Dewdney P R Holland and A Kyprianidis ldquoA causalaccount of nonlocal Einstein-Podolsky-Rosen spin correla-tionsrdquo Journal of Physics A Mathematical and General vol 20no 14 pp 4717ndash4732 1987

[67] C Dewdney P R Holland A Kyprianidis and J P Vigier ldquoSpinand non-locality in quantum mechanicsrdquo Nature vol 336 no6199 pp 536ndash544 1988

[68] M Gondran and A Gondran ldquoA new causal interpretationof EPR-B experimentrdquo in Proceedings of the InternationalConference QuantumTheory Reconsideration of Foundations-6(QTRF 12) pp 370ndash375 Vaxjo Sweden June 2012

[69] A Einstein Ether and the Theory of Relativity University ofLeyden 1920

[70] K Popper Quantum Theory and the Schism in Physics FromThe Postscript to the Logic of Scientific Discovery HutchinsonsLondon UK edited by W W Bartley III 1982

[71] E Schrodinger ldquoDer stetige Ubergang von der Mikro- zurMakromechanikrdquo Die Naturwissenschaften vol 14 no 28 pp664ndash666 1926

[72] R J Glauber C deWitt A Blandin and C Cohen-Tanoudji inQuantum Optics and Electronics Les Houches Lectures 1964 p63 Gordon and Breach New York NY USA 1965

[73] I Bialynicki-Birula M Kalinski and J H Eberly ldquoLagrangeequilibrium points in celestial mechanics and nonspreadingwave packets for strongly driven rydberg electronsrdquo PhysicalReview Letters vol 73 no 13 pp 1777ndash1780 1994

[74] A Buchleitner and D Delande ldquoNondispersive electronic wavepackets in multiphoton processesrdquo Physical Review Letters vol75 no 8 pp 1487ndash1490 1995

[75] A Buchleitner D Delande and J Zakrzewski ldquoNon-dispersivewave packets in periodically driven quantum systemsrdquo PhysicsReports vol 368 no 5 pp 409ndash547 2002

[76] H Maeda and T F Gallagher ldquoNondispersing wave packetsrdquoPhysical Review Letters vol 92 Article ID 133004 2004

[77] L de Broglie and J L Andrade e Silva La Reinterpretation de laMecanique Ondulatoire Gauthier-Villars 1971

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

More than thirty years after John Bellrsquos 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 Jonsson in 1961with electrons [7] The method of Feynman path integralsallows us to calculate the time-dependent wave functionThe 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 particlesFinally 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 4we present an analytic expression of the wave function inthe Stern-Gerlach experiment This explicit solution requiresthe calculation of a Pauli spinor with a spatial extensionThis 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 particlesrsquo 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 119905)

solution to the Schrodinger equation

119894ℏ

120597Ψ (x 119905)120597119905

= minus

ℏ

2

2119898

ΔΨ (x 119905) + 119881 (x) Ψ (x 119905) (1)

Ψ (x 0) = Ψ0(x) (2)

With the variable change Ψ(x 119905) = radic120588

ℏ(x 119905) exp (119894(119878ℏ(x 119905)

ℏ)) the Schrodinger equation can be decomposed intoMadelung equations [8] (1926)

120597119878

ℏ(x 119905)120597119905

+

1

2119898

(nabla119878

ℏ(x 119905))

2

+ 119881 (x) minus ℏ

2

2119898

Δradic120588

ℏ(x 119905)

radic120588

ℏ(x 119905)

= 0

(3)

120597120588

ℏ(x 119905)120597119905

+ div(120588ℏ(x 119905) nabla119878

ℏ(x 119905)119898

) = 0 (4)

with initial conditions

120588

ℏ(x 0) = 120588ℏ

0(x) 119878

ℏ(x 0) = 119878ℏ

0(x) (5)

Madelung equations correspond to a set of noninteractingquantum particles all prepared in the same way (same 120588ℏ

0(x)

and 119878ℏ0(x))

A quantum particle is said to be statistically prepared ifits initial probability density 120588ℏ

0(x) and its initial action 119878ℏ

0(x)

converge when ℏ rarr 0 to nonsingular functions 1205880(x) and

119878

0(x) It is the case of an electronic or119862

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 120588ℏ

(x 119905) and the action 119878ℏ(x 119905) solutions tothe Madelung equations (3) (4) and (5) converge when ℏ rarr

0 to the classical density 120588(x 119905) and the classical action 119878(x 119905)solutions to the statistical Hamilton-Jacobi equations

120597119878 (x 119905)120597119905

+

1

2119898

(nabla119878 (x 119905))2 + 119881 (x 119905) = 0 (6)

119878 (x 0) = 1198780(x) (7)

120597120588 (x 119905)120597119905

+ div (120588 (x 119905) nabla119878 (x 119905)119898

) = 0(8)

120588 (x 0) = 1205880(x) (9)

We give some indications on the demonstration of thistheorem when the wave function Ψ(x 119905) is written as afunction of the initial wave function Ψ

0(x) by the Feynman

paths integral [11]

Ψ (x 119905) = int119865 (119905 ℏ) exp ( 119894ℏ

119878

119888119897(x 119905 x

0))Ψ

0(x

0) 119889x

0 (10)

where 119865(119905 ℏ) is an independent function of x and of x0 For a

statistically prepared quantum particle the wave function is



0(x

0) exp((119894ℏ)(119878ℏ

0(x

0)+119878

119888119897(x 119905

x0)))119889119909



119888119897(x 119905 x


ℎ(x 119905)


119878 (x 119905) = minx0(119878

0(x

0) + 119878

119888119897(x 119905 x

0)) (11)





0(x) and 119878






119889X (119905)119889119905

=

Jℎ(x 119905)120588

ℎ(x 119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816x=X(119905)

=

nabla119878

ℎ(x 119905)119898

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=X(119905)

(12)

where

Jℎ (x 119905) = ℏ

2119898119894

(Ψ


(x 119905))(13)






0(x) and the




119875 [X (0) = x] equiv 119875 (x 0) = 1003816100381610038161003816

Ψ

0(x)1003816100381610038161003816

2

= 120588

ℎ

0(x) (14)





119875 [X (119905) = x] equiv 119875 (x 119905) = |Ψ (x 119905)|2 = 120588ℎ(x 119905) (15)






60and 119862




02 120583m02 120583m

08 120583m10120583m

35 cm35 cm

y

z

x

S



minus119910221205732






2= 35 cm



0(119910)


1= 119889

1V ≃ 210

minus11 slt 119905 lt 119905

1+ 119889



119860(119910 119905) + Ψ

119861(119910 119905)

70 cm

10120583m


35 cm

3120583m


with Ψ

119860(119910 119905) = int

119860119870(119910 119905 119910

119886 119905

1)Ψ(119910

119886 119905

1)119889119910

119886 Ψ

119861(119910 119905) =

int

119861119870(119910 119905 119910

119887 119905

1)Ψ(119910

119887 119905

1)119889119910

119887 and 119870(119910 119905 119910

120572 119905

1) = (1198982119894ℏ(119905 minus

119905

1))

12119890

119894119898(119910minus119910120572)22ℏ(119905minus1199051)



119860+ Ψ

119861|


119860|

2+ |Ψ

119861|

2)


minus1 minus05

(120583m)0 05 1

(a) 035mm

minus1 minus05

(120583m)0 05 1

(b) 35mm

minus2 minus1

(120583m)0 1 2

(c) 35 cm

minus10 minus5

(120583m)0 5 10

(d) 35 cm


119861|


2+ |Ψ

119861|











minus4

minus3

minus10

minus2

minus1

(120583m)

(cm)0 10 20 30 35

0

1

2

3

4


minus08

minus06

minus04

minus02

minus1

(120583m)

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

0

02

04

06

08

1











Ψ

0(119911) = (2120587120590

2

0)

minus12

119890

minus119911241205902

0(

cos120579

0

2

119890

minus119894(12059302)

sin120579

0

2

119890

119894(12059302)

) (16)


Ψ

0= (

cos120579

0

2

119890

minus119894(12059302)

sin120579

0

2

119890

119894(12059302)

) (17)




minus08

minus06

minus04

minus02

minus1

(120583m)


h10 h100

h1000 h10000

0 10 20 30

0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1


x

y

z

y = t

D

A1

P1

Δl( = 500ms)T = 1000∘K

E

TN+

Nminus





1




0= 10





120593





0and the 120593


0is




x

y

z

1205930

1205790

|minus⟩

|+⟩


0are the





119894ℏ(

120597120595

+

120597119905

120597120595

minus

120597119905

) = minus

ℏ

2

2119898

Δ(

120595

+

120595

minus

) + 120583

119861B120590(120595+

120595

minus

) (18)

where 120583

119861= 119890ℏ2119898


120590 = (120590

119909 120590

119910 120590



119909= 119861

1015840

0119909 119861

119910= 0 and 119861

119911= 119861

0minus 119861

1015840

0119911 with

119861

0= 5 Tesla 1198611015840

0= |120597119861120597119911| = 10





Ψ (119911 119905) ≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911minus(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3+1205831198611198610119905+(ℏ12059302))ℏ)

119894 sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((minus1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3minus1205831198611198610119905minus(ℏ12059302))ℏ)

) (19)


Ψ (119911 119905 + Δ119905)

≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890


0119890

119894((119898119906119911+ℏ120593+)ℏ)

sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+119911Δ+119906119905)241205902

0119890

119894((minus119898119906119911+ℏ120593minus)ℏ)

)

(20)

where

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

= 10

minus5 m 119906 =

120583

119861119861

1015840

0(Δ119905)

119898

= 1ms(21)



electromagnet

120588

1205790(119911 119905 + Δ119905) ≃ (2120587120590

2

0)

minus12

(cos2120579

0

2

119890


0

+sin2 1205790

2

119890

minus(119911+119911Δ+119906119905)221205902

0)

(22)






119905

119863≃

3120590

0minus 119911

Δ

119906

= 3 times 10

minus4 s (23)


120588

119878(119905) = (

int

1003816

1003816

1003816

1003816

120595

+(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911 int120595

+(119911 119905) 120595

lowast

minus(119911 119905) 119889119911

int120595

minus(119911 119905) 120595

lowast

+(119911 119905) 119889119911 int

1003816

1003816

1003816

1003816

120595

minus(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911

)

(24)


minus06

(mm)minus06

(mm)minus06

(mm)minus06

(mm)

0 cm 6 cm 16 cm 21 cm

0 06 0 06 0 06 0 06



minus1

z(m

m)

x (mm)0 1 2 3 4 5

0

1N+

Nminus


For 119905 ge 119905


+(119911 119905 + Δ119905)120595

minus(119911 119905 + Δ119905) is null


120588

119878(119905 + Δ119905) = (2120587120590

2

0)

minus1

(

cos2120579

0

2

0

0 sin2120579

0

2

) (25)


1


+ the term 120595



0) one of the


119911 Ψ( 119905 +

Δ119905) ≃ (2120587120590

2

0)

minus14 cos(12057902)119890


0119890

119894((1198981199061+ℏ120593+)ℏ)(

1

0)

Then we have 119878119911Ψ = (ℏ2)120590

119911Ψ = +(ℏ2)Ψ

If isin 119873




1) the


(2120587120590

2

0)

minus14 sin(12057902)119890

minus(2+119911Δ+119906119905)241205902

0119890

119894((minus1198981199062+ℏ120593minus)ℏ)(

0

1) Then

we have 119878

119911Ψ = (ℏ2)120590




sin2(120579





1




1 at this


1










0= 1205873 120593






119905


0in the spinor



1205790 and minus1205872 if 1199110lt

119911

1205790 with

119911

1205790= 120590

0119865

minus1(sin2 1205790

2

) (26)




0 5 10 15 20

0

02

04

06

08

minus04

minus02

y (cm)

z(m

m)






0 120593





5 EPR-B Experiment


0 5 10 15 20

0

02

04

06

minus06

minus04

minus02

y (cm)

z(m

m)


0 120593





Ψ

0(r

119860 r

119861) =

1

radic2

119891 (r119860) 119891 (r

119861) (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(27)


Ψ

0(r

119860 r

119861) =

1

radic2

(

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩) (28)





x

y

z

z

z

z998400

x

z z998400

x

yz

x998400z998400

120575


O Atom AAtom B

EAEB



119860and E

119861




(2120587120590

2

0)

minus12119890

minus(1199092+1199112)41205902




and 120590119911119861 120590

119911119860|plusmn


119860⟩ 120590

119911119861|plusmn

119861⟩ =

plusmn|plusmn




119860 r

119861 119905) of



119860and




Ψ

119886119887(r

119860 r


page 417]

119894ℏ

120597Ψ

119886119887

120597119905

= (minus

ℏ

2

2119898

Δ

119860minus

ℏ

2

2119898

Δ

119861)Ψ

119886119887+ 120583119861

E119860119895(120590

119895)

119886

119888Ψ

119888119887

+ 120583119861

E119861119895(120590

119895)

119887

119889Ψ

119886119889

(29)


Ψ

119886119887(r

119860 r

119861 0) = Ψ

119886119887

0(r

119860 r

119861)

(30)

where Ψ119886119887

0(r

119860 r










[68]

Ψ (r119860 r

119861 119905

0+ Δ119905 + 119905)

=

1

radic2

119891 (r119861)

times (119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus 119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(31)

with

119891

plusmn(r 119905) ≃ 119891 (119909 119911 ∓ 119911

Δ∓ 119906119905) 119890

119894((plusmn119898119906119911ℏ)+120593plusmn(119905))

(32)



119861 119905

0+ Δ119905 + 119905) is found by


119860 r

119861 119905

0+ Δ119905 + 119905)Ψ(r

119860 r

119861 119905

0+ Δ119905 + 119905) on 119909

119860

and 119909119861

120588 (119911

119860 119911

119861 119905

0+ Δ119905 + 119905)

= ((2120587120590

2

0)

minus12

119890

minus(119911119861)221205902

0)

times ((2120587120590

2

0)

minus12

times

1

2

(119890


0+ 119890

minus(119911119860+119911Δ+119906119905)221205902

0))

(33)







0and 119905

0+Δ119905+

119905



E119860during 119905

0+ Δ119905 + 119905

119863and 119905

0+ 2(Δ119905 + 119905

119863)


0+ Δ119905 + 119905

119863


0+Δ119905+119905



Ψ

119861plusmn119860(r

119861 119905

0+ Δ119905 + 119905

1) = 119891 (r

119861)

1003816

1003816

1003816

1003816

∓

119861⟩ (34)


119863) yields



0(r

119860 120579

119860

0 120593

119860

0) = 119891(r

119860)

(cos(12057911986002)|+

119860⟩ + sin(120579119860

02)119890

119894120593119860

0|minus

119860⟩) and Ψ

119861

0(r

119861 120579

119861

0 120593

119861

0) =

119891(r119861)(cos(120579119861

02)|+

119861⟩ + sin(120579119861

02)119890

119894120593119861

0|minus

119861⟩) with 120579119861

0= 120587 minus 120579

119860

0

and 120593119861

0= 120593

119860



Ψ

0(r

119860 120579

119860 120593

119860 r

119861 120579

119861 120593

119861)

= minus119890

119894120593119860

119891 (r119860) 119891 (r

119861) times (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(35)



119860

0(r

119860 120579

119860

0 120593

119860

0) and Ψ119861

0(r

119861 120579

119861

0 120593

119861



119860 120579

119860 120593

119860 r

119861 120579

119861 120593





Ψ

119860(r

119860 119905

0+ Δ119905 + 119905)

= cos120579

119860

0

2

119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩ + sin

120579

119860

0

2

119890

119894120593119860

0119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

(36)


0 120593

119860

0) For


119860


0)



119860(119905) 119905)) [48]


119861(119905) = V

119910119905 119911

119861(119905) = 119911

119861

0 and 119909

119861(119905) = 119909

119861

0 By


119860(119911

119860(119905) 119905) and 120593

119861(119905) =

120593(119911



Ψ

119861(r

119861 119905

0+ Δ119905 + 119905)

= 119891 (r119861) (cos 120579

119861(119905)

2

1003816

1003816

1003816

1003816

+

119861⟩ + sin 120579

119861(119905)

2

119890

119894120593119861(119905) 10038161003816

1003816

1003816

minus

119861⟩)

(37)



119863


119863) = 120587



1015840)


1015840119861(119905

0+ Δ119905 + 119905





0+ 2(Δ119905 + 119905

119863) gives









6 Conclusion







Appendix



119909 0 119861




component

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) plusmn 120583

119861(119861

0minus 119861

1015840

0119911)120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905)

(A1)


120595


119894120583

119861119861

0119905

ℏ

)120595

plusmn(119909 119911 119905)

(A2)

(A1) becomes

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905) exp(plusmn119894

2120583

119861119861

0119905

ℏ

)

(A3)


119871= 2120583

119861119861

0ℏ = 1 4 times 10

11 sminus1 Since |1198610| ≫ |119861

1015840

0119911|

and |1198610| ≫ |119861

1015840



119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905) = minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

(A4)


0



on 119909 120595plusmn(119909 119911 119905) equiv 120595



5 sWe obtain

120595

+(119911 119905) = 120595

119870(119911 119905) cos

120579

0

2

119890

119894(12059302) 119870 = minus120583

119861119861

1015840

0

120595

minus(119911 119905) = 120595

119870(119911 119905) 119894 sin

120579

0

2

119890

minus119894(12059302) 119870 = +120583

119861119861

1015840

0

(A5)

120590

2

119905= 120590

2

0+ (ℏ1199052119898120590

0)

2and

120595

119870(119911 119905) = (2120587120590

2

119905)

minus14

119890

minus(119911+11987011990522119898)2

41205902

119905

times exp 119894

ℏ

[

[

minus

ℏ

2

tanminus1(

ℏ119905

2119898120590

2

0

) minus 119870119905119911 minus

119870

2119905

3

6119898

+

(119911 + 119870119905

22119898)

2

ℏ

2119905

2

8119898120590

2

0120590

2

119905

]

]

(A6)



0= 4 times

10

minus11 m ≪ 120590

0= 10


119905≃ 120590

0and

120595

119870(119911 119905)

≃ (2120587120590

2

0)

minus14

119890

minus(119911+11987011990522119898)2

41205902

0 exp 119894

ℏ

[minus119870119905119911 minus

119870

2119905

3

6119898

]

(A7)


Ψ (119911 Δ119905) = (

120595

+(119911 Δ119905)

120595

minus(119911 Δ119905)

) (A8)

with

120595

+(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911minus119911Δ)241205902

0)+(119894ℏ)119898119906119911 cos

120579

0

2

119890

119894120593+

120595

minus(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911+119911Δ)241205902

0)minus(119894ℏ)119898119906119911

119894 sin120579

0

2

119890

119894120593minus

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

119906 =

120583

0119861

1015840

0(Δ119905)

119898

120593

+=

120593

0

2

minus

120583

119861119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

120593

minus= minus

120593

0

2

+

120583

0119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

(A9)




minus


plusmn(119909 119911 119905) =

120595

119909(119909 119905)120595


have 120595119909(119909 119905) ≃ (2120587120590

2

0)

minus14119890

minus119909241205902

0 and

120595

+(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

cos120579

0

2


0+(119894ℏ)(119898119906119911minus(12)119898119906

2119905+ℏ120593+)

120595

minus(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

119894 sin120579

0

2

times expminus(119911+119911Δ+119906119905)241205902

0+(119894ℏ)(minus119898119906119911minus(12)119898119906

2119905+ℏ120593minus)

(A10)




References



























60moleculesrdquo



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





0(x

0) exp((119894ℏ)(119878ℏ

0(x

0)+119878

119888119897(x 119905

x0)))119889119909



119888119897(x 119905 x


ℎ(x 119905)


119878 (x 119905) = minx0(119878

0(x

0) + 119878

119888119897(x 119905 x

0)) (11)





0(x) and 119878






119889X (119905)119889119905

=

Jℎ(x 119905)120588

ℎ(x 119905)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816x=X(119905)

=

nabla119878

ℎ(x 119905)119898

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816119909=X(119905)

(12)

where

Jℎ (x 119905) = ℏ

2119898119894

(Ψ


(x 119905))(13)






0(x) and the




119875 [X (0) = x] equiv 119875 (x 0) = 1003816100381610038161003816

Ψ

0(x)1003816100381610038161003816

2

= 120588

ℎ

0(x) (14)





119875 [X (119905) = x] equiv 119875 (x 119905) = |Ψ (x 119905)|2 = 120588ℎ(x 119905) (15)






60and 119862




02 120583m02 120583m

08 120583m10120583m

35 cm35 cm

y

z

x

S



minus119910221205732






2= 35 cm



0(119910)


1= 119889

1V ≃ 210

minus11 slt 119905 lt 119905

1+ 119889



119860(119910 119905) + Ψ

119861(119910 119905)

70 cm

10120583m


35 cm

3120583m


with Ψ

119860(119910 119905) = int

119860119870(119910 119905 119910

119886 119905

1)Ψ(119910

119886 119905

1)119889119910

119886 Ψ

119861(119910 119905) =

int

119861119870(119910 119905 119910

119887 119905

1)Ψ(119910

119887 119905

1)119889119910

119887 and 119870(119910 119905 119910

120572 119905

1) = (1198982119894ℏ(119905 minus

119905

1))

12119890

119894119898(119910minus119910120572)22ℏ(119905minus1199051)



119860+ Ψ

119861|


119860|

2+ |Ψ

119861|

2)


minus1 minus05

(120583m)0 05 1

(a) 035mm

minus1 minus05

(120583m)0 05 1

(b) 35mm

minus2 minus1

(120583m)0 1 2

(c) 35 cm

minus10 minus5

(120583m)0 5 10

(d) 35 cm


119861|


2+ |Ψ

119861|











minus4

minus3

minus10

minus2

minus1

(120583m)

(cm)0 10 20 30 35

0

1

2

3

4


minus08

minus06

minus04

minus02

minus1

(120583m)

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

0

02

04

06

08

1











Ψ

0(119911) = (2120587120590

2

0)

minus12

119890

minus119911241205902

0(

cos120579

0

2

119890

minus119894(12059302)

sin120579

0

2

119890

119894(12059302)

) (16)


Ψ

0= (

cos120579

0

2

119890

minus119894(12059302)

sin120579

0

2

119890

119894(12059302)

) (17)




minus08

minus06

minus04

minus02

minus1

(120583m)


h10 h100

h1000 h10000

0 10 20 30

0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1


x

y

z

y = t

D

A1

P1

Δl( = 500ms)T = 1000∘K

E

TN+

Nminus





1




0= 10





120593





0and the 120593


0is




x

y

z

1205930

1205790

|minus⟩

|+⟩


0are the





119894ℏ(

120597120595

+

120597119905

120597120595

minus

120597119905

) = minus

ℏ

2

2119898

Δ(

120595

+

120595

minus

) + 120583

119861B120590(120595+

120595

minus

) (18)

where 120583

119861= 119890ℏ2119898


120590 = (120590

119909 120590

119910 120590



119909= 119861

1015840

0119909 119861

119910= 0 and 119861

119911= 119861

0minus 119861

1015840

0119911 with

119861

0= 5 Tesla 1198611015840

0= |120597119861120597119911| = 10





Ψ (119911 119905) ≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911minus(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3+1205831198611198610119905+(ℏ12059302))ℏ)

119894 sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((minus1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3minus1205831198611198610119905minus(ℏ12059302))ℏ)

) (19)


Ψ (119911 119905 + Δ119905)

≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890


0119890

119894((119898119906119911+ℏ120593+)ℏ)

sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+119911Δ+119906119905)241205902

0119890

119894((minus119898119906119911+ℏ120593minus)ℏ)

)

(20)

where

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

= 10

minus5 m 119906 =

120583

119861119861

1015840

0(Δ119905)

119898

= 1ms(21)



electromagnet

120588

1205790(119911 119905 + Δ119905) ≃ (2120587120590

2

0)

minus12

(cos2120579

0

2

119890


0

+sin2 1205790

2

119890

minus(119911+119911Δ+119906119905)221205902

0)

(22)






119905

119863≃

3120590

0minus 119911

Δ

119906

= 3 times 10

minus4 s (23)


120588

119878(119905) = (

int

1003816

1003816

1003816

1003816

120595

+(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911 int120595

+(119911 119905) 120595

lowast

minus(119911 119905) 119889119911

int120595

minus(119911 119905) 120595

lowast

+(119911 119905) 119889119911 int

1003816

1003816

1003816

1003816

120595

minus(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911

)

(24)


minus06

(mm)minus06

(mm)minus06

(mm)minus06

(mm)

0 cm 6 cm 16 cm 21 cm

0 06 0 06 0 06 0 06



minus1

z(m

m)

x (mm)0 1 2 3 4 5

0

1N+

Nminus


For 119905 ge 119905


+(119911 119905 + Δ119905)120595

minus(119911 119905 + Δ119905) is null


120588

119878(119905 + Δ119905) = (2120587120590

2

0)

minus1

(

cos2120579

0

2

0

0 sin2120579

0

2

) (25)


1


+ the term 120595



0) one of the


119911 Ψ( 119905 +

Δ119905) ≃ (2120587120590

2

0)

minus14 cos(12057902)119890


0119890

119894((1198981199061+ℏ120593+)ℏ)(

1

0)

Then we have 119878119911Ψ = (ℏ2)120590

119911Ψ = +(ℏ2)Ψ

If isin 119873




1) the


(2120587120590

2

0)

minus14 sin(12057902)119890

minus(2+119911Δ+119906119905)241205902

0119890

119894((minus1198981199062+ℏ120593minus)ℏ)(

0

1) Then

we have 119878

119911Ψ = (ℏ2)120590




sin2(120579





1




1 at this


1










0= 1205873 120593






119905


0in the spinor



1205790 and minus1205872 if 1199110lt

119911

1205790 with

119911

1205790= 120590

0119865

minus1(sin2 1205790

2

) (26)




0 5 10 15 20

0

02

04

06

08

minus04

minus02

y (cm)

z(m

m)






0 120593





5 EPR-B Experiment


0 5 10 15 20

0

02

04

06

minus06

minus04

minus02

y (cm)

z(m

m)


0 120593





Ψ

0(r

119860 r

119861) =

1

radic2

119891 (r119860) 119891 (r

119861) (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(27)


Ψ

0(r

119860 r

119861) =

1

radic2

(

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩) (28)





x

y

z

z

z

z998400

x

z z998400

x

yz

x998400z998400

120575


O Atom AAtom B

EAEB



119860and E

119861




(2120587120590

2

0)

minus12119890

minus(1199092+1199112)41205902




and 120590119911119861 120590

119911119860|plusmn


119860⟩ 120590

119911119861|plusmn

119861⟩ =

plusmn|plusmn




119860 r

119861 119905) of



119860and




Ψ

119886119887(r

119860 r


page 417]

119894ℏ

120597Ψ

119886119887

120597119905

= (minus

ℏ

2

2119898

Δ

119860minus

ℏ

2

2119898

Δ

119861)Ψ

119886119887+ 120583119861

E119860119895(120590

119895)

119886

119888Ψ

119888119887

+ 120583119861

E119861119895(120590

119895)

119887

119889Ψ

119886119889

(29)


Ψ

119886119887(r

119860 r

119861 0) = Ψ

119886119887

0(r

119860 r

119861)

(30)

where Ψ119886119887

0(r

119860 r










[68]

Ψ (r119860 r

119861 119905

0+ Δ119905 + 119905)

=

1

radic2

119891 (r119861)

times (119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus 119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(31)

with

119891

plusmn(r 119905) ≃ 119891 (119909 119911 ∓ 119911

Δ∓ 119906119905) 119890

119894((plusmn119898119906119911ℏ)+120593plusmn(119905))

(32)



119861 119905

0+ Δ119905 + 119905) is found by


119860 r

119861 119905

0+ Δ119905 + 119905)Ψ(r

119860 r

119861 119905

0+ Δ119905 + 119905) on 119909

119860

and 119909119861

120588 (119911

119860 119911

119861 119905

0+ Δ119905 + 119905)

= ((2120587120590

2

0)

minus12

119890

minus(119911119861)221205902

0)

times ((2120587120590

2

0)

minus12

times

1

2

(119890


0+ 119890

minus(119911119860+119911Δ+119906119905)221205902

0))

(33)







0and 119905

0+Δ119905+

119905



E119860during 119905

0+ Δ119905 + 119905

119863and 119905

0+ 2(Δ119905 + 119905

119863)


0+ Δ119905 + 119905

119863


0+Δ119905+119905



Ψ

119861plusmn119860(r

119861 119905

0+ Δ119905 + 119905

1) = 119891 (r

119861)

1003816

1003816

1003816

1003816

∓

119861⟩ (34)


119863) yields



0(r

119860 120579

119860

0 120593

119860

0) = 119891(r

119860)

(cos(12057911986002)|+

119860⟩ + sin(120579119860

02)119890

119894120593119860

0|minus

119860⟩) and Ψ

119861

0(r

119861 120579

119861

0 120593

119861

0) =

119891(r119861)(cos(120579119861

02)|+

119861⟩ + sin(120579119861

02)119890

119894120593119861

0|minus

119861⟩) with 120579119861

0= 120587 minus 120579

119860

0

and 120593119861

0= 120593

119860



Ψ

0(r

119860 120579

119860 120593

119860 r

119861 120579

119861 120593

119861)

= minus119890

119894120593119860

119891 (r119860) 119891 (r

119861) times (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(35)



119860

0(r

119860 120579

119860

0 120593

119860

0) and Ψ119861

0(r

119861 120579

119861

0 120593

119861



119860 120579

119860 120593

119860 r

119861 120579

119861 120593





Ψ

119860(r

119860 119905

0+ Δ119905 + 119905)

= cos120579

119860

0

2

119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩ + sin

120579

119860

0

2

119890

119894120593119860

0119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

(36)


0 120593

119860

0) For


119860


0)



119860(119905) 119905)) [48]


119861(119905) = V

119910119905 119911

119861(119905) = 119911

119861

0 and 119909

119861(119905) = 119909

119861

0 By


119860(119911

119860(119905) 119905) and 120593

119861(119905) =

120593(119911



Ψ

119861(r

119861 119905

0+ Δ119905 + 119905)

= 119891 (r119861) (cos 120579

119861(119905)

2

1003816

1003816

1003816

1003816

+

119861⟩ + sin 120579

119861(119905)

2

119890

119894120593119861(119905) 10038161003816

1003816

1003816

minus

119861⟩)

(37)



119863


119863) = 120587



1015840)


1015840119861(119905

0+ Δ119905 + 119905





0+ 2(Δ119905 + 119905

119863) gives









6 Conclusion







Appendix



119909 0 119861




component

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) plusmn 120583

119861(119861

0minus 119861

1015840

0119911)120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905)

(A1)


120595


119894120583

119861119861

0119905

ℏ

)120595

plusmn(119909 119911 119905)

(A2)

(A1) becomes

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905) exp(plusmn119894

2120583

119861119861

0119905

ℏ

)

(A3)


119871= 2120583

119861119861

0ℏ = 1 4 times 10

11 sminus1 Since |1198610| ≫ |119861

1015840

0119911|

and |1198610| ≫ |119861

1015840



119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905) = minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

(A4)


0



on 119909 120595plusmn(119909 119911 119905) equiv 120595



5 sWe obtain

120595

+(119911 119905) = 120595

119870(119911 119905) cos

120579

0

2

119890

119894(12059302) 119870 = minus120583

119861119861

1015840

0

120595

minus(119911 119905) = 120595

119870(119911 119905) 119894 sin

120579

0

2

119890

minus119894(12059302) 119870 = +120583

119861119861

1015840

0

(A5)

120590

2

119905= 120590

2

0+ (ℏ1199052119898120590

0)

2and

120595

119870(119911 119905) = (2120587120590

2

119905)

minus14

119890

minus(119911+11987011990522119898)2

41205902

119905

times exp 119894

ℏ

[

[

minus

ℏ

2

tanminus1(

ℏ119905

2119898120590

2

0

) minus 119870119905119911 minus

119870

2119905

3

6119898

+

(119911 + 119870119905

22119898)

2

ℏ

2119905

2

8119898120590

2

0120590

2

119905

]

]

(A6)



0= 4 times

10

minus11 m ≪ 120590

0= 10


119905≃ 120590

0and

120595

119870(119911 119905)

≃ (2120587120590

2

0)

minus14

119890

minus(119911+11987011990522119898)2

41205902

0 exp 119894

ℏ

[minus119870119905119911 minus

119870

2119905

3

6119898

]

(A7)


Ψ (119911 Δ119905) = (

120595

+(119911 Δ119905)

120595

minus(119911 Δ119905)

) (A8)

with

120595

+(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911minus119911Δ)241205902

0)+(119894ℏ)119898119906119911 cos

120579

0

2

119890

119894120593+

120595

minus(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911+119911Δ)241205902

0)minus(119894ℏ)119898119906119911

119894 sin120579

0

2

119890

119894120593minus

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

119906 =

120583

0119861

1015840

0(Δ119905)

119898

120593

+=

120593

0

2

minus

120583

119861119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

120593

minus= minus

120593

0

2

+

120583

0119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

(A9)




minus


plusmn(119909 119911 119905) =

120595

119909(119909 119905)120595


have 120595119909(119909 119905) ≃ (2120587120590

2

0)

minus14119890

minus119909241205902

0 and

120595

+(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

cos120579

0

2


0+(119894ℏ)(119898119906119911minus(12)119898119906

2119905+ℏ120593+)

120595

minus(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

119894 sin120579

0

2

times expminus(119911+119911Δ+119906119905)241205902

0+(119894ℏ)(minus119898119906119911minus(12)119898119906

2119905+ℏ120593minus)

(A10)




References



























60moleculesrdquo



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




02 120583m02 120583m

08 120583m10120583m

35 cm35 cm

y

z

x

S



minus119910221205732






2= 35 cm



0(119910)


1= 119889

1V ≃ 210

minus11 slt 119905 lt 119905

1+ 119889



119860(119910 119905) + Ψ

119861(119910 119905)

70 cm

10120583m


35 cm

3120583m


with Ψ

119860(119910 119905) = int

119860119870(119910 119905 119910

119886 119905

1)Ψ(119910

119886 119905

1)119889119910

119886 Ψ

119861(119910 119905) =

int

119861119870(119910 119905 119910

119887 119905

1)Ψ(119910

119887 119905

1)119889119910

119887 and 119870(119910 119905 119910

120572 119905

1) = (1198982119894ℏ(119905 minus

119905

1))

12119890

119894119898(119910minus119910120572)22ℏ(119905minus1199051)



119860+ Ψ

119861|


119860|

2+ |Ψ

119861|

2)


minus1 minus05

(120583m)0 05 1

(a) 035mm

minus1 minus05

(120583m)0 05 1

(b) 35mm

minus2 minus1

(120583m)0 1 2

(c) 35 cm

minus10 minus5

(120583m)0 5 10

(d) 35 cm


119861|


2+ |Ψ

119861|











minus4

minus3

minus10

minus2

minus1

(120583m)

(cm)0 10 20 30 35

0

1

2

3

4


minus08

minus06

minus04

minus02

minus1

(120583m)

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

0

02

04

06

08

1











Ψ

0(119911) = (2120587120590

2

0)

minus12

119890

minus119911241205902

0(

cos120579

0

2

119890

minus119894(12059302)

sin120579

0

2

119890

119894(12059302)

) (16)


Ψ

0= (

cos120579

0

2

119890

minus119894(12059302)

sin120579

0

2

119890

119894(12059302)

) (17)




minus08

minus06

minus04

minus02

minus1

(120583m)


h10 h100

h1000 h10000

0 10 20 30

0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1


x

y

z

y = t

D

A1

P1

Δl( = 500ms)T = 1000∘K

E

TN+

Nminus





1




0= 10





120593





0and the 120593


0is




x

y

z

1205930

1205790

|minus⟩

|+⟩


0are the





119894ℏ(

120597120595

+

120597119905

120597120595

minus

120597119905

) = minus

ℏ

2

2119898

Δ(

120595

+

120595

minus

) + 120583

119861B120590(120595+

120595

minus

) (18)

where 120583

119861= 119890ℏ2119898


120590 = (120590

119909 120590

119910 120590



119909= 119861

1015840

0119909 119861

119910= 0 and 119861

119911= 119861

0minus 119861

1015840

0119911 with

119861

0= 5 Tesla 1198611015840

0= |120597119861120597119911| = 10





Ψ (119911 119905) ≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911minus(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3+1205831198611198610119905+(ℏ12059302))ℏ)

119894 sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((minus1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3minus1205831198611198610119905minus(ℏ12059302))ℏ)

) (19)


Ψ (119911 119905 + Δ119905)

≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890


0119890

119894((119898119906119911+ℏ120593+)ℏ)

sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+119911Δ+119906119905)241205902

0119890

119894((minus119898119906119911+ℏ120593minus)ℏ)

)

(20)

where

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

= 10

minus5 m 119906 =

120583

119861119861

1015840

0(Δ119905)

119898

= 1ms(21)



electromagnet

120588

1205790(119911 119905 + Δ119905) ≃ (2120587120590

2

0)

minus12

(cos2120579

0

2

119890


0

+sin2 1205790

2

119890

minus(119911+119911Δ+119906119905)221205902

0)

(22)






119905

119863≃

3120590

0minus 119911

Δ

119906

= 3 times 10

minus4 s (23)


120588

119878(119905) = (

int

1003816

1003816

1003816

1003816

120595

+(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911 int120595

+(119911 119905) 120595

lowast

minus(119911 119905) 119889119911

int120595

minus(119911 119905) 120595

lowast

+(119911 119905) 119889119911 int

1003816

1003816

1003816

1003816

120595

minus(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911

)

(24)


minus06

(mm)minus06

(mm)minus06

(mm)minus06

(mm)

0 cm 6 cm 16 cm 21 cm

0 06 0 06 0 06 0 06



minus1

z(m

m)

x (mm)0 1 2 3 4 5

0

1N+

Nminus


For 119905 ge 119905


+(119911 119905 + Δ119905)120595

minus(119911 119905 + Δ119905) is null


120588

119878(119905 + Δ119905) = (2120587120590

2

0)

minus1

(

cos2120579

0

2

0

0 sin2120579

0

2

) (25)


1


+ the term 120595



0) one of the


119911 Ψ( 119905 +

Δ119905) ≃ (2120587120590

2

0)

minus14 cos(12057902)119890


0119890

119894((1198981199061+ℏ120593+)ℏ)(

1

0)

Then we have 119878119911Ψ = (ℏ2)120590

119911Ψ = +(ℏ2)Ψ

If isin 119873




1) the


(2120587120590

2

0)

minus14 sin(12057902)119890

minus(2+119911Δ+119906119905)241205902

0119890

119894((minus1198981199062+ℏ120593minus)ℏ)(

0

1) Then

we have 119878

119911Ψ = (ℏ2)120590




sin2(120579





1




1 at this


1










0= 1205873 120593






119905


0in the spinor



1205790 and minus1205872 if 1199110lt

119911

1205790 with

119911

1205790= 120590

0119865

minus1(sin2 1205790

2

) (26)




0 5 10 15 20

0

02

04

06

08

minus04

minus02

y (cm)

z(m

m)






0 120593





5 EPR-B Experiment


0 5 10 15 20

0

02

04

06

minus06

minus04

minus02

y (cm)

z(m

m)


0 120593





Ψ

0(r

119860 r

119861) =

1

radic2

119891 (r119860) 119891 (r

119861) (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(27)


Ψ

0(r

119860 r

119861) =

1

radic2

(

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩) (28)





x

y

z

z

z

z998400

x

z z998400

x

yz

x998400z998400

120575


O Atom AAtom B

EAEB



119860and E

119861




(2120587120590

2

0)

minus12119890

minus(1199092+1199112)41205902




and 120590119911119861 120590

119911119860|plusmn


119860⟩ 120590

119911119861|plusmn

119861⟩ =

plusmn|plusmn




119860 r

119861 119905) of



119860and




Ψ

119886119887(r

119860 r


page 417]

119894ℏ

120597Ψ

119886119887

120597119905

= (minus

ℏ

2

2119898

Δ

119860minus

ℏ

2

2119898

Δ

119861)Ψ

119886119887+ 120583119861

E119860119895(120590

119895)

119886

119888Ψ

119888119887

+ 120583119861

E119861119895(120590

119895)

119887

119889Ψ

119886119889

(29)


Ψ

119886119887(r

119860 r

119861 0) = Ψ

119886119887

0(r

119860 r

119861)

(30)

where Ψ119886119887

0(r

119860 r










[68]

Ψ (r119860 r

119861 119905

0+ Δ119905 + 119905)

=

1

radic2

119891 (r119861)

times (119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus 119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(31)

with

119891

plusmn(r 119905) ≃ 119891 (119909 119911 ∓ 119911

Δ∓ 119906119905) 119890

119894((plusmn119898119906119911ℏ)+120593plusmn(119905))

(32)



119861 119905

0+ Δ119905 + 119905) is found by


119860 r

119861 119905

0+ Δ119905 + 119905)Ψ(r

119860 r

119861 119905

0+ Δ119905 + 119905) on 119909

119860

and 119909119861

120588 (119911

119860 119911

119861 119905

0+ Δ119905 + 119905)

= ((2120587120590

2

0)

minus12

119890

minus(119911119861)221205902

0)

times ((2120587120590

2

0)

minus12

times

1

2

(119890


0+ 119890

minus(119911119860+119911Δ+119906119905)221205902

0))

(33)







0and 119905

0+Δ119905+

119905



E119860during 119905

0+ Δ119905 + 119905

119863and 119905

0+ 2(Δ119905 + 119905

119863)


0+ Δ119905 + 119905

119863


0+Δ119905+119905



Ψ

119861plusmn119860(r

119861 119905

0+ Δ119905 + 119905

1) = 119891 (r

119861)

1003816

1003816

1003816

1003816

∓

119861⟩ (34)


119863) yields



0(r

119860 120579

119860

0 120593

119860

0) = 119891(r

119860)

(cos(12057911986002)|+

119860⟩ + sin(120579119860

02)119890

119894120593119860

0|minus

119860⟩) and Ψ

119861

0(r

119861 120579

119861

0 120593

119861

0) =

119891(r119861)(cos(120579119861

02)|+

119861⟩ + sin(120579119861

02)119890

119894120593119861

0|minus

119861⟩) with 120579119861

0= 120587 minus 120579

119860

0

and 120593119861

0= 120593

119860



Ψ

0(r

119860 120579

119860 120593

119860 r

119861 120579

119861 120593

119861)

= minus119890

119894120593119860

119891 (r119860) 119891 (r

119861) times (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(35)



119860

0(r

119860 120579

119860

0 120593

119860

0) and Ψ119861

0(r

119861 120579

119861

0 120593

119861



119860 120579

119860 120593

119860 r

119861 120579

119861 120593





Ψ

119860(r

119860 119905

0+ Δ119905 + 119905)

= cos120579

119860

0

2

119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩ + sin

120579

119860

0

2

119890

119894120593119860

0119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

(36)


0 120593

119860

0) For


119860


0)



119860(119905) 119905)) [48]


119861(119905) = V

119910119905 119911

119861(119905) = 119911

119861

0 and 119909

119861(119905) = 119909

119861

0 By


119860(119911

119860(119905) 119905) and 120593

119861(119905) =

120593(119911



Ψ

119861(r

119861 119905

0+ Δ119905 + 119905)

= 119891 (r119861) (cos 120579

119861(119905)

2

1003816

1003816

1003816

1003816

+

119861⟩ + sin 120579

119861(119905)

2

119890

119894120593119861(119905) 10038161003816

1003816

1003816

minus

119861⟩)

(37)



119863


119863) = 120587



1015840)


1015840119861(119905

0+ Δ119905 + 119905





0+ 2(Δ119905 + 119905

119863) gives









6 Conclusion







Appendix



119909 0 119861




component

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) plusmn 120583

119861(119861

0minus 119861

1015840

0119911)120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905)

(A1)


120595


119894120583

119861119861

0119905

ℏ

)120595

plusmn(119909 119911 119905)

(A2)

(A1) becomes

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905) exp(plusmn119894

2120583

119861119861

0119905

ℏ

)

(A3)


119871= 2120583

119861119861

0ℏ = 1 4 times 10

11 sminus1 Since |1198610| ≫ |119861

1015840

0119911|

and |1198610| ≫ |119861

1015840



119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905) = minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

(A4)


0



on 119909 120595plusmn(119909 119911 119905) equiv 120595



5 sWe obtain

120595

+(119911 119905) = 120595

119870(119911 119905) cos

120579

0

2

119890

119894(12059302) 119870 = minus120583

119861119861

1015840

0

120595

minus(119911 119905) = 120595

119870(119911 119905) 119894 sin

120579

0

2

119890

minus119894(12059302) 119870 = +120583

119861119861

1015840

0

(A5)

120590

2

119905= 120590

2

0+ (ℏ1199052119898120590

0)

2and

120595

119870(119911 119905) = (2120587120590

2

119905)

minus14

119890

minus(119911+11987011990522119898)2

41205902

119905

times exp 119894

ℏ

[

[

minus

ℏ

2

tanminus1(

ℏ119905

2119898120590

2

0

) minus 119870119905119911 minus

119870

2119905

3

6119898

+

(119911 + 119870119905

22119898)

2

ℏ

2119905

2

8119898120590

2

0120590

2

119905

]

]

(A6)



0= 4 times

10

minus11 m ≪ 120590

0= 10


119905≃ 120590

0and

120595

119870(119911 119905)

≃ (2120587120590

2

0)

minus14

119890

minus(119911+11987011990522119898)2

41205902

0 exp 119894

ℏ

[minus119870119905119911 minus

119870

2119905

3

6119898

]

(A7)


Ψ (119911 Δ119905) = (

120595

+(119911 Δ119905)

120595

minus(119911 Δ119905)

) (A8)

with

120595

+(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911minus119911Δ)241205902

0)+(119894ℏ)119898119906119911 cos

120579

0

2

119890

119894120593+

120595

minus(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911+119911Δ)241205902

0)minus(119894ℏ)119898119906119911

119894 sin120579

0

2

119890

119894120593minus

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

119906 =

120583

0119861

1015840

0(Δ119905)

119898

120593

+=

120593

0

2

minus

120583

119861119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

120593

minus= minus

120593

0

2

+

120583

0119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

(A9)




minus


plusmn(119909 119911 119905) =

120595

119909(119909 119905)120595


have 120595119909(119909 119905) ≃ (2120587120590

2

0)

minus14119890

minus119909241205902

0 and

120595

+(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

cos120579

0

2


0+(119894ℏ)(119898119906119911minus(12)119898119906

2119905+ℏ120593+)

120595

minus(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

119894 sin120579

0

2

times expminus(119911+119911Δ+119906119905)241205902

0+(119894ℏ)(minus119898119906119911minus(12)119898119906

2119905+ℏ120593minus)

(A10)




References



























60moleculesrdquo



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus1 minus05

(120583m)0 05 1

(a) 035mm

minus1 minus05

(120583m)0 05 1

(b) 35mm

minus2 minus1

(120583m)0 1 2

(c) 35 cm

minus10 minus5

(120583m)0 5 10

(d) 35 cm


119861|


2+ |Ψ

119861|











minus4

minus3

minus10

minus2

minus1

(120583m)

(cm)0 10 20 30 35

0

1

2

3

4


minus08

minus06

minus04

minus02

minus1

(120583m)

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

0

02

04

06

08

1











Ψ

0(119911) = (2120587120590

2

0)

minus12

119890

minus119911241205902

0(

cos120579

0

2

119890

minus119894(12059302)

sin120579

0

2

119890

119894(12059302)

) (16)


Ψ

0= (

cos120579

0

2

119890

minus119894(12059302)

sin120579

0

2

119890

119894(12059302)

) (17)




minus08

minus06

minus04

minus02

minus1

(120583m)


h10 h100

h1000 h10000

0 10 20 30

0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1


x

y

z

y = t

D

A1

P1

Δl( = 500ms)T = 1000∘K

E

TN+

Nminus





1




0= 10





120593





0and the 120593


0is




x

y

z

1205930

1205790

|minus⟩

|+⟩


0are the





119894ℏ(

120597120595

+

120597119905

120597120595

minus

120597119905

) = minus

ℏ

2

2119898

Δ(

120595

+

120595

minus

) + 120583

119861B120590(120595+

120595

minus

) (18)

where 120583

119861= 119890ℏ2119898


120590 = (120590

119909 120590

119910 120590



119909= 119861

1015840

0119909 119861

119910= 0 and 119861

119911= 119861

0minus 119861

1015840

0119911 with

119861

0= 5 Tesla 1198611015840

0= |120597119861120597119911| = 10





Ψ (119911 119905) ≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911minus(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3+1205831198611198610119905+(ℏ12059302))ℏ)

119894 sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((minus1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3minus1205831198611198610119905minus(ℏ12059302))ℏ)

) (19)


Ψ (119911 119905 + Δ119905)

≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890


0119890

119894((119898119906119911+ℏ120593+)ℏ)

sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+119911Δ+119906119905)241205902

0119890

119894((minus119898119906119911+ℏ120593minus)ℏ)

)

(20)

where

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

= 10

minus5 m 119906 =

120583

119861119861

1015840

0(Δ119905)

119898

= 1ms(21)



electromagnet

120588

1205790(119911 119905 + Δ119905) ≃ (2120587120590

2

0)

minus12

(cos2120579

0

2

119890


0

+sin2 1205790

2

119890

minus(119911+119911Δ+119906119905)221205902

0)

(22)






119905

119863≃

3120590

0minus 119911

Δ

119906

= 3 times 10

minus4 s (23)


120588

119878(119905) = (

int

1003816

1003816

1003816

1003816

120595

+(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911 int120595

+(119911 119905) 120595

lowast

minus(119911 119905) 119889119911

int120595

minus(119911 119905) 120595

lowast

+(119911 119905) 119889119911 int

1003816

1003816

1003816

1003816

120595

minus(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911

)

(24)


minus06

(mm)minus06

(mm)minus06

(mm)minus06

(mm)

0 cm 6 cm 16 cm 21 cm

0 06 0 06 0 06 0 06



minus1

z(m

m)

x (mm)0 1 2 3 4 5

0

1N+

Nminus


For 119905 ge 119905


+(119911 119905 + Δ119905)120595

minus(119911 119905 + Δ119905) is null


120588

119878(119905 + Δ119905) = (2120587120590

2

0)

minus1

(

cos2120579

0

2

0

0 sin2120579

0

2

) (25)


1


+ the term 120595



0) one of the


119911 Ψ( 119905 +

Δ119905) ≃ (2120587120590

2

0)

minus14 cos(12057902)119890


0119890

119894((1198981199061+ℏ120593+)ℏ)(

1

0)

Then we have 119878119911Ψ = (ℏ2)120590

119911Ψ = +(ℏ2)Ψ

If isin 119873




1) the


(2120587120590

2

0)

minus14 sin(12057902)119890

minus(2+119911Δ+119906119905)241205902

0119890

119894((minus1198981199062+ℏ120593minus)ℏ)(

0

1) Then

we have 119878

119911Ψ = (ℏ2)120590




sin2(120579





1




1 at this


1










0= 1205873 120593






119905


0in the spinor



1205790 and minus1205872 if 1199110lt

119911

1205790 with

119911

1205790= 120590

0119865

minus1(sin2 1205790

2

) (26)




0 5 10 15 20

0

02

04

06

08

minus04

minus02

y (cm)

z(m

m)






0 120593





5 EPR-B Experiment


0 5 10 15 20

0

02

04

06

minus06

minus04

minus02

y (cm)

z(m

m)


0 120593





Ψ

0(r

119860 r

119861) =

1

radic2

119891 (r119860) 119891 (r

119861) (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(27)


Ψ

0(r

119860 r

119861) =

1

radic2

(

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩) (28)





x

y

z

z

z

z998400

x

z z998400

x

yz

x998400z998400

120575


O Atom AAtom B

EAEB



119860and E

119861




(2120587120590

2

0)

minus12119890

minus(1199092+1199112)41205902




and 120590119911119861 120590

119911119860|plusmn


119860⟩ 120590

119911119861|plusmn

119861⟩ =

plusmn|plusmn




119860 r

119861 119905) of



119860and




Ψ

119886119887(r

119860 r


page 417]

119894ℏ

120597Ψ

119886119887

120597119905

= (minus

ℏ

2

2119898

Δ

119860minus

ℏ

2

2119898

Δ

119861)Ψ

119886119887+ 120583119861

E119860119895(120590

119895)

119886

119888Ψ

119888119887

+ 120583119861

E119861119895(120590

119895)

119887

119889Ψ

119886119889

(29)


Ψ

119886119887(r

119860 r

119861 0) = Ψ

119886119887

0(r

119860 r

119861)

(30)

where Ψ119886119887

0(r

119860 r










[68]

Ψ (r119860 r

119861 119905

0+ Δ119905 + 119905)

=

1

radic2

119891 (r119861)

times (119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus 119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(31)

with

119891

plusmn(r 119905) ≃ 119891 (119909 119911 ∓ 119911

Δ∓ 119906119905) 119890

119894((plusmn119898119906119911ℏ)+120593plusmn(119905))

(32)



119861 119905

0+ Δ119905 + 119905) is found by


119860 r

119861 119905

0+ Δ119905 + 119905)Ψ(r

119860 r

119861 119905

0+ Δ119905 + 119905) on 119909

119860

and 119909119861

120588 (119911

119860 119911

119861 119905

0+ Δ119905 + 119905)

= ((2120587120590

2

0)

minus12

119890

minus(119911119861)221205902

0)

times ((2120587120590

2

0)

minus12

times

1

2

(119890


0+ 119890

minus(119911119860+119911Δ+119906119905)221205902

0))

(33)







0and 119905

0+Δ119905+

119905



E119860during 119905

0+ Δ119905 + 119905

119863and 119905

0+ 2(Δ119905 + 119905

119863)


0+ Δ119905 + 119905

119863


0+Δ119905+119905



Ψ

119861plusmn119860(r

119861 119905

0+ Δ119905 + 119905

1) = 119891 (r

119861)

1003816

1003816

1003816

1003816

∓

119861⟩ (34)


119863) yields



0(r

119860 120579

119860

0 120593

119860

0) = 119891(r

119860)

(cos(12057911986002)|+

119860⟩ + sin(120579119860

02)119890

119894120593119860

0|minus

119860⟩) and Ψ

119861

0(r

119861 120579

119861

0 120593

119861

0) =

119891(r119861)(cos(120579119861

02)|+

119861⟩ + sin(120579119861

02)119890

119894120593119861

0|minus

119861⟩) with 120579119861

0= 120587 minus 120579

119860

0

and 120593119861

0= 120593

119860



Ψ

0(r

119860 120579

119860 120593

119860 r

119861 120579

119861 120593

119861)

= minus119890

119894120593119860

119891 (r119860) 119891 (r

119861) times (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(35)



119860

0(r

119860 120579

119860

0 120593

119860

0) and Ψ119861

0(r

119861 120579

119861

0 120593

119861



119860 120579

119860 120593

119860 r

119861 120579

119861 120593





Ψ

119860(r

119860 119905

0+ Δ119905 + 119905)

= cos120579

119860

0

2

119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩ + sin

120579

119860

0

2

119890

119894120593119860

0119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

(36)


0 120593

119860

0) For


119860


0)



119860(119905) 119905)) [48]


119861(119905) = V

119910119905 119911

119861(119905) = 119911

119861

0 and 119909

119861(119905) = 119909

119861

0 By


119860(119911

119860(119905) 119905) and 120593

119861(119905) =

120593(119911



Ψ

119861(r

119861 119905

0+ Δ119905 + 119905)

= 119891 (r119861) (cos 120579

119861(119905)

2

1003816

1003816

1003816

1003816

+

119861⟩ + sin 120579

119861(119905)

2

119890

119894120593119861(119905) 10038161003816

1003816

1003816

minus

119861⟩)

(37)



119863


119863) = 120587



1015840)


1015840119861(119905

0+ Δ119905 + 119905





0+ 2(Δ119905 + 119905

119863) gives









6 Conclusion







Appendix



119909 0 119861




component

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) plusmn 120583

119861(119861

0minus 119861

1015840

0119911)120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905)

(A1)


120595


119894120583

119861119861

0119905

ℏ

)120595

plusmn(119909 119911 119905)

(A2)

(A1) becomes

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905) exp(plusmn119894

2120583

119861119861

0119905

ℏ

)

(A3)


119871= 2120583

119861119861

0ℏ = 1 4 times 10

11 sminus1 Since |1198610| ≫ |119861

1015840

0119911|

and |1198610| ≫ |119861

1015840



119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905) = minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

(A4)


0



on 119909 120595plusmn(119909 119911 119905) equiv 120595



5 sWe obtain

120595

+(119911 119905) = 120595

119870(119911 119905) cos

120579

0

2

119890

119894(12059302) 119870 = minus120583

119861119861

1015840

0

120595

minus(119911 119905) = 120595

119870(119911 119905) 119894 sin

120579

0

2

119890

minus119894(12059302) 119870 = +120583

119861119861

1015840

0

(A5)

120590

2

119905= 120590

2

0+ (ℏ1199052119898120590

0)

2and

120595

119870(119911 119905) = (2120587120590

2

119905)

minus14

119890

minus(119911+11987011990522119898)2

41205902

119905

times exp 119894

ℏ

[

[

minus

ℏ

2

tanminus1(

ℏ119905

2119898120590

2

0

) minus 119870119905119911 minus

119870

2119905

3

6119898

+

(119911 + 119870119905

22119898)

2

ℏ

2119905

2

8119898120590

2

0120590

2

119905

]

]

(A6)



0= 4 times

10

minus11 m ≪ 120590

0= 10


119905≃ 120590

0and

120595

119870(119911 119905)

≃ (2120587120590

2

0)

minus14

119890

minus(119911+11987011990522119898)2

41205902

0 exp 119894

ℏ

[minus119870119905119911 minus

119870

2119905

3

6119898

]

(A7)


Ψ (119911 Δ119905) = (

120595

+(119911 Δ119905)

120595

minus(119911 Δ119905)

) (A8)

with

120595

+(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911minus119911Δ)241205902

0)+(119894ℏ)119898119906119911 cos

120579

0

2

119890

119894120593+

120595

minus(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911+119911Δ)241205902

0)minus(119894ℏ)119898119906119911

119894 sin120579

0

2

119890

119894120593minus

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

119906 =

120583

0119861

1015840

0(Δ119905)

119898

120593

+=

120593

0

2

minus

120583

119861119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

120593

minus= minus

120593

0

2

+

120583

0119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

(A9)




minus


plusmn(119909 119911 119905) =

120595

119909(119909 119905)120595


have 120595119909(119909 119905) ≃ (2120587120590

2

0)

minus14119890

minus119909241205902

0 and

120595

+(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

cos120579

0

2


0+(119894ℏ)(119898119906119911minus(12)119898119906

2119905+ℏ120593+)

120595

minus(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

119894 sin120579

0

2

times expminus(119911+119911Δ+119906119905)241205902

0+(119894ℏ)(minus119898119906119911minus(12)119898119906

2119905+ℏ120593minus)

(A10)




References



























60moleculesrdquo



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





minus4

minus3

minus10

minus2

minus1

(120583m)

(cm)0 10 20 30 35

0

1

2

3

4


minus08

minus06

minus04

minus02

minus1

(120583m)

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

0

02

04

06

08

1











Ψ

0(119911) = (2120587120590

2

0)

minus12

119890

minus119911241205902

0(

cos120579

0

2

119890

minus119894(12059302)

sin120579

0

2

119890

119894(12059302)

) (16)


Ψ

0= (

cos120579

0

2

119890

minus119894(12059302)

sin120579

0

2

119890

119894(12059302)

) (17)




minus08

minus06

minus04

minus02

minus1

(120583m)


h10 h100

h1000 h10000

0 10 20 30

0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1


x

y

z

y = t

D

A1

P1

Δl( = 500ms)T = 1000∘K

E

TN+

Nminus





1




0= 10





120593





0and the 120593


0is




x

y

z

1205930

1205790

|minus⟩

|+⟩


0are the





119894ℏ(

120597120595

+

120597119905

120597120595

minus

120597119905

) = minus

ℏ

2

2119898

Δ(

120595

+

120595

minus

) + 120583

119861B120590(120595+

120595

minus

) (18)

where 120583

119861= 119890ℏ2119898


120590 = (120590

119909 120590

119910 120590



119909= 119861

1015840

0119909 119861

119910= 0 and 119861

119911= 119861

0minus 119861

1015840

0119911 with

119861

0= 5 Tesla 1198611015840

0= |120597119861120597119911| = 10





Ψ (119911 119905) ≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911minus(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3+1205831198611198610119905+(ℏ12059302))ℏ)

119894 sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((minus1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3minus1205831198611198610119905minus(ℏ12059302))ℏ)

) (19)


Ψ (119911 119905 + Δ119905)

≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890


0119890

119894((119898119906119911+ℏ120593+)ℏ)

sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+119911Δ+119906119905)241205902

0119890

119894((minus119898119906119911+ℏ120593minus)ℏ)

)

(20)

where

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

= 10

minus5 m 119906 =

120583

119861119861

1015840

0(Δ119905)

119898

= 1ms(21)



electromagnet

120588

1205790(119911 119905 + Δ119905) ≃ (2120587120590

2

0)

minus12

(cos2120579

0

2

119890


0

+sin2 1205790

2

119890

minus(119911+119911Δ+119906119905)221205902

0)

(22)






119905

119863≃

3120590

0minus 119911

Δ

119906

= 3 times 10

minus4 s (23)


120588

119878(119905) = (

int

1003816

1003816

1003816

1003816

120595

+(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911 int120595

+(119911 119905) 120595

lowast

minus(119911 119905) 119889119911

int120595

minus(119911 119905) 120595

lowast

+(119911 119905) 119889119911 int

1003816

1003816

1003816

1003816

120595

minus(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911

)

(24)


minus06

(mm)minus06

(mm)minus06

(mm)minus06

(mm)

0 cm 6 cm 16 cm 21 cm

0 06 0 06 0 06 0 06



minus1

z(m

m)

x (mm)0 1 2 3 4 5

0

1N+

Nminus


For 119905 ge 119905


+(119911 119905 + Δ119905)120595

minus(119911 119905 + Δ119905) is null


120588

119878(119905 + Δ119905) = (2120587120590

2

0)

minus1

(

cos2120579

0

2

0

0 sin2120579

0

2

) (25)


1


+ the term 120595



0) one of the


119911 Ψ( 119905 +

Δ119905) ≃ (2120587120590

2

0)

minus14 cos(12057902)119890


0119890

119894((1198981199061+ℏ120593+)ℏ)(

1

0)

Then we have 119878119911Ψ = (ℏ2)120590

119911Ψ = +(ℏ2)Ψ

If isin 119873




1) the


(2120587120590

2

0)

minus14 sin(12057902)119890

minus(2+119911Δ+119906119905)241205902

0119890

119894((minus1198981199062+ℏ120593minus)ℏ)(

0

1) Then

we have 119878

119911Ψ = (ℏ2)120590




sin2(120579





1




1 at this


1










0= 1205873 120593






119905


0in the spinor



1205790 and minus1205872 if 1199110lt

119911

1205790 with

119911

1205790= 120590

0119865

minus1(sin2 1205790

2

) (26)




0 5 10 15 20

0

02

04

06

08

minus04

minus02

y (cm)

z(m

m)






0 120593





5 EPR-B Experiment


0 5 10 15 20

0

02

04

06

minus06

minus04

minus02

y (cm)

z(m

m)


0 120593





Ψ

0(r

119860 r

119861) =

1

radic2

119891 (r119860) 119891 (r

119861) (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(27)


Ψ

0(r

119860 r

119861) =

1

radic2

(

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩) (28)





x

y

z

z

z

z998400

x

z z998400

x

yz

x998400z998400

120575


O Atom AAtom B

EAEB



119860and E

119861




(2120587120590

2

0)

minus12119890

minus(1199092+1199112)41205902




and 120590119911119861 120590

119911119860|plusmn


119860⟩ 120590

119911119861|plusmn

119861⟩ =

plusmn|plusmn




119860 r

119861 119905) of



119860and




Ψ

119886119887(r

119860 r


page 417]

119894ℏ

120597Ψ

119886119887

120597119905

= (minus

ℏ

2

2119898

Δ

119860minus

ℏ

2

2119898

Δ

119861)Ψ

119886119887+ 120583119861

E119860119895(120590

119895)

119886

119888Ψ

119888119887

+ 120583119861

E119861119895(120590

119895)

119887

119889Ψ

119886119889

(29)


Ψ

119886119887(r

119860 r

119861 0) = Ψ

119886119887

0(r

119860 r

119861)

(30)

where Ψ119886119887

0(r

119860 r










[68]

Ψ (r119860 r

119861 119905

0+ Δ119905 + 119905)

=

1

radic2

119891 (r119861)

times (119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus 119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(31)

with

119891

plusmn(r 119905) ≃ 119891 (119909 119911 ∓ 119911

Δ∓ 119906119905) 119890

119894((plusmn119898119906119911ℏ)+120593plusmn(119905))

(32)



119861 119905

0+ Δ119905 + 119905) is found by


119860 r

119861 119905

0+ Δ119905 + 119905)Ψ(r

119860 r

119861 119905

0+ Δ119905 + 119905) on 119909

119860

and 119909119861

120588 (119911

119860 119911

119861 119905

0+ Δ119905 + 119905)

= ((2120587120590

2

0)

minus12

119890

minus(119911119861)221205902

0)

times ((2120587120590

2

0)

minus12

times

1

2

(119890


0+ 119890

minus(119911119860+119911Δ+119906119905)221205902

0))

(33)







0and 119905

0+Δ119905+

119905



E119860during 119905

0+ Δ119905 + 119905

119863and 119905

0+ 2(Δ119905 + 119905

119863)


0+ Δ119905 + 119905

119863


0+Δ119905+119905



Ψ

119861plusmn119860(r

119861 119905

0+ Δ119905 + 119905

1) = 119891 (r

119861)

1003816

1003816

1003816

1003816

∓

119861⟩ (34)


119863) yields



0(r

119860 120579

119860

0 120593

119860

0) = 119891(r

119860)

(cos(12057911986002)|+

119860⟩ + sin(120579119860

02)119890

119894120593119860

0|minus

119860⟩) and Ψ

119861

0(r

119861 120579

119861

0 120593

119861

0) =

119891(r119861)(cos(120579119861

02)|+

119861⟩ + sin(120579119861

02)119890

119894120593119861

0|minus

119861⟩) with 120579119861

0= 120587 minus 120579

119860

0

and 120593119861

0= 120593

119860



Ψ

0(r

119860 120579

119860 120593

119860 r

119861 120579

119861 120593

119861)

= minus119890

119894120593119860

119891 (r119860) 119891 (r

119861) times (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(35)



119860

0(r

119860 120579

119860

0 120593

119860

0) and Ψ119861

0(r

119861 120579

119861

0 120593

119861



119860 120579

119860 120593

119860 r

119861 120579

119861 120593





Ψ

119860(r

119860 119905

0+ Δ119905 + 119905)

= cos120579

119860

0

2

119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩ + sin

120579

119860

0

2

119890

119894120593119860

0119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

(36)


0 120593

119860

0) For


119860


0)



119860(119905) 119905)) [48]


119861(119905) = V

119910119905 119911

119861(119905) = 119911

119861

0 and 119909

119861(119905) = 119909

119861

0 By


119860(119911

119860(119905) 119905) and 120593

119861(119905) =

120593(119911



Ψ

119861(r

119861 119905

0+ Δ119905 + 119905)

= 119891 (r119861) (cos 120579

119861(119905)

2

1003816

1003816

1003816

1003816

+

119861⟩ + sin 120579

119861(119905)

2

119890

119894120593119861(119905) 10038161003816

1003816

1003816

minus

119861⟩)

(37)



119863


119863) = 120587



1015840)


1015840119861(119905

0+ Δ119905 + 119905





0+ 2(Δ119905 + 119905

119863) gives









6 Conclusion







Appendix



119909 0 119861




component

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) plusmn 120583

119861(119861

0minus 119861

1015840

0119911)120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905)

(A1)


120595


119894120583

119861119861

0119905

ℏ

)120595

plusmn(119909 119911 119905)

(A2)

(A1) becomes

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905) exp(plusmn119894

2120583

119861119861

0119905

ℏ

)

(A3)


119871= 2120583

119861119861

0ℏ = 1 4 times 10

11 sminus1 Since |1198610| ≫ |119861

1015840

0119911|

and |1198610| ≫ |119861

1015840



119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905) = minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

(A4)


0



on 119909 120595plusmn(119909 119911 119905) equiv 120595



5 sWe obtain

120595

+(119911 119905) = 120595

119870(119911 119905) cos

120579

0

2

119890

119894(12059302) 119870 = minus120583

119861119861

1015840

0

120595

minus(119911 119905) = 120595

119870(119911 119905) 119894 sin

120579

0

2

119890

minus119894(12059302) 119870 = +120583

119861119861

1015840

0

(A5)

120590

2

119905= 120590

2

0+ (ℏ1199052119898120590

0)

2and

120595

119870(119911 119905) = (2120587120590

2

119905)

minus14

119890

minus(119911+11987011990522119898)2

41205902

119905

times exp 119894

ℏ

[

[

minus

ℏ

2

tanminus1(

ℏ119905

2119898120590

2

0

) minus 119870119905119911 minus

119870

2119905

3

6119898

+

(119911 + 119870119905

22119898)

2

ℏ

2119905

2

8119898120590

2

0120590

2

119905

]

]

(A6)



0= 4 times

10

minus11 m ≪ 120590

0= 10


119905≃ 120590

0and

120595

119870(119911 119905)

≃ (2120587120590

2

0)

minus14

119890

minus(119911+11987011990522119898)2

41205902

0 exp 119894

ℏ

[minus119870119905119911 minus

119870

2119905

3

6119898

]

(A7)


Ψ (119911 Δ119905) = (

120595

+(119911 Δ119905)

120595

minus(119911 Δ119905)

) (A8)

with

120595

+(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911minus119911Δ)241205902

0)+(119894ℏ)119898119906119911 cos

120579

0

2

119890

119894120593+

120595

minus(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911+119911Δ)241205902

0)minus(119894ℏ)119898119906119911

119894 sin120579

0

2

119890

119894120593minus

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

119906 =

120583

0119861

1015840

0(Δ119905)

119898

120593

+=

120593

0

2

minus

120583

119861119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

120593

minus= minus

120593

0

2

+

120583

0119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

(A9)




minus


plusmn(119909 119911 119905) =

120595

119909(119909 119905)120595


have 120595119909(119909 119905) ≃ (2120587120590

2

0)

minus14119890

minus119909241205902

0 and

120595

+(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

cos120579

0

2


0+(119894ℏ)(119898119906119911minus(12)119898119906

2119905+ℏ120593+)

120595

minus(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

119894 sin120579

0

2

times expminus(119911+119911Δ+119906119905)241205902

0+(119894ℏ)(minus119898119906119911minus(12)119898119906

2119905+ℏ120593minus)

(A10)




References



























60moleculesrdquo



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus08

minus06

minus04

minus02

minus1

(120583m)


h10 h100

h1000 h10000

0 10 20 30

0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1

minus08

minus06

minus04

minus02

minus1

(120583m)


0

02

04

06

08

1


x

y

z

y = t

D

A1

P1

Δl( = 500ms)T = 1000∘K

E

TN+

Nminus





1




0= 10





120593





0and the 120593


0is




x

y

z

1205930

1205790

|minus⟩

|+⟩


0are the





119894ℏ(

120597120595

+

120597119905

120597120595

minus

120597119905

) = minus

ℏ

2

2119898

Δ(

120595

+

120595

minus

) + 120583

119861B120590(120595+

120595

minus

) (18)

where 120583

119861= 119890ℏ2119898


120590 = (120590

119909 120590

119910 120590



119909= 119861

1015840

0119909 119861

119910= 0 and 119861

119911= 119861

0minus 119861

1015840

0119911 with

119861

0= 5 Tesla 1198611015840

0= |120597119861120597119911| = 10





Ψ (119911 119905) ≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911minus(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3+1205831198611198610119905+(ℏ12059302))ℏ)

119894 sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((minus1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3minus1205831198611198610119905minus(ℏ12059302))ℏ)

) (19)


Ψ (119911 119905 + Δ119905)

≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890


0119890

119894((119898119906119911+ℏ120593+)ℏ)

sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+119911Δ+119906119905)241205902

0119890

119894((minus119898119906119911+ℏ120593minus)ℏ)

)

(20)

where

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

= 10

minus5 m 119906 =

120583

119861119861

1015840

0(Δ119905)

119898

= 1ms(21)



electromagnet

120588

1205790(119911 119905 + Δ119905) ≃ (2120587120590

2

0)

minus12

(cos2120579

0

2

119890


0

+sin2 1205790

2

119890

minus(119911+119911Δ+119906119905)221205902

0)

(22)






119905

119863≃

3120590

0minus 119911

Δ

119906

= 3 times 10

minus4 s (23)


120588

119878(119905) = (

int

1003816

1003816

1003816

1003816

120595

+(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911 int120595

+(119911 119905) 120595

lowast

minus(119911 119905) 119889119911

int120595

minus(119911 119905) 120595

lowast

+(119911 119905) 119889119911 int

1003816

1003816

1003816

1003816

120595

minus(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911

)

(24)


minus06

(mm)minus06

(mm)minus06

(mm)minus06

(mm)

0 cm 6 cm 16 cm 21 cm

0 06 0 06 0 06 0 06



minus1

z(m

m)

x (mm)0 1 2 3 4 5

0

1N+

Nminus


For 119905 ge 119905


+(119911 119905 + Δ119905)120595

minus(119911 119905 + Δ119905) is null


120588

119878(119905 + Δ119905) = (2120587120590

2

0)

minus1

(

cos2120579

0

2

0

0 sin2120579

0

2

) (25)


1


+ the term 120595



0) one of the


119911 Ψ( 119905 +

Δ119905) ≃ (2120587120590

2

0)

minus14 cos(12057902)119890


0119890

119894((1198981199061+ℏ120593+)ℏ)(

1

0)

Then we have 119878119911Ψ = (ℏ2)120590

119911Ψ = +(ℏ2)Ψ

If isin 119873




1) the


(2120587120590

2

0)

minus14 sin(12057902)119890

minus(2+119911Δ+119906119905)241205902

0119890

119894((minus1198981199062+ℏ120593minus)ℏ)(

0

1) Then

we have 119878

119911Ψ = (ℏ2)120590




sin2(120579





1




1 at this


1










0= 1205873 120593






119905


0in the spinor



1205790 and minus1205872 if 1199110lt

119911

1205790 with

119911

1205790= 120590

0119865

minus1(sin2 1205790

2

) (26)




0 5 10 15 20

0

02

04

06

08

minus04

minus02

y (cm)

z(m

m)






0 120593





5 EPR-B Experiment


0 5 10 15 20

0

02

04

06

minus06

minus04

minus02

y (cm)

z(m

m)


0 120593





Ψ

0(r

119860 r

119861) =

1

radic2

119891 (r119860) 119891 (r

119861) (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(27)


Ψ

0(r

119860 r

119861) =

1

radic2

(

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩) (28)





x

y

z

z

z

z998400

x

z z998400

x

yz

x998400z998400

120575


O Atom AAtom B

EAEB



119860and E

119861




(2120587120590

2

0)

minus12119890

minus(1199092+1199112)41205902




and 120590119911119861 120590

119911119860|plusmn


119860⟩ 120590

119911119861|plusmn

119861⟩ =

plusmn|plusmn




119860 r

119861 119905) of



119860and




Ψ

119886119887(r

119860 r


page 417]

119894ℏ

120597Ψ

119886119887

120597119905

= (minus

ℏ

2

2119898

Δ

119860minus

ℏ

2

2119898

Δ

119861)Ψ

119886119887+ 120583119861

E119860119895(120590

119895)

119886

119888Ψ

119888119887

+ 120583119861

E119861119895(120590

119895)

119887

119889Ψ

119886119889

(29)


Ψ

119886119887(r

119860 r

119861 0) = Ψ

119886119887

0(r

119860 r

119861)

(30)

where Ψ119886119887

0(r

119860 r










[68]

Ψ (r119860 r

119861 119905

0+ Δ119905 + 119905)

=

1

radic2

119891 (r119861)

times (119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus 119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(31)

with

119891

plusmn(r 119905) ≃ 119891 (119909 119911 ∓ 119911

Δ∓ 119906119905) 119890

119894((plusmn119898119906119911ℏ)+120593plusmn(119905))

(32)



119861 119905

0+ Δ119905 + 119905) is found by


119860 r

119861 119905

0+ Δ119905 + 119905)Ψ(r

119860 r

119861 119905

0+ Δ119905 + 119905) on 119909

119860

and 119909119861

120588 (119911

119860 119911

119861 119905

0+ Δ119905 + 119905)

= ((2120587120590

2

0)

minus12

119890

minus(119911119861)221205902

0)

times ((2120587120590

2

0)

minus12

times

1

2

(119890


0+ 119890

minus(119911119860+119911Δ+119906119905)221205902

0))

(33)







0and 119905

0+Δ119905+

119905



E119860during 119905

0+ Δ119905 + 119905

119863and 119905

0+ 2(Δ119905 + 119905

119863)


0+ Δ119905 + 119905

119863


0+Δ119905+119905



Ψ

119861plusmn119860(r

119861 119905

0+ Δ119905 + 119905

1) = 119891 (r

119861)

1003816

1003816

1003816

1003816

∓

119861⟩ (34)


119863) yields



0(r

119860 120579

119860

0 120593

119860

0) = 119891(r

119860)

(cos(12057911986002)|+

119860⟩ + sin(120579119860

02)119890

119894120593119860

0|minus

119860⟩) and Ψ

119861

0(r

119861 120579

119861

0 120593

119861

0) =

119891(r119861)(cos(120579119861

02)|+

119861⟩ + sin(120579119861

02)119890

119894120593119861

0|minus

119861⟩) with 120579119861

0= 120587 minus 120579

119860

0

and 120593119861

0= 120593

119860



Ψ

0(r

119860 120579

119860 120593

119860 r

119861 120579

119861 120593

119861)

= minus119890

119894120593119860

119891 (r119860) 119891 (r

119861) times (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(35)



119860

0(r

119860 120579

119860

0 120593

119860

0) and Ψ119861

0(r

119861 120579

119861

0 120593

119861



119860 120579

119860 120593

119860 r

119861 120579

119861 120593





Ψ

119860(r

119860 119905

0+ Δ119905 + 119905)

= cos120579

119860

0

2

119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩ + sin

120579

119860

0

2

119890

119894120593119860

0119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

(36)


0 120593

119860

0) For


119860


0)



119860(119905) 119905)) [48]


119861(119905) = V

119910119905 119911

119861(119905) = 119911

119861

0 and 119909

119861(119905) = 119909

119861

0 By


119860(119911

119860(119905) 119905) and 120593

119861(119905) =

120593(119911



Ψ

119861(r

119861 119905

0+ Δ119905 + 119905)

= 119891 (r119861) (cos 120579

119861(119905)

2

1003816

1003816

1003816

1003816

+

119861⟩ + sin 120579

119861(119905)

2

119890

119894120593119861(119905) 10038161003816

1003816

1003816

minus

119861⟩)

(37)



119863


119863) = 120587



1015840)


1015840119861(119905

0+ Δ119905 + 119905





0+ 2(Δ119905 + 119905

119863) gives









6 Conclusion







Appendix



119909 0 119861




component

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) plusmn 120583

119861(119861

0minus 119861

1015840

0119911)120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905)

(A1)


120595


119894120583

119861119861

0119905

ℏ

)120595

plusmn(119909 119911 119905)

(A2)

(A1) becomes

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905) exp(plusmn119894

2120583

119861119861

0119905

ℏ

)

(A3)


119871= 2120583

119861119861

0ℏ = 1 4 times 10

11 sminus1 Since |1198610| ≫ |119861

1015840

0119911|

and |1198610| ≫ |119861

1015840



119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905) = minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

(A4)


0



on 119909 120595plusmn(119909 119911 119905) equiv 120595



5 sWe obtain

120595

+(119911 119905) = 120595

119870(119911 119905) cos

120579

0

2

119890

119894(12059302) 119870 = minus120583

119861119861

1015840

0

120595

minus(119911 119905) = 120595

119870(119911 119905) 119894 sin

120579

0

2

119890

minus119894(12059302) 119870 = +120583

119861119861

1015840

0

(A5)

120590

2

119905= 120590

2

0+ (ℏ1199052119898120590

0)

2and

120595

119870(119911 119905) = (2120587120590

2

119905)

minus14

119890

minus(119911+11987011990522119898)2

41205902

119905

times exp 119894

ℏ

[

[

minus

ℏ

2

tanminus1(

ℏ119905

2119898120590

2

0

) minus 119870119905119911 minus

119870

2119905

3

6119898

+

(119911 + 119870119905

22119898)

2

ℏ

2119905

2

8119898120590

2

0120590

2

119905

]

]

(A6)



0= 4 times

10

minus11 m ≪ 120590

0= 10


119905≃ 120590

0and

120595

119870(119911 119905)

≃ (2120587120590

2

0)

minus14

119890

minus(119911+11987011990522119898)2

41205902

0 exp 119894

ℏ

[minus119870119905119911 minus

119870

2119905

3

6119898

]

(A7)


Ψ (119911 Δ119905) = (

120595

+(119911 Δ119905)

120595

minus(119911 Δ119905)

) (A8)

with

120595

+(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911minus119911Δ)241205902

0)+(119894ℏ)119898119906119911 cos

120579

0

2

119890

119894120593+

120595

minus(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911+119911Δ)241205902

0)minus(119894ℏ)119898119906119911

119894 sin120579

0

2

119890

119894120593minus

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

119906 =

120583

0119861

1015840

0(Δ119905)

119898

120593

+=

120593

0

2

minus

120583

119861119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

120593

minus= minus

120593

0

2

+

120583

0119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

(A9)




minus


plusmn(119909 119911 119905) =

120595

119909(119909 119905)120595


have 120595119909(119909 119905) ≃ (2120587120590

2

0)

minus14119890

minus119909241205902

0 and

120595

+(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

cos120579

0

2


0+(119894ℏ)(119898119906119911minus(12)119898119906

2119905+ℏ120593+)

120595

minus(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

119894 sin120579

0

2

times expminus(119911+119911Δ+119906119905)241205902

0+(119894ℏ)(minus119898119906119911minus(12)119898119906

2119905+ℏ120593minus)

(A10)




References



























60moleculesrdquo



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




x

y

z

1205930

1205790

|minus⟩

|+⟩


0are the





119894ℏ(

120597120595

+

120597119905

120597120595

minus

120597119905

) = minus

ℏ

2

2119898

Δ(

120595

+

120595

minus

) + 120583

119861B120590(120595+

120595

minus

) (18)

where 120583

119861= 119890ℏ2119898


120590 = (120590

119909 120590

119910 120590



119909= 119861

1015840

0119909 119861

119910= 0 and 119861

119911= 119861

0minus 119861

1015840

0119911 with

119861

0= 5 Tesla 1198611015840

0= |120597119861120597119911| = 10





Ψ (119911 119905) ≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911minus(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3+1205831198611198610119905+(ℏ12059302))ℏ)

119894 sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+(1205831198611198611015840

02119898)119905

2)2

41205902

0119890

119894((minus1205831198611198611015840

0119905119911minus(120583

2

11986111986110158402

06119898)119905

3minus1205831198611198610119905minus(ℏ12059302))ℏ)

) (19)


Ψ (119911 119905 + Δ119905)

≃ (

cos120579

0

2

(2120587120590

2

0)

minus12

119890


0119890

119894((119898119906119911+ℏ120593+)ℏ)

sin120579

0

2

(2120587120590

2

0)

minus12

119890

minus(119911+119911Δ+119906119905)241205902

0119890

119894((minus119898119906119911+ℏ120593minus)ℏ)

)

(20)

where

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

= 10

minus5 m 119906 =

120583

119861119861

1015840

0(Δ119905)

119898

= 1ms(21)



electromagnet

120588

1205790(119911 119905 + Δ119905) ≃ (2120587120590

2

0)

minus12

(cos2120579

0

2

119890


0

+sin2 1205790

2

119890

minus(119911+119911Δ+119906119905)221205902

0)

(22)






119905

119863≃

3120590

0minus 119911

Δ

119906

= 3 times 10

minus4 s (23)


120588

119878(119905) = (

int

1003816

1003816

1003816

1003816

120595

+(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911 int120595

+(119911 119905) 120595

lowast

minus(119911 119905) 119889119911

int120595

minus(119911 119905) 120595

lowast

+(119911 119905) 119889119911 int

1003816

1003816

1003816

1003816

120595

minus(119911 119905)

1003816

1003816

1003816

1003816

2

119889119911

)

(24)


minus06

(mm)minus06

(mm)minus06

(mm)minus06

(mm)

0 cm 6 cm 16 cm 21 cm

0 06 0 06 0 06 0 06



minus1

z(m

m)

x (mm)0 1 2 3 4 5

0

1N+

Nminus


For 119905 ge 119905


+(119911 119905 + Δ119905)120595

minus(119911 119905 + Δ119905) is null


120588

119878(119905 + Δ119905) = (2120587120590

2

0)

minus1

(

cos2120579

0

2

0

0 sin2120579

0

2

) (25)


1


+ the term 120595



0) one of the


119911 Ψ( 119905 +

Δ119905) ≃ (2120587120590

2

0)

minus14 cos(12057902)119890


0119890

119894((1198981199061+ℏ120593+)ℏ)(

1

0)

Then we have 119878119911Ψ = (ℏ2)120590

119911Ψ = +(ℏ2)Ψ

If isin 119873




1) the


(2120587120590

2

0)

minus14 sin(12057902)119890

minus(2+119911Δ+119906119905)241205902

0119890

119894((minus1198981199062+ℏ120593minus)ℏ)(

0

1) Then

we have 119878

119911Ψ = (ℏ2)120590




sin2(120579





1




1 at this


1










0= 1205873 120593






119905


0in the spinor



1205790 and minus1205872 if 1199110lt

119911

1205790 with

119911

1205790= 120590

0119865

minus1(sin2 1205790

2

) (26)




0 5 10 15 20

0

02

04

06

08

minus04

minus02

y (cm)

z(m

m)






0 120593





5 EPR-B Experiment


0 5 10 15 20

0

02

04

06

minus06

minus04

minus02

y (cm)

z(m

m)


0 120593





Ψ

0(r

119860 r

119861) =

1

radic2

119891 (r119860) 119891 (r

119861) (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(27)


Ψ

0(r

119860 r

119861) =

1

radic2

(

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩) (28)





x

y

z

z

z

z998400

x

z z998400

x

yz

x998400z998400

120575


O Atom AAtom B

EAEB



119860and E

119861




(2120587120590

2

0)

minus12119890

minus(1199092+1199112)41205902




and 120590119911119861 120590

119911119860|plusmn


119860⟩ 120590

119911119861|plusmn

119861⟩ =

plusmn|plusmn




119860 r

119861 119905) of



119860and




Ψ

119886119887(r

119860 r


page 417]

119894ℏ

120597Ψ

119886119887

120597119905

= (minus

ℏ

2

2119898

Δ

119860minus

ℏ

2

2119898

Δ

119861)Ψ

119886119887+ 120583119861

E119860119895(120590

119895)

119886

119888Ψ

119888119887

+ 120583119861

E119861119895(120590

119895)

119887

119889Ψ

119886119889

(29)


Ψ

119886119887(r

119860 r

119861 0) = Ψ

119886119887

0(r

119860 r

119861)

(30)

where Ψ119886119887

0(r

119860 r










[68]

Ψ (r119860 r

119861 119905

0+ Δ119905 + 119905)

=

1

radic2

119891 (r119861)

times (119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus 119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(31)

with

119891

plusmn(r 119905) ≃ 119891 (119909 119911 ∓ 119911

Δ∓ 119906119905) 119890

119894((plusmn119898119906119911ℏ)+120593plusmn(119905))

(32)



119861 119905

0+ Δ119905 + 119905) is found by


119860 r

119861 119905

0+ Δ119905 + 119905)Ψ(r

119860 r

119861 119905

0+ Δ119905 + 119905) on 119909

119860

and 119909119861

120588 (119911

119860 119911

119861 119905

0+ Δ119905 + 119905)

= ((2120587120590

2

0)

minus12

119890

minus(119911119861)221205902

0)

times ((2120587120590

2

0)

minus12

times

1

2

(119890


0+ 119890

minus(119911119860+119911Δ+119906119905)221205902

0))

(33)







0and 119905

0+Δ119905+

119905



E119860during 119905

0+ Δ119905 + 119905

119863and 119905

0+ 2(Δ119905 + 119905

119863)


0+ Δ119905 + 119905

119863


0+Δ119905+119905



Ψ

119861plusmn119860(r

119861 119905

0+ Δ119905 + 119905

1) = 119891 (r

119861)

1003816

1003816

1003816

1003816

∓

119861⟩ (34)


119863) yields



0(r

119860 120579

119860

0 120593

119860

0) = 119891(r

119860)

(cos(12057911986002)|+

119860⟩ + sin(120579119860

02)119890

119894120593119860

0|minus

119860⟩) and Ψ

119861

0(r

119861 120579

119861

0 120593

119861

0) =

119891(r119861)(cos(120579119861

02)|+

119861⟩ + sin(120579119861

02)119890

119894120593119861

0|minus

119861⟩) with 120579119861

0= 120587 minus 120579

119860

0

and 120593119861

0= 120593

119860



Ψ

0(r

119860 120579

119860 120593

119860 r

119861 120579

119861 120593

119861)

= minus119890

119894120593119860

119891 (r119860) 119891 (r

119861) times (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(35)



119860

0(r

119860 120579

119860

0 120593

119860

0) and Ψ119861

0(r

119861 120579

119861

0 120593

119861



119860 120579

119860 120593

119860 r

119861 120579

119861 120593





Ψ

119860(r

119860 119905

0+ Δ119905 + 119905)

= cos120579

119860

0

2

119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩ + sin

120579

119860

0

2

119890

119894120593119860

0119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

(36)


0 120593

119860

0) For


119860


0)



119860(119905) 119905)) [48]


119861(119905) = V

119910119905 119911

119861(119905) = 119911

119861

0 and 119909

119861(119905) = 119909

119861

0 By


119860(119911

119860(119905) 119905) and 120593

119861(119905) =

120593(119911



Ψ

119861(r

119861 119905

0+ Δ119905 + 119905)

= 119891 (r119861) (cos 120579

119861(119905)

2

1003816

1003816

1003816

1003816

+

119861⟩ + sin 120579

119861(119905)

2

119890

119894120593119861(119905) 10038161003816

1003816

1003816

minus

119861⟩)

(37)



119863


119863) = 120587



1015840)


1015840119861(119905

0+ Δ119905 + 119905





0+ 2(Δ119905 + 119905

119863) gives









6 Conclusion







Appendix



119909 0 119861




component

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) plusmn 120583

119861(119861

0minus 119861

1015840

0119911)120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905)

(A1)


120595


119894120583

119861119861

0119905

ℏ

)120595

plusmn(119909 119911 119905)

(A2)

(A1) becomes

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905) exp(plusmn119894

2120583

119861119861

0119905

ℏ

)

(A3)


119871= 2120583

119861119861

0ℏ = 1 4 times 10

11 sminus1 Since |1198610| ≫ |119861

1015840

0119911|

and |1198610| ≫ |119861

1015840



119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905) = minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

(A4)


0



on 119909 120595plusmn(119909 119911 119905) equiv 120595



5 sWe obtain

120595

+(119911 119905) = 120595

119870(119911 119905) cos

120579

0

2

119890

119894(12059302) 119870 = minus120583

119861119861

1015840

0

120595

minus(119911 119905) = 120595

119870(119911 119905) 119894 sin

120579

0

2

119890

minus119894(12059302) 119870 = +120583

119861119861

1015840

0

(A5)

120590

2

119905= 120590

2

0+ (ℏ1199052119898120590

0)

2and

120595

119870(119911 119905) = (2120587120590

2

119905)

minus14

119890

minus(119911+11987011990522119898)2

41205902

119905

times exp 119894

ℏ

[

[

minus

ℏ

2

tanminus1(

ℏ119905

2119898120590

2

0

) minus 119870119905119911 minus

119870

2119905

3

6119898

+

(119911 + 119870119905

22119898)

2

ℏ

2119905

2

8119898120590

2

0120590

2

119905

]

]

(A6)



0= 4 times

10

minus11 m ≪ 120590

0= 10


119905≃ 120590

0and

120595

119870(119911 119905)

≃ (2120587120590

2

0)

minus14

119890

minus(119911+11987011990522119898)2

41205902

0 exp 119894

ℏ

[minus119870119905119911 minus

119870

2119905

3

6119898

]

(A7)


Ψ (119911 Δ119905) = (

120595

+(119911 Δ119905)

120595

minus(119911 Δ119905)

) (A8)

with

120595

+(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911minus119911Δ)241205902

0)+(119894ℏ)119898119906119911 cos

120579

0

2

119890

119894120593+

120595

minus(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911+119911Δ)241205902

0)minus(119894ℏ)119898119906119911

119894 sin120579

0

2

119890

119894120593minus

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

119906 =

120583

0119861

1015840

0(Δ119905)

119898

120593

+=

120593

0

2

minus

120583

119861119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

120593

minus= minus

120593

0

2

+

120583

0119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

(A9)




minus


plusmn(119909 119911 119905) =

120595

119909(119909 119905)120595


have 120595119909(119909 119905) ≃ (2120587120590

2

0)

minus14119890

minus119909241205902

0 and

120595

+(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

cos120579

0

2


0+(119894ℏ)(119898119906119911minus(12)119898119906

2119905+ℏ120593+)

120595

minus(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

119894 sin120579

0

2

times expminus(119911+119911Δ+119906119905)241205902

0+(119894ℏ)(minus119898119906119911minus(12)119898119906

2119905+ℏ120593minus)

(A10)




References



























60moleculesrdquo



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus06

(mm)minus06

(mm)minus06

(mm)minus06

(mm)

0 cm 6 cm 16 cm 21 cm

0 06 0 06 0 06 0 06



minus1

z(m

m)

x (mm)0 1 2 3 4 5

0

1N+

Nminus


For 119905 ge 119905


+(119911 119905 + Δ119905)120595

minus(119911 119905 + Δ119905) is null


120588

119878(119905 + Δ119905) = (2120587120590

2

0)

minus1

(

cos2120579

0

2

0

0 sin2120579

0

2

) (25)


1


+ the term 120595



0) one of the


119911 Ψ( 119905 +

Δ119905) ≃ (2120587120590

2

0)

minus14 cos(12057902)119890


0119890

119894((1198981199061+ℏ120593+)ℏ)(

1

0)

Then we have 119878119911Ψ = (ℏ2)120590

119911Ψ = +(ℏ2)Ψ

If isin 119873




1) the


(2120587120590

2

0)

minus14 sin(12057902)119890

minus(2+119911Δ+119906119905)241205902

0119890

119894((minus1198981199062+ℏ120593minus)ℏ)(

0

1) Then

we have 119878

119911Ψ = (ℏ2)120590




sin2(120579





1




1 at this


1










0= 1205873 120593






119905


0in the spinor



1205790 and minus1205872 if 1199110lt

119911

1205790 with

119911

1205790= 120590

0119865

minus1(sin2 1205790

2

) (26)




0 5 10 15 20

0

02

04

06

08

minus04

minus02

y (cm)

z(m

m)






0 120593





5 EPR-B Experiment


0 5 10 15 20

0

02

04

06

minus06

minus04

minus02

y (cm)

z(m

m)


0 120593





Ψ

0(r

119860 r

119861) =

1

radic2

119891 (r119860) 119891 (r

119861) (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(27)


Ψ

0(r

119860 r

119861) =

1

radic2

(

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩) (28)





x

y

z

z

z

z998400

x

z z998400

x

yz

x998400z998400

120575


O Atom AAtom B

EAEB



119860and E

119861




(2120587120590

2

0)

minus12119890

minus(1199092+1199112)41205902




and 120590119911119861 120590

119911119860|plusmn


119860⟩ 120590

119911119861|plusmn

119861⟩ =

plusmn|plusmn




119860 r

119861 119905) of



119860and




Ψ

119886119887(r

119860 r


page 417]

119894ℏ

120597Ψ

119886119887

120597119905

= (minus

ℏ

2

2119898

Δ

119860minus

ℏ

2

2119898

Δ

119861)Ψ

119886119887+ 120583119861

E119860119895(120590

119895)

119886

119888Ψ

119888119887

+ 120583119861

E119861119895(120590

119895)

119887

119889Ψ

119886119889

(29)


Ψ

119886119887(r

119860 r

119861 0) = Ψ

119886119887

0(r

119860 r

119861)

(30)

where Ψ119886119887

0(r

119860 r










[68]

Ψ (r119860 r

119861 119905

0+ Δ119905 + 119905)

=

1

radic2

119891 (r119861)

times (119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus 119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(31)

with

119891

plusmn(r 119905) ≃ 119891 (119909 119911 ∓ 119911

Δ∓ 119906119905) 119890

119894((plusmn119898119906119911ℏ)+120593plusmn(119905))

(32)



119861 119905

0+ Δ119905 + 119905) is found by


119860 r

119861 119905

0+ Δ119905 + 119905)Ψ(r

119860 r

119861 119905

0+ Δ119905 + 119905) on 119909

119860

and 119909119861

120588 (119911

119860 119911

119861 119905

0+ Δ119905 + 119905)

= ((2120587120590

2

0)

minus12

119890

minus(119911119861)221205902

0)

times ((2120587120590

2

0)

minus12

times

1

2

(119890


0+ 119890

minus(119911119860+119911Δ+119906119905)221205902

0))

(33)







0and 119905

0+Δ119905+

119905



E119860during 119905

0+ Δ119905 + 119905

119863and 119905

0+ 2(Δ119905 + 119905

119863)


0+ Δ119905 + 119905

119863


0+Δ119905+119905



Ψ

119861plusmn119860(r

119861 119905

0+ Δ119905 + 119905

1) = 119891 (r

119861)

1003816

1003816

1003816

1003816

∓

119861⟩ (34)


119863) yields



0(r

119860 120579

119860

0 120593

119860

0) = 119891(r

119860)

(cos(12057911986002)|+

119860⟩ + sin(120579119860

02)119890

119894120593119860

0|minus

119860⟩) and Ψ

119861

0(r

119861 120579

119861

0 120593

119861

0) =

119891(r119861)(cos(120579119861

02)|+

119861⟩ + sin(120579119861

02)119890

119894120593119861

0|minus

119861⟩) with 120579119861

0= 120587 minus 120579

119860

0

and 120593119861

0= 120593

119860



Ψ

0(r

119860 120579

119860 120593

119860 r

119861 120579

119861 120593

119861)

= minus119890

119894120593119860

119891 (r119860) 119891 (r

119861) times (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(35)



119860

0(r

119860 120579

119860

0 120593

119860

0) and Ψ119861

0(r

119861 120579

119861

0 120593

119861



119860 120579

119860 120593

119860 r

119861 120579

119861 120593





Ψ

119860(r

119860 119905

0+ Δ119905 + 119905)

= cos120579

119860

0

2

119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩ + sin

120579

119860

0

2

119890

119894120593119860

0119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

(36)


0 120593

119860

0) For


119860


0)



119860(119905) 119905)) [48]


119861(119905) = V

119910119905 119911

119861(119905) = 119911

119861

0 and 119909

119861(119905) = 119909

119861

0 By


119860(119911

119860(119905) 119905) and 120593

119861(119905) =

120593(119911



Ψ

119861(r

119861 119905

0+ Δ119905 + 119905)

= 119891 (r119861) (cos 120579

119861(119905)

2

1003816

1003816

1003816

1003816

+

119861⟩ + sin 120579

119861(119905)

2

119890

119894120593119861(119905) 10038161003816

1003816

1003816

minus

119861⟩)

(37)



119863


119863) = 120587



1015840)


1015840119861(119905

0+ Δ119905 + 119905





0+ 2(Δ119905 + 119905

119863) gives









6 Conclusion







Appendix



119909 0 119861




component

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) plusmn 120583

119861(119861

0minus 119861

1015840

0119911)120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905)

(A1)


120595


119894120583

119861119861

0119905

ℏ

)120595

plusmn(119909 119911 119905)

(A2)

(A1) becomes

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905) exp(plusmn119894

2120583

119861119861

0119905

ℏ

)

(A3)


119871= 2120583

119861119861

0ℏ = 1 4 times 10

11 sminus1 Since |1198610| ≫ |119861

1015840

0119911|

and |1198610| ≫ |119861

1015840



119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905) = minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

(A4)


0



on 119909 120595plusmn(119909 119911 119905) equiv 120595



5 sWe obtain

120595

+(119911 119905) = 120595

119870(119911 119905) cos

120579

0

2

119890

119894(12059302) 119870 = minus120583

119861119861

1015840

0

120595

minus(119911 119905) = 120595

119870(119911 119905) 119894 sin

120579

0

2

119890

minus119894(12059302) 119870 = +120583

119861119861

1015840

0

(A5)

120590

2

119905= 120590

2

0+ (ℏ1199052119898120590

0)

2and

120595

119870(119911 119905) = (2120587120590

2

119905)

minus14

119890

minus(119911+11987011990522119898)2

41205902

119905

times exp 119894

ℏ

[

[

minus

ℏ

2

tanminus1(

ℏ119905

2119898120590

2

0

) minus 119870119905119911 minus

119870

2119905

3

6119898

+

(119911 + 119870119905

22119898)

2

ℏ

2119905

2

8119898120590

2

0120590

2

119905

]

]

(A6)



0= 4 times

10

minus11 m ≪ 120590

0= 10


119905≃ 120590

0and

120595

119870(119911 119905)

≃ (2120587120590

2

0)

minus14

119890

minus(119911+11987011990522119898)2

41205902

0 exp 119894

ℏ

[minus119870119905119911 minus

119870

2119905

3

6119898

]

(A7)


Ψ (119911 Δ119905) = (

120595

+(119911 Δ119905)

120595

minus(119911 Δ119905)

) (A8)

with

120595

+(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911minus119911Δ)241205902

0)+(119894ℏ)119898119906119911 cos

120579

0

2

119890

119894120593+

120595

minus(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911+119911Δ)241205902

0)minus(119894ℏ)119898119906119911

119894 sin120579

0

2

119890

119894120593minus

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

119906 =

120583

0119861

1015840

0(Δ119905)

119898

120593

+=

120593

0

2

minus

120583

119861119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

120593

minus= minus

120593

0

2

+

120583

0119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

(A9)




minus


plusmn(119909 119911 119905) =

120595

119909(119909 119905)120595


have 120595119909(119909 119905) ≃ (2120587120590

2

0)

minus14119890

minus119909241205902

0 and

120595

+(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

cos120579

0

2


0+(119894ℏ)(119898119906119911minus(12)119898119906

2119905+ℏ120593+)

120595

minus(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

119894 sin120579

0

2

times expminus(119911+119911Δ+119906119905)241205902

0+(119894ℏ)(minus119898119906119911minus(12)119898119906

2119905+ℏ120593minus)

(A10)




References



























60moleculesrdquo



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0 5 10 15 20

0

02

04

06

08

minus04

minus02

y (cm)

z(m

m)






0 120593





5 EPR-B Experiment


0 5 10 15 20

0

02

04

06

minus06

minus04

minus02

y (cm)

z(m

m)


0 120593





Ψ

0(r

119860 r

119861) =

1

radic2

119891 (r119860) 119891 (r

119861) (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(27)


Ψ

0(r

119860 r

119861) =

1

radic2

(

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩) (28)





x

y

z

z

z

z998400

x

z z998400

x

yz

x998400z998400

120575


O Atom AAtom B

EAEB



119860and E

119861




(2120587120590

2

0)

minus12119890

minus(1199092+1199112)41205902




and 120590119911119861 120590

119911119860|plusmn


119860⟩ 120590

119911119861|plusmn

119861⟩ =

plusmn|plusmn




119860 r

119861 119905) of



119860and




Ψ

119886119887(r

119860 r


page 417]

119894ℏ

120597Ψ

119886119887

120597119905

= (minus

ℏ

2

2119898

Δ

119860minus

ℏ

2

2119898

Δ

119861)Ψ

119886119887+ 120583119861

E119860119895(120590

119895)

119886

119888Ψ

119888119887

+ 120583119861

E119861119895(120590

119895)

119887

119889Ψ

119886119889

(29)


Ψ

119886119887(r

119860 r

119861 0) = Ψ

119886119887

0(r

119860 r

119861)

(30)

where Ψ119886119887

0(r

119860 r










[68]

Ψ (r119860 r

119861 119905

0+ Δ119905 + 119905)

=

1

radic2

119891 (r119861)

times (119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus 119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(31)

with

119891

plusmn(r 119905) ≃ 119891 (119909 119911 ∓ 119911

Δ∓ 119906119905) 119890

119894((plusmn119898119906119911ℏ)+120593plusmn(119905))

(32)



119861 119905

0+ Δ119905 + 119905) is found by


119860 r

119861 119905

0+ Δ119905 + 119905)Ψ(r

119860 r

119861 119905

0+ Δ119905 + 119905) on 119909

119860

and 119909119861

120588 (119911

119860 119911

119861 119905

0+ Δ119905 + 119905)

= ((2120587120590

2

0)

minus12

119890

minus(119911119861)221205902

0)

times ((2120587120590

2

0)

minus12

times

1

2

(119890


0+ 119890

minus(119911119860+119911Δ+119906119905)221205902

0))

(33)







0and 119905

0+Δ119905+

119905



E119860during 119905

0+ Δ119905 + 119905

119863and 119905

0+ 2(Δ119905 + 119905

119863)


0+ Δ119905 + 119905

119863


0+Δ119905+119905



Ψ

119861plusmn119860(r

119861 119905

0+ Δ119905 + 119905

1) = 119891 (r

119861)

1003816

1003816

1003816

1003816

∓

119861⟩ (34)


119863) yields



0(r

119860 120579

119860

0 120593

119860

0) = 119891(r

119860)

(cos(12057911986002)|+

119860⟩ + sin(120579119860

02)119890

119894120593119860

0|minus

119860⟩) and Ψ

119861

0(r

119861 120579

119861

0 120593

119861

0) =

119891(r119861)(cos(120579119861

02)|+

119861⟩ + sin(120579119861

02)119890

119894120593119861

0|minus

119861⟩) with 120579119861

0= 120587 minus 120579

119860

0

and 120593119861

0= 120593

119860



Ψ

0(r

119860 120579

119860 120593

119860 r

119861 120579

119861 120593

119861)

= minus119890

119894120593119860

119891 (r119860) 119891 (r

119861) times (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(35)



119860

0(r

119860 120579

119860

0 120593

119860

0) and Ψ119861

0(r

119861 120579

119861

0 120593

119861



119860 120579

119860 120593

119860 r

119861 120579

119861 120593





Ψ

119860(r

119860 119905

0+ Δ119905 + 119905)

= cos120579

119860

0

2

119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩ + sin

120579

119860

0

2

119890

119894120593119860

0119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

(36)


0 120593

119860

0) For


119860


0)



119860(119905) 119905)) [48]


119861(119905) = V

119910119905 119911

119861(119905) = 119911

119861

0 and 119909

119861(119905) = 119909

119861

0 By


119860(119911

119860(119905) 119905) and 120593

119861(119905) =

120593(119911



Ψ

119861(r

119861 119905

0+ Δ119905 + 119905)

= 119891 (r119861) (cos 120579

119861(119905)

2

1003816

1003816

1003816

1003816

+

119861⟩ + sin 120579

119861(119905)

2

119890

119894120593119861(119905) 10038161003816

1003816

1003816

minus

119861⟩)

(37)



119863


119863) = 120587



1015840)


1015840119861(119905

0+ Δ119905 + 119905





0+ 2(Δ119905 + 119905

119863) gives









6 Conclusion







Appendix



119909 0 119861




component

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) plusmn 120583

119861(119861

0minus 119861

1015840

0119911)120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905)

(A1)


120595


119894120583

119861119861

0119905

ℏ

)120595

plusmn(119909 119911 119905)

(A2)

(A1) becomes

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905) exp(plusmn119894

2120583

119861119861

0119905

ℏ

)

(A3)


119871= 2120583

119861119861

0ℏ = 1 4 times 10

11 sminus1 Since |1198610| ≫ |119861

1015840

0119911|

and |1198610| ≫ |119861

1015840



119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905) = minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

(A4)


0



on 119909 120595plusmn(119909 119911 119905) equiv 120595



5 sWe obtain

120595

+(119911 119905) = 120595

119870(119911 119905) cos

120579

0

2

119890

119894(12059302) 119870 = minus120583

119861119861

1015840

0

120595

minus(119911 119905) = 120595

119870(119911 119905) 119894 sin

120579

0

2

119890

minus119894(12059302) 119870 = +120583

119861119861

1015840

0

(A5)

120590

2

119905= 120590

2

0+ (ℏ1199052119898120590

0)

2and

120595

119870(119911 119905) = (2120587120590

2

119905)

minus14

119890

minus(119911+11987011990522119898)2

41205902

119905

times exp 119894

ℏ

[

[

minus

ℏ

2

tanminus1(

ℏ119905

2119898120590

2

0

) minus 119870119905119911 minus

119870

2119905

3

6119898

+

(119911 + 119870119905

22119898)

2

ℏ

2119905

2

8119898120590

2

0120590

2

119905

]

]

(A6)



0= 4 times

10

minus11 m ≪ 120590

0= 10


119905≃ 120590

0and

120595

119870(119911 119905)

≃ (2120587120590

2

0)

minus14

119890

minus(119911+11987011990522119898)2

41205902

0 exp 119894

ℏ

[minus119870119905119911 minus

119870

2119905

3

6119898

]

(A7)


Ψ (119911 Δ119905) = (

120595

+(119911 Δ119905)

120595

minus(119911 Δ119905)

) (A8)

with

120595

+(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911minus119911Δ)241205902

0)+(119894ℏ)119898119906119911 cos

120579

0

2

119890

119894120593+

120595

minus(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911+119911Δ)241205902

0)minus(119894ℏ)119898119906119911

119894 sin120579

0

2

119890

119894120593minus

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

119906 =

120583

0119861

1015840

0(Δ119905)

119898

120593

+=

120593

0

2

minus

120583

119861119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

120593

minus= minus

120593

0

2

+

120583

0119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

(A9)




minus


plusmn(119909 119911 119905) =

120595

119909(119909 119905)120595


have 120595119909(119909 119905) ≃ (2120587120590

2

0)

minus14119890

minus119909241205902

0 and

120595

+(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

cos120579

0

2


0+(119894ℏ)(119898119906119911minus(12)119898119906

2119905+ℏ120593+)

120595

minus(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

119894 sin120579

0

2

times expminus(119911+119911Δ+119906119905)241205902

0+(119894ℏ)(minus119898119906119911minus(12)119898119906

2119905+ℏ120593minus)

(A10)




References



























60moleculesrdquo



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




x

y

z

z

z

z998400

x

z z998400

x

yz

x998400z998400

120575


O Atom AAtom B

EAEB



119860and E

119861




(2120587120590

2

0)

minus12119890

minus(1199092+1199112)41205902




and 120590119911119861 120590

119911119860|plusmn


119860⟩ 120590

119911119861|plusmn

119861⟩ =

plusmn|plusmn




119860 r

119861 119905) of



119860and




Ψ

119886119887(r

119860 r


page 417]

119894ℏ

120597Ψ

119886119887

120597119905

= (minus

ℏ

2

2119898

Δ

119860minus

ℏ

2

2119898

Δ

119861)Ψ

119886119887+ 120583119861

E119860119895(120590

119895)

119886

119888Ψ

119888119887

+ 120583119861

E119861119895(120590

119895)

119887

119889Ψ

119886119889

(29)


Ψ

119886119887(r

119860 r

119861 0) = Ψ

119886119887

0(r

119860 r

119861)

(30)

where Ψ119886119887

0(r

119860 r










[68]

Ψ (r119860 r

119861 119905

0+ Δ119905 + 119905)

=

1

radic2

119891 (r119861)

times (119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus 119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(31)

with

119891

plusmn(r 119905) ≃ 119891 (119909 119911 ∓ 119911

Δ∓ 119906119905) 119890

119894((plusmn119898119906119911ℏ)+120593plusmn(119905))

(32)



119861 119905

0+ Δ119905 + 119905) is found by


119860 r

119861 119905

0+ Δ119905 + 119905)Ψ(r

119860 r

119861 119905

0+ Δ119905 + 119905) on 119909

119860

and 119909119861

120588 (119911

119860 119911

119861 119905

0+ Δ119905 + 119905)

= ((2120587120590

2

0)

minus12

119890

minus(119911119861)221205902

0)

times ((2120587120590

2

0)

minus12

times

1

2

(119890


0+ 119890

minus(119911119860+119911Δ+119906119905)221205902

0))

(33)







0and 119905

0+Δ119905+

119905



E119860during 119905

0+ Δ119905 + 119905

119863and 119905

0+ 2(Δ119905 + 119905

119863)


0+ Δ119905 + 119905

119863


0+Δ119905+119905



Ψ

119861plusmn119860(r

119861 119905

0+ Δ119905 + 119905

1) = 119891 (r

119861)

1003816

1003816

1003816

1003816

∓

119861⟩ (34)


119863) yields



0(r

119860 120579

119860

0 120593

119860

0) = 119891(r

119860)

(cos(12057911986002)|+

119860⟩ + sin(120579119860

02)119890

119894120593119860

0|minus

119860⟩) and Ψ

119861

0(r

119861 120579

119861

0 120593

119861

0) =

119891(r119861)(cos(120579119861

02)|+

119861⟩ + sin(120579119861

02)119890

119894120593119861

0|minus

119861⟩) with 120579119861

0= 120587 minus 120579

119860

0

and 120593119861

0= 120593

119860



Ψ

0(r

119860 120579

119860 120593

119860 r

119861 120579

119861 120593

119861)

= minus119890

119894120593119860

119891 (r119860) 119891 (r

119861) times (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(35)



119860

0(r

119860 120579

119860

0 120593

119860

0) and Ψ119861

0(r

119861 120579

119861

0 120593

119861



119860 120579

119860 120593

119860 r

119861 120579

119861 120593





Ψ

119860(r

119860 119905

0+ Δ119905 + 119905)

= cos120579

119860

0

2

119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩ + sin

120579

119860

0

2

119890

119894120593119860

0119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

(36)


0 120593

119860

0) For


119860


0)



119860(119905) 119905)) [48]


119861(119905) = V

119910119905 119911

119861(119905) = 119911

119861

0 and 119909

119861(119905) = 119909

119861

0 By


119860(119911

119860(119905) 119905) and 120593

119861(119905) =

120593(119911



Ψ

119861(r

119861 119905

0+ Δ119905 + 119905)

= 119891 (r119861) (cos 120579

119861(119905)

2

1003816

1003816

1003816

1003816

+

119861⟩ + sin 120579

119861(119905)

2

119890

119894120593119861(119905) 10038161003816

1003816

1003816

minus

119861⟩)

(37)



119863


119863) = 120587



1015840)


1015840119861(119905

0+ Δ119905 + 119905





0+ 2(Δ119905 + 119905

119863) gives









6 Conclusion







Appendix



119909 0 119861




component

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) plusmn 120583

119861(119861

0minus 119861

1015840

0119911)120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905)

(A1)


120595


119894120583

119861119861

0119905

ℏ

)120595

plusmn(119909 119911 119905)

(A2)

(A1) becomes

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905) exp(plusmn119894

2120583

119861119861

0119905

ℏ

)

(A3)


119871= 2120583

119861119861

0ℏ = 1 4 times 10

11 sminus1 Since |1198610| ≫ |119861

1015840

0119911|

and |1198610| ≫ |119861

1015840



119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905) = minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

(A4)


0



on 119909 120595plusmn(119909 119911 119905) equiv 120595



5 sWe obtain

120595

+(119911 119905) = 120595

119870(119911 119905) cos

120579

0

2

119890

119894(12059302) 119870 = minus120583

119861119861

1015840

0

120595

minus(119911 119905) = 120595

119870(119911 119905) 119894 sin

120579

0

2

119890

minus119894(12059302) 119870 = +120583

119861119861

1015840

0

(A5)

120590

2

119905= 120590

2

0+ (ℏ1199052119898120590

0)

2and

120595

119870(119911 119905) = (2120587120590

2

119905)

minus14

119890

minus(119911+11987011990522119898)2

41205902

119905

times exp 119894

ℏ

[

[

minus

ℏ

2

tanminus1(

ℏ119905

2119898120590

2

0

) minus 119870119905119911 minus

119870

2119905

3

6119898

+

(119911 + 119870119905

22119898)

2

ℏ

2119905

2

8119898120590

2

0120590

2

119905

]

]

(A6)



0= 4 times

10

minus11 m ≪ 120590

0= 10


119905≃ 120590

0and

120595

119870(119911 119905)

≃ (2120587120590

2

0)

minus14

119890

minus(119911+11987011990522119898)2

41205902

0 exp 119894

ℏ

[minus119870119905119911 minus

119870

2119905

3

6119898

]

(A7)


Ψ (119911 Δ119905) = (

120595

+(119911 Δ119905)

120595

minus(119911 Δ119905)

) (A8)

with

120595

+(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911minus119911Δ)241205902

0)+(119894ℏ)119898119906119911 cos

120579

0

2

119890

119894120593+

120595

minus(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911+119911Δ)241205902

0)minus(119894ℏ)119898119906119911

119894 sin120579

0

2

119890

119894120593minus

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

119906 =

120583

0119861

1015840

0(Δ119905)

119898

120593

+=

120593

0

2

minus

120583

119861119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

120593

minus= minus

120593

0

2

+

120583

0119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

(A9)




minus


plusmn(119909 119911 119905) =

120595

119909(119909 119905)120595


have 120595119909(119909 119905) ≃ (2120587120590

2

0)

minus14119890

minus119909241205902

0 and

120595

+(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

cos120579

0

2


0+(119894ℏ)(119898119906119911minus(12)119898119906

2119905+ℏ120593+)

120595

minus(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

119894 sin120579

0

2

times expminus(119911+119911Δ+119906119905)241205902

0+(119894ℏ)(minus119898119906119911minus(12)119898119906

2119905+ℏ120593minus)

(A10)




References



























60moleculesrdquo



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






0and 119905

0+Δ119905+

119905



E119860during 119905

0+ Δ119905 + 119905

119863and 119905

0+ 2(Δ119905 + 119905

119863)


0+ Δ119905 + 119905

119863


0+Δ119905+119905



Ψ

119861plusmn119860(r

119861 119905

0+ Δ119905 + 119905

1) = 119891 (r

119861)

1003816

1003816

1003816

1003816

∓

119861⟩ (34)


119863) yields



0(r

119860 120579

119860

0 120593

119860

0) = 119891(r

119860)

(cos(12057911986002)|+

119860⟩ + sin(120579119860

02)119890

119894120593119860

0|minus

119860⟩) and Ψ

119861

0(r

119861 120579

119861

0 120593

119861

0) =

119891(r119861)(cos(120579119861

02)|+

119861⟩ + sin(120579119861

02)119890

119894120593119861

0|minus

119861⟩) with 120579119861

0= 120587 minus 120579

119860

0

and 120593119861

0= 120593

119860



Ψ

0(r

119860 120579

119860 120593

119860 r

119861 120579

119861 120593

119861)

= minus119890

119894120593119860

119891 (r119860) 119891 (r

119861) times (

1003816

1003816

1003816

1003816

+

119860⟩

1003816

1003816

1003816

1003816

minus

119861⟩ minus

1003816

1003816

1003816

1003816

minus

119860⟩

1003816

1003816

1003816

1003816

+

119861⟩)

(35)



119860

0(r

119860 120579

119860

0 120593

119860

0) and Ψ119861

0(r

119861 120579

119861

0 120593

119861



119860 120579

119860 120593

119860 r

119861 120579

119861 120593





Ψ

119860(r

119860 119905

0+ Δ119905 + 119905)

= cos120579

119860

0

2

119891

+(r

119860 119905)

1003816

1003816

1003816

1003816

+

119860⟩ + sin

120579

119860

0

2

119890

119894120593119860

0119891

minus(r

119860 119905)

1003816

1003816

1003816

1003816

minus

119860⟩

(36)


0 120593

119860

0) For


119860


0)



119860(119905) 119905)) [48]


119861(119905) = V

119910119905 119911

119861(119905) = 119911

119861

0 and 119909

119861(119905) = 119909

119861

0 By


119860(119911

119860(119905) 119905) and 120593

119861(119905) =

120593(119911



Ψ

119861(r

119861 119905

0+ Δ119905 + 119905)

= 119891 (r119861) (cos 120579

119861(119905)

2

1003816

1003816

1003816

1003816

+

119861⟩ + sin 120579

119861(119905)

2

119890

119894120593119861(119905) 10038161003816

1003816

1003816

minus

119861⟩)

(37)



119863


119863) = 120587



1015840)


1015840119861(119905

0+ Δ119905 + 119905





0+ 2(Δ119905 + 119905

119863) gives









6 Conclusion







Appendix



119909 0 119861




component

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) plusmn 120583

119861(119861

0minus 119861

1015840

0119911)120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905)

(A1)


120595


119894120583

119861119861

0119905

ℏ

)120595

plusmn(119909 119911 119905)

(A2)

(A1) becomes

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905) exp(plusmn119894

2120583

119861119861

0119905

ℏ

)

(A3)


119871= 2120583

119861119861

0ℏ = 1 4 times 10

11 sminus1 Since |1198610| ≫ |119861

1015840

0119911|

and |1198610| ≫ |119861

1015840



119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905) = minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

(A4)


0



on 119909 120595plusmn(119909 119911 119905) equiv 120595



5 sWe obtain

120595

+(119911 119905) = 120595

119870(119911 119905) cos

120579

0

2

119890

119894(12059302) 119870 = minus120583

119861119861

1015840

0

120595

minus(119911 119905) = 120595

119870(119911 119905) 119894 sin

120579

0

2

119890

minus119894(12059302) 119870 = +120583

119861119861

1015840

0

(A5)

120590

2

119905= 120590

2

0+ (ℏ1199052119898120590

0)

2and

120595

119870(119911 119905) = (2120587120590

2

119905)

minus14

119890

minus(119911+11987011990522119898)2

41205902

119905

times exp 119894

ℏ

[

[

minus

ℏ

2

tanminus1(

ℏ119905

2119898120590

2

0

) minus 119870119905119911 minus

119870

2119905

3

6119898

+

(119911 + 119870119905

22119898)

2

ℏ

2119905

2

8119898120590

2

0120590

2

119905

]

]

(A6)



0= 4 times

10

minus11 m ≪ 120590

0= 10


119905≃ 120590

0and

120595

119870(119911 119905)

≃ (2120587120590

2

0)

minus14

119890

minus(119911+11987011990522119898)2

41205902

0 exp 119894

ℏ

[minus119870119905119911 minus

119870

2119905

3

6119898

]

(A7)


Ψ (119911 Δ119905) = (

120595

+(119911 Δ119905)

120595

minus(119911 Δ119905)

) (A8)

with

120595

+(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911minus119911Δ)241205902

0)+(119894ℏ)119898119906119911 cos

120579

0

2

119890

119894120593+

120595

minus(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911+119911Δ)241205902

0)minus(119894ℏ)119898119906119911

119894 sin120579

0

2

119890

119894120593minus

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

119906 =

120583

0119861

1015840

0(Δ119905)

119898

120593

+=

120593

0

2

minus

120583

119861119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

120593

minus= minus

120593

0

2

+

120583

0119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

(A9)




minus


plusmn(119909 119911 119905) =

120595

119909(119909 119905)120595


have 120595119909(119909 119905) ≃ (2120587120590

2

0)

minus14119890

minus119909241205902

0 and

120595

+(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

cos120579

0

2


0+(119894ℏ)(119898119906119911minus(12)119898119906

2119905+ℏ120593+)

120595

minus(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

119894 sin120579

0

2

times expminus(119911+119911Δ+119906119905)241205902

0+(119894ℏ)(minus119898119906119911minus(12)119898119906

2119905+ℏ120593minus)

(A10)




References



























60moleculesrdquo



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics









6 Conclusion







Appendix



119909 0 119861




component

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) plusmn 120583

119861(119861

0minus 119861

1015840

0119911)120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905)

(A1)


120595


119894120583

119861119861

0119905

ℏ

)120595

plusmn(119909 119911 119905)

(A2)

(A1) becomes

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905) exp(plusmn119894

2120583

119861119861

0119905

ℏ

)

(A3)


119871= 2120583

119861119861

0ℏ = 1 4 times 10

11 sminus1 Since |1198610| ≫ |119861

1015840

0119911|

and |1198610| ≫ |119861

1015840



119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905) = minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

(A4)


0



on 119909 120595plusmn(119909 119911 119905) equiv 120595



5 sWe obtain

120595

+(119911 119905) = 120595

119870(119911 119905) cos

120579

0

2

119890

119894(12059302) 119870 = minus120583

119861119861

1015840

0

120595

minus(119911 119905) = 120595

119870(119911 119905) 119894 sin

120579

0

2

119890

minus119894(12059302) 119870 = +120583

119861119861

1015840

0

(A5)

120590

2

119905= 120590

2

0+ (ℏ1199052119898120590

0)

2and

120595

119870(119911 119905) = (2120587120590

2

119905)

minus14

119890

minus(119911+11987011990522119898)2

41205902

119905

times exp 119894

ℏ

[

[

minus

ℏ

2

tanminus1(

ℏ119905

2119898120590

2

0

) minus 119870119905119911 minus

119870

2119905

3

6119898

+

(119911 + 119870119905

22119898)

2

ℏ

2119905

2

8119898120590

2

0120590

2

119905

]

]

(A6)



0= 4 times

10

minus11 m ≪ 120590

0= 10


119905≃ 120590

0and

120595

119870(119911 119905)

≃ (2120587120590

2

0)

minus14

119890

minus(119911+11987011990522119898)2

41205902

0 exp 119894

ℏ

[minus119870119905119911 minus

119870

2119905

3

6119898

]

(A7)


Ψ (119911 Δ119905) = (

120595

+(119911 Δ119905)

120595

minus(119911 Δ119905)

) (A8)

with

120595

+(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911minus119911Δ)241205902

0)+(119894ℏ)119898119906119911 cos

120579

0

2

119890

119894120593+

120595

minus(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911+119911Δ)241205902

0)minus(119894ℏ)119898119906119911

119894 sin120579

0

2

119890

119894120593minus

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

119906 =

120583

0119861

1015840

0(Δ119905)

119898

120593

+=

120593

0

2

minus

120583

119861119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

120593

minus= minus

120593

0

2

+

120583

0119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

(A9)




minus


plusmn(119909 119911 119905) =

120595

119909(119909 119905)120595


have 120595119909(119909 119905) ≃ (2120587120590

2

0)

minus14119890

minus119909241205902

0 and

120595

+(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

cos120579

0

2


0+(119894ℏ)(119898119906119911minus(12)119898119906

2119905+ℏ120593+)

120595

minus(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

119894 sin120579

0

2

times expminus(119911+119911Δ+119906119905)241205902

0+(119894ℏ)(minus119898119906119911minus(12)119898119906

2119905+ℏ120593minus)

(A10)




References



























60moleculesrdquo



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




component

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) plusmn 120583

119861(119861

0minus 119861

1015840

0119911)120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905)

(A1)


120595


119894120583

119861119861

0119905

ℏ

)120595

plusmn(119909 119911 119905)

(A2)

(A1) becomes

119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905)

= minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

∓ 119894120583

119861119861

1015840

0119909120595

∓(119909 119911 119905) exp(plusmn119894

2120583

119861119861

0119905

ℏ

)

(A3)


119871= 2120583

119861119861

0ℏ = 1 4 times 10

11 sminus1 Since |1198610| ≫ |119861

1015840

0119911|

and |1198610| ≫ |119861

1015840



119894ℏ

120597120595

plusmn

120597119905

(119909 119911 119905) = minus

ℏ

2

2119898

nabla

2120595

plusmn(119909 119911 119905) ∓ 120583

119861119861

1015840

0119911120595

plusmn(119909 119911 119905)

(A4)


0



on 119909 120595plusmn(119909 119911 119905) equiv 120595



5 sWe obtain

120595

+(119911 119905) = 120595

119870(119911 119905) cos

120579

0

2

119890

119894(12059302) 119870 = minus120583

119861119861

1015840

0

120595

minus(119911 119905) = 120595

119870(119911 119905) 119894 sin

120579

0

2

119890

minus119894(12059302) 119870 = +120583

119861119861

1015840

0

(A5)

120590

2

119905= 120590

2

0+ (ℏ1199052119898120590

0)

2and

120595

119870(119911 119905) = (2120587120590

2

119905)

minus14

119890

minus(119911+11987011990522119898)2

41205902

119905

times exp 119894

ℏ

[

[

minus

ℏ

2

tanminus1(

ℏ119905

2119898120590

2

0

) minus 119870119905119911 minus

119870

2119905

3

6119898

+

(119911 + 119870119905

22119898)

2

ℏ

2119905

2

8119898120590

2

0120590

2

119905

]

]

(A6)



0= 4 times

10

minus11 m ≪ 120590

0= 10


119905≃ 120590

0and

120595

119870(119911 119905)

≃ (2120587120590

2

0)

minus14

119890

minus(119911+11987011990522119898)2

41205902

0 exp 119894

ℏ

[minus119870119905119911 minus

119870

2119905

3

6119898

]

(A7)


Ψ (119911 Δ119905) = (

120595

+(119911 Δ119905)

120595

minus(119911 Δ119905)

) (A8)

with

120595

+(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911minus119911Δ)241205902

0)+(119894ℏ)119898119906119911 cos

120579

0

2

119890

119894120593+

120595

minus(119911 Δ119905) = (2120587120590

2

0)

minus14

119890

minus((119911+119911Δ)241205902

0)minus(119894ℏ)119898119906119911

119894 sin120579

0

2

119890

119894120593minus

119911

Δ=

120583

119861119861

1015840

0(Δ119905)

2

2119898

119906 =

120583

0119861

1015840

0(Δ119905)

119898

120593

+=

120593

0

2

minus

120583

119861119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

120593

minus= minus

120593

0

2

+

120583

0119861

0Δ119905

ℏ

minus

119870

2(Δ119905)

3

6119898ℏ

(A9)




minus


plusmn(119909 119911 119905) =

120595

119909(119909 119905)120595


have 120595119909(119909 119905) ≃ (2120587120590

2

0)

minus14119890

minus119909241205902

0 and

120595

+(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

cos120579

0

2


0+(119894ℏ)(119898119906119911minus(12)119898119906

2119905+ℏ120593+)

120595

minus(119911 119905 + Δ119905)

≃ (2120587120590

2

0)

minus14

119894 sin120579

0

2

times expminus(119911+119911Δ+119906119905)241205902

0+(119894ℏ)(minus119898119906119911minus(12)119898119906

2119905+ℏ120593minus)

(A10)




References



























60moleculesrdquo



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






References



























60moleculesrdquo



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Research Article Measurement in the de Broglie-Bohm...

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Transcript of Research Article Measurement in the de Broglie-Bohm...