particle trajectories and interference in a time-dependent model of neutron single crystal...

Volume 109A, number 8 PHYSICS LETTERS 17 June 1985

PARTICLE TRAJECTORIES AND INTERFERENCE IN A TIME-DEPENDENT MODEL OF NEUTRON SINGLE CRYSTAL INTERFEROMETRY

C. DEWDNEY

lnstitut Henri Poincarb, Laboratoire de Physique Thborique, 11, rue P. et M. Curie, 75231 Paris Cedex 05, France

Received 12 April 1985; accepted for publication 22 April 1985

The de Broglie-Bohm interpretation of quantum mechanics is shown to provide an explanation of the observed spatial interference in neutron single crystal interferometers in terms of well-defined individual particle trajectories with continuously variable energy.

The recent single crystal neutron interferometry experiments [ 1,2] have emphasised the fact that compared with the very successful application of quantum mechanics to the statistical prediction of the results of experiments since the 1920's, our understanding of the processes giving rise to these results has hardly progressed at all. In fact very little attempt has been made to develop such an understanding or explanation of quantum phenomena since Bohr pro- nounced such a project impossible [3]. This proscription of the possible in quantum physics is derived from arbitrary philosophical assumptions and does not follow from the existence and success of quantum mechanics [4]. The possibility of causal explanations of quantum phenomena in terms of well-defined individual processes cannot be excluded a priori, as was demonstrated by de Broglie [5] and Bohm [6] who produced such models.

In a series of recent papers [7] these questions have been discussed within the context of neutron interferometry. It has emerged from these discussions that if Bohr's position is set aside, that is if the task of physics is held to consist not only of the attempt to predict the statistical frequency of results in ensembles of similar experiments but also of that to provide explanations and descriptions of the individual processes involved between source and detection, then the only known manner in which this can be done at present, without leading to ambiguity and

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contradiction, is through the causal stochastic interpretation of de Broglie [5], Bohm [6] and Vigier [8]. No problems arise in the analysis of the apparatus plus system (which for Bohr form an unanalyzable whole) into constituent parts, as the essential feature of the unity of quantum phenomena is now manifested by the quantum potential which ~rises from the non- locally correlated stochastic fluctuations of the under- lying Dirac covariant ether [9].

Wheeler's delayed choice experiments only serve to illustrate the ambiguities which arise if the phenomena are analysed arbitrarily into constituent parts [10]. Further, as Bohr himself argued [11], wave particle duality cannot be taken as a serious physical conception of individual quantum processes and only leads to contradiction, as is clearly seen in the interpretation of the time dependent spin-flip neutron interferometry experiments [4,7].

In this contribution the purpose is to propose a simple model of the neutron interferometer and to demonstrate exactly how the causal stochastic interpretation of quantum mechanics could provide a description of the spatial interference figure in terms of definite, individual particle trajectories.

The neutron interferometer is constructed by cut- ting accurately a perfect single crystal of silicon (~10 cm) to produce three slabs of perfectly aligned crystal planes supported by the remaining part of the crystal. Each set of crystal planes splits the neutron beam

377


Fig. 1. The neutron interferometer showing positions of semi- transparent surfaces M1, M2, M2', M3.

into a forward and a deviated beam by Laue diffraction. The forward and deviated beams that emerge behind the interferometer each contain two components, one from each path through the interferometer. These are superposed and the relative intensity of the emerging beams is measured by counters (see fig. 1) at C and D. An interference figure is produced by introducing a phase shift X between the two beams. This is done by inserting a piece of aluminium, which can be rotated to increase the phase shift of one beam relative to the other. As the amount of phase shift varies the relative intensities of the two emerging beams also vary. The correlated count rates in detectors C and D oscillate in anti-phase as the beam phase shift × varies [2] :

I c = 3 , - t ~ c o s x , I D = ~ ( l + c o s X ) ,

where a and ? are constants. As Greenberger [12] has suggested the essential

conceptual features of the experiment can be understood by replacing the three sets of crystal planes by semi-reflecting mirrors in suitable positions, see fig. 1. Furthermore, it is shown that the essential features of the phenomena occur with respect to the part of the motion perpendicular to the mirror surfaces.

In order to solve the time dependent Schr6dinger equation in this context, the semi-transparent mirrors are represented by square potential barriers. An algo- rithm for the numerical solution of the time dependent Schrt'~dinger equation in the presence of potentials has been given by Goldberg, Schey and Schwartz [ 13]. This has been suitably adapted to enable a causal interpretation of all quantum features associated with the scattering of gaussian wave packets from square potentials, in terms of well-defined, continuous, individual particle trajectories with continuously variable energy [14].

A gaussian profile is a good representation of the neutron beam characteristics [2] and the above description can be simply adapted to describe what happens at the first set of crystal planes. A gaussian wave packet scatters from a square potential to produce transmitted and reflected wave packets each of gaussian form. The transmission ratio depends on the magnitude V B and widtha of the potential, the average incident energy E 0 and the inital form of the wave packet ff0(x). For given values of VB, a and ff0(x)a value E 0 can be found which gives a transmission ratio of one half in the numerical calculation. Whether any individual particle enters the transmitted or reflected wave packet depends on its initial position in the wave packet xi0.

The details of the numerical integration technique are given in refs. [13,14], the results of which are presented in fig. 2 *1. Fig. 2a shows the effective potential (that is the classical barrier potential plus the quantum potential) and fig. 2b the associated particle trajectories from a suitably chosen gaussian distribution of initial positions at t = 0. Of course which tra- jectory an individual neutron occupies, although in principle perfectly determined, is in fact uncontrollable. The probability of occupation is given by I~0(xi0)l 2. The centre of the packet is x = 0.5 at t = 0, ~k0(x ) = exp [ - (x - 0.5)2/2o 2 ] exp(ik0x ) in our arbitrary units (~l = 1, m = 0.5). With V B --- (50~r) 2, a = 0.016, o 0 = 0.050 and k 0 = 1.0793 X 501r, the transmission ratio is one half. Clearly any particle which adopts an initial position within the rear half of the wave packet is reflected by the large quantum potential peaks which arise at the time the centre of the incident packet reaches the barrier edge. (The bunching of the trajectories on the peaks produces Wiener fringes.) Any particle which adopts an initial position within the front half of the packet is transmitted. The fate of the neutron at the first set of crystal planes is thus determined and, contrary to the arbitrary descriptions of the proponents of the delayed choice experiments,

,1 A more dynamic representation of these results has been generated in the form of a motion picture. This shows the quantum potential and probability density along with particle positions and energies at each moment of the scattering. It provides a more complete picture since it was not possible to plot all the computed points in the quantum potentials given here.

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o 80

X

o 70

I b j I J I I I [ 0 50 0 . 0 0 2 . 0 0 4 . 0 0 6 . 0 0 8 . 0 0 1 0 . 0 0 12 .00 1 4 , 0 0

T ×I 0 - '

NEUTRON TRAJECTORIES SINPAC

Fig. 2. (a) The effective potential (quantum potential plus classical potential) for a single gaussian wave packet incident on a square potential barrier (M1), in the region x = 0.6 to 0.9, t = 4.0 X 10 -4 to 12.0 X 10 -4. (b) The trajectories associated with (a) generated from a gaussian distribution of initial positions at t = 0. Transmission ratio is one half.

379


X

T

NEUTRON TRAJECTORIES'PHSHIO ×1 0 -"

Fig. 3. (a) The effective potential for one wave packet incident from either side of the semi transparent surface (M3) with no phase difference, X = 0. (b) Trajectories associated with (a).

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I I I I I I

0 . 8 0

X

0.TO

I I I I I I I b 0 .30 0.00 2 . 0 0 4 . 0 0 6 . 0 0 8 , 0 0 1 0 . 0 0 1 2 . 0 0

T x l 0 -" NEUTRON TRAJECTORIES PHSH15

1 4 . 0 0

Fig. 4. (a) The effective potential for two incident packets with phase difference X = ~r/2. (b) Trajectories associated with (b).

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is not affected by what we choose to measure at a later time.

The reflected and transmitted wave packets form the beams 4I and VII in the interferometer, only one of which actually contains the neutron. These two wave packets scatter from the two mirrors M2, M2' in exactly the same way as already described, being split into parts.

The reflected wave packets from M2 and M2' con- verge on the last semi-transparent mirror M3. In order to simulate this stage of the process the same technique as before can be employed with an altered initial wave function distribution

q,O(X) = @01(X) + @02(X),

~01(X) = exp [--(x -- 0.5)2/2002 ] exp(ikox),

~02(X) = exp [ - (x - 1.0)2/2o21

× exp [- i(k0x + ¢)1 exp(ix),

representing the two converging wave packets with X the phase shift applied to beam II and ¢ a constant phase factor to symmetrize the two waves with respect to the barrier potential. With these initial conditions the results of the numerical calculation show that varying the phase shift factor X between 0 and 2rr produces the correct type of interference figure. The contrast of the interference fringes depends crucially on certain factors. The transmission ratio for single wave packet scattering from the same potential must be one half. The width of the potential barrier a must be small compared with the half width of the wave packet, large values of a reduce the contrast.

When × = 0, rr, 2rt the quantum potential, as shown in fig. 3a is symmetric about the barrier centre. A series of violent oscillations develop on each side of the barrier potential. These arise when each incident wave interferes with the combination of its own reflected wave and the transmitted wave from the other side. In this case the amplitude of the outgoing wave will be enhanced, this increases the magnitude of the oscillations in the quantum potential and a greater proportion of the incoming trajectories will be turned around outside the barrier. The remaining trajectories, which entered the region of the potential before the oscillations in the quantum potential outside became sufficient to reflect them, are turned around inside

the potential region by the quantum potential peak which occurs there due to the interference of the two transmitted waves in this region. This peak ensures that all the neutron trajectories in the lower (upper) beam emerge from the interferometer in the lower (upper) beam. Comparison with fig. 2b shows that it is these trajectories, that would have been transmitted in the single packet case, that are reflected within the barrier in this case.

With X = zr/2 the situation is very different. In this case the quantum potential oscillations are greatly reduced on one side of the potential barrier, at the time when the particle is passing (in the limit they disappear) as shown in the upper section of fig. 4a. In the lower section notice that the maxima of the quantum potential oscillations occurr at an earlier time en- suring that all the trajectories, see fig. 4b, constituting beam II are reflected. Those constituting beam I now enter the potential barrier and emerge after the reflec- tion of those in beam II, both forming the single emerging beam. In this case the reflected wave from beam I is (almost completely) cancelled by the antiphase transmitted wave from beam II.

When X = 31r/2 the situation is essentially reversed (see fig. 5), all the trajectories and any neutron emerging in the upper section. The few trajectories which do not follow the others come from the extreme tails of the packets and so have very low probability, here they represent the effect of a f'mite potential width.

One interesting feature of this approach is that it allows an explanation of why the contrast of the interference figure does not depend on how much the wave packets spread during the elapsed time to the third set of crystal planes. The contrast depends on the reflected wave from one beam and the transmitted wave from the other having the same amplitude in the region of interference. If one of the beams has a time delay introduced say by inserting aluminium slides in one beam path [15] then these two packets can never interfere destructively to cancel each other completely. This reduces the contrast since if the outgoing wave is not completely cancelled it interferes with the incoming wave giving rise to quantum potential oscillations which reflect neutron trajectories. Since the spreading of the packets in beam one and two is the same this important amplitude matching depends only on the separation of the packet centres caused by the time delay and not the degree of spreading. In this model

382


x

T NEUTRON TRAJECTORIES PHSH05

×10 -'

Fig. 5. (a) The effective potential for two incident packets with phase difference X = 3,r/2. (b) Trajectories associated with (a).

383


the contrast depends on o 0, the initial half width, since this influences the relative intensity of each o f the t ransmit ted and reflected packets.

This simple model o f spatial interference in neutron interferometry clearly shows that Bohr's proscription is an arbitrary assumption and not a necessary conse- quence o f quantum mechanics. I t is possible to analyze quantum phenomena into consti tuent parts, but this does not represent a return to a simple classical model since the quantum potent ial which arises is not like a classical potential. It is a potent ial which depends on the state o f the system as a whole and as such goes beyond the simple mechanical potentials of classical physics.

Bohr was correct to emphasise the fundamental wholeness o f quantum phenomena, but unanalyzabil i ty does not follow. Quantum phenomena may be anal- yzed consistently in the quantum potent ial approach. There is then no contradict ion between the existence o f def'mite neutron trajectories, the neutron travels along path I or path II in the interferometer, and the existence of interference. I f the neutron is to be described between source and detect ion then in order for the existence of neutron trajectories to be compatible with observed results the quantum potential must exist to determine their distr ibution and hence their wave- like properties. Any approach which does not involve such a feature can only lead to contradict ion and paradox, as has been discussed elsewhere [3,6,9].

In this paper spatial interference only is discussed, more exciting questions arise in the case in which time dependent spin-flip coils are introduced to invert the neutron spin by exchange of energy [ 16]. These experiments may also be subject to a similar analysis [7] and detailed calculations will be published later.

The author wishes to thank the Royal Society for a European Exchange Fellowship during tenure of which this work was done and the Insti tut Henri Poincar6 for their hospitality.

References

[1 ] H. Ranch, Test of quantum mechanics by matter wave interferometry, in: Intern. Syrup. on the Foundations of quantum mechanics (Tokyo, August 1983), J. Phys. Soc. Japan and references therein.

[2] S.A. Werner and A.G. Klein, in: Neutron scattering, eds. D.H. Price and K. Sk~51d (Academic Press, New York, 1984) and references therein.

[3] N. Bohr, Atomic theory and the description of nature (CUP, 1934).

[4] C. Dewdney, Quantum interference, the quantum potential and complementarity, in: Proc. Second Intern. Conf. on Epistemology (Athens, 1984), to be published.

[5] L. de Broglie, Nonlinear wave mechanics (Elsevier, Amsterdam, 1960).

[6] D. Bohm, Phys. Rev. 85 (1952) 166, 180. [7] C. Dewdney, Ph. Gueret, A. Kyprianidis and J.P. Vigier,

Phys. Lett. 102A (1984) 291; C. Dewdney, A. Kyprianidis, A. Garuccio, Ph. Gueret and J.P. Vigier, Lett. Nuovo Cimento 40 (1984) 481; C. Dewdney, A. Garuecio, A. Kyprianidis and J.P. Vigier, Phys. Lett. 104A (1984) 325.

[8] D. Bohm and J.P. Vigier, Phys. Rev. 96 (1954) 208. [9] J.P. Vigier, Astr. Nachr. 303 (1982) 61.

[10] D. Bohm, C. Dewdney and B.J. Hiley, The Wheeler delayed choice experiment understood through the quantum potential appraoch, Nature, to be published.

[11] N. Bohr, in: Albert Einstein: philosopher-scientist (Evanston, 1949).

[12] D.M. Greenberger, Rev. Mod. Phys. 55 (1983) 875. [13] A. Goldberg, H.M. Schey and J.H. Schwartz, Am. J.

Phys. 35 (1967). [14] C. Dewdney and B.J. Hliey, Found. Phys. 12 (1982) 27. [15] H. Kaiser, S.A. Werner and E.A. George, Phys. Rev.

Lett. 50 (1983) 560. [16] G. Badurek, H. Ranch and J. Summhammer, Phys. Rev.

Lett. 51 (1983) 1015.

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