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Foundations oI" Ph)'sics, Vol. 25. No. 1. 1995

Compatible Statistical Interpretation of a Wave Packet

Mirjana Bo~i6 ~ and Zvonko M a r k I

Received Januar_t' 4. 1994

A compatible statistical interpretation oJ' a wave packet is proposed. De Broglian probabilities whidt unite wave and particle features q f quantons are evahtated for fi'ee wave packets and Jbr a supelposition o f wave packets. The obtaflled expressions provide a vet T plausible and ph)'sicallr appealing explanation o[" coherence ht apparently #woherent beams and o1" the characteristk" too&clarion o f the momentum distribution, found recentl.r fit netttron httetferomett T combflled with spectral filtering. Certahl conchtsions abottt dualism and objectivity ht quantum domaht are also derived.

1. I N T R O D U C T I O N

Since the birth of quantum theory, its central conceptual and theoretical problem has been the understanding of various aspects of a dualism of wave and corpuscular properties of quantum particles. The famous de Broglie relation~

;~ = / , /p (1)

synthesized, in an ingenious way, corpuscular (p), wave (;~), and quantum (/I) properties of microscopic objects. In order to stress that in the quantum domain we are dealing with a new kind of objects for which classical names are not appropriate, Levy Leblond ~2~ proposed to give to quantum objects the common name quantons.

Schr6dinger's equation, ~3~ which determines the dynamics of quantons, is also a particular synthesis of mechanical laws and of the laws of wave motion. In de Broglie's relation, clearly 2 is a wave property, and p is a

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160 Bo~i~ and Mari~

mechanical property. Schr6dinger's wave equation and consequently wave function synthesize, in a nondirect way, mechanical and wave properties. This implies numerous controversies between two schools of thought; the first which was founded by de Broglie, ~4~ Schr6dinger, ~-~ and Einstein ~('~ and the second founded by Bohr, c7~ Heisenberg, (s~ and von Neumann. ~9~

Both schools agree that modulus square of the wave function I~,(r, t)[ 2 is a probability of the particle presence, and therefore it reflects particle property. But, the interpretation of ~p(r, t) itself is a subject of disagreements. According to Schr6dinger, ~~ de BroglieJ 4~ Bohm and Vigier,' I I~ Selleri,~l,-~ Vigier ~ ~3~ and Petroni, '~4~ a wave function describes objectively an existing wave which accompanies a particle. The physical nature of this wave has not been understood yet. On the other hand, Bohr, ~7~ Heisenberg, '~ and von Neumann '9~ refused to associate any physical reality with a wave function ~b(r, t).

The view of two schools about duality and objectivity may be analyzed and confronted in a new way by enriching the probabilistic scheme of the standard quantum mechanics with a new kind of probabilities, called de Broglian probabilities.']4'15~ This enlarged probabilistic interpretation, called compatible statistical interpretation (CSI), provides new understanding of single-particle interference ']~'~6~ as well as of two- particle interference'~7' phenomena.

In the present paper we expose CSI for a wave packet in light of recent interference experiments with spectral filtering.'~s'tg~ In Sec. 2 are outlined the basic elements of CSI, in general. In Sec. 3 is given a short summery of physical pictures about wave packets and of new experimental data about the nature of wave packets. De Broglian probabilities for a Gaussian wave packet and for a superposition of two Gaussian wave packets are used in Sec. 4 to explain the characteristic modulation of the momentum distribution of the outgoing beam from the interferometer. Certain conclusions about dualism and objectivity in quantum domain are also derived.

2. OBJECTIVITY A N D PROBABILITIES IN Q U A N T U M M E C H A N I C S

Any analysis of objectivity in quantum domain has to start with the Schr6dinger equation and its solutions. Stationary solutions ~b,,,(r, t) of the Schr6dinger equation,

2n~ V- + V(r)} ~bm(r, t) = ill ~t ~bm(r' t) (2)

Statistical Interpretation of a Wave Packet 161

are the particular solutions of this equation having the form

ip,,,(r, t) = e - iE"'//'~0,,,(r) (3)

where q~,,,(r) satisfies the stationary Schr6dinger equation

- h~- V 2 + V(r)} cp,,,(r) = E,,,cp,,,(r) 2m

(4)

The index m of energies and states consists in general of a set of quantum numbers. In both the above-mentioned interpretations of the theory, quantum numbers in stationary states are regarded as objectively existing characteristics of quantons in these states. It follows from the assertion that the probability to find a particular eigenvalue when one measures the corresponding quantity in this state is equal to one. Since the corresponding operators are derived from the appropriate mechanical quantities, their eigenvalues are evidently of the mechanical nature.

The use of the superposition principle in the case of a general solution of the Schr6dinger equation.

~b(r. t ) = ~ c,,~bm(r, t) (5) m

makes the association of mechanical quantities not so simple. Proponents of the two mentioned schools of thought have very different approaches concerning this subject.

Quantum mechanics avoids considering eventual objective physical quantities in the state (5). It refuses to attribute the properties to quantons in this state (5) before the measurement. The quantum theory of measurement attributes reality only to the results of measurements. This theory even prohibits one from thinking about objectively existing properties of quantons before the measurement. Instead, it deals with probabilities Jc,,,r 2 of finding a value m in the measurement of the observable M (or a set of observables M). De Broglie called those probabilities predicted (because they predict the results of measurements) and insisted ~4'2~ on the fact that for their measurement special preparation (spectral filtering) is necessary. The predicted probabilities are not of classical kind since they do not satisfy axioms of the classical theory of probability. The probabilities J~(r, t)J-" d3r which determine the space distribution of particles in the state ~b(r, t) are called present probabilities since they may be determined without any preparation of the physical system.

New kinds of probabilities, based on the concepts of wave mechanics, have been associated recently with the state (5). The probabilities P,,(r, t),

162 Bo~i~ and Mari~

called de Broglian probabilities, permit one to attribute mechanical quantities to particles in those states. By definition P,,,(r, t) d~r is the probability that a particle in the state (5) is at r at the moment t and that its quantum numbers are m. De Broglian probabilities are of classical kind since they satisfy axioms of the classical theory of probability. They are determined from the following set of equations ~ J5 ~7).

I@(r, t)l 2 = ~ , P,,,(r, t) (6) m

P,,,(r, t) Ic.,I 2 I@,,,(r, t)l-'

P,,(r, t) It.I-" Ir t)l 2 (7)

Equation (6) is the generalized Selleri-Tarozzi relation ~-'~) which says that the event that a particle is at r and has quantum numbers m is independent of the event that a particle is at r and has a quantum numbers m'. Equation (7) is written as a hypothesis. It represents a generalization of the analogous (but of a very plausible) relation written for Mach-Zehnder and neutron interferometersJ ~5~

From Eqs. (6) and (7) follow the expressions for de Broglian probabilities:

{ ' (,, (,,, r (r, t) r t i t e, ,(r , t ) = Ic,,l" I~',,,(r, t)l 2 1 + ~ '"" '* ' * Z,, Ic,,I 2 [~.(r, t)l 2 (8)

If m is continuous, instead of the series (5) we have the integral

~p(r, t) = f dm c(m) ~b,,,(r, t) (9)

and the associated de Broglian probabilities P(m, r, t) read

P(m, r, t ) = Ic(m)l 2 I~,,,(r, t)l 2 ~ d n d . ' c*(n) c(n ') ff,,*(r, t) ~p,,,(r, t) an [c(n)[ -~ [~, ,(r, t)[ -~

(10)

The probabilities Pro(r, t) describe an objectively existing distribution in space, of the particle characterized by the set of quantum numbers m, surrounded by the wave if(r, t). This distribution evidently differs from the distribution I~,,,(r)l-' = I~0,,,(r)l-" in the state ~p,,,(r, t).

The proposed interpretation is not in contradiction with the experimen- tally established fact that Ic,,,l" is a probability to find after a measurement a particle in the state r t) with a set of eigenvalues 17l. There is no


contradiction because the set of functions ~,,,(r, t) are the possible states after the measurement, whereas q~(r, t) is a (real) state before the measurement. All particles in the state qJ(r, t) are surrounded by the same wave, ~,(r, t). The coordinate distribution of a particle with quantum number m is P , , ( r , t ) , This distribution is determined by a l l ~b,,(r, t) in ~O(r, t) and not only by ~,,,,(r, t). Quantum measurements, as it is claimed, change the quantum state ~(r, t) through the procedure of spectral filtering, or any other interaction with apparatus, into various states ~p,,,(r, t). To which channel a particle will enter depends on the values of its mechanical quantities.

3. I N T E R F E R O M E T R Y WITH WAVE PACKETS

Despite the above-mentioned prohibition imposed by the standard interpretation, many physicists have used quantum measurements to make pictures of objects, states, and events in a quantum domain. This is best seen on the example of a wave packet. --

At the very beginning of the development of quantum mechanics, attempts were made to identify quantum particles with wave packets. But, when it became clear that wave packets are spreading with time, (23) this picture met severe difficulty. In 1953 Schr6dinger wrote '5~ that this difficulty was, for him and for de Broglie, the main reason to give up (for a long period) opposing the standard interpretation of quantum mechanics. He explained: "As long as a particle, an electron or proton, etc., was still believed to be a permanent, individually identifiable entity, it could not adequately be pictured in our mind as a wave parcel. For, as a rule, apart from artificially constructed and therefore irrelevant exceptions, no wave parcel can be indicated which does not eventually disperse into an ever increasing volume of space."

The importance of wave packets comes above all from the fact that wave packets are general solutions of the free-particle Schr6dinger equation (V(r) =0) . The stationary states of this equation have the form

where

1 ~//k(r, t) -- (27~)3/2 e - i ' ' ' ' e i k ' r ( 1 1 )

h2k 2 h m - - E k= 2 m ' p = h k (12)

164 Bo~i~: and Mari~

and a wave packet is any linear superposition of those states which is usually written in the form

t~,(r, t) = (2rc)3/~ g ( k ) e - i ' " ' . e i k ' r d 3 k (13)

The physical importance of wave packets is due to the fact that they are normalizable to one, which is not the case with plane waves ( 11 ), The most interesting wave packet is the Gaussian one-dimensional wave packet

1 ~(_x, t ) = ~ 2 ~ - , - g(k) e ilk . . . . . . ] dk (14)

where

N///• - - r52q k - - k o 12/4 g(k) = (2~)1/4 e ( 15 )

For a Gaussian wave packet one writes the superposition (14) in the closed form~24~:

- - " e ik ' ' 'exp - 5_~ (16) ~t(X, / ) = (~4 + 4h2t2/m2)l/4 + 2iht/m J

where cp is real and independent of x:

hk o 2ht ~o = - ,9 - 2n~ t, with tan 2,9 = (17) ma 2

Physical pictures associated with a Gaussian wave packet are of two types: one is based on the form (14), and the other is given by expression (16).

Most often, the use of wave packets is related to the proper ty of particle beams not to be monochromatic. Namely, the usual particle beams are characterized by a certain spread Ap = h Ak of momenta. Because of it, with the form (14) in mind, there is a tendency to associate a wave packet with a beam of particles and not with an individual particle. In such an approach one often regards particle beams as consisting of particles having different momenta and being in the corresponding plane wave states.

The shortcoming of that picture is the unnormalizabil i ty of a plane wave exp{ i ( k x - rot)} which, by assumption, is a wave function of a particle (from a beam) with momentum p = hk. Namely, it seems very plausible that wave functions of individual particles should be normalizable. Another shortcoming of such a picture is that it is difficult to understand what coherence in such a beam means physically.


/

/ \ / \

/ \ / / \ \

0 It

Fig. 1. Scheme of the interferometer.

In fact, in explaining coherence of neutron beams, one usually uses a picture [based on the expression (16)] in which the wave packet is iden- tified with a localized particle. Kaiser etal. describe ~ts~ that as follows: "When we say that there is a neutron wave packet of coherence length Ax, we often think semiclassically that the wave packet "is" the localized neutron particle. If the two wave packets traversing paths I and II in the interferometer overlap within a distance Ax in the region of recombination, we picture the neutron overlapping with itself, giving rise to "interference" (see Fig. 1 ). But, if the spatial shift of one wave packet with respect to the other is sufficiently large (greater than the coherence length, which is proportional to the width of the Gaussian wave packet), they no longer overlap in the region of a superposition. ~25~ Because of it, the disappearance of the contrast of the interferogram, when the order of interference increases, has usually been attributed to the spatial shifts of wave packets from two paths.

However, recent interference experiments, which are combined with spectral filtering, make questionable and doubtful the above picture of a particle and the explanation of the cause of disappearance of interference. In those experiments the initial Gaussian wave packet is split by the first plate of the interferometer into two beams; 7ti and ~uu, associated with paths I and II, as shown in Fig. 2. The beam on path II passes through the "sample" of thickness D. The nuclear potential inside the sample causes the

825 25 1-12

166 Bo~i6 and Marit~

o k

r

/

/ N / \

/ /

\ filter

0

\ R

and the order of interference

D m = - - ( 2 0 )

Da

(ko is the value of the wave vector at the maximum of the Gaussian distribution), the phase shift takes the form

D k o Zk(D) = --2re - - - - (21)

D;. k

Fig. 2. Scheme of the interferometer with a filter.

change of wave vector and consequently induces the phase shift, x d D ) , of each monochromat ic component of the wave function. The phase shift is given by

2~ Zk(D) = -- ~ - Nb, .D (18)

where b,. is the nuclear coherent scattering length and N is the particle density. By defining the s.c. ) , -- thickness

ko D~.= (19)

Nb,.


From the third plate of the interferometer emerge two beams: one in the O direction and the other in the H direction. In the ideal interferometer (the real is very close to the ideal), the wave function at the exit of the interferometer in the O direction is the sum of the wave functions associated with paths I and II:

~Uo(X, t) = tPl(x, t) + ~Pv(x, t) (22)

Under the above-described condit ions there exists the following relation among the monochromat ic components ~ki(X, t) and ~Uku(x, t) of ~U~(X, t) and ~'t i | (X , t):

I//kll(X, t) = ~uki(x, t) e izk~~ (23)

By taking into account that the initial wave function has the form (14), that the plates split a beam into two equal sub-beams, and that between ~ku(X, t) and 7"kt(X, t) there exists the relation (23), we conclude that the wave packet in the direction O is given by

o r

I f + ~ e i [ k x - - o Jr ] IPo(X, t)=~-----~ - ~ g(k) dk

l_...~ f + z ei[kx ,,,t+q, kID~] + g(k) dh" 2 x / ~ --,-

t//O(.X', t ) = ~ [ g(k) e ilk . . . . . . . . ][1 +ei'pk(D)]dk (24)

In the absence of the filter it is found that the detected number of neutrons show almost no variat ion with increasing width of the sample, when the order of interference is large. In presence of the filter, the detected number of neutrons with a determined value of k shows a variation with D, even for large values ofm.

Researchers concluded that their results show the importance of different plane wave components coming from two beam paths I and II. Therefore, it appears that we have to come back again to the picture of a beam made up of independent particles in plane wave states. In fact, Kaiser etal. conclude ~jS~ that "matter waves are not particles, and we have no right to think of them as such, even in a semiclassical way . . . . We know that the neutron is a particle when it is emitted, and again when it is

168 Bo2i~ and Mari~

detected, but between these two times, the physical connection between the neutron particle and the wave packet remains hidden, no matter how diligently we try to analyze the quantum questions with our classical tools."

We agree with those conclusions. Moreover, we are going to show that the above-described phenomena are more understandable if one considers waves and particles as two distinct entities and if one couples mechanics of particles and the laws of wave motion as they are coupled in compatible statistical interpretation.

4. DE B R O G L I A N PROBABILITIES FOR WAVE PACKETS

In the compatible statistical h~terpretation a wave packet ( I 1 ) describes a wave which accompanies each particle in the beam, independently of the value of its momentum. The probability density for a particle to be at r and to have a momentum k when its wave function is (11) is determined by de Broglian probability:

P(k,r,t)=lg(k)lZjd3kd3k' g*(k)g(k')~(r,t)r (25)

The latter expression is obtained fi'om (10) by taking into account that [e-J'"'. eJk"l = 1 as well as the Parseval identity. The integral of de Broglian probabilities over k is equal to IqJ(r, t)12:

f d3k P(k, r, t ) = ItP(r, t)i-' (26)

In this interpretation the wave packet q;(x, t) describes a wave which sur- rounds any particle in the beam of particles under consideration. Different particles have different momenta. The objective space distribution of a particle with momentum p is determined by P(k, r, t).

The proposed interpretation of a wave packet has something in common with, and something different from each of the two above-analyzed physical pictures. It associates the whole wave packet with a particle, but it does not identify a wave packet with a particle. A particle and a wave packet are two different entities. Compatible statistical interpretation attributes definite values of momenta to each particle but denies that the distribution over coordinate r of a particle with given momentum is determined by a plane wave. Instead it affirms that the space distribution is determined by the whole wave function.


In a similar way, we may associate with the one-dimensional Gaussian wave packet (24) at the exit of the neutron interferometer de Broglian probability densities PD(k, x , t). According to compatible statistical interpretation, P D ( k , x , t) is the probability density that particle has wave vector k and to be at x at the moment t:

P D ( k , x , t ) = ~ l g ( k ) [ 2 l + c o s _ D~. k J

[~ +_ ~ g(k ) e iEk ........ 1[ 1 + e i~ek'~ dk] 2

+ ~ Ig(k)[-' [ 1 + cos 2n(D/D~.)(ko/k)] dk (27)

The above expression contains two distinct parts: the first depends on k, and the second is independent of k but depends on x and t. The k-dependent factor

1 E ~:k~D(k) = ~ Ig(k)l z 1 +cos2z~ D k ~ D;, k J (28)

is equal to the probability of the presence of a particle with wave vector k in the beam at the exit in the O direction. It is important to point out that the latter probability is identical with the probability (in the theory of measurement) to find after measurement of k the particle with wave vector k. This is a specific characteristics of a superposition over a plane wave and is due to the fact that the modulus of exp i ( o ~ t - k x ) , [exp i ( t o t - k x ) [ = 1, is independent o f x and k. Thanks to this property, in (27) there exist only terms which depend on k and terms which do not depend on k. There are no terms which depend both on k and on x.

The second part is common to all probabilities PD(k, x, t) (since it is independent of k) and determines the distribution over x of a particle with any momentum p = hk at the exit of the interferometer in the O direction�9

The results of recent neutron and photon experiments in which the spectral analyzer is put at the exit of the interferometer but before the detector may be understood consistently using the above concepts�9

In CSI as well as in the standard interpretation the detected number of neutrons Io(D) in the absence of a filter (in a long time interval) is proportional to

' "~rt dk (29) Io = --~ d t l~go(X , t ) ] -= ]g(k)[ z l + c o s - D~. k J

170 Bo~i~ and Mark

2.50

2.00

1.50

1.00

0.50

Io(O)

o.oo o.oo l o . o o 2o.oo 5o.oo 40.00 50.00

U/O~ Fig. 3. The dependence of intensity I. on the thickness D. The values of the parameters are: Ak/k = 0.002, i.e.. 5 = 1.87007 10-'qm k~ = 2.6737 10" ffm.

The dependence of Io on D is illustrated in Fig. 3. We see that Io tends to a constant with increasing D, when D,> D;., in agreement with Rauch's calculationd2S' based on experimental results.

According to CSI, the role of the spectral filter is to select the particles with momentum p = hk. The number of selected particles with momentum p during a (theoretically) infinite time interval should be equal to the number of such particles which were present behind the interferometer in the O direction. Therefore, the detected number of filtered neutrons with momentum p = hk should be propor t ional to

IkD = PD(k, x, t) dt /30)

The factor 13D(k) in PD(k, x, t) is time independent. It is easy to see that the integral over t of the second factor in PD(k,x , t) is equal to Dr. Therefore,

[ 27r D k ~ I m = l g ( k ) ] 2 l + c o s D~. k J (31)


According to the SCI the variation of IUD with k (illustrated in Fig. 4) is due to the different behavior of particles with different k when they leave the third plate of the interferometer. Whether a particle with given momentum p = hk will join the O beam (or the H beam) is determined above all by the value of the factor [ 1 + cos 2rc(D/D~.)(ko/k)]. It is true that the two wave packets associated with paths I and II at the exit from the third plate are separated ~t8) in space by Al=2rcNb,.D/k 2, but this fact does not influence the choice among the two directions which each particle "makes"

o.oo / , k o.oo / 2.45 2.90 2.45 2.90

10o.oo ] loo.oo

0.00 t , 0.00 r , ~ . . . . ~ , , 2.4a 2.9o 2.4s 2.9o

2 .45 2 .90 2 .45 2 .90

ooo': 1 2 .45 2.90 2 .45 2 .90

Fig. 4. The variation of lkD with k for different values of m = D/Dz. The unit along the Ikn axis is 10-9; tile unit along the k axis is l0 s 1/cm.

172 Bo$i~ and Mari~

leaving the th i rd plate. T he s e p a r a t i o n of wave packe t s affects the second

t e rm in P o ( k , r, t) which is independent ofk .

T h e a b o v e c o n s i d e r a t i o n s are the pa r t i cu l a r rea l iza t ion of o u r view on

the n a t u r e of p robab i l i t i e s in q u a n t u m m e c h a n i c s as a syn thes i s of wave

and par t ic le proper t ies . We h a v e e l a b o r a t e d these ideas in a long a n d close c o l l a b o r a t i o n wi th Prof. J e a n - P i e r r e Vigier. His p e r m a n e n t ins is tence on

the i m p o r t a n c e of new expe r imen t s , by seeing in t h e m very often an

implic i t t o u c h wi th the f o u n d a t i o n a l p r o b l e m s of q u a n t u m mechan ic s , has

been en l i gh t en ing a n d insp i r ing for us. His i m a g i n a t i o n , by which exis t ing

expe r imen t s cou ld be s u p p l e m e n t e d so as to b e c o m e crucia l ones for the

old con t rove r s i e s in q u a n t u m mechan ic s , has o p e n e d to us a new h o r i z o n

for theore t ica l inquiry . This ar t icle is ou r m o d e s t c o n t r i b u t i o n to his

jubilee.

R E F E R E N C E S

I. L. de Broglie, Recherches sur la thborie des quanta (Th~se. Paris, 1924): reprinted in Fond. Louis de Broglie 17, 1 (1992).

2. J. M. Levy-Leblond, Physica B 151, 314 (1988). 3. E. Schr6dinger, Phys. Rev. 28, 1049 (1926). 4. L. de Broglie, Etude critique des bases de I'interprbtation actuelle de la mkcanique

ondulatoire (Gauthiers-Villars, Paris, 1963 ). 5. E. Schrfdinger, "The meaning of wave mechanics," in Lo,is de Broglie, ph)'sicien et

penseur ( Gauthiers-Villars, Paris, 1963). 6. A. Einstein, in: Rapports et discussions du cinq,ddme conseil de physique de I'bastitut biter-

national de Physique Solvay, Brussels, 1927 (Gauthiers-Villars, Paris, 1928), p. 253. 7. N. Bohr, in: Albert Einstein, Philosopher-Scientist, P.A. Schilpp, ed. (Open Court,

Evanston, 1949), p. 200. 8. W. Heisenberg, Tbe Physical Prhwiples t~f the Quantum TbeoiT (Dover. New York, 1930). 9. J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton, New

Jersey, 1955). 10. L. de Broglie, Etude critique des bases de I'hlterprdtation act,tel& de la mkcanique

ondulatoire (Gauthiers-Villars. Paris, 1963). 11. D. Bohm and J. P. Vigier, Pbys. Rev. 96. 208 (1954). 12. F. Selleri, Found. Phys. 12, 1087 (1982). 13. J. P. Vigier, Physica B 151, 386 (1988). 14. N. C. Petroni, Pbys. Lett. A 14. 370 (1989); 160, 107 (1991). 15. M. Bo~i~ and Z. Mari~, Pbys. Left. A 158, 33 (1991). 16. M. Bo~i~:, Z. Mari6, and J. P. Vigier, Found. Pbys. 2L 1325 (1992). 17. M. Bo~i~: and Z. Mari~, "Probability and interference," to be published in Courants,

amers, et ecueils en microphysique, G. Lochak and P. Luchak, eds. (Fond. Louis de Broglie, Paris, 1994), p. 89.

18. H. Kaiser. R. Clothier, S. A. Werner, H. Rauch, and H. W61witsch, Phys. Rev. A 45, 31 (1992).

19. H. Rauch, "State selection and spectral modelling in neutron interferometry,'" in Quantum Physics and Universe, M. Namiki, ed. (Waseda Univ. Tokyo, 1992).


20. G. Lochak, "De Broglie initial conception of de Broglie wave," in The Wa~,e-Particle Dualism. S. Diner, D. Fargue, G. Lochak, and F. Selleri, eds. (Reidel, Dordrecht, 1984).

21. F. Selleri and G, Tarozzi. Nuovo C#~zento 43, 31 (1978). 22. A. G. Klein, G. I. Opat, and W. A. Hamilton, Phys. Rel,. Lett. 50, 563 [1983). 23. E. Schr6dinger, Sitzungsber. Preuss. Akad. Iu 296 (1930). 24. C. Cohen-Tannoudji, Bernard Diu, and Franck Laloe, Mbcanique quantique I I Hennann,

Paris, 1977). 25. H. Rauch, "Scope of neutron interferometry," in Ne~m'on inte~ferometrv U. Bonse and

H. Rauch, eds. [Clarendon Press, Oxford. 1979).