CHAPTER 11

SOME ARGUMENTS AGAINST

THE EXISTENCE OF DE BROGLIE WAVES

WOLFGANG MOCKENHEIM

1. DISTINGUISHING SCHRODINGER'S FROM DE BROGLIE'S WAVES

The degree of physical reality to be attributed to Schrodinger's wave function depends strongly on the degree of philosophical realism the "observer" is endowed with. The corresponding opinions span a wide range. The pure realist considers an electron as just a small particle, with the wave function being only a mathematical construct in his mind. The pure instrumentalist, on the other hand, refuses to talk about such notions as "reality" but accepts the wave function as the only entity corresponding to his idea of an electron (because a small particle would possess definite position and momentum). Of course, there are many epistemo­logical niches between these extreme positions. However, if we dare to talk about an observer-independent reality (as we wish to do in this chapter), then we cannot accept Schrodinger's wave function as a part of this reality because its shape depends on our knowledge, changing (i.e., collapsing instantaneously) with every additional bit of information we accumulate. *

This feature of the wave function, which makes realists reluctant to add it to the inventory of their world, is just supplying that flexibility which is required to prevent quantum theory from internal contradictions and paradoxes.

*To give an example, the only wave function appropriate to describe a neutron emitted by a nucleus is a spherical wave. The additional knowledge (gained by a nondemolition measurement) that the neutron propagates close to a fixed direction leads to a plane wave. If further the decay time can be approxi­mated (e. g., by detecting the recoiling nucleus) we have to construct a Gaussian wave packet. All these constructs describe one and the same neutron which has not suffered any physical interaction and which, after all, can be detected as a tiny particle with fixed mass and spin.

WOLFGANG MOCKENHEIM • Landshuter Allee 1, D-8903 Bobingen, Germany.

Wave-Particle Duality, edited by Franco Selleri. Plenum Press, New York, 1992.

187

F. Selleri (ed.), Wave-Particle Duality© Plenum Press, New York 1992

188 WOLFGANG MOcKENHEIM

This advantage is not shared by a different entity which, as a special structure of reality that can be described in terms of a formalism developed for waves, is a genuine part of reality. This structure has been known for a long time in the case of light, and it was predicted by de Broglie to apply to material particles too. Various experiments, starting with Young's double slit via Davisson-Germer's electron inter­ferences to recent interference experiments involving neutrons and heavier parti­cles, have proven the reality of these wave structures by presenting evidence in the form of distinct interference structures which could never be generated by ambig­uous mathematical constructs existing merely in the brains of the experimenters.

2. DE BROGLIE'S INITIAL CONCEPT OF DE BROGLIE WAVES

For more than 50 years, de Broglie's initial ideas have been discussed and modified, with the unfortunate result that, today, it is not at all clear what we should understand by de Broglie waves, also called matter waves, empty waves, pilot waves, or phase waves. It is not necessary for our purposes to exhaustively discuss all features contained in the original concept* but we need to clarify some essential points to supply a common basis for the subsequent discussion. In order to remain as authentic as possible, we quote a few original statements.

After pointing out that there is a significant difference to Schrodinger's statistical wave function, de Broglie states that his wave "has a very low amplitude and does not carry energy, at least not in a noticeable manner. The particle is a very small zone of highly concentrated energy incorporated in the wave, in which it constitutes a sort of generally mobile singUlarity. "(2) "Only its phase, related directly to the motion of the particle, seemed to me of fundamental signifi­cance .... "(3) " ••• we think that in wave mechanics one must introduce precise hidden parameters, which are the positions and velocities of the corpus­cles .... "(4) "On con~oit alors l'onde continue comme guidant Ie mouvement de la particule. C'est une onde pilote. "t(5) The reality attached by de Broglie to his wave becomes particularly evident by the following gedankenexperiment, set up to derive Bohr's postulate. "Passons maintenant au cas d'un electron decrivant d'une vitesse uniforme sensiblement inferieur a c une trajectoire fermee. Au temps t = 0, Ie mobile est en un pointO. L'onde fictive associee, partantalors de 0 et decrivant toute la trajectoire avec la vitesse c/~, rattrape l'electron au temps T en un point 0' tel que 00' = ~CT. "+(6) There is scarcely any statement which more

*The interested reader is referred to a brief but competent review by de Broglie's former assistant G. Lochak.(I)

tThe continuous wave is considered as guiding the motion of the particle. It is a pilot wave. tLet us consider now the case of an electron which describes a closed loop at a constant velocity significantly lower than c. At time t = 0 the electron is at point O. The fictitious wave associated with the electron starts at 0 and describes the same loop with the velocity c/j3, overtaking the electron at time T at a point O· with 00' = j3CT.

SOME ARGUMENTS AGAINST DE BROGUE WAVES 189

eloquently can underline the reality of this wave, propagating with phase velocity c113, overtaking the particle and interacting with it.

Hence, we have to deal with a real wave without (measurable) energy or momentum, which is connected to the particle by

E = mc2 = hv

p = mv = hi}"

(1)

(2)

with E and p energy and momentum of the particle, respectively. The particle's velocity (in free space)

v = dE/dp = d\I (moc2)2 + (pc)2/dp = p/m (3)

is equal to the wave's group velocity

v = dv/d(lI}") (4)

The phase velocity, given by

v = }..v = c2/v (5)

is always larger than the speed of light for particles with nonvanishing rest mass mo'

As mentioned above, there is overwhelming experimental evidence showing that certain parts of reality behave in a way that can be described by assuming the existence of de Broglie waves. The big question is, does it really make sense to assume the existence of such waves? And, if so, what is the oscillating medium? In particular, is this medium present everywhere and always (similar to the pre­relativistic ether) or does it emerge and vanish out of nothing into nothing (like the visual impression of a vapor trail), being created continuously by every particle?

3. DISCUSSION OF MODELS

If we prefer to stick to the opinion that reality consists of small particles which are guided by waves, we seem to have only the two models mentioned above. (This is dictated by the tertium non datur too: Either the undulatory medium is a permanent one or it is not.)

Let us for a moment assume the latter case. Then the medium of de Broglie waves has to be void of energy, because it is continuously created by particles which do not lose energy. The idea of something possessing nothing but "reality" but nevertheless influencing a particle by "guiding" it, is very hard to accept. Moreover, applying it to de Broglie's derivation of Bohr's postulate, we are forced to assume a wave-front propagation (or propagation of any part of the wave) with

190 WOLFGANG MDCKENHEIM

superluminal velocity V. If, further, the empty wave created by this process can influence a particle's motion or act in any other physically detectable way, this entails the possibility of sending signals with superluminal velocity. Hence, there would be a contradiction to special relativity. (If the wave cannot act in any physically detectable way, we need not further talk about it at all.) This verdict is strong enough to drop the "vapor trail" model. Consequently, the negative result of an experiment to detect the empty wave of a bunch of photons, (7) though not conclusive, (8,9) agrees with our conclusion.

The alternative to the vapor trail model is that of a permanent medium, i.e., the ether. But regardless of the problems which that notion raises in connection with questions concerning the isotropy of space independent of the state of the particle's motion, the same argument as given above holds also in this case. If a wave can "overtake" a particle in circular motion by propagating with superlumi­nal velocity, and if this wave can act in a physical way, then we have the same contradiction with special relativity.

Hence, we can conclude that de Broglie's example, (6) fails. In order to save the existence of de Broglie waves, one can take the view(8) that the Fourier components form a wave packet such that outside a small bounded region the components cancel to zero. But it is easy to show that this view is in contradiction with de Broglie's initial concept. Let us consider a single particle. As quoted above, this is, for de Broglie, incorporated in the wave(2) and has precise position and velocity. (4)

Hence, from Eqs. (1) and (2) we can deduce precise wave parameters A and v, leading to only one Fourier component. No superposition is possible. If, in order to remove this problem, the quantum mechanical uncertainty of energy and momentum is introduced, we end with Schr6dinger's wave packet and de Broglie's initial concept is destroyed.

Although a single particle can be prepared and handled experimentally, we need not restrict our argument to this case but we can extend it to a bunch of several particles. Due to the independent existence of the de Broglie wave of each particle, it can by no means be expected that outside of the bunch these waves cancel each other. Should this be the case for a special bunch of particles (due to their energies, velocities, and relative locations), this must be considered an accidental and most improbable event. Likewise, there could be a bunch with parameters such that the amplitude of the common wave has the maximum possible amplitude outside of the narrow location containing the particles.

Only the Fourier components of a Schr6dinger wave packet always will can­cel outside of this region because they are chosen according to this requirement.

4. CONCLUSION

According to the discussion above it will be very difficult if not impossible to maintain de Broglie's picture of the independent existence of wave and particle.

SOME ARGUMENTS AGAINST DE BROGLIE WAVES 191

On the other hand, there is so much experimental evidence for some wavelike structure of reality (as well as for its particlelike structure) that we cannot avoid to accept the" dualistic" view: Reality consists of entities inaccessible to our present means of reception which, according to our questions, answer in a way that can be interpreted as and be dealt with mathematical tools developed for waves or particles. But we cannot expect to grasp the complete reality by either of these notions.

REFERENCES

1. G. LoCHAK, in: The Wave-Particle Dualism (S. DiNER, D. FARGUE, G. LoCHAK, and F. SELLERI,

eds.), pp. 1-25, Reidel, Dordrecht (1984). 2. L. DE BROGLIE and 1. ANDRADE E SILVA, Phys. Rev. 172, 1284-1285 (1968). 3. L. DE BROGLIE, Non-linear Wave Mechanics, Elsevier, Amsterdam (1960), Preface. 4. L. DE BROGLIE, G. LocHAK, 1. A. BESWICK, and 1. VASSALO-PEREIRA, Found. Phys. 6, 3-14

(1976). 5. L. DE BROGLIE, 1. Phys. Radium 8, 225-241 (1927). 6. L. DE BROGLIE, C. R. 177, 507-510 (1923). 7. W MDcKENHEIM, P. LoKAI, and B. BURGHARDT, Phys. Lett. A 127,387-390 (1988). 8. F. SELLERI, Phys. Lett. A 132, 72-74 (1988). 9. W MDcKENHEIM, Phys. Lett. A 132, 75-76 (1988).