Volume 132, number 2,3 PHYSICS LETTERS A 26 September 1988

CAN DISPERSION EXPERIMENTS TEACH US ANYTHING ABOUT DE BROGLIE WAVES?

W. MtiCKENHEIM

Stiring IO, D-3400 Giittingen. FRG

Received 26 May 1988; accepted for publication 2 1 July 1988 Communicated by J.P. Vigier

De Broglie’s view of de Broglie waves is briefly analyzed. It is shown that the wave attached to a particle must not be mixed up with the Fourier components of a Schrijdinger wave packet. Hence, dispersion experiments are capable of delivering new facts.

In the preceding Letter Selleri [ 1 ] has criticized our experiment to investigate a possible effect re- lated to the phase velocity of de Broglie waves [ 2 1. His arguing is based on an assumption which holds strictly whenever the particle (or the material struc- ture) is understood as being nothing else but another name for the result of interference of (monochro- matic) waves. But, as we will see below, this as- sumption is not compatible with de Broglie’s own ideas which we tried to test in our experiment.

For this purpose we have to analyze de Broglie’s own ideas about his waves, which do not carry ob- servable amounts of energy and momentum and, therefore, do not fit into the common picture of elec- tromagnetic waves. Similar to Schtiinger’s proba- bility waves (in a hidden-variables theorist’s interpretation) a particle of very small volume is embedded in a spatially extended wave. But as de Broglie has emphasized on several occasions (cf. e.g. ref. [ 3]), his waves are real entities (i.e. indepen- dent of an observer) and have to be sharply distin- guished from the VI waves. Moreover, they can even exist without the corresponding particle. “For him the fundamental quantities were not the wave-length but the frequencies of the wave and the clock, and the phase and group velocity . ..” [ 41. Most essential is the next paraphrase, providing insight in de Broglie’s mode of reasoning: “He said the following: Consider a circular Bohr orbit. The electron starts from a point 0 and describes the orbit with a ve- locity vo, while the wave starts at the same moment, but with the much greater phase velocity r+=c’v,,

so that, at certain time r the wave will overtake the particle in a point O’.” [ 41.

Nothing can more eloquently underline the reality of this wave, propagating, overtaking the particle and interacting with it.

If de Broglie’s picture is to be taken seriously, it must have the same predictive power as orthodox quantum mechanics. That means, it must remain correct for an electron’s motion in high Rydberg states as well as the motion of more complex parti- cles, composed of many elementary particles. From this we conclude that a de Broglie wave with phase velocity ~+=E/P has to be attributed to any particle of energy E and momentum p. This wave is a part of reality, it propagates with its phase velocity, and its domain of existence is not restricted to the near surrounding of the particle.

This situation is different from ordinary quantum mechanics. There a particle can be described just in the appropriate language. If an interference experi- ment is performed, a few plane waves are utilized, while a narrow wave packet is constructed if the po- sition of the particle has to be determined. Of course, the monochromatic waves chosen to construct the wave packet cannot be considered part of reality, and, in fact, interference effects cease under such condi- tions. But the big advantage of ordinary quantum mechanics is that no reality is claimed for these mathematical constructs, but only for the phenom- ena observed.

This advantage is lost with de Broglie waves. Due to the reality ascribed to these waves, it cannot da

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Volume 132, number 2,3 PHYSICS LETTERS A 26 September 1988

pend on the intended experiment, which form they will assume. Once and for all there is the wave prop- agating with its phase velocity r+.. In case of a bunch of N particles which independently from one another go the same way, there is no reason why their waves should cancel each other outside of the region, which in ordinary quantum mechanics is described by a wave packet. If interfering with each other, at least a number of waves of the order N ‘I2 should survive, but this statistical destructive interference would equally well occur within “the wave packet”. On the other hand, in case of a coherent superposition, i.e. a particle formed by many others, we have a wave again, moving (if the constituents have identical en- ergy and momentum) with the same phase velocity up = CE/ Cp as the individual waves.

Alas, we do not know of experimental equipment to verify this idea with material systems. Here pho- tonic de Broglie waves might induce stimulated emission in laser media. We performed a corre- sponding experiment [ 21, aiming to strip the pho- tons off the wave in a quartz fiber. According to the arguments given above, there are two reasons why one cannot exclude a positive result (if believing in the existence of de Broglie waves and of the pro- posed effect) :

( 1) If the bunch of roughly N= 1014 photons forms a coherent system, then a single wave can be attrib- uted to it, preceding it with the same phase velocity as the individual waves of the single photons (be- cause the common one must be composed thereof).

(2) If the bunch of photons consists of N inde- pendent individuals, then a number of the order N ‘I2 will survive in the worst case. Since the strength of the effect is unknown, the surviving fraction may suffer to detect it.

Finally, in case we have neither mode ( 1) nor mode (2 ), a mixture of both should work equally well.

I cannot agree with Selleri, that stimulating and inhibiting effects of two or more independent waves necessarily cancel. For de Broglie waves this can merely be a hypothesis, because the waves them- selves are only of hypothetical character. But I com- pletely agree with him that entirely new quantities should be looked for in this field. And this provokes me to mention a third (highly speculative) idea be- hind our experiment, which I would not have men-

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tioned without seeing Selleri’s Letter. The big question is: What kind of stuff do de

Broglie waves consist of? As they most probably do not possess energy, but are meaningless without pos- sessing reality, it must be a very strange one. This raises the next question: Is the stuff a persistent me- dium (similar to the prerelativistic ether), being present everywhere and always? Or does it vanish and emerge? The latter would not violate known conservation law. But, if true, the stuff of the Broglie waves propagates like a jet stream, and it cannot a priori be excluded that its mere presence, even if not oscillating, might cause some observable effect, pro- vided no energy or momentum transfer is required. After all, also a calm water surface is sufficient to cause wet feet.

The negative result of our experiment [ 21 has been interpreted by us as evidence that de Broglie waves do not induce stimulated emission in laser media. However, as for other experimental results, in par- ticular in such a speculative field, different interpre- tations are possible. For instance, the effect caused by either the common wave of the bunch or the sur- viving individual waves may be too weak to be de- tected. Or the common wave, due to its short wavelength A= h/Cp, does not propagate with the phase velocity of the individual waves in a refractive medium. Or the individual waves are not as real as de Broglie thought, in other words, these waves do not exist.

But given the case that they do exist, though not inducing stimulated emission, then further disper- sion experiments (preferably utilizing shorter pulses and longer ways than in ref. [ 21) may be capable of detecting them by looking for other effects like in- terference between a bunch of photons and its pre- ceding empty wave.

References

[ 11 F. Selleri, Phys. Lett. A 132 (1988) 72. [2] W. Mtickenheim, P. Lokai, B. Burghardt, Phys. Lett. A 127

(1988) 387. [ 3 ] L. de Broglie and J. Andrade e Silva, Phys. Rev. 172 ( 1968)

1284. [ 4 ] G. Lochak, De Broglie’s initial conception of de Broglie waves,

in: The wave-particle dualism, eds. S. Diner et al. (Reidel, Dordrecht, 1984) p. 1.

[ 5 I F. Selleri, Found. Phys. 12 ( 1982) 1087.