3 Wave-Particle Duality

newton.jpg The changing of bodies into light, and light into bodies, is very conformable to the course of Nature, which seems delighted with transmutations
Isaac Newton (1642-1727).

Wave-Particle duality is often characterized by slogans like:

The first of these statements is rather general and needs further explanation. The second is in a sense more concrete, but also more mysterious. Let us begin with this second statement, which could be referred to as ’naive wave-particle duality’.

The validity of the ’naive wave-particle duality’ is closely related to the validity of the IUP discussed in section 2. Recall that the uncertainties Δx and Δp appearing in the IUP describe position- and momentum uncertainties of a single particle. In ’modern versions’ of the IUP these measurement uncertainties are transferred, by means of positivistic philosophical techniques, into intrinsic properties of all matter ’particles’ (sometimes called ’quantons’). As a consequence, particles become quantons, which have neither exact positions nor exact momenta. The IUP then says that position is well-defined if momentum is ill-defined and vice-versa. The most familiar mathematical representation of this complementary wave-particle behavior is given by the widths of wave-packets in direct and Fourier-transformed space.

heisenberg_4.jpg "The problem of quantum theory centers on the fact that the particle picture and the wave picture are merely two different aspects of one and the same physical reality."
Werner Heisenberg (1901-1976)
The Physical Principles of the Quantum Theory.

As discussed in numerous publications as well as in section 2, there is no reason to assume that something like the IUP is realized in nature. Instead, valid uncertainty relations take the form (4), where the quantities Δx and Δp [denoted by          sep
( △x   ) and         sep
( △p   ) in (4)] describe standard deviations in an ensemble, and the reciprocity between widths of wave-packets mentioned above, refers to limitations in the preparation of quantum mechanical ensembles. This simple and important fact, which is still not accepted by all members of the scientific community, leads to the breakdown of the idea that a wave-particle duality (in its naive form) could exist. Note also the important role played by earlier experimental data, with very low resolution, for the creation of this idea.

What then, if any, is the meaning of the ’general wave-particle duality’ as expressed by the first of the above slogans? The definite answer to this question may be found in the results of modern experiments: Wave-like behavior occurs only as a consequence of the collective behavior of a large number of elementary particles and has never been observed by looking at individual events.

As a complement to the last point let me quote the following passage from Ballentine’s book on quantum mechanics (p. 4 of [12]), where one of the historical roots - namely imprecise early measurements - of the wave-particle misconception is pointed out:

Are ”particles” really ”waves”? In the early experiments, the diffraction patterns were detected holistically by means of a photographic plate, which could not detect individual particles. As a result, the notion grew that particle and wave properties were mutually incompatible, or complementary, in the sense that different measurement apparatuses would be required to observe them. That idea, however, was only an unfortunate generalization from a technological limitation. Today it is possible to detect the arrival of individual electrons, and to see the diffraction pattern emerge as a statistical pattern made up of many small spots (Tonomura et al., 1989).

doubleslite_interference.jpg Akira Tonomura's beautiful double-slit interference pattern created by means of a large number of electrons. See the video at the Hitachi website.

In Akira Tonomura’s video clip (wmv) one can see the step-by-step development of a double-slit interference pattern by means of a large number of ’independent’ single electron events. The final pattern, as described by the quantum-mechanical wave function, corresponds to an ensemble (of identically prepared) individual particles. This beautiful pioneering experiment can be interpreted as a visual, and rather convincing, demonstration of the statistical interpretation of QT. A detailed description of his experiment has been published [150] (pdf) by Tonomura. A detailed explanation and interpretation of this experiment has also been published by Silverman [143], who writes:

”The manifestations of wave-like behavior are statistical in nature and always emerge from the collective outcome of many electron events. In the present experiment nothing wave-like is discernible in the arrival of single electrons at the observation plane. It is only after the arrival of perhaps tens of thousands of electrons that a pattern interpretable as wave-like interference emerges.”

Needless to say that this behavior is not restricted to a particular class (electrons) of elementary particles; similar results have been obtained with photons [58], neutrons [129] , and atoms [122]. Statistical (quantum mechanical) correlations which may lead to wave-like behavior of ensembles are a general feature of microscopic particles.

So, coming back to the first of the above statements, there is in fact something like a wave-particle duality; however individual particles are always particles and never waves. Wave-like behavior can only be observed in experimental results determined by a large number of individual particles. The mixing-up of individual particles with ensembles - which consist of a virtually infinite number of individual particles - is closely related to the mixing-up of the two corresponding interpretations of the wave function (the latter error implies the first and is therefore more fundamental47 ). Thus, the term ”wave-particle duality” is a misnomer even if understood in the sense of the first of the above points.48

The idea of ”wave-particle duality” is related to the more general philosophical concept of ”complementarity”, whose importance has been emphatically stressed by Bohr [25]:

bohr.jpg In fact this new feature of natural philosophy [i.e., complementarity] means a radical revision of our attitude as regards physical reality....
Niels Bohr (1885-1962).

The fact that true wave-particle duality does not exist may also be read off directly from the structure of QT, as show many years ago by Popper [119], Landé [89], [90] and others. I quote the following comment from Poppers book (p. 217 of [119])

”The paradox of the equivalence of two so fundamentally different images as those of particle and wave was resolved by Borns statistical interpretation of the two theories. He showed that the wave theory too can be taken as a particle theory; for Schroedingers wave equation can be interpreted in such a way that it gives us the probability of finding the particle within any given region of space.”

lande_1.jpg "Today, after endless repetition, a dual nature of matter may seem as obvious and indisputable to the experts as the immobility of the Earth seemed to Galileo's learned colleagues who refused to look through his telescope because it might make them dizzy."
Alfred Lande (1888-1976).

The miraculous ”wave-particle duality” continues to flourish in popular texts and elementary text books. However, the rate of appearance of this term in scientific works has been decreasing in recent years (the same is true for Bohr’s notion of complementarity49 ). This is mainly due to experimental data like the one reported by Tonomura et al. [150], which clearly reveal the many-particle origin of wave-like phenomena.