# Wave-particle duality

In physics, wave-particle duality holds that light and matter can exhibit properties of both waves and of particles. This concept is a key part of quantum mechanics.

In the usual formulations of classical mechanics a given object is either a particle or a wave. For example, an electron is a particle (because they are observed to behave in particle-like ways), and light is a wave (because it behaves in wave-like ways, such as interference: see below). This categorisation was applied even to objects below the scale of direct observation, essentially by analogy with macroscopic phenomena.

However, problems emerge with the viewpoint: electrons too can be made to interfere and thus appear wave-like; light (especially in the photoelectric effect, as analysed in 1905 by Albert Einstein) can possess particle-like properties. Quantum mechanics emphasises the primacy of measurement and not attributing properties to objects beyond what can be measured. Hence the concept of wave-particle duality arose: it is not necessary, or useful, to say that an electron is a particle - or a wave - just that in certain circumstances it behaves like a wave, and in others like a particle.

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## Fresnel, Maxwell, and Young

In the early 1800s, the double-slit experiments by Young and Fresnel provided evidence for Huygens' theories: these experiments showed that when light is sent through a grid, a characteristic interference pattern is observed, very similar to the pattern resulting from the interference of water waves; the wavelength of light can be computed from such patterns. Maxwell, during the late-1800s, explained light as the propagation of electromagnetic waves with the Maxwell equations. These equations were verified by experiment, and Huygens' view became widely accepted.

## Einstein and photons

In 1905, Einstein reconciled Huygens' view with that of Newton; he explained the photoelectric effect (an effect in which light did not seem to act as a wave) by postulating the existence of photons, quanta of energy with particulate qualities. Einstein postulated that light's frequency, ν, is related to the energy, E, of its photons:

[itex]E = h \nu [itex],

where h is Planck's constant (6.626 x 10-34 J seconds).

## De Broglie

In 1924, de Broglie claimed that all matter has a wave-like nature; he related wavelength, λ, and momentum, p:

[itex]\lambda = \frac{h}{p}[itex].

This is a generalization of Einstein's equation above since the momentum of a photon is given by p = E / c where c is the speed of light in vacuum, and λ = c / ν.

De Broglie's formula was confirmed three years later by Clinton Joseph Davisson and Lester Halbert Germer, by guiding a beam of electrons (which have rest mass) through a crystalline grid and observing the predicted interference patterns. Similar experiments have since been conducted with neutrons and protons. Authors of similar recent experiments with atoms and molecules claim that these larger particles also act like waves. The most famous experiments are those of Estermann and Otto Stern in 1929, and the diffraction of fullerene C60 by researchers from the University of Vienna 1 in 1999; in the later case, the wavelength of de Broglie is 2.5 pm whereas the diameter of the molecule is about 1 nm, i.e. about 400 times larger.

This is still a controversial subject because these experimenters have assumed arguments of wave-particle duality and have assumed the validity of deBroglie's equation in their argument.

The Planck constant h is extremely small and that explains why we don't perceive a wave-like quality of everyday objects: their wavelengths are exceedingly small. The fact that matter can have very short wavelengths is exploited in electron microscopy.

In quantum mechanics, the wave-particle duality is explained as follows: every system and particle is described by state functions which encode the probability distributions of all measurable variables. The position of the particle is one such variable. Before an observation is made the position of the particle is described in terms of probability waves which can interfere with each other.

In quantum electrodynamics, Richard Feynman shows the wave-particle duality of photons and electrons is an illusion. In his view, photons and electrons obey rules that share some qualities of both particles and waves. They are neither particle nor wave, but some generalized object with no direct macroscopic analog.

An intriguingly simple experiment, the double-slit experiment, summarizes the duality: aim an electron gun at a screen with two slits and record their positions of detection at a detector behind the screen. You will observe an interference pattern just like the one produced by diffraction of a light or water wave at two slits. This pattern will even appear if you slow down the electron source so that only one electron's worth of charge per second comes through. "Classically speaking", every electron is a point particle and must either travel through the first or through the second slit. So we should be able to produce the same interference pattern if we ran the experiment twice as long, closing slit number one for the first half, then closing slit number two for the second half. But the same pattern does not emerge. Furthermore, if we build detectors around the slits in order to determine which path a particular electron takes, this very measurement destroys the interference pattern as well. But this is a classical explanation and something much more profound is taking place.

The interference pattern can be explained as a result of the charge wave being diffracted by both slits and interfering with itself. In quantum mechanics, the state function is a complex-valued function of space and time. The square of the magnitude of this function describes the probability of finding the electron at a given location at a given time. Interference is due to the fact that the square of the magnitude of the sum of two complex numbers may be different from the sum of the squares of their magnitudes.

The experiment also illustrates an interesting feature of quantum mechanics. Until an observation is made the position of a particle is described in terms of probability waves, but after the particle is observed, it is described as a fixed value. How to conceptualize the process of measurement is one of the great unresolved questions of quantum mechanics. The standard interpretation is the Copenhagen interpretation which leads to interesting thought experiments such as Schrödinger's cat. Due to this confusion, some theorists (including Stephen Hawking and Murray Gell-Mann) believe the many-worlds interpretation is true. However, there is currently some doubt over the validity of both the Copenhagen interpretation and the many-worlds interpretation, due to the controversial Shahriar Afshar's experiment  (http://www.irims.org/quant-ph/030503/), a variation of the two-pin-hole "which way" experiment.

Notes

Note 1: Wave-particle duality of C60, M. Arndt , O. Nairz, J. Voss-Andreae, C. Keller, G. van der Zouw, A. Zeilinger, Nature 401, 680-682, 14 October 1999

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