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Wave–particle duality is the concept in quantum mechanics that quantum entities can have both particle and a wave properties according to the experimental circumstances. It expresses the inability of the classical concepts "particle" or "wave" to fully describe the behavior of quantum objects. As Albert Einstein wrote:

"It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do."

Waves are familiar in many systems, for instance water on a lake. Particles are also common, for instance projectiles from a gun. Both of these examples are at the macroscopic scale of meters. Quantum systems are for too small to see but either wave or particle behaviors can be observed depending on our method of measurement. Duality does not tell us what quantum systems "really are"; duality is not an interpretation of quantum mechanics, instead it says that no language from classical macroscopic physics can explain the totality of quantum behavior; quantum entities are not classical waves, classical particles, and they are not both.

Foundation
Wave-particle duality as a physical model derives from Niels Bohr's 1926 presentation in Como, Italy, at a scientific celebration of the work of Alessandro Volta 100 years previous. Bohr's subject was complementarity, the idea that measurements of quantum events provide complementary information through seemingly contradictory results. Bohr's presentation crystallize the issues ultimately leading to the modern wave-particle duality concept.

The modern approach is to side-step at first the question of whether there are waves or particles, and think instead in terms of probabilities. Taking as an example a single electron, it would be described in terms of a wave function $$\psi(r)$$ as a function of position $$r$$. There will a corresponding probability of detecting the electron at a position $$r$$ of $$|\psi(r)|^2$$, for instance by the light it produces at a detector. At any position this value is a fraction of one, with the sum (integral) over all positions one. One electron can be detected anywhere; however if there is some very large number such as a million, approximately a million times $$|\psi(r)|^2$$ will be detected at each position. Because it is all probabilities, it won't be exact, but the more electrons are collected the closer it will become. One electron only produces light at a specific location, so when it is detected it is effectively a particle. However, millions can either be detected at many positions, equivalent statistically to a wave, or if the probability is only large in a small region, all the detections will be essentially the same so it is effectively a particle. Whether the electrons are behaving as particles or a waves depending on how they are being detected.

This can be illustrated by experimental data. Single-particle interferometry, specifically the double-slit experiment with electrons at low count rates has become a classic this purpose. Richard Feynman called this experiment "a phenomenon which is impossible [...] to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery [of quantum mechanics]. Both particle and wave aspects are seen in different parts of this experiment. Note that exactly the same holds for photons and atoms, with the caveat of coherence which will be mentioned later.

Observing electron as waves
An electron double slit experiment is shown schematically in the figure below, together with results for each slit individually and the two slit combination, all at high electron intensity so many millions of electrons are being detected. The schematic distorts the geometry to illustrate the concept. In practice the source was 30.5cm from the slits; the two slits were 65nm (one nm is one billionth of a meter) and 0.3 micrometers apart (shown magnified in the upper left corner). Therefore the slits are very tiny and far from the source. The entire experiment is performed inside of a stainless steel vacuum chamber.

Two different kinds of detectors can be used. First a sensitive electrical current mater can be scanned across the detection plane; the current versus distance gives the image. Second an electron lens can magnify the detection plane onto microchannel plate detector giving the full image at once. The methods give the same results verifying that we are measuring electrons.

As shown on the right of the figure, with one or the other slit open a smooth peak shows up. When both slits are open the image is dramatically different. The pattern oscillates, characteristic of wave interference. For qualitative comparison, the image below from a ripple tank shows double slit water wave interference.

Before they reach the slits the electrons would be described as a plane wave. There is an equal probability of the electrons passing through either of the two slits. Some distance beyond the slits another obstacle acts as a detector, For electrons this detector will amplify incoming electrons and accelerate them into a phosphor screen, creating a dot for each electron. Millions of these provide the continuous image

Observing electrons as particles
Having observed wave behavior, now change the experiment, lowering the intensity of the electron source until only one or two are detected per second, appearing as individual dots. As shown in the movie clip below, the dots on the detector seem at first to be random. After some time a pattern emerges. Eventually we see an alternating pattern of light and dark bands shown in the image at the top.

Probabilistic connection
The electron double-slit experiment demonstrates particle behavior when viewing individual events, and wave behavior when events are summed up. Wave-particle duality asserts that separate experiments are always needed. Identifying these separate behaviors with an underlying model of nature leads to quantum waves and probability.

The Schrodinger equation accurately predicts the wave interference. Unlike water waves, the modulus square of the quantum wave amplitudes is a probability distribution for the observed electron counts. The electrons observed act like samples from a continuous pattern predicted by the quantum wave equation.

Coherence or density matrix
The above is equally valid for electrons, photons or atoms with one caveat: coherence. Consider the two slits at $$r_1$$and $$r_2$$. If we write

$$\psi(r_1)=|\psi(r_1)|\exp(i\phi_1)$$ $$\psi(r_2)=|\psi(r_2)|\exp(i\phi_2)$$

with phase terms $$\psi_1$$ and $$\psi_2$$, the mutual coherence is defined as

$$\Gamma(r_1,r_2)=\psi(r_1)\psi^*(r_2) =
 * \psi(r_1)\psi(r_2)|\exp(i(\phi_1-\phi_2))=|\psi(r_1)\psi(r_2)|\exp(i\Delta\phi)$$

The question now is whether the phase difference $$\Delta\phi$$ is a constant for every single electron (or photon, atom) or not when we consider millions being detected. If it is constant, ideally $$\Delta\phi=0$$, then the system is coherent and the oscillating fringes detailed above are observed. If, instead, the phase difference is statistically random then the system is incoherent and the fringes are not observed, just a smoothly varying intensity. The same type of approach can also be used within quantum mechanics, and is called the density matrix formalism.

The distance over which there is coherence is called the coherence length in classical electromagnetic theory, and in quantum mechanics the quantum coherence. The light from a laser is highly coherent, with a coherence length of centimeters or more; that from a normal light bulb is almost completely incoherent. When the coherence length is very small then there is no interference behavior, and effectively pure particle behavior; when it is large then wave-like behaviour dominates. The coherence length of the detector also matters, since in most cases it is very small, for instance the "dots" in the video above. Whether the behavior is that of particles or a waves depends on experimental details, the wave-particle duality.

Observing photons as particles
In the photoelectric effect, wherein electrons are emitted from atoms when they absorb energy from light. According to the classical theory of light and matter, the strength or amplitude of a light wave was in proportion to its brightness: a bright light should have been easily strong enough to create a large current. In 1902, Philipp Lenard discovered that the energy of these ejected electrons did not depend on the intensity of the incoming light, but instead on its frequency. Albert Einstein explained this enigma by postulating that electrons can receive energy from an electromagnetic field only in discrete units (quanta or photons): an amount of energy E that was related to the frequency f of the light by


 * $$E=hf$$

where h is the Planck constant (6.626×10−34 J⋅s). Only photons of a high enough frequency (above a certain threshold value) could knock an electron free. For example, photons of blue light had sufficient energy to free an electron from the metal, but photons of red light did not. One photon of light above the threshold frequency could release only one electron; the higher the frequency of a photon, the higher the kinetic energy of the emitted electron, but no amount of light below the threshold frequency could release an electron. Despite confirmation by various experimental observations, the photon theory (as it came to be called later) remained controversial until Arthur Compton performed a series of experiments from 1922 to 1924 demonstrating the momentum of light.

History
Around the year 1900 it was understood that light was a wave, and electrons as well as atoms were particles. There were a few pieces of experimental evidence that hinted at something deeper. Over the next quarter century there was a major change in scientific thinking with acceptance of quantization of light as well as wave behavior of electrons, all of which led to the concept of wave-particle duality.

The wave theory of light, broadly successful for over a hundred years, had been challenged by Planck's 1901 model of blackbody radiation and Einstein's 1905 interpretation of the photoelectric effect. These theoretical models used discrete energy, a quantum, to describe the interaction of light with matter. Despite confirmation by various experimental observations, the photon theory (as it came to be called) remained controversial until Arthur Compton performed a series of experiments from 1922 to 1924 demonstrating the momentum of light. The experimental evidence of particle-like momentum seemingly contradicted other experiments demonstrating the wave-like interference of light.

The contradictory evidence from electrons arrived in the opposite order. Many experiments by J. J. Thompson, Robert Millikan,  and Charles Wilson   among others had shown that free electrons had particle properties. However in 1924 Louis de Broglie proposed that electrons had an associated wave and Schrödinger demonstrated that wave equations accurately account for electron properties in atoms. Again some experiments showed particle properties and others wave properties.

Bohr's resolution of these contradictions is to accept them. Wave-particle duality as a physical model derives from Niels Bohr's 1926 presentation in Como, Italy, at a scientific celebration of the work of Alessandro Volta 100 years previous. Bohr's subject was complementarity, the idea that measurements of quantum events provide complementary information through seemingly contradictory results. In his Como lecture he says: "our interpretation of the experimental material rests essentially upon the classical concepts." Direct observation being impossible, observations of quantum effects are necessarily classical. Whatever the nature of quantum events, our only information will arrive via classical results. If experiments sometimes produce wave results and sometimes particle results, that is the nature of light and of the ultimate constituents of matter. While Bohr's presentation was not well received at that time, it did crystallize the issues ultimately leading to the modern wave-particle duality concept.

Interferometry
Interference of atom matter waves was first observed by Immanuel Estermann and Otto Stern in 1930, when a Na beam was diffracted off a surface of NaCl.

In 1974, the Italian physicists Pier Giorgio Merli, Gian Franco Missiroli, and Giulio Pozzi performed the first experiment of single particle interferometry using electrons and a biprism (instead of slits), showing that the statistics of electron detection leads to interference patterns as predicted by quantum theory.

In 1999, the diffraction of C60 fullerenes by researchers from the University of Vienna was reported. Fullerenes are comparatively large and massive objects, having an atomic mass of about 720 u. The de Broglie wavelength of the incident beam was about 2.5 pm, whereas the diameter of the molecule is about 1 nm, about 400 times larger. In the following years, experiments were extended to tetraphenylporphyrin, phthalocyanine molecules and a mixture of molecules beyond 10,000 u has been demonstrated.

In 2018, single particle interference has been first demonstrated for antimatter in the Positron Laboratory (L-NESS) of Rafael Ferragut in Como (Italy), by a group led by Marco Giammarchi.

Coincidence
While energy of ejected electrons reflected Planck's constant, the existence of photons was not explicitly proven until the discovery of the photon antibunching effect. This refers to the observation that once a single emitter (an atom, molecule, solid state emitter, etc.) radiates a detectable light signal, it cannot immediately release a second signal until after the emitter has been re-excited. This leads to a statistically quantifiable time delay between light emissions, so detection of multiple signals becomes increasingly unlikely as the observation time dips under the excited-state lifetime of the emitter. The effect can be demonstrated in an undergraduate-level lab.

This phenomenon could only be explained via photons. Einstein's "light quanta" would not be called photons until 1925, but even in 1905 they represented the quintessential example of wave–particle duality. Electromagnetic radiation propagates following linear wave equations, but can only be emitted or absorbed as discrete elements, thus acting as a wave and a particle simultaneously.

Complementarity and the quantum eraser
Immediately after Bohr's introduction of complementarity, Einstein and others sought to define experiments which could detect particle paths in wave-interference experiments. Feynman described one variant in 1965: use light to "look" electrons coming out of the slits in a double slit experiment; according to Bohr's complementarity, the electrons would no longer interfere.

Experiments which directly interact with particles leave questions about the origin of duality: could the light hitting Feynman's electrons alter their wave behavior? Marlan Scully and Kai Druhl addressed this concern in 1982 by proposing a new type of experiment they called a "quantum eraser". Various forms of this sophisticated experiment have been performed; they all use two sources or two slits with some "marker" like polarization for one of the two. When marked the particle origin can be known, but experimentally no interference is observed. Using coincidence detection, the marks can be ignored and interference restored. Analysis of these experiments is closely related to the analysis of quantum entanglement.

The Afshar claim
In 2007 Afshar et al. published an experiment claiming that it is possible to simultaneously observe both wave and particle properties of photons. The experiment was a variation of double slit interference. At the location of the detector in the traditional double slit experiment, physical grid was placed to coincide with the dark areas of destructive interference. Behind the grid, the light from the two slits was focused on to two detectors. Comparisons of intensity in each detectors was interpreted as providing information about the particle nature of light emerging from the corresponding slit. The analysis claims that the interference pattern at the location of the grid remains while particles are detected from specific slits. This claim has been disputed by multiple other scientists. While not a successful challenge to wave-particle duality, the experiment and the numerous responses show the continued importance of this concept.