Double-slit experiment

1230: 2631:(measuring device) and the object being observed (physically interacted with), not any absolute property possessed by the object. In the case of an electron, if it is initially "observed" at a particular slit, then the observer–particle (photon–electron) interaction includes information about the electron's position. This partially constrains the particle's eventual location at the screen. If it is "observed" (measured with a photon) not at a particular slit but rather at the screen, then there is no "which path" information as part of the interaction, so the electron's "observed" position on the screen is determined strictly by its probability function. This makes the resulting pattern on the screen the same as if each individual electron had passed through both slits. 2731: 1519: 2747: 1475:. If one sets polarizers before each slit with their axes orthogonal to each other, the interference pattern will be eliminated. The polarizers can be considered as introducing which-path information to each beam. Introducing a third polarizer in front of the detector with an axis of 45° relative to the other polarizers "erases" this information, allowing the interference pattern to reappear. This can also be accounted for by considering the light to be a classical wave, and also when using circular polarizers and single photons. Implementations of the polarizers using 1408:

probability zero. It is interesting to consider what would happen if the photon were definitely in either of paths between the beam splitters. This can be accomplished by blocking one of the paths, or equivalently by detecting the presence of a photon there. In both cases there will be no interference between the paths anymore, and both photodetectors will be hit with probability 1/2. From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it is in a genuine quantum superposition of the two paths.

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physicists who played an important role in the establishment of quantum mechanics, and who were collaborators of Bohr's at his Institute or took part in the discussions during the crucial years. On closer inspection, one sees quite easily that these ideas are divergent in detail and that in particular the views of Bohr, the spiritual leader of the school, form a separate entity which can now be understood only by a thorough study of as many as possible of the relevant publications by Bohr himself.

1639: 8085: 8059: 1066: 1099: 1010:, producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles. However, the light is always found to be absorbed at the screen at discrete points, as individual particles (not waves); the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected 1622:, changing it from transparent to reflective for around 200 femtoseconds long where a subsequent probe laser beam hitting the ITO screen would then see this temporary change in optical properties as a slit in time and two of them as a double slit with a phase difference adding up destructively or constructively on each frequency component resulting in an interference pattern. Similar results have been obtained classically on water waves. 2183: 48: 8121: 1553:

vacuum. The interference pattern between the two electron waves could then be observed. In 2017, researchers performed the double-slit experiment using light-induced field electron emitters. With this technique, emission sites can be optically selected on a scale of ten nanometers. By selectively deactivating (closing) one of the two emissions (slits), researchers were able to show that the interference pattern disappeared.

2719: 34: 2591:, and others. The term "Copenhagen interpretation" was apparently coined by Heisenberg during the 1950s to refer to ideas developed in the 1925–1927 period, glossing over his disagreements with Bohr. Consequently, there is no definitive historical statement of what the interpretation entails. Features common across versions of the Copenhagen interpretation include the idea that quantum mechanics is intrinsically 8145: 1647: 8097: 8133: 2611:

probability distribution. The particles are discrete and identical; many are needed to build up the full interference pattern. The results from some of the which-way experiments are described as observations of complementarity: modifying the experiment to monitor the slit suppresses the interference pattern. Other which-way experiments make no mention of complementarity in their analysis.

8109: 1502:. Weak measurement followed by post-selection did not allow simultaneous position and momentum measurements for each individual particle, but rather allowed measurement of the average trajectory of the particles that arrived at different positions. In other words, the experimenters were creating a statistical map of the full trajectory landscape. 2667:

slit de Broglie-Bohm trajectories were first calculated by Chris Dewdney while working with Chris Philippidis and Basil Hiley at Birkbeck College (London). The de Broglie-Bohm theory produces the same statistical results as standard quantum mechanics, but dispenses with many of its conceptual difficulties by adding complexity through an

1388: 2607:. A particular experiment can demonstrate particle behavior (passing through a definite slit) or wave behavior (interference), but not both at the same time. Copenhagen-type interpretations hold that quantum descriptions are objective, in that they are independent of physicists' personal beliefs and other arbitrary mental factors. 1429:), technically feasible realizations of this experiment were not proposed until the 1970s. (Naive implementations of the textbook thought experiment are not possible because photons cannot be detected without absorbing the photon.) Currently, multiple experiments have been performed illustrating various aspects of complementarity. 1631: 2666:

states that particles also have precise locations at all times, and that their velocities are defined by the wave-function. So while a single particle will travel through one particular slit in the double-slit experiment, the so-called "pilot wave" that influences it will travel through both. The two

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argues that the way to understand the double-slit experiment is that in each universe the particle travels through a specific slit, but its motion is affected by the interference with particles in other universes. This creates the observable fringes. David Wallace, another advocate of the many-worlds

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have been developed that can recreate various aspects of quantum mechanical systems, including single-particle interference through a double-slit. A silicone oil droplet, bouncing along the surface of a liquid, self-propels via resonant interactions with its own wave field. The droplet gently sloshes

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In a highly publicized experiment in 2012, researchers claimed to have identified the path each particle had taken without any adverse effects at all on the interference pattern generated by the particles. In order to do this, they used a setup such that particles coming to the screen were not from a

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An important version of this experiment involves single particle detection. Illuminating the double-slit with a low intensity results in single particles being detected as white dots on the screen. Remarkably, however, an interference pattern emerges when these particles are allowed to build up one

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particles, and these particles were fired in a straight line through a slit and allowed to strike a screen on the other side, we would expect to see a pattern corresponding to the size and shape of the slit. However, when this "single-slit experiment" is actually performed, the pattern on the screen

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The experiment belongs to a general class of "double path" experiments, in which a wave is split into two separate waves (the wave is typically made of many photons and better referred to as a wave front, not to be confused with the wave properties of the individual photon) that later combine into a

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Frabboni, Stefano; Gabrielli, Alessandro; Carlo Gazzadi, Gian; Giorgi, Filippo; Matteucci, Giorgio; Pozzi, Giulio; Cesari, Nicola Semprini; Villa, Mauro; Zoccoli, Antonio (May 2012). "The Young-Feynman two-slits experiment with single electrons: Build-up of the interference pattern and arrival-time

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equation, which implies that as the plane of observation gets closer to the plane in which the slits are located, the diffraction patterns associated with each slit decrease in size, so that the area in which interference occurs is reduced, and may vanish altogether when there is no overlap in the

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The Mach–Zehnder interferometer can be seen as a simplified version of the double-slit experiment. Instead of propagating through free space after the two slits, and hitting any position in an extended screen, in the interferometer the photons can only propagate via two paths, and hit two discrete

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If one illuminates two parallel slits, the light from the two slits again interferes. Here the interference is a more pronounced pattern with a series of alternating light and dark bands. The width of the bands is a property of the frequency of the illuminating light. (See the bottom photograph to

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Numerical simulation of the double-slit experiment with electrons. Figure on the left: evolution (from left to right) of the intensity of the electron beam at the exit of the slits (left) up to the detection screen located 10 cm after the slits (right). The higher the intensity, the more the

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Where d is the distance between the two slits. When the two waves are in phase, i.e. the path difference is equal to an integral number of wavelengths, the summed amplitude, and therefore the summed intensity is maximum, and when they are in anti-phase, i.e. the path difference is equal to half a

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In the double-slit experiment, the two slits are illuminated by the quasi-monochromatic light of a single laser. If the width of the slits is small enough (much less than the wavelength of the laser light), the slits diffract the light into cylindrical waves. These two cylindrical wavefronts are

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In 2002, an electron field emission source was used to demonstrate the double-slit experiment. In this experiment, a coherent electron wave was emitted from two closely located emission sites on the needle apex, which acted as double slits, splitting the wave into two coherent electron waves in a

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In 2012, Stefano Frabboni and co-workers sent single electrons onto nanofabricated slits (about 100 nm wide) and, by detecting the transmitted electrons with a single-electron detector, they could show the build-up of a double-slit interference pattern. Many related experiments involving the

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performed a related experiment using single electrons from a coherent source and a biprism beam splitter, showing the statistical nature of the buildup of the interference pattern, as predicted by quantum theory. In 2002, the single-electron version of the experiment was voted "the most beautiful

2643:. The unifying theme is that physical reality is identified with a wavefunction, and this wavefunction always evolves unitarily, i.e., following the Schrödinger equation with no collapses. Consequently, there are many parallel universes, which only interact with each other through interference. 2610:

The results from the most basic double slit experiment, the observation of an interference pattern, is explained by wave interference from the two paths to the screen from each of the two slits. The single-particle results show that the waves are probability amplitudes which square to produce a

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It was shown experimentally in 1972 that in a double-slit system where only one slit was open at any time, interference was nonetheless observed provided the path difference was such that the detected photon could have come from either slit. The experimental conditions were such that the photon

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In 2005, E. R. Eliel presented an experimental and theoretical study of the optical transmission of a thin metal screen perforated by two subwavelength slits, separated by many optical wavelengths. The total intensity of the far-field double-slit pattern is shown to be reduced or enhanced as a

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in which the light is spread out. The smaller the slit, the greater the angle of spread. The top portion of the image shows the central portion of the pattern formed when a red laser illuminates a slit and, if one looks carefully, two faint side bands. More bands can be seen with a more highly

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Copenhagen interpretation as a unified and consistent logical structure. Terms such as "Copenhagen interpretation" or "Copenhagen school" are based on the history of the development of quantum mechanics; they form a simplified and often convenient way of referring to the ideas of a number of

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A photon emitted by the laser hits the first beam splitter and is then in a superposition between the two possible paths. In the second beam splitter these paths interfere, causing the photon to hit the photodetector on the right with probability one, and the photodetector on the bottom with

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superimposed, and the amplitude, and therefore the intensity, at any point in the combined wavefronts depends on both the magnitude and the phase of the two wavefronts. The difference in phase between the two waves is determined by the difference in the distance travelled by the two waves.

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Behaviors mimicked via this hydrodynamic pilot-wave system include quantum single particle diffraction, tunneling, quantized orbits, orbital level splitting, spin, and multimodal statistics. It is also possible to infer uncertainty relations and exclusion principles. Videos are available

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While there is no doubt that Young's demonstration of optical interference, using sunlight, pinholes and cards, played a vital part in the acceptance of the wave theory of light, there is some question as to whether he ever actually performed a double-slit interference experiment.

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Feynman was fond of saying that all of quantum mechanics can be gleaned from carefully thinking through the implications of this single experiment. He also proposed (as a thought experiment) that if detectors were placed before each slit, the interference pattern would disappear.

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However, more complicated systems that involve two or more particles in superposition are not amenable to such a simple, classically intuitive explanation. Accordingly, no hydrodynamic analog of entanglement has been developed. Nevertheless, optical analogs are possible.

5590:-dimensional configuration space or 'phase space'. It is difficult to visualize a reality comprising imaginary functions in an abstract, multi-dimensional space. No difficulty arises, however, if the imaginary functions are not to be given a real interpretation.") 1591:, causes it to exhibit behaviors previously thought to be peculiar to elementary particles – including behaviors customarily taken as evidence that elementary particles are spread through space like waves, without any specific location, until they are measured. 1048:

The double-slit experiment (and its variations) has become a classic for its clarity in expressing the central puzzles of quantum mechanics. Because it demonstrates the fundamental limitation of the ability of the observer to predict experimental results,

1568:, using new instruments that allowed control of the transmission of the two slits and the monitoring of single-electron detection events. Electrons were fired by an electron gun and passed through one or two slits of 62 nm wide × 4 μm tall. 1235:

Experimental electron double slit diffraction pattern. Across the middle of the image at the top, the intensity alternates from high to low, showing interference in the signal from the two slits. Bottom: movie of the pattern being built up dot-by-dot.

2603:, which states that objects have certain pairs of complementary properties that cannot all be observed or measured simultaneously. Moreover, the act of "observing" or "measuring" an object is irreversible, and no truth can be attributed to an object, 2122: 1432:

An experiment performed in 1987 produced results that demonstrated that partial information could be obtained regarding which path a particle had taken without destroying the interference altogether. This "wave-particle trade-off" takes form of an

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of quantum mechanics provided by Feynman. The path integral formulation replaces the classical notion of a single, unique trajectory for a system, with a sum over all possible trajectories. The trajectories are added together by using

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In 1999, a quantum interference experiment (using a diffraction grating, rather than two slits) was successfully performed with buckyball molecules (each of which comprises 60 carbon atoms). A buckyball is large enough (diameter about

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principle that photons can behave as either particles or waves, but cannot be observed as both at the same time. Despite the importance of this thought experiment in the history of quantum mechanics (for example, see the discussion on

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It seems that light passes through one slit or the other in the form of photons if we set up an experiment to detect which slit the photon passes, but passes through both slits in the form of a wave if we perform an interference

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demonstrated that under different circumstances, light can behave as if it is composed of discrete particles. These seemingly contradictory discoveries made it necessary to go beyond classical physics and take into account the

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beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to

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The experiment can be done with entities much larger than electrons and photons, although it becomes more difficult as size increases. The largest entities for which the double-slit experiment has been performed were

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in 1909, by reducing the level of incident light until photon emission/absorption events were mostly non-overlapping. A slit interference experiment was not performed with anything other than light until 1961, when

1029:, are found to exhibit the same behavior when fired towards a double slit. Additionally, the detection of individual discrete impacts is observed to be inherently probabilistic, which is inexplicable using 1254:

of detecting the particle at a specific place on the screen giving a statistical interference pattern. This phenomenon has been shown to occur with photons, electrons, atoms, and even some molecules: with

1110:(1773–1829) first demonstrated this phenomenon, it indicated that light consists of waves, as the distribution of brightness can be explained by the alternately additive and subtractive interference of 7225:

Movie showing single electron events build up to form an interference pattern in double-slit experiments. Several versions with and without narration (File size = 3.6 to 10.4 MB) (Movie Length = 1m 8s)

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In 2013, a quantum interference experiment (using diffraction gratings, rather than two slits) was successfully performed with molecules that each comprised 810 atoms (whose total mass was over 10,000

1073:(the faint spots on either side of the main band) forms due to the nonzero width of the slit. This diffraction pattern is also seen in the double-slit image, but with many smaller interference fringes. 2309: 1814: 3303:...if in a double-slit experiment, the detectors which register outcoming photons are placed immediately behind the diaphragm with two slits: A photon is registered in one detector, not in both... 1987: 3449:

Yaakov Y. Fein; Philipp Geyer; Patrick Zwick; Filip Kiałka; Sebastian Pedalino; Marcel Mayor; Stefan Gerlich; Markus Arndt (September 2019). "Quantum superposition of molecules beyond 25 kDa".

4690:"However, the 'wave-particle trade-off is now expressed in terms of an inequality, known as Englert-Greenberger duality or simply wave-particle duality relation". See also ref 24 in this work. 1250:, which states that all matter exhibits both wave and particle properties: The particle is measured as a single pulse at a single position, while the modulus squared of the wave describes the 1420:

predicts that if particle detectors are positioned at the slits, showing through which slit a photon goes, the interference pattern will disappear. This which-way experiment illustrates the

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and his research student Alexander Reid demonstrated that electrons show the same behavior, which was later extended to atoms and molecules. Thomas Young's experiment with light was part of

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interpretation, writes that in the familiar setup of the double-slit experiment the two paths are not sufficiently separated for a description in terms of parallel universes to make sense.

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observed by the double-slit experiment. Feynman stressed that his formulation is merely a mathematical description, not an attempt to describe a real process that we can measure.

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A simulation that runs in Mathematica Player, in which the number of quantum particles, the frequency of the particles, and the slit separation can be independently varied

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the liquid with every bounce. At the same time, ripples from past bounces affect its course. The droplet's interaction with its own ripples, which form what is known as a

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point-like source, but from a source with two intensity maxima. However, commentators such as Svensson have pointed out that there is in fact no conflict between the

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to the action along each path. The differences in the cumulative action along the different paths (and thus the relative phases of the contributions) produces the

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demonstrate that extracting "which path" information after a particle passes through the slits can seem to retroactively alter its previous behavior at the slits.

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Camilleri, K.; Schlosshauer, M. (2015). "Niels Bohr as Philosopher of Experiment: Does Decoherence Theory Challenge Bohr's Doctrine of Classical Concepts?".

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Arndt, Markus; Nairz, Olaf; Vos-Andreae, Julian; Keller, Claudia; Van Der Zouw, Gerbrand; Zeilinger, Anton (1999). "Wave–particle duality of C60 molecules".

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is one such model; it states that each point on a wavefront generates a secondary wavelet, and that the disturbance at any subsequent point can be found by

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Mermin, N. David (1 January 2017). "Why QBism Is Not the Copenhagen Interpretation and What John Bell Might Have Thought of It". In Bertlmann, Reinhold;

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demonstrate that particles do not form the interference pattern if one detects which slit they pass through. These results demonstrate the principle of

7303: 1685:), the phase difference can be found using the geometry shown in the figure below right. The path difference between two waves travelling at an angle 3026:

Eibenberger, Sandra; et al. (2013). "Matter-wave interference with particles selected from a molecular library with masses exceeding 10000 amu".

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Bartell, L. (1980). "Complementarity in the double-slit experiment: On simple realizable systems for observing intermediate particle-wave behavior".

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color is light blue – Figure in the center: impacts of the electrons observed on the screen – Figure on the right: intensity of the electrons in the

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experiments demonstrate that wave behavior can be restored by erasing or otherwise making permanently unavailable the "which path" information.

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approximation (on the screen). Numerical data from Claus Jönsson's experiment (1961). Photons, atoms and molecules follow a similar evolution.

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A diagram of Wheeler's delayed choice experiment, showing the principle of determining the path of the photon after it passes through the slit

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performed it with coherent electron beams and multiple slits. In 1974, the Italian physicists Pier Giorgio Merli, Gian Franco Missiroli, and

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Sillitto, R.M.; Wykes, Catherine (1972). "An interference experiment with light beams modulated in anti-phase by an electro-optic shutter".

1114:. Young's experiment, performed in the early 1800s, played a crucial role in the understanding of the wave theory of light, vanquishing the 6393:(February 2015). "Does it Make Sense to Speak of Self-Locating Uncertainty in the Universal Wave Function? Remarks on Sebens and Carroll". 7230:

Freeview video 'Electron Waves Unveil the Microcosmos' A Royal Institution Discourse by Akira Tonomura provided by the Vega Science Trust

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Yanagisawa, Hirofumi; Ciappina, Marcelo; Hafner, Christian; Schötz, Johannes; Osterwalder, Jürg; Kling, Matthias F. (4 October 2017).

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100 trajectories guided by the wave function. In De Broglie-Bohm's theory, a particle is represented, at any time, by a wave function

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coherent interference have been performed; they are the basis of modern electron diffraction, microscopy and high resolution imaging.

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Nairz, Olaf; Brezger, Björn; Arndt, Markus; Zeilinger, Anton (2001). "Diffraction of Complex Molecules by Structures Made of Light".

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function represents the fine structure, and the coarser structure represents diffraction by the individual slits as described by the

1367:) in 2011, and with molecules of up to 2000 atoms in 2019. In addition interference patterns built up from single particles, up to 4 4140: 3782: 2154:

function in this equation, and the second figure shows the combined intensity of the light diffracted from the two slits, where the

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photodetectors. This makes it possible to describe it via simple linear algebra in dimension 2, rather than differential equations.

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wavelength, one and a half wavelengths, etc., then the two waves cancel and the summed intensity is zero. This effect is known as

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Oshima, C.; Mastuda, K.; Kona, T.; Mogami, Y.; Komaki, M.; Murata, Y.; Yamashita, T.; Kuzumaki, T.; Horiike, Y. (4 January 2002).

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as well as other complicated combinations of de Broglie and Compton waves. To date there is no evidence that these are useful.

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This is illustrated in the figure above, where the first pattern is the diffraction pattern of a single slit, given by the

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Svensson, Bengt E. Y. (2013). "Pedagogical Review of Quantum Measurement Theory with an Emphasis on Weak Measurements".

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Website with the movie and other information from the first single electron experiment by Merli, Missiroli, and Pozzi.

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Niels Bohr and the Development of Physics: Essays Dedicated to Niels Bohr on the Occasion of his Seventieth Birthday

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P. Mittelstaedt; A. Prieur; R. Schieder (1987). "Unsharp particle-wave duality in a photon split-beam experiment".

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Chiao, R. Y.; P. G. Kwiat; Steinberg, A. M. (1995). "Quantum non-locality in two-photon experiments at Berkeley".

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provides a detailed treatment of the mathematics of double-slit interference in the context of quantum mechanics.

7994: 7703: 7698: 7421: 3395: 3201: 2796: 2535: 1891: 1392: 1382: 1007: 977: 426: 406: 274: 120: 6261: 5199:"Optical Control of Young's Type Double-slit Interferometer for Laser-induced Electron Emission from a Nano-tip" 4915: 4710: 3923:

Sala, S.; Ariga, A.; Ereditato, A.; Ferragut, R.; Giammarchi, M.; Leone, M.; Pistillo, C.; Scampoli, P. (2019).

2627:, observations such as those in the double-slit experiment result specifically from the interaction between the 1069:

Same double-slit assembly (0.7 mm between slits); in top image, one slit is closed. In the single-slit image, a

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passes through one slit (as would a classical particle), and not through both slits (as would a wave). However,

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In 1967, Pfleegor and Mandel demonstrated two-source interference using two separate lasers as light sources.

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Near-field intensity distribution patterns for plasmonic slits with equal widths (A) and non-equal widths (B).

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demonstrates that light and matter can satisfy the seemingly incongruous classical definitions for both waves

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Merli, P G; Missiroli, G F; Pozzi, G (1976). "On the statistical aspect of electron interference phenomena".

3419: 2781: 2553:, the double-slit experiment is often used to highlight the differences and similarities between the various 2531: 1655: 1446: 706: 444: 344: 2117:{\displaystyle {\begin{aligned}I(\theta )&\propto \cos ^{2}\left~\mathrm {sinc} ^{2}\left\end{aligned}}} 8170: 7718: 7652: 7647: 7618: 7331: 3593:

Jönsson, Claus (1 August 1961). "Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten".

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the contributions of the individual wavelets at that point. This summation needs to take into account the

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C.S. Peirce (July 1879). "Note on the Progress of Experiments for Comparing a Wave-length with a Meter".

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a position (center of mass). This is a kind of augmented reality compared to the standard interpretation.

2604: 1115: 676: 421: 411: 6289:"Elastic and inelastic electrons in the double-slit experiment: A variant of Feynman's which-way set-up" 8075: 8062: 7824: 7632: 7603: 6871:

Mukhopadhyay, P. (1986). "A correlation between the compton wavelength and the de Broglie wavelength".

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D.M. Greenberger and A. Yasin, "Simultaneous wave and particle knowledge in a neutron interferometer",

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relating the visibility of the interference pattern and the distinguishability of the which-way paths.

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A laboratory double-slit assembly; distance between top posts is approximately 2.5 cm (one inch).

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Philippidis, C.; Dewdney, C.; Hiley, B. J. (1979). "Quantum interference and the quantum potential".

2776: 2600: 2572: 2566: 2195: 1421: 1247: 1019: 627: 588: 541: 516: 439: 299: 93: 2431:{\displaystyle p(x,y,z,t)\propto \left\vert \int _{\text{all paths}}e^{iS(x,y,z,t)}\right\vert ^{2}} 1158: 7913: 7893: 7883: 7873: 7829: 7404: 7200: 4397: 3981: 1396: 1107: 1057:, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery ." 711: 77: 7160:

Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics

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Bacot, Vincent; Labousse, Matthieu; Eddi, Antonin; Fink, Mathias; Fort, Emmanuel (November 2016).

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Through Two Doors at Once: The Elegant Experiment That Captures the Enigma of Our Quantum Reality

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Through Two Doors at Once: The Elegant Experiment That Captures the Enigma of Our Quantum Reality

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Steeds, John; Merli, Pier Giorgio; Pozzi, Giulio; Missiroli, GianFranco; Tonomura, Akira (2003).

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A simple do-it-at-home illustration of the quantum eraser phenomenon was given in an article in

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Since that time a number of related experiments have been published, with a little controversy.

556: 454: 220: 7933: 7708: 7613: 7481: 5037: 3799: 3571: 3538: 3532: 3491: 3403::Quantum Mechanics p.1-1 "There is one lucky break, however— electrons behave just like light". 2786: 2754: 2730: 2200: 2186:

One of an infinite number of equally likely paths used in the Feynman path integral (see also:

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particles. This ambiguity is considered evidence for the fundamentally probabilistic nature of

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The Elegant Universe: Super Strings, Hidden Dimensions, and the Quest for the Ultimate Theory

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Each path is considered equally likely, and thus contributes the same amount. However, the

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In 2023, an experiment was reported recreating an interference pattern in time by shining a

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explains the pattern as being the result of the interference of light waves from the slit.

671: 583: 309: 266: 116: 100: 85: 3902:(Fourth edition, first published in paperback ed.). Oxford: Oxford University Press. 915: 39:

Photons or matter (like electrons) produce an interference pattern when two slits are used

8: 8149: 7974: 7943: 7888: 7868: 7776: 7733: 7588: 7514: 7441: 7431: 7343: 5560: 5535: 5516: 5143: 4351: 4138: 3778: 3010: 2831: 2811: 2170: 1671: 1546: 1411: 1122:, which had been the accepted model of light propagation in the 17th and 18th centuries. 1083: 1078: 1070: 1054: 1030: 998: 770: 578: 496: 324: 304: 256: 53:

Light from a green laser passing through two slits 0.4 mm wide and 0.1 mm apart

7131: 6927: 6884: 6841: 6755: 6547: 6416: 6360: 6222: 5935: 5800: 5742: 5687: 5617: 5578:. New York: Oxford University Press. pp. 76. ("The wavefunction of a system containing 5551: 5481: 5425: 5417: 5372: 5364: 5293: 5224: 5159: 5104: 5053: 5006: 4951: 4889: 4777: 4722: 4614: 4579: 4544: 4367: 4310: 4256: 4198: 4110: 4015: 4007: 3940: 3739: 3700: 3657: 3606: 3494:

first proposed the use of this effect as an artifact-independent reference standard for

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The probability distribution of the outcome is the normalized square of the norm of the

16:

Physics experiment, showing light and matter can be modelled by both waves and particles

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Frabboni, Stefano; Gazzadi, Gian Carlo; Grillo, Vincenzo; Pozzi, Giulio (1 July 2015).

6242: 6118: 6080: 6046: 6018: 5947: 5921: 5820: 5762: 5707: 5673: 5634: 5601: 5500: 5465: 5376: 5350: 5323: 5276:; Alkemade, P.F.A.; Blok, H.; Hooft, G.W.; Lenstra, D.; Eliel, E.R. (7 February 2005). 5249: 5210: 5198: 5124: 4856: 4838: 4789: 4763: 4681: 4626: 4379: 4327: 4294: 4218: 4184: 4027: 3993: 3957: 3924: 3756: 3723: 3626: 3474: 3432: 3071: 3037: 2989: 2932: 2885: 2550: 2166: 1417: 491: 416: 349: 261: 97: 6685: 5341:

Bach, Roger; et al. (March 2013). "Controlled double-slit electron diffraction".

4730: 4097:

Donati, O; Missiroli, G F; Pozzi, G (1973). "An Experiment on Electron Interference".

1674:

of a light field can be measured—this is proportional to the square of the amplitude.

8044: 7953: 7923: 7851: 7814: 7809: 7791: 7756: 7746: 7461: 7426: 7409: 7312: 7163: 7135: 7108: 7086: 7079: 7063: 7040: 7012: 7006: 6991: 6983: 6969: 6939: 6935: 6896: 6892: 6853: 6849: 6814: 6802: 6767: 6763: 6722: 6701: 6689: 6650: 6559: 6514: 6489: 6483: 6440: 6428: 6316: 6308: 6267: 6234: 6185: 6157: 6147: 6110: 6084: 6050: 6036: 5983: 5973: 5866: 5824: 5812: 5784: 5766: 5754: 5726: 5699: 5639: 5315: 5254: 5236: 5179: 5171: 5116: 5065: 5018: 4959: 4874:

Pfleegor, R. L.; Mandel, L. (July 1967). "Interference of Independent Photon Beams".

4860: 4793: 4673: 4630: 4428: 4418: 4383: 4332: 4210: 4077: 4052: 4019: 3962: 3903: 3878: 3855: 3819: 3761: 3669: 3630: 3618: 3542: 3509: 3478: 3374: 3341: 3314: 3282: 3207: 3163: 3133: 3104: 3063: 2981: 2936: 2924: 2889: 2841: 2816: 2702: 2683: 2584: 2576: 1572: 1533:

In 1991, Carnal and Mlynek performed the classic Young's double slit experiment with

1131: 1042: 925: 900: 840: 835: 735: 701: 681: 279: 138: 89: 73: 6914:

Elbaz, Claude (1985). "On de Broglie waves and Compton waves of massive particles".

6571: 6458: 6376: 6098: 5951: 5890: 5711: 5327: 4785: 4505: 4222: 3815: 3448: 8101: 7771: 7766: 7623: 7519: 6931: 6888: 6845: 6794: 6759: 6681: 6642: 6551: 6420: 6364: 6304: 6300: 6246: 6226: 6072: 6028: 5939: 5804: 5746: 5691: 5629: 5621: 5555: 5495: 5485: 5429: 5421: 5368: 5305: 5297: 5244: 5228: 5163: 5128: 5108: 5057: 5010: 4955: 4893: 4848: 4781: 4726: 4618: 4583: 4548: 4371: 4322: 4314: 4260: 4202: 4114: 4011: 3952: 3944: 3851: 3847: 3811: 3751: 3743: 3704: 3661: 3610: 3466: 3428: 3075: 3055: 2993: 2971: 2916: 2875: 2528: 2318: 2211: 1615: 1495: 1488: 930: 920: 910: 810: 790: 775: 745: 613: 501: 5380: 5301: 5167: 4206: 4031: 2905:"Electron diffraction chez Thomson: early responses to quantum physics in Britain" 2724:

Trajectories of particles in De Broglie–Bohm theory in the double-slit experiment.

1976:

equation is needed to determine the intensity of the diffracted light as follows:

1154: 8001: 7928: 7908: 7878: 7841: 7836: 7741: 7565: 7207: 6716: 6462: 6010: 5602:"Classical hypercorrelation and wave-optics analogy of quantum superdense coding" 4876: 4163: 4144: 3786: 3127: 1654:

Much of the behaviour of light can be modelled using classical wave theory. The

1618:

which would alter the properties of the electrons within the material due to the

1565: 1387: 1050: 955: 825: 805: 551: 391: 7235:

Hitachi website that provides background on Tonomura video and link to the video

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Li, Pengyun; Sun, Yifan; Yang, Zhenwei; Song, Xinbing; Zhang, Xiangdong (2016).

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Quantum and Semiclassical Optics: Journal of the European Optical Society Part B

4482: 1681:

If the viewing distance is large compared with the separation of the slits (the

8089: 7979: 7948: 7938: 7560: 7550: 7384: 6961: 6670:"Testing the limits of quantum mechanics: motivation, state of play, prospects" 6017:. The Frontiers Collection. Springer International Publishing. pp. 83–93. 5965: 5808: 5750: 5232: 4973: 4587: 2864:"The Bakerian lecture. Experiments and calculation relative to physical optics" 2697:

More complex variants of this type of approach have appeared, for instance the

2187: 1465: 1065: 890: 850: 830: 800: 780: 730: 696: 546: 536: 329: 61: 7256: 6585: 6424: 4375: 3747: 3506:

World in the Balance: The historic quest for an absolute system of measurement

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Davisson, C. J (1928). "The diffraction of electrons by a crystal of nickel".

2920: 8159: 7898: 7751: 7642: 7476: 7446: 7399: 6943: 6900: 6857: 6806: 6771: 6693: 6654: 6563: 6432: 6334: 6312: 6238: 6161: 6114: 5987: 5758: 5703: 5240: 5175: 4988: 4897: 4677: 4552: 4432: 4410: 4023: 3980:

Bach, Roger; Pope, Damian; Liou, Sy-Hwang; Batelaan, Herman (13 March 2013).

3859: 3823: 3673: 3622: 2985: 2928: 2691: 2644: 2624: 2592: 2207: 2128: 1663: 1534: 1180:

In 2018, single particle interference was demonstrated for antimatter in the

1168: 950: 945: 875: 845: 815: 686: 632: 359: 334: 212:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle } 124: 80:

in 1801, as a demonstration of the wave behavior of visible light. In 1927,

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Jenkins FA and White HE, Fundamentals of Optics, 1967, McGraw Hill, New York

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of this contribution at any given point along the path is determined by the

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Zeilinger, A. (1999). "Experiment and the foundations of quantum physics".

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Jönsson, Claus (1 January 1974). "Electron Diffraction at Multiple Slits".

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Longhurst RS, Physical and Geometrical Optics, 1967, 2nd Edition, Longmans

4912:"Interference of Independent Photon Beams: The Pfleegor-Mandel Experiment" 7819: 6454: 6390: 6351: 6205:

Scully, Marian O.; Englert, Berthold-Georg; Walther, Herbert (May 1991).

6175: 5856: 5038:"Young's Double-Slit Experiment with Atoms: A Simple Atom Interferometer" 4989:"Young's Double-Slit Experiment with Atoms: A Simple Atom Interferometer" 4768: 4189: 3310:

Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral

2806: 2441: 1619: 1088: 905: 860: 795: 750: 112: 6122: 5434: 4685: 4661: 4448:"Complementarity and the Copenhagen Interpretation of Quantum Mechanics" 6798: 6555: 6368: 4809:"Disentangling the wave–particle duality in the double-slit experiment" 4706: 4622: 4475:"Quantum Mechanics 1925–1927: Triumph of the Copenhagen Interpretation" 4318: 3614: 3059: 2662:

An alternative to the standard understanding of quantum mechanics, the

2580: 1820: 1650:

Two slits are illuminated by a plane wave, showing the path difference.

1630: 1611: 1588: 1181: 895: 865: 785: 760: 755: 740: 8084: 7271: 6646: 5695: 5625: 5310: 5278:"Plasmon-Assisted Two-Slit Transmission: Young's Experiment Revisited" 4264: 4118: 3877:. North Holland personal library (3rd ed.). Amsterdam: Elsevier. 3665: 3199: 2321:, to get the probability distribution for the position of a particle: 1510: 7673: 7369: 6230: 5536:"Probabilities and trajectories in a classical wave–particle duality" 5273: 3973: 3724:"The Merli–Missiroli–Pozzi Two-Slit Electron-Interference Experiment" 2976: 2951: 2596: 1682: 1667: 1545:, nearly half a million times larger than a proton) to be seen in an 1542: 1111: 1026: 386: 7281: 5517:"Have We Been Interpreting Quantum Mechanics Wrong This Whole Time?" 4137:(Introduction, subscription needed for full text, quoted in full in 3836: 3708: 2182: 1564:

performed the double-slit experiment with electrons as described by

1098: 92:

long before the development of quantum mechanics and the concept of

7260:: Yves Couder . Explains Wave/Particle Duality via Silicon Droplets 6611:(Fall 2021 ed.), Metaphysics Research Lab, Stanford University 5926: 5886: 5678: 5215: 4600: 2588: 1038: 765: 111:

single wave. Changes in the path-lengths of both waves result in a

6407: 6023: 5355: 5112: 4843: 3998: 3042: 3021: 3019: 1537:

helium atoms passing through micrometer-scale slits in gold foil.

47: 6718:

The Road to Reality: A Complete Guide to the Laws of the Universe

4048:

The Fabric of the Cosmos: Space, Time, and the Texture of Reality

2524:, over all paths from the point of origin to the final point, of 4753: 3274: 1498:

performed in this variant of the double-slit experiment and the

7247:"Single-particle interference observed for macroscopic objects" 4135:

New Scientist: Quantum wonders: Corpuscles and buckyballs, 2010

3495: 3200:

Feynman, Richard P.; Robert B. Leighton; Matthew Sands (1965).

3016: 1440: 1053:

called it "a phenomenon which is impossible to explain in any

1011: 5196: 5144:"Young's Interference of Electrons in Field Emission Patterns" 4937: 2718: 1395:

exhibit wave-like interference and particle-like detection at

1148:

A low-intensity double-slit experiment was first performed by

33: 5662:"Time reversal and holography with spacetime transformations" 1646: 1575:). The record was raised to 2000 atoms (25,000 amu) in 2019. 1193: 1002: 6447: 1412:"Which-way" experiments and the principle of complementarity 1041:

that each comprised 2000 atoms (whose total mass was 25,000

103:

was correct, and his experiment is sometimes referred to as

6742:

Horodecki, R. (1981). "De broglie wave and its dual wave".

5865:(2nd ed.). Princeton University Press. pp. 2–16. 5090: 3444: 3442: 3155: 2525: 1189: 8108: 6286: 5271: 4510:

15th UK and European Meeting on the Foundations of Physics

4174: 3922: 2511:{\displaystyle \iiint _{\text{all space}}p(x,y,z,t)\,dV=1} 6785:

Horodecki, R. (1983). "Superluminal singular dual wave".

4873: 3797: 2868:

Philosophical Transactions of the Royal Society of London

2313:

All these contributions are then added together, and the

1204: 5568: 3439: 3275:

Leon Lederman; Christopher T. Hill (27 September 2011).

2304:{\displaystyle A_{\text{path}}(x,y,z,t)=e^{iS(x,y,z,t)}} 1947:

For example, if two slits are separated by 0.5 mm (

1809:{\displaystyle ~d\theta _{n}=n\lambda ,~n=0,1,2,\ldots } 1634:

Two-slit diffraction pattern with an incident plane wave

6533: 5141: 4238:"Quantum interference experiments with large molecules" 3925:"First demonstration of antimatter wave interferometry" 2639:

As with Copenhagen, there are multiple variants of the

1953:), and are illuminated with a 0.6 μm wavelength laser ( 1557:

function of the wavelength of the incident light beam.

7272:

Java demonstration of Young's double slit interference

6828:

Das, S.N. (1984). "De Broglie wave and Compton wave".

6592:. Metaphysics Research Lab, Stanford University. 2017. 5272:

Schouten, H.F.; Kuzmin, N.; Dubois, G.; Visser, T.D.;

19:"Slit experiment" redirects here. For other uses, see 8073: 5909: 5659: 4398:"Entangled photons show interference and bilocation." 2454: 2327: 2220: 1985: 1894: 1841: 1747: 1698: 150: 6204: 4986: 4096: 3979: 2541: 1625: 5913:

Studies in History and Philosophy of Modern Physics

4235: 3838:distribution using a fast-readout pixel detector". 3686: 2855: 1963:), the spacing of the fringes will be 1.2 mm. 7078: 7055: 4503: 3800:"The double-slit experiment with single electrons" 3574:(1909). "Interference Fringes with Feeble Light". 2605:except according to the results of its measurement 2510: 2430: 2303: 2116: 1936: 1868: 1823:of the light. The angular spacing of the fringes, 1808: 1738:. The interference fringe maxima occur at angles 1722: 1642:Photo of the double-slit interference of sunlight. 1578: 211: 6459:"Many-Worlds Interpretation of Quantum Mechanics" 5724: 5514: 4504:Boscá Díaz-Pintado, María C. (29–31 March 2007). 4352:"Quantum superposition of molecules beyond 25kDa" 4295:"Quantum interference of large organic molecules" 3373:. UK: Cambridge University Press. pp. 9–10. 3129:History of the Principle of Interference of Light 3011:Physicists Smash Record For Wave–Particle Duality 1605: 76:. This type of experiment was first performed by 8157: 6513:. Oxford: Oxford University Press. p. 382. 5899:. Metaphysics Research Lab, Stanford University. 5891:"Copenhagen Interpretation of Quantum Mechanics" 5725:Rodríguez-Fortuño, Francisco J. (3 April 2023). 4704: 4662:"The uncertainty relations in quantum mechanics" 2694:have criticized it for not adding anything new. 4824: 4822: 4481:. American Institute of Physics. Archived from 3337:Niels Bohr and Complementarity: An Introduction 3156:Lederman, Leon M.; Christopher T. Hill (2011). 1972:is appreciable compared to the wavelength, the 1199: 7213: 7105:Q is for Quantum: Particle Physics from A to Z 6184:. Kluwer Academic Publishers. pp. 36–39. 6071:. Princeton University Press. pp. 41–54. 5785:"Light waves squeezed through 'slits in time'" 5527: 4700: 4698: 4292: 3306: 3295: 2909:The British Journal for the History of Science 2621:relational interpretation of quantum mechanics 2575:is a collection of views about the meaning of 2194:The double-slit experiment can illustrate the 1595:illustrating various features of this system. 1376: 7297: 5903: 5879: 5599: 5586:position coordinates and is a function in a 3 4717:. Vol. 296, no. 5. pp. 90–95. 4559: 4130: 4128: 3982:"Controlled double-slit electron diffraction" 3504:, as referenced by Crease, Robert P. (2011). 3281:. Prometheus Books, Publishers. p. 109. 2634: 1869:{\displaystyle \theta _{f}\approx \lambda /d} 1723:{\displaystyle d\sin \theta \approx d\theta } 1371:photons can also show interference patterns. 978: 7252:Pilot-Wave Hydrodynamics: Supplemental Video 7034: 6982: 6870: 6339:International Journal of Theoretical Physics 5782: 5035: 4819: 4749: 4747: 4594: 4396:Hessmo, B., M. W. Mitchell, and P. Walther. 4293:Stefan Gerlich; et al. (5 April 2011). 4071: 4065: 3570: 3005: 3003: 2952:"Diffraction of Cathode Rays by a Thin Film" 2614: 2560: 2177: 1530:density in the system was much less than 1. 1479:photon pairs have no classical explanation. 1441:Delayed choice and quantum eraser variations 1238:Click on the thumbnail to enlarge the movie. 206: 180: 7008:QED: The Strange Theory of Light and Matter 6002: 5727:"An optical double-slit experiment in time" 5392: 5390: 4695: 4524: 4415:Introduction to Quantum Information Science 4349: 3521: 3025: 2949: 2674:While the model is in many ways similar to 1937:{\displaystyle ~w=z\theta _{f}=z\lambda /d} 1077:If light consisted strictly of ordinary or 997:In the basic version of this experiment, a 7304: 7290: 6960: 6628: 6207:"Quantum optical tests of complementarity" 5576:The Quantum Story: A History in 40 Moments 5533: 4350:Yaakov Fein; et al. (December 2019). 4125: 3333: 3327: 3162:. US: Prometheus Books. pp. 102–111. 3151: 3149: 2671:quantum potential to guide the particles. 2595:, with probabilities calculated using the 1271:) in 2001, with 2 molecules of 430 atoms ( 985: 971: 7194: 6784: 6741: 6602: 6406: 6350: 6097: 6067:(1999). "The Copenhagen Interpretation". 6022: 5970:The Logical Analysis of Quantum Mechanics 5925: 5677: 5633: 5559: 5499: 5489: 5433: 5354: 5309: 5248: 5214: 4842: 4767: 4744: 4565: 4326: 4236:Nairz, O; Arndt, M; Zeilinger, A (2003). 4188: 3997: 3956: 3755: 3195: 3193: 3191: 3189: 3187: 3185: 3183: 3181: 3179: 3041: 3000: 2975: 2879: 2651: 2495: 1879:The spacing of the fringes at a distance 6629:Heisenberg, W. (1956). Pauli, W (ed.). " 6337:(1996). "Relational Quantum Mechanics". 6280: 6266:. North Holland, John Wiley & Sons. 6134: 6132: 6101:(1953). "Strife about Complementarity". 5457: 5387: 4828: 4445: 3416: 3092: 2181: 1645: 1637: 1629: 1517: 1509: 1450: 1386: 1097: 1064: 7265: 7099: 7035:French, A.P.; Taylor, Edwin F. (1978). 7004: 6714: 6667: 6609:The Stanford Encyclopedia of Philosophy 6590:The Stanford Encyclopedia of Philosophy 6508: 6481: 6453: 6333: 6259: 6253: 6144:The Interpretation of Quantum Mechanics 5964: 4806: 4530: 4472: 3643: 3592: 3235: 3203:The Feynman Lectures on Physics, Vol. 3 3146: 2902: 2682:. Many authors such as nobel laureates 2440:As is always the case when calculating 1196:), by a group led by Marco Giammarchi. 96:. He believed it demonstrated that the 8158: 7240: 7157: 7076: 7053: 6008: 4409: 4403: 4044: 3897: 3872: 3527: 3176: 3125: 1205:Interference from individual particles 7311: 7285: 7148: 7025: 6968:. London: Weidenfeld & Nicolson. 6913: 6174: 6168: 6138: 6129: 6063: 5783:Castelvecchi, Davide (3 April 2023). 5778: 5776: 5655: 5653: 5540:Journal of Physics: Conference Series 3370:Quantum Physics: Illusion Or Reality? 2861: 1670:of the individual wavelets. Only the 1427:Einstein's version of this experiment 1025:Other atomic-scale entities, such as 6674:Journal of Physics: Condensed Matter 6389: 6181:Quantum Theory: Concepts and Methods 5885: 5463: 5396: 5340: 4506:"Updating the wave–particle duality" 3721: 3313:. US: World Scientific. p. 14. 3245:The Internet Encyclopedia of Science 3206:. Addison-Wesley. pp. 1.1–1.8. 2555:interpretations of quantum mechanics 1460:Wheeler's delayed-choice experiments 1143:Englert–Greenberger duality relation 1125:However, the later discovery of the 7121: 6827: 6488:. London: Penguin. pp. 40–53. 6468:Stanford Encyclopedia of Philosophy 5896:Stanford Encyclopedia of Philosophy 5855: 5426:10.1146/annurev-fluid-010814-014506 4987:Carnal, O.; Mlynek, J. (May 1991). 4659: 3900:High-resolution electron microscopy 3366: 3354: 3099:. New York, NY: Pi Press. pp. 3029:Physical Chemistry Chemical Physics 1596: 1505: 1482: 13: 7037:An Introduction to Quantum Physics 6966:Quantum: A Guide for the Perplexed 6954: 5862:Quantum Field Theory in a Nutshell 5773: 5650: 5515:Natalie Wolchover (30 June 2014). 3537:. New York: W.W. Norton. pp. 3433:10.1002/j.1538-7305.1928.tb00342.x 2068: 2065: 2062: 2059: 517:Sum-over-histories (path integral) 203: 177: 133:Part of a series of articles about 14: 8187: 7177: 5445:from the original on 21 June 2015 4731:10.1038/scientificamerican0507-90 4711:"A do-it-yourself quantum eraser" 4454:. Dept. of Physics, U. of Toronto 3508:. New York: W.W. Norton. p. 317. 3307:Müller-Kirsten, H. J. W. (2006). 2950:Thomson, G. P.; Reid, A. (1927). 2542:Interpretations of the experiment 1626:Classical wave-optics formulation 154: 8143: 8131: 8119: 8107: 8095: 8083: 8058: 8057: 7184:Double slit interference lecture 6907: 6864: 6821: 6778: 6735: 6708: 6661: 6622: 5992:here is no point in looking for 5406:Annual Review of Fluid Mechanics 4807:Francis, Matthew (21 May 2012). 4051:. Random House LLC. p. 90. 3340:. US: Springer. pp. 75–76. 3096:The Last Man Who Knew Everything 2792:Dual-polarization interferometry 2745: 2729: 2717: 1500:Heisenberg uncertainty principle 1228: 1217: 46: 32: 21:Slit experiment (disambiguation) 7162:(5th ed.). W. H. Freeman. 6596: 6578: 6527: 6502: 6475: 6383: 6327: 6198: 6091: 6069:Understanding Quantum Mechanics 6057: 5958: 5849: 5840: 5831: 5718: 5593: 5508: 5334: 5265: 5190: 5135: 5084: 5036:Carnal, O.; Mlynek, J. (1991). 5029: 4980: 4966: 4931: 4904: 4867: 4800: 4653: 4637: 4497: 4466: 4439: 4390: 4343: 4286: 4229: 4168: 4150: 4090: 4038: 3916: 3891: 3866: 3830: 3791: 3779:"The most beautiful experiment" 3772: 3715: 3680: 3637: 3586: 3564: 3555: 3485: 3406: 3396:The Feynman Lectures on Physics 3387: 3268: 3259: 3229: 3220: 2827:Young's interference experiment 1579:Hydrodynamic pilot wave analogs 123:, which splits the beam with a 8007:Relativistic quantum mechanics 7149:Sears, Francis Weston (1949). 7126:. Cambridge University Press. 7011:. Princeton University Press. 6305:10.1016/j.ultramic.2015.03.006 6146:. Princeton University Press. 5561:10.1088/1742-6596/361/1/012001 5466:"Quantum mechanics writ large" 3852:10.1016/j.ultramic.2012.03.017 3119: 3082: 2943: 2896: 2492: 2468: 2412: 2388: 2355: 2331: 2296: 2272: 2255: 2231: 1999: 1993: 1957:), then at a distance of 1 m ( 1606:Double-slit experiment on time 1562:University of Nebraska–Lincoln 1210:by one (see the image below). 1102:Young's drawing of diffraction 1015: 667:Relativistic quantum mechanics 199: 192: 173: 1: 7985:Quantum statistical mechanics 7762:Quantum differential calculus 7684:Delayed-choice quantum eraser 7467:Symmetry in quantum mechanics 7107:. Weidenfeld & Nicolson. 6607:, in Zalta, Edward N. (ed.), 5893:. In Zalta, Edward N. (ed.). 5534:Couder, Y.; Fort, E. (2012). 5373:10.1088/1367-2630/15/3/033018 5302:10.1103/physrevlett.94.053901 5168:10.1103/PhysRevLett.88.038301 4207:10.1103/physrevlett.87.160401 4016:10.1088/1367-2630/15/3/033018 3992:(3). IOP Publishing: 033018. 3420:Bell System Technical Journal 3247:. The Worlds of David Darling 2848: 2782:Delayed-choice quantum eraser 2165:Similar calculations for the 1447:Delayed-choice quantum eraser 707:Quantum statistical mechanics 8166:Foundational quantum physics 7005:Feynman, Richard P. (1988). 6936:10.1016/0375-9601(85)90379-2 6893:10.1016/0375-9601(86)90200-8 6850:10.1016/0375-9601(84)90291-3 6764:10.1016/0375-9601(81)90571-5 4960:10.1016/0375-9601(72)91015-8 4157:Wave Particle Duality of C60 2837:Hydrodynamic quantum analogs 2579:, stemming from the work of 2169:can be made by applying the 1614:pulse at a screen coated in 1560:In 2012, researchers at the 1200:Variations of the experiment 7: 7787:Quantum stochastic calculus 7777:Quantum measurement problem 7699:Mach–Zehnder interferometer 7214:Single particle experiments 6686:10.1088/0953-8984/14/15/201 6603:Goldstein, Sheldon (2021), 6033:10.1007/978-3-319-38987-5_4 5944:10.1016/j.shpsb.2015.01.005 5062:10.1103/PhysRevLett.66.2689 5015:10.1103/PhysRevLett.66.2689 4417:. Oxford University Press. 4245:American Journal of Physics 4099:American Journal of Physics 4072:Ananthaswamy, Anil (2018). 3898:Spence, John C. H. (2017). 3689:American Journal of Physics 3646:American Journal of Physics 3572:Sir Geoffrey, Ingram Taylor 3502:American Journal of Science 3367:Rae, Alastair I.M. (2004). 3334:Plotnitsky, Arkady (2012). 2797:Elitzur–Vaidman bomb tester 2764: 2444:, the results must then be 1885:from the slits is given by 1393:Mach–Zehnder interferometer 1383:Mach–Zehnder interferometer 1377:Mach-Zehnder interferometer 1116:corpuscular theory of light 1060: 677:Quantum information science 121:Mach–Zehnder interferometer 10: 8192: 5809:10.1038/d41586-023-00968-4 5751:10.1038/s41567-023-02026-2 5399:"Pilot-wave hydrodynamics" 5233:10.1038/s41598-017-12832-3 4588:10.1103/RevModPhys.71.S288 3414:Davisson–Germer experiment 2655: 2641:many-worlds interpretation 2635:Many-worlds interpretation 2564: 1966:If the width of the slits 1486: 1444: 1380: 1166:experiment" by readers of 18: 8053: 8015: 7967: 7847:Quantum complexity theory 7825:Quantum cellular automata 7800: 7732: 7666: 7579: 7543: 7530:Path integral formulation 7497: 7362: 7319: 6425:10.1007/s10701-014-9862-5 6103:Science Progress (1933– ) 5464:Bush, John W. M. (2010). 4786:10.1088/1355-5111/7/3/006 4568:Reviews of Modern Physics 4376:10.1038/s41567-019-0663-9 4143:25 September 2017 at the 3816:10.1088/2058-7058/16/5/24 3748:10.1007/s00016-011-0079-0 3471:10.1038/s41567-019-0663-9 3278:Quantum Physics for Poets 3159:Quantum Physics for Poets 3093:Robinson, Andrew (2006). 2921:10.1017/S0007087410000026 2777:Complementarity (physics) 2615:Relational interpretation 2573:Copenhagen interpretation 2567:Copenhagen interpretation 2561:Copenhagen interpretation 2196:path integral formulation 2178:Path-integral formulation 2174:two diffracted patterns. 1656:Huygens–Fresnel principle 1597:(See the External links.) 119:. Another version is the 7914:Quantum machine learning 7894:Quantum key distribution 7884:Quantum image processing 7874:Quantum error correction 7724:Wheeler's delayed choice 7201:Huygens and interference 7124:The New Quantum Universe 7081:The Fabric of the Cosmos 6787:Lettere al Nuovo Cimento 6260:Messiah, Albert (1966). 4898:10.1103/PhysRev.159.1084 4553:10.1103/PhysRevD.21.1698 4446:Harrison, David (2002). 4400:CERN Courier (2004): 11. 3561:Feynman, 1965, chapter 3 3132:. Springer. p. 65. 3126:Kipnis, Naum S. (1991). 1188:) of Rafael Ferragut in 712:Quantum machine learning 465:Wheeler's delayed-choice 7830:Quantum finite automata 7206:28 October 2007 at the 7026:Frank, Philipp (1957). 6715:Penrose, Roger (2004). 6511:The Emergent Multiverse 6509:Wallace, David (2012). 6482:Deutsch, David (1998). 5491:10.1073/pnas.1012399107 5148:Physical Review Letters 5042:Physical Review Letters 4994:Physical Review Letters 4974:""To a light particle"" 4853:10.12743/quanta.v2i1.12 4473:Cassidy, David (2008). 4076:. Penguin. p. 63. 3241:"Wave–Particle Duality" 2903:Navarro, Jaume (2010). 2599:, and the principle of 2317:of the final result is 1397:single-photon detectors 422:Leggett–Garg inequality 7934:Quantum neural network 7195:Interactive animations 7077:Greene, Brian (2005). 7054:Greene, Brian (2000). 6395:Foundations of Physics 6077:10.2307/j.ctv173f2pm.9 5582:particles depends on 3 5397:Bush, John WM (2015). 5343:New Journal of Physics 4603:Foundations of Physics 4045:Greene, Brian (2007). 3986:New Journal of Physics 3949:10.1126/sciadv.aav7610 3873:Cowley, J. M. (1995). 3728:Physics in Perspective 3595:Zeitschrift für Physik 3492:Charles Sanders Peirce 2881:10.1098/rstl.1804.0001 2862:Young, Thomas (1804). 2787:Diffraction from slits 2664:De Broglie–Bohm theory 2658:de Broglie–Bohm theory 2652:De Broglie–Bohm theory 2512: 2432: 2305: 2201:functional integration 2191: 2147:≠ 0, and sinc(0) = 1. 2118: 1974:Fraunhofer diffraction 1938: 1870: 1810: 1724: 1651: 1643: 1635: 1616:indium tin oxide (ITO) 1523: 1515: 1456: 1400: 1246:This demonstrates the 1159:University of Tübingen 1103: 1074: 213: 66:double-slit experiment 7959:Quantum teleportation 7487:Wave–particle duality 7158:Tipler, Paul (2004). 7028:Philosophy of Science 6668:Leggett, A J (2002). 6485:The Fabric of Reality 6015:Quantum Speakables II 5574:Baggott, Jim (2011). 4299:Nature Communications 4162:31 March 2012 at the 3781:. Physics World 2002 3265:Feynman, 1965, p. 1.7 3226:Feynman, 1965, p. 1.5 2802:N-slit interferometer 2699:three wave hypothesis 2688:Anthony James Leggett 2513: 2433: 2306: 2185: 2119: 1939: 1871: 1811: 1725: 1649: 1641: 1633: 1521: 1513: 1454: 1390: 1248:wave–particle duality 1186:Politecnico di Milano 1101: 1068: 1020:wave–particle duality 999:coherent light source 407:Elitzur–Vaidman 397:Davisson–Germer 214: 94:wave–particle duality 7990:Quantum field theory 7919:Quantum metamaterial 7864:Quantum cryptography 7594:Consistent histories 7266:Computer simulations 7258:Through the Wormhole 7058:The Elegant Universe 3576:Prof. Cam. Phil. Soc 2772:Aharonov-Bohm effect 2711:Bohmian trajectories 2680:quantum field theory 2676:Schrödinger equation 2623:, first proposed by 2536:interference pattern 2452: 2325: 2218: 1983: 1892: 1839: 1745: 1696: 1584:Hydrodynamic analogs 1257:buckminsterfullerene 1127:photoelectric effect 672:Quantum field theory 584:Consistent histories 221:Schrödinger equation 148: 117:interference pattern 101:wave theory of light 86:George Paget Thomson 8171:Physics experiments 7975:Quantum fluctuation 7944:Quantum programming 7904:Quantum logic gates 7889:Quantum information 7869:Quantum electronics 7344:Classical mechanics 7241:Hydrodynamic analog 7132:2003nqu..book.....H 6928:1985PhLA..109....7E 6885:1986PhLA..114..179M 6842:1984PhLA..102..338D 6756:1981PhLA...87...95H 6605:"Bohmian Mechanics" 6586:"Bohmian Mechanics" 6548:1979NCimB..52...15P 6417:2015FoPh...45..211K 6361:1996IJTP...35.1637R 6223:1991Natur.351..111S 5936:2015SHPMP..49...73C 5801:2023Natur.616..230C 5743:2023NatPh..19..929R 5688:2016NatPh..12..972B 5618:2015NatSR...518574L 5552:2012JPhCS.361a2001C 5482:2010PNAS..10717455B 5476:(41): 17455–17456. 5418:2015AnRFM..47..269B 5365:2013NJPh...15c3018B 5294:2005PhRvL..94e3901S 5225:2017NatSR...712661Y 5160:2002PhRvL..88c8301O 5105:1999Natur.401..680A 5054:1991PhRvL..66.2689C 5007:1991PhRvL..66.2689C 4952:1972PhLA...39..333S 4890:1967PhRv..159.1084P 4778:1995QuSOp...7..259C 4723:2007SciAm.296e..90H 4715:Scientific American 4615:1987FoPh...17..891M 4580:1999RvMPS..71..288Z 4545:1980PhRvD..21.1698B 4368:2019NatPh..15.1242F 4311:2011NatCo...2..263G 4257:2003AmJPh..71..319N 4199:2001PhRvL..87p0401N 4111:1973AmJPh..41..639D 4008:2013NJPh...15c3018B 3941:2019SciA....5.7610S 3875:Diffraction physics 3785:24 May 2021 at the 3740:2012PhP....14..178R 3701:1976AmJPh..44..306M 3658:1974AmJPh..42....4J 3607:1961ZPhy..161..454J 3463:2019NatPh..15.1242F 3052:2013PCCP...1514696E 3036:(35): 14696–14700. 2968:1927Natur.119Q.890T 2832:Measurement problem 2812:Photon polarization 2171:Fresnel diffraction 2131:is defined as sinc( 1547:electron microscope 1473:Scientific American 1182:Positron Laboratory 1087:refined apparatus. 1084:diffraction pattern 1071:diffraction pattern 1031:classical mechanics 460:Stern–Gerlach 257:Classical mechanics 98:Christiaan Huygens' 84:and, independently 82:Davisson and Germer 8028:in popular culture 7810:Quantum algorithms 7658:Von Neumann–Wigner 7638:Objective collapse 7349:Old quantum theory 7122:Hey, Tony (2003). 6990:. Dutton/Penguin. 6984:Ananthaswamy, Anil 6799:10.1007/bf02817964 6556:10.1007/bf02743566 6536:Il Nuovo Cimento B 6369:10.1007/BF02302261 5972:. Pergamon Press. 5606:Scientific Reports 5203:Scientific Reports 4623:10.1007/BF00734319 4485:on 14 January 2016 4319:10.1038/ncomms1263 3615:10.1007/BF01342460 3060:10.1039/C3CP51500A 2551:thought experiment 2508: 2428: 2301: 2192: 2114: 2112: 1934: 1866: 1806: 1720: 1652: 1644: 1636: 1524: 1516: 1457: 1418:thought experiment 1401: 1104: 1075: 648:Von Neumann–Wigner 628:Objective-collapse 427:Mach–Zehnder 417:Leggett inequality 412:Franck–Hertz 262:Old quantum theory 209: 107:or Young's slits. 105:Young's experiment 8071: 8070: 8045:Quantum mysticism 8023:Schrödinger's cat 7954:Quantum simulator 7924:Quantum metrology 7852:Quantum computing 7815:Quantum amplifier 7792:Quantum spacetime 7757:Quantum cosmology 7747:Quantum chemistry 7462:Scattering theory 7410:Zero-point energy 7405:Degenerate levels 7313:Quantum mechanics 7169:978-0-7167-0810-0 7153:. Addison Wesley. 7141:978-0-521-56457-1 7114:978-0-7538-0685-2 7092:978-0-375-72720-7 7069:978-0-375-70811-4 7046:978-0-393-09106-9 7018:978-0-691-02417-2 6997:978-1-101-98609-7 6975:978-0-297-84305-4 6916:Physics Letters A 6873:Physics Letters A 6830:Physics Letters A 6744:Physics Letters A 6728:978-0-224-04447-9 6680:(15): R415–R451. 6647:10.1063/1.3060063 6520:978-0-19-954696-1 6495:978-0-14-014690-5 6457:(5 August 2021). 6263:Quantum Mechanics 6217:(6322): 111–116. 6153:978-0-691-03669-4 6042:978-3-319-38985-1 5979:978-0-08-017158-6 5872:978-0-691-14034-6 5696:10.1038/nphys3810 5626:10.1038/srep18574 5099:(6754): 680–682. 5048:(21): 2689–2692. 5001:(21): 2689–2694. 4940:Physics Letters A 4918:on 3 January 2011 4533:Physical Review D 4479:Werner Heisenberg 4424:978-0-19-921570-6 4362:(12): 1242–1245. 4265:10.1119/1.1531580 4119:10.1119/1.1987321 4083:978-1-101-98611-0 4058:978-0-307-42853-0 3909:978-0-19-879583-4 3884:978-0-444-82218-5 3666:10.1119/1.1987592 3548:978-0-393-04688-5 3514:978-0-393-07298-3 3457:(12): 1242–1245. 3380:978-1-139-45527-5 3347:978-1-4614-4517-3 3288:978-1-61614-281-0 3213:978-0-201-02118-9 3169:978-1-61614-281-0 3139:978-0-8176-2316-6 3110:978-0-13-134304-7 2842:Pilot wave theory 2822:Schrödinger's cat 2817:Quantum coherence 2703:Ryszard Horodecki 2684:Werner Heisenberg 2619:According to the 2585:Werner Heisenberg 2577:quantum mechanics 2548:Schrödinger's cat 2462: 2374: 2228: 2104: 2056: 2048: 1897: 1778: 1750: 1573:atomic mass units 1496:weak measurements 1134:nature of light. 1043:atomic mass units 995: 994: 702:Scattering theory 682:Quantum computing 455:Schrödinger's cat 387:Bell's inequality 195: 170: 139:Quantum mechanics 90:classical physics 74:quantum mechanics 8183: 8148: 8147: 8146: 8136: 8135: 8134: 8124: 8123: 8122: 8112: 8111: 8100: 8099: 8098: 8088: 8087: 8079: 8061: 8060: 7772:Quantum geometry 7767:Quantum dynamics 7624:Superdeterminism 7520:Matrix mechanics 7375:Bra–ket notation 7306: 7299: 7292: 7283: 7282: 7173: 7154: 7145: 7118: 7096: 7084: 7073: 7061: 7050: 7031: 7030:. 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Rev. Lett 5270: 5266: 5195: 5191: 5140: 5136: 5089: 5085: 5074: 5072: 5034: 5030: 4985: 4981: 4972: 4971: 4967: 4936: 4932: 4921: 4919: 4910: 4909: 4905: 4877:Physical Review 4872: 4868: 4827: 4820: 4805: 4801: 4752: 4745: 4735: 4733: 4703: 4696: 4666:Current Science 4658: 4654: 4650:, 391–4 (1988). 4645:Physics Letters 4642: 4638: 4599: 4595: 4564: 4560: 4529: 4525: 4515: 4513: 4502: 4498: 4488: 4486: 4471: 4467: 4457: 4455: 4444: 4440: 4425: 4408: 4404: 4395: 4391: 4348: 4344: 4291: 4287: 4277: 4275: 4271: 4240: 4234: 4230: 4177:Phys. Rev. Lett 4173: 4169: 4164:Wayback Machine 4155: 4151: 4145:Wayback Machine 4133: 4126: 4095: 4091: 4084: 4070: 4066: 4059: 4043: 4039: 3978: 3974: 3935:(5): eaav7610. 3921: 3917: 3910: 3896: 3892: 3885: 3871: 3867: 3840:Ultramicroscopy 3835: 3831: 3796: 3792: 3787:Wayback Machine 3777: 3773: 3720: 3716: 3709:10.1119/1.10184 3685: 3681: 3642: 3638: 3591: 3587: 3569: 3565: 3560: 3556: 3549: 3526: 3522: 3490: 3486: 3447: 3440: 3411: 3407: 3392: 3388: 3381: 3359: 3355: 3348: 3332: 3328: 3321: 3320:978-981-2566911 3300: 3296: 3289: 3273: 3269: 3264: 3260: 3250: 3248: 3234: 3230: 3225: 3221: 3214: 3198: 3177: 3170: 3154: 3147: 3140: 3124: 3120: 3111: 3087: 3083: 3024: 3017: 3008: 3001: 2948: 2944: 2901: 2897: 2860: 2856: 2851: 2846: 2767: 2762: 2761: 2760: 2759: 2758: 2750: 2742: 2741: 2734: 2726: 2725: 2722: 2713: 2712: 2660: 2654: 2637: 2617: 2601:complementarity 2593:indeterministic 2569: 2563: 2544: 2459: 2455: 2453: 2450: 2449: 2422: 2381: 2377: 2371: 2367: 2366: 2362: 2361: 2326: 2323: 2322: 2265: 2261: 2225: 2221: 2219: 2216: 2215: 2180: 2159: 2155: 2151: 2111: 2110: 2084: 2082: 2078: 2072: 2058: 2057: 2028: 2026: 2022: 2013: 2009: 2002: 1986: 1984: 1981: 1980: 1967: 1958: 1954: 1948: 1926: 1911: 1907: 1893: 1890: 1889: 1880: 1858: 1846: 1842: 1840: 1837: 1836: 1830: 1824: 1819:where λ is the 1758: 1754: 1746: 1743: 1742: 1697: 1694: 1693: 1686: 1666:as well as the 1628: 1608: 1581: 1566:Richard Feynman 1508: 1491: 1485: 1449: 1443: 1422:complementarity 1414: 1385: 1379: 1363: 1360: 1359: 1358: 1354: 1351: 1350: 1349: 1345: 1342: 1341: 1340: 1336: 1333: 1332: 1331: 1327: 1324: 1323: 1322: 1318: 1315: 1314: 1313: 1311: 1306: 1303: 1302: 1301: 1297: 1294: 1293: 1292: 1288: 1285: 1284: 1283: 1279: 1276: 1275: 1274: 1272: 1267: 1264: 1263: 1262: 1260: 1244: 1243: 1242: 1241: 1240: 1233: 1224: 1223: 1222: 1207: 1202: 1063: 1051:Richard Feynman 991: 962: 961: 960: 725: 717: 716: 662: 661:Advanced topics 654: 653: 652: 604:Hidden-variable 594:de Broglie–Bohm 573: 571:Interpretations 563: 562: 561: 531: 523: 522: 521: 479: 471: 470: 469: 436: 392:CHSH inequality 381: 373: 372: 371: 300:Complementarity 294: 286: 285: 284: 252: 223: 198: 187: 186: 172: 162: 157: 149: 146: 145: 58: 57: 56: 55: 54: 51: 42: 41: 40: 37: 24: 17: 12: 11: 5: 8189: 8179: 8178: 8176:Wave mechanics 8173: 8168: 8153: 8152: 8140: 8128: 8116: 8104: 8092: 8069: 8068: 8066: 8065: 8054: 8051: 8050: 8048: 8047: 8042: 8037: 8032: 8031: 8030: 8019: 8017: 8013: 8012: 8010: 8009: 8004: 7999: 7998: 7997: 7987: 7982: 7980:Casimir effect 7977: 7971: 7969: 7965: 7964: 7962: 7961: 7956: 7951: 7946: 7941: 7939:Quantum optics 7936: 7931: 7926: 7921: 7916: 7911: 7906: 7901: 7896: 7891: 7886: 7881: 7876: 7871: 7866: 7861: 7860: 7859: 7849: 7844: 7839: 7834: 7833: 7832: 7822: 7817: 7812: 7806: 7804: 7798: 7797: 7795: 7794: 7789: 7784: 7779: 7774: 7769: 7764: 7759: 7754: 7749: 7744: 7738: 7736: 7730: 7729: 7727: 7726: 7721: 7716: 7714:Quantum eraser 7711: 7706: 7701: 7696: 7691: 7686: 7681: 7676: 7670: 7668: 7664: 7663: 7661: 7660: 7655: 7650: 7645: 7640: 7635: 7630: 7629: 7628: 7627: 7626: 7611: 7606: 7601: 7596: 7591: 7585: 7583: 7577: 7576: 7574: 7573: 7568: 7563: 7558: 7553: 7547: 7545: 7541: 7540: 7538: 7537: 7532: 7527: 7522: 7517: 7512: 7507: 7501: 7499: 7495: 7494: 7492: 7491: 7490: 7489: 7484: 7474: 7469: 7464: 7459: 7454: 7449: 7444: 7439: 7434: 7429: 7424: 7419: 7414: 7413: 7412: 7407: 7402: 7397: 7387: 7385:Density matrix 7382: 7377: 7372: 7366: 7364: 7360: 7359: 7357: 7356: 7351: 7346: 7341: 7340: 7339: 7329: 7323: 7321: 7317: 7316: 7309: 7308: 7301: 7294: 7286: 7280: 7279: 7274: 7267: 7264: 7263: 7262: 7254: 7249: 7242: 7239: 7238: 7237: 7232: 7227: 7222: 7215: 7212: 7211: 7210: 7196: 7193: 7192: 7191: 7179: 7178:External links 7176: 7175: 7174: 7168: 7155: 7146: 7140: 7119: 7113: 7097: 7091: 7074: 7068: 7051: 7045: 7032: 7023: 7017: 7002: 6996: 6980: 6974: 6956: 6953: 6950: 6949: 6906: 6879:(4): 179–182. 6863: 6836:(8): 338–339. 6820: 6777: 6734: 6727: 6707: 6660: 6621: 6595: 6577: 6526: 6519: 6501: 6494: 6474: 6446: 6401:(2): 211–217. 6382: 6335:Rovelli, Carlo 6326: 6279: 6272: 6252: 6197: 6190: 6167: 6152: 6128: 6090: 6056: 6041: 6001: 5978: 5957: 5902: 5878: 5871: 5848: 5839: 5830: 5772: 5737:(7): 929–930. 5731:Nature Physics 5717: 5666:Nature Physics 5649: 5592: 5567: 5526: 5507: 5456: 5412:(1): 269–292. 5386: 5333: 5264: 5189: 5134: 5083: 5028: 4979: 4965: 4946:(4): 333–334. 4930: 4903: 4866: 4818: 4799: 4762:(3): 259–278. 4743: 4694: 4672:(2): 203–218. 4652: 4636: 4609:(9): 891–903. 4593: 4558: 4523: 4496: 4465: 4438: 4423: 4411:Vedral, Vlatko 4402: 4389: 4356:Nature Physics 4342: 4285: 4274:on 4 June 2015 4251:(4): 319–325. 4228: 4183:(16): 160401. 4167: 4149: 4124: 4105:(5): 639–644. 4089: 4082: 4064: 4057: 4037: 3972: 3915: 3908: 3890: 3883: 3865: 3829: 3790: 3771: 3734:(2): 178–194. 3714: 3695:(3): 306–307. 3679: 3636: 3601:(4): 454–474. 3585: 3563: 3554: 3547: 3520: 3518: 3517: 3484: 3451:Nature Physics 3438: 3405: 3386: 3379: 3353: 3346: 3326: 3319: 3294: 3287: 3267: 3258: 3237:Darling, David 3228: 3219: 3212: 3175: 3168: 3145: 3138: 3118: 3116: 3115: 3109: 3081: 3015: 2999: 2942: 2915:(2): 245–275. 2895: 2853: 2852: 2850: 2847: 2845: 2844: 2839: 2834: 2829: 2824: 2819: 2814: 2809: 2804: 2799: 2794: 2789: 2784: 2779: 2774: 2768: 2766: 2763: 2751: 2744: 2743: 2735: 2728: 2727: 2723: 2716: 2715: 2714: 2710: 2709: 2708: 2707: 2656:Main article: 2653: 2650: 2636: 2633: 2616: 2613: 2565:Main article: 2562: 2559: 2543: 2540: 2532:proportionally 2507: 2504: 2501: 2498: 2494: 2491: 2488: 2485: 2482: 2479: 2476: 2473: 2470: 2467: 2458: 2425: 2420: 2414: 2411: 2408: 2405: 2402: 2399: 2396: 2393: 2390: 2387: 2384: 2380: 2370: 2365: 2360: 2357: 2354: 2351: 2348: 2345: 2342: 2339: 2336: 2333: 2330: 2298: 2295: 2292: 2289: 2286: 2283: 2280: 2277: 2274: 2271: 2268: 2264: 2260: 2257: 2254: 2251: 2248: 2245: 2242: 2239: 2236: 2233: 2224: 2188:Wiener process 2179: 2176: 2125: 2124: 2108: 2103: 2099: 2096: 2093: 2090: 2087: 2081: 2075: 2070: 2067: 2064: 2061: 2052: 2047: 2043: 2040: 2037: 2034: 2031: 2025: 2021: 2016: 2012: 2008: 2005: 2003: 2001: 1998: 1995: 1992: 1989: 1988: 1945: 1944: 1933: 1929: 1925: 1922: 1919: 1914: 1910: 1906: 1903: 1900: 1877: 1876: 1865: 1861: 1857: 1854: 1849: 1845: 1832:, is given by 1826: 1817: 1816: 1805: 1802: 1799: 1796: 1793: 1790: 1787: 1784: 1781: 1775: 1772: 1769: 1766: 1761: 1757: 1753: 1731: 1730: 1719: 1716: 1713: 1710: 1707: 1704: 1701: 1627: 1624: 1607: 1604: 1580: 1577: 1507: 1504: 1487:Main article: 1484: 1481: 1466:Quantum eraser 1445:Main article: 1442: 1439: 1413: 1410: 1381:Main article: 1378: 1375: 1361: 1352: 1343: 1334: 1325: 1316: 1304: 1295: 1286: 1277: 1265: 1234: 1227: 1226: 1225: 1216: 1215: 1214: 1213: 1212: 1206: 1203: 1201: 1198: 1062: 1059: 993: 992: 990: 989: 982: 975: 967: 964: 963: 959: 958: 953: 948: 943: 938: 933: 928: 923: 918: 913: 908: 903: 898: 893: 888: 883: 878: 873: 868: 863: 858: 853: 848: 843: 838: 833: 828: 823: 818: 813: 808: 803: 798: 793: 788: 783: 778: 773: 768: 763: 758: 753: 748: 743: 738: 733: 727: 726: 723: 722: 719: 718: 715: 714: 709: 704: 699: 697:Density matrix 694: 689: 684: 679: 674: 669: 663: 660: 659: 656: 655: 651: 650: 645: 640: 635: 630: 625: 620: 619: 618: 617: 616: 601: 596: 591: 586: 581: 575: 574: 569: 568: 565: 564: 560: 559: 554: 549: 544: 539: 533: 532: 529: 528: 525: 524: 520: 519: 514: 509: 504: 499: 494: 488: 487: 486: 480: 477: 476: 473: 472: 468: 467: 462: 457: 451: 450: 449: 448: 447: 445:Delayed-choice 440:Quantum eraser 435: 434: 429: 424: 419: 414: 409: 404: 399: 394: 389: 383: 382: 379: 378: 375: 374: 370: 369: 368: 367: 357: 352: 347: 342: 337: 332: 330:Quantum number 327: 322: 317: 312: 307: 302: 296: 295: 292: 291: 288: 287: 283: 282: 277: 271: 270: 269: 264: 259: 253: 250: 249: 246: 245: 244: 243: 238: 233: 225: 224: 219: 208: 205: 201: 194: 191: 185: 182: 179: 175: 168: 165: 161: 156: 153: 142: 141: 135: 134: 115:, creating an 71: 62:modern physics 52: 45: 44: 43: 38: 31: 30: 29: 28: 27: 15: 9: 6: 4: 3: 2: 8188: 8177: 8174: 8172: 8169: 8167: 8164: 8163: 8161: 8151: 8141: 8139: 8129: 8127: 8117: 8115: 8110: 8105: 8103: 8093: 8091: 8086: 8081: 8080: 8077: 8064: 8056: 8055: 8052: 8046: 8043: 8041: 8038: 8036: 8033: 8029: 8026: 8025: 8024: 8021: 8020: 8018: 8014: 8008: 8005: 8003: 8000: 7996: 7993: 7992: 7991: 7988: 7986: 7983: 7981: 7978: 7976: 7973: 7972: 7970: 7966: 7960: 7957: 7955: 7952: 7950: 7947: 7945: 7942: 7940: 7937: 7935: 7932: 7930: 7927: 7925: 7922: 7920: 7917: 7915: 7912: 7910: 7907: 7905: 7902: 7900: 7899:Quantum logic 7897: 7895: 7892: 7890: 7887: 7885: 7882: 7880: 7877: 7875: 7872: 7870: 7867: 7865: 7862: 7858: 7855: 7854: 7853: 7850: 7848: 7845: 7843: 7840: 7838: 7835: 7831: 7828: 7827: 7826: 7823: 7821: 7818: 7816: 7813: 7811: 7808: 7807: 7805: 7803: 7799: 7793: 7790: 7788: 7785: 7783: 7780: 7778: 7775: 7773: 7770: 7768: 7765: 7763: 7760: 7758: 7755: 7753: 7752:Quantum chaos 7750: 7748: 7745: 7743: 7740: 7739: 7737: 7735: 7731: 7725: 7722: 7720: 7719:Stern–Gerlach 7717: 7715: 7712: 7710: 7707: 7705: 7702: 7700: 7697: 7695: 7692: 7690: 7687: 7685: 7682: 7680: 7677: 7675: 7672: 7671: 7669: 7665: 7659: 7656: 7654: 7653:Transactional 7651: 7649: 7646: 7644: 7643:Quantum logic 7641: 7639: 7636: 7634: 7631: 7625: 7622: 7621: 7620: 7617: 7616: 7615: 7612: 7610: 7607: 7605: 7602: 7600: 7597: 7595: 7592: 7590: 7587: 7586: 7584: 7582: 7578: 7572: 7569: 7567: 7564: 7562: 7559: 7557: 7554: 7552: 7549: 7548: 7546: 7542: 7536: 7533: 7531: 7528: 7526: 7523: 7521: 7518: 7516: 7513: 7511: 7508: 7506: 7503: 7502: 7500: 7496: 7488: 7485: 7483: 7480: 7479: 7478: 7477:Wave function 7475: 7473: 7470: 7468: 7465: 7463: 7460: 7458: 7455: 7453: 7452:Superposition 7450: 7448: 7447:Quantum state 7445: 7443: 7440: 7438: 7435: 7433: 7430: 7428: 7425: 7423: 7420: 7418: 7415: 7411: 7408: 7406: 7403: 7401: 7400:Excited state 7398: 7396: 7393: 7392: 7391: 7388: 7386: 7383: 7381: 7378: 7376: 7373: 7371: 7368: 7367: 7365: 7361: 7355: 7352: 7350: 7347: 7345: 7342: 7338: 7335: 7334: 7333: 7330: 7328: 7325: 7324: 7322: 7318: 7314: 7307: 7302: 7300: 7295: 7293: 7288: 7287: 7284: 7278: 7275: 7273: 7270: 7269: 7261: 7259: 7255: 7253: 7250: 7248: 7245: 7244: 7236: 7233: 7231: 7228: 7226: 7223: 7221: 7218: 7217: 7209: 7205: 7202: 7199: 7198: 7189: 7185: 7182: 7181: 7171: 7165: 7161: 7156: 7152: 7147: 7143: 7137: 7133: 7129: 7125: 7120: 7116: 7110: 7106: 7102: 7101:Gribbin, John 7098: 7094: 7088: 7083: 7082: 7075: 7071: 7065: 7060: 7059: 7052: 7048: 7042: 7038: 7033: 7029: 7024: 7020: 7014: 7010: 7009: 7003: 6999: 6993: 6989: 6985: 6981: 6977: 6971: 6967: 6963: 6959: 6958: 6945: 6941: 6937: 6933: 6929: 6925: 6921: 6917: 6910: 6902: 6898: 6894: 6890: 6886: 6882: 6878: 6874: 6867: 6859: 6855: 6851: 6847: 6843: 6839: 6835: 6831: 6824: 6816: 6812: 6808: 6804: 6800: 6796: 6792: 6788: 6781: 6773: 6769: 6765: 6761: 6757: 6753: 6749: 6745: 6738: 6730: 6724: 6720: 6719: 6711: 6703: 6699: 6695: 6691: 6687: 6683: 6679: 6675: 6671: 6664: 6656: 6652: 6648: 6644: 6640: 6636: 6635:Physics Today 6632: 6625: 6610: 6606: 6599: 6591: 6587: 6581: 6573: 6569: 6565: 6561: 6557: 6553: 6549: 6545: 6541: 6537: 6530: 6522: 6516: 6512: 6505: 6497: 6491: 6487: 6486: 6478: 6470: 6469: 6464: 6460: 6456: 6450: 6442: 6438: 6434: 6430: 6426: 6422: 6418: 6414: 6409: 6404: 6400: 6396: 6392: 6386: 6378: 6374: 6370: 6366: 6362: 6358: 6353: 6348: 6344: 6340: 6336: 6330: 6322: 6318: 6314: 6310: 6306: 6302: 6298: 6294: 6290: 6283: 6275: 6273:0-486-40924-4 6269: 6265: 6264: 6256: 6248: 6244: 6240: 6236: 6232: 6228: 6224: 6220: 6216: 6212: 6208: 6201: 6193: 6191:0-7923-2549-4 6187: 6183: 6182: 6177: 6171: 6163: 6159: 6155: 6149: 6145: 6141: 6135: 6133: 6124: 6120: 6116: 6112: 6108: 6104: 6100: 6099:Rosenfeld, L. 6094: 6086: 6082: 6078: 6074: 6070: 6066: 6065:Omnès, Roland 6060: 6052: 6048: 6044: 6038: 6034: 6030: 6025: 6020: 6016: 6012: 6005: 5998: 5995: 5989: 5985: 5981: 5975: 5971: 5967: 5961: 5953: 5949: 5945: 5941: 5937: 5933: 5928: 5923: 5919: 5915: 5914: 5906: 5898: 5897: 5892: 5888: 5882: 5874: 5868: 5864: 5863: 5858: 5852: 5843: 5834: 5826: 5822: 5818: 5814: 5810: 5806: 5802: 5798: 5795:(7956): 230. 5794: 5790: 5786: 5779: 5777: 5768: 5764: 5760: 5756: 5752: 5748: 5744: 5740: 5736: 5732: 5728: 5721: 5713: 5709: 5705: 5701: 5697: 5693: 5689: 5685: 5680: 5675: 5671: 5667: 5663: 5656: 5654: 5645: 5641: 5636: 5631: 5627: 5623: 5619: 5615: 5611: 5607: 5603: 5596: 5589: 5585: 5581: 5577: 5571: 5562: 5557: 5553: 5549: 5546:(1): 012001. 5545: 5541: 5537: 5530: 5522: 5518: 5511: 5502: 5497: 5492: 5487: 5483: 5479: 5475: 5471: 5467: 5460: 5441: 5436: 5431: 5427: 5423: 5419: 5415: 5411: 5407: 5400: 5393: 5391: 5382: 5378: 5374: 5370: 5366: 5362: 5357: 5352: 5349:(3): 033018. 5348: 5344: 5337: 5329: 5325: 5321: 5317: 5312: 5307: 5303: 5299: 5295: 5291: 5288:(5): 053901. 5287: 5283: 5279: 5275: 5268: 5260: 5256: 5251: 5246: 5242: 5238: 5234: 5230: 5226: 5222: 5217: 5212: 5208: 5204: 5200: 5193: 5185: 5181: 5177: 5173: 5169: 5165: 5161: 5157: 5154:(3): 038301. 5153: 5149: 5145: 5138: 5130: 5126: 5122: 5118: 5114: 5113:10.1038/44348 5110: 5106: 5102: 5098: 5094: 5087: 5071: 5067: 5063: 5059: 5055: 5051: 5047: 5043: 5039: 5032: 5024: 5020: 5016: 5012: 5008: 5004: 5000: 4996: 4995: 4990: 4983: 4975: 4969: 4961: 4957: 4953: 4949: 4945: 4941: 4934: 4917: 4913: 4907: 4899: 4895: 4891: 4887: 4883: 4879: 4878: 4870: 4862: 4858: 4854: 4850: 4845: 4840: 4836: 4832: 4825: 4823: 4814: 4810: 4803: 4795: 4791: 4787: 4783: 4779: 4775: 4770: 4765: 4761: 4757: 4750: 4748: 4732: 4728: 4724: 4720: 4716: 4712: 4708: 4705:Hillmer, R.; 4701: 4699: 4691: 4687: 4683: 4679: 4675: 4671: 4667: 4663: 4656: 4649: 4646: 4640: 4632: 4628: 4624: 4620: 4616: 4612: 4608: 4604: 4597: 4589: 4585: 4581: 4577: 4573: 4569: 4562: 4554: 4550: 4546: 4542: 4538: 4534: 4527: 4511: 4507: 4500: 4484: 4480: 4476: 4469: 4453: 4449: 4442: 4434: 4430: 4426: 4420: 4416: 4412: 4406: 4399: 4393: 4385: 4381: 4377: 4373: 4369: 4365: 4361: 4357: 4353: 4346: 4338: 4334: 4329: 4324: 4320: 4316: 4312: 4308: 4304: 4300: 4296: 4289: 4270: 4266: 4262: 4258: 4254: 4250: 4246: 4239: 4232: 4224: 4220: 4216: 4212: 4208: 4204: 4200: 4196: 4191: 4186: 4182: 4178: 4171: 4165: 4161: 4158: 4153: 4146: 4142: 4139: 4136: 4131: 4129: 4120: 4116: 4112: 4108: 4104: 4100: 4093: 4085: 4079: 4075: 4068: 4060: 4054: 4050: 4049: 4041: 4033: 4029: 4025: 4021: 4017: 4013: 4009: 4005: 4000: 3995: 3991: 3987: 3983: 3976: 3968: 3964: 3959: 3954: 3950: 3946: 3942: 3938: 3934: 3930: 3926: 3919: 3911: 3905: 3901: 3894: 3886: 3880: 3876: 3869: 3861: 3857: 3853: 3849: 3845: 3841: 3833: 3825: 3821: 3817: 3813: 3809: 3805: 3804:Physics World 3801: 3794: 3788: 3784: 3780: 3775: 3767: 3763: 3758: 3753: 3749: 3745: 3741: 3737: 3733: 3729: 3725: 3718: 3710: 3706: 3702: 3698: 3694: 3690: 3683: 3675: 3671: 3667: 3663: 3659: 3655: 3651: 3647: 3640: 3632: 3628: 3624: 3620: 3616: 3612: 3608: 3604: 3600: 3597:(in German). 3596: 3589: 3581: 3577: 3573: 3567: 3558: 3550: 3544: 3540: 3536: 3535: 3530: 3529:Greene, Brian 3524: 3515: 3511: 3507: 3503: 3499: 3498: 3497: 3493: 3488: 3480: 3476: 3472: 3468: 3464: 3460: 3456: 3452: 3445: 3443: 3434: 3430: 3426: 3422: 3421: 3415: 3409: 3402: 3398: 3397: 3390: 3382: 3376: 3372: 3371: 3364: 3357: 3349: 3343: 3339: 3338: 3330: 3322: 3316: 3312: 3311: 3304: 3298: 3290: 3284: 3280: 3279: 3271: 3262: 3246: 3242: 3238: 3232: 3223: 3215: 3209: 3205: 3204: 3196: 3194: 3192: 3190: 3188: 3186: 3184: 3182: 3180: 3171: 3165: 3161: 3160: 3152: 3150: 3141: 3135: 3131: 3130: 3122: 3112: 3106: 3102: 3098: 3097: 3091: 3090: 3085: 3077: 3073: 3069: 3065: 3061: 3057: 3053: 3049: 3044: 3039: 3035: 3031: 3030: 3022: 3020: 3012: 3006: 3004: 2995: 2991: 2987: 2983: 2978: 2973: 2969: 2965: 2962:(3007): 890. 2961: 2957: 2953: 2946: 2938: 2934: 2930: 2926: 2922: 2918: 2914: 2910: 2906: 2899: 2891: 2887: 2882: 2877: 2873: 2869: 2865: 2858: 2854: 2843: 2840: 2838: 2835: 2833: 2830: 2828: 2825: 2823: 2820: 2818: 2815: 2813: 2810: 2808: 2805: 2803: 2800: 2798: 2795: 2793: 2790: 2788: 2785: 2783: 2780: 2778: 2775: 2773: 2770: 2769: 2756: 2748: 2739: 2732: 2720: 2706: 2704: 2700: 2695: 2693: 2692:Roger Penrose 2689: 2685: 2681: 2677: 2672: 2670: 2665: 2659: 2649: 2646: 2645:David Deutsch 2642: 2632: 2630: 2626: 2625:Carlo Rovelli 2622: 2612: 2608: 2606: 2602: 2598: 2594: 2590: 2586: 2582: 2578: 2574: 2568: 2558: 2556: 2552: 2549: 2539: 2537: 2533: 2530: 2527: 2523: 2522:superposition 2518: 2505: 2502: 2499: 2496: 2489: 2486: 2483: 2480: 2477: 2474: 2471: 2465: 2456: 2448:by imposing: 2447: 2443: 2438: 2423: 2418: 2409: 2406: 2403: 2400: 2397: 2394: 2391: 2385: 2382: 2378: 2368: 2363: 2358: 2352: 2349: 2346: 2343: 2340: 2337: 2334: 2328: 2320: 2316: 2311: 2293: 2290: 2287: 2284: 2281: 2278: 2275: 2269: 2266: 2262: 2258: 2252: 2249: 2246: 2243: 2240: 2237: 2234: 2222: 2213: 2209: 2204: 2202: 2197: 2189: 2184: 2175: 2172: 2168: 2163: 2148: 2146: 2142: 2138: 2134: 2130: 2129:sinc function 2106: 2101: 2097: 2094: 2091: 2088: 2085: 2079: 2073: 2050: 2045: 2041: 2038: 2035: 2032: 2029: 2023: 2019: 2014: 2010: 2006: 2004: 1996: 1990: 1979: 1978: 1977: 1975: 1970: 1964: 1961: 1951: 1931: 1927: 1923: 1920: 1917: 1912: 1908: 1904: 1901: 1898: 1888: 1887: 1886: 1883: 1863: 1859: 1855: 1852: 1847: 1843: 1835: 1834: 1833: 1829: 1822: 1803: 1800: 1797: 1794: 1791: 1788: 1785: 1782: 1779: 1773: 1770: 1767: 1764: 1759: 1755: 1751: 1741: 1740: 1739: 1737: 1717: 1714: 1711: 1708: 1705: 1702: 1699: 1692: 1691: 1690: 1689:is given by: 1684: 1679: 1675: 1673: 1669: 1665: 1661: 1657: 1648: 1640: 1632: 1623: 1621: 1617: 1613: 1603: 1599: 1598: 1592: 1590: 1585: 1576: 1574: 1569: 1567: 1563: 1558: 1554: 1550: 1548: 1544: 1538: 1536: 1531: 1527: 1520: 1512: 1503: 1501: 1497: 1490: 1480: 1478: 1474: 1469: 1467: 1463: 1461: 1453: 1448: 1438: 1436: 1430: 1428: 1423: 1419: 1416:A well-known 1409: 1405: 1398: 1394: 1391:Photons in a 1389: 1384: 1374: 1372: 1370: 1258: 1253: 1249: 1239: 1231: 1220: 1211: 1197: 1195: 1191: 1187: 1183: 1178: 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Leeds, UK 3652:(1): 4–11. 3363:experiment. 2807:Matter wave 2529:propagating 2442:probability 1620:Kerr effect 1252:probability 1089:Diffraction 931:von Neumann 916:Schrödinger 692:EPR paradox 623:Many-worlds 557:Schrödinger 512:Schrödinger 507:Phase-space 497:Interaction 402:Double-slit 380:Experiments 355:Uncertainty 325:Nonlocality 320:Measurement 305:Decoherence 275:Hamiltonian 113:phase shift 8160:Categories 7968:Extensions 7802:Technology 7648:Relational 7599:Copenhagen 7510:Heisenberg 7457:Tunnelling 7320:Background 7039:. Norton. 5927:1502.06547 5679:1510.01277 5311:1887/71482 5216:1710.02216 4736:11 January 3427:: 90–105. 3251:18 October 2849:References 2581:Niels Bohr 2446:normalized 2167:near field 2162:function. 2127:where the 1821:wavelength 1612:pump laser 1589:pilot wave 1535:metastable 1435:inequality 1112:wavefronts 926:Sommerfeld 841:Heisenberg 836:Gutzwiller 776:de Broglie 724:Scientists 638:Relational 589:Copenhagen 492:Heisenberg 350:Tunnelling 251:Background 8102:Astronomy 7674:Bell test 7544:Equations 7370:Born rule 6944:0375-9601 6901:0375-9601 6858:0375-9601 6815:120784358 6807:1827-613X 6772:0375-9601 6702:250911999 6694:0953-8984 6655:0031-9228 6641:(8): 12. 6615:14 August 6564:1826-9877 6441:118471198 6433:0015-9018 6408:1408.1944 6313:0304-3991 6299:: 49–56. 6239:0028-0836 6162:439453957 6140:Omnès, R. 6115:0036-8504 6085:203390914 6051:118458259 6024:1409.2454 5988:799397091 5920:: 73–83. 5887:Faye, Jan 5825:257922697 5767:257945438 5759:1745-2481 5704:1745-2481 5612:: 18574. 5356:1210.6243 5241:2045-2322 5176:0031-9007 4861:119242577 4844:1202.5148 4794:118987962 4707:Kwiat, P. 4678:0011-3891 4631:122856271 4433:442351498 4384:203638258 4024:1367-2630 3999:1210.6243 3860:0304-3991 3846:: 73–76. 3824:0953-8585 3674:0002-9505 3631:121659705 3623:0044-3328 3516:. p. 203. 3479:203638258 3393:Feynman, 3043:1310.8343 2986:0028-0836 2937:171025814 2929:0007-0874 2890:110408369 2755:far field 2597:Born rule 2546:Like the 2461:all space 2457:∭ 2373:all paths 2369:∫ 2359:∝ 2315:magnitude 2102:λ 2098:θ 2095:⁡ 2086:π 2046:λ 2042:θ 2039:⁡ 2030:π 2020:⁡ 2007:∝ 1997:θ 1924:λ 1909:θ 1856:λ 1853:≈ 1844:θ 1804:… 1771:λ 1756:θ 1718:θ 1712:≈ 1709:θ 1706:⁡ 1683:far field 1672:intensity 1668:amplitude 1541:0.7 1477:entangled 1369:entangled 1184:(L-NESS, 1079:classical 1039:molecules 1027:electrons 1008:interfere 956:Zeilinger 801:Ehrenfest 530:Equations 207:⟩ 204:Ψ 193:^ 181:⟩ 178:Ψ 155:ℏ 8063:Category 7857:Timeline 7609:Ensemble 7589:Bayesian 7482:Collapse 7354:Glossary 7337:Timeline 7204:Archived 7103:(1999). 6986:(2018). 6964:(2003). 6572:53575967 6377:16325959 6321:25799917 6178:(1995). 6142:(1994). 6123:43414997 6013:(eds.). 5968:(1973). 5952:27697360 5889:(2019). 5859:(2010). 5817:37012471 5712:53536274 5644:26689679 5440:Archived 5328:19197175 5320:15783641 5274:Gbur, G. 5259:28978914 5184:11801091 5121:18494170 5075:20 March 5070:10043591 5023:10043591 4709:(2007). 4686:24103129 4413:(2006). 4337:21468015 4223:21547361 4215:11690188 4160:Archived 4141:Archived 3967:31058223 3783:Archived 3766:26525832 3531:(1999). 3239:(2007). 3068:23900710 2874:: 1–16. 2765:See also 2690:and Sir 2629:observer 2589:Max Born 2135:) = sin( 1061:Overview 881:Millikan 806:Einstein 791:Davisson 746:Blackett 731:Aharonov 599:Ensemble 579:Bayesian 484:Overview 365:Collapse 345:Symmetry 236:Glossary 8090:Science 8076:Portals 8016:Related 7995:History 7734:Science 7566:Rydberg 7332:History 7128:Bibcode 6924:Bibcode 6881:Bibcode 6838:Bibcode 6752:Bibcode 6544:Bibcode 6465:(ed.). 6413:Bibcode 6357:Bibcode 6247:4311842 6219:Bibcode 5932:Bibcode 5797:Bibcode 5739:Bibcode 5684:Bibcode 5635:4686973 5614:Bibcode 5548:Bibcode 5501:2955131 5478:Bibcode 5449:21 June 5414:Bibcode 5361:Bibcode 5290:Bibcode 5250:5627254 5221:Bibcode 5156:Bibcode 5129:4424892 5101:Bibcode 5050:Bibcode 5003:Bibcode 4948:Bibcode 4922:16 June 4886:Bibcode 4774:Bibcode 4719:Bibcode 4611:Bibcode 4576:Bibcode 4541:Bibcode 4516:21 June 4489:21 June 4458:21 June 4452:UPSCALE 4364:Bibcode 4328:3104521 4307:Bibcode 4305:: 263. 4253:Bibcode 4195:Bibcode 4107:Bibcode 4004:Bibcode 3958:6499593 3937:Bibcode 3757:4617474 3736:Bibcode 3697:Bibcode 3654:Bibcode 3603:Bibcode 3459:Bibcode 3101:123–124 3076:3944699 3048:Bibcode 2994:4122313 2964:Bibcode 2319:squared 1660:summing 1157:of the 1132:quantum 921:Simmons 911:Rydberg 876:Moseley 856:Kramers 846:Hilbert 831:Glauber 826:Feynman 811:Everett 781:Compton 552:Rydberg 241:History 7709:Popper 7190:of MIT 7166: 7151:Optics 7138: 7111: 7089: 7066: 7043: 7015: 6994: 6972: 6942: 6899: 6856: 6813: 6805: 6770: 6725: 6700: 6692: 6653: 6570: 6562: 6517: 6492: 6439: 6431: 6375: 6319: 6311: 6270: 6245: 6237: 6211:Nature 6188: 6160: 6150: 6121: 6113: 6083: 6049: 6039: 5986: 5976: 5950: 5869: 5823: 5815: 5789:Nature 5765: 5757: 5710: 5702: 5642: 5632: 5498: 5381:832961 5379: 5326: 5318: 5257: 5247: 5239: 5182: 5174: 5127: 5119: 5093:Nature 5068: 5021: 4859: 4831:Quanta 4792: 4684: 4676: 4629: 4431: 4421: 4382: 4335: 4325: 4278:4 June 4221: 4213: 4080: 4055: 4032:832961 4030: 4022: 3965: 3955: 3906: 3881: 3858: 3822: 3764: 3754: 3672: 3629: 3621: 3582:: 114. 3545: 3539:97–109 3512: 3496:length 3477: 3377: 3344: 3317: 3285: 3210: 3166: 3136: 3107: 3074: 3066: 2992: 2984: 2956:Nature 2935: 2927: 2888: 2686:, Sir 2669:ad hoc 2212:action 2055: 1896: 1777: 1749: 1012:photon 951:Zeeman 946:Wigner 896:Planck 866:Landau 851:Jordan 502:Matrix 432:Popper 64:, the 8114:Stars 7619:Local 7561:Pauli 7551:Dirac 6811:S2CID 6698:S2CID 6568:S2CID 6461:. 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