Double-slit experiment: Wikis


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Quantum mechanics
\Delta x\, \Delta p \ge \frac{\hbar}{2}
Uncertainty principle
Introduction · Mathematical formulation
Double-slit experiment
Davisson–Germer experiment
Stern–Gerlach experiment
Bell's inequality experiment
Popper's experiment
Schrödinger's cat
Elitzur–Vaidman bomb-tester
Quantum eraser
Same double-slit assembly (0.7mm between slits); in top image, one slit is closed. Note that the single-slit diffraction pattern — the faint spots on either side of the main band — is also seen in the double-slit image, but at twice the intensity and with the addition of many smaller interference fringes.
A laboratory double-slit assembly; distance between top posts approximately one inch.
Single-slit diffraction pattern
Double-slit diffraction and interference pattern

In quantum mechanics, the double-slit experiment (often referred to as Young's experiment) demonstrates the inseparability of the wave and particle natures of light and other quantum particles. A coherent light source illuminates a thin plate with two parallel slits cut in it, and the light passing through the slits strikes a screen behind them. The wave nature of light causes the light waves passing through both slits to interfere, creating an interference pattern of bright and dark bands on the screen. However, at the screen, the light is always found to be absorbed as though it were made of discrete particles, called photons.[1][2]

If the light travels from the source to the screen as particles, then on the basis of a classical reasoning the number that strike any particular point on the screen is expected to be equal to the sum of those that go through the left slit and those that go through the right slit. In other words, according to classical particle physics the brightness at any point should be the sum of the brightness when the right slit is blocked and the brightness when the left slit is blocked. However, it is found that unblocking both slits makes some points on the screen brighter, and other points darker. This can only be explained by the alternately additive and subtractive interference of waves, not the exclusively additive nature of particles, so we know that light must have some particle-wave duality.[3]

Any modification of the apparatus that can determine which slit a photon passes through destroys the interference pattern,[3] illustrating the complementarity principle; that the light can demonstrate both particle and wave characteristics, but not both at the same time.[4][5][6]. However, an experiment performed in 1987[7] produced results that demonstrated that which-path information could be obtained without destroying the possibility of interference. This showed the effect of measurements that disturbed the particles in transit to a lesser degree and thereby influenced the interference pattern only to a comparable extent.

The double slit experiment can also be performed (using different apparatus) with particles of matter such as electrons with the same results, demonstrating that they also show particle-wave duality.



Normally, when only one slit is open, the pattern on the screen is a diffraction pattern, a fairly narrow central band with dimmer bands parallel to it on each side. (See the top photograph to the right.) When both slits are open, the pattern displayed becomes very much more detailed and at least four times as wide. (See the bottom photograph to the right.) These facts were elucidated by Thomas Young in a paper entitled "Experiments and Calculations Relative to Physical Optics," published in 1803. To a very high degree of success, these results could be explained by the method of Huygens–Fresnel principle that based its calculations on the hypothesis that light consists of waves propagated through some medium. However, discovery of the photoelectric effect made it necessary to go beyond classical physics and take the quantum nature of light and/or matter into account.

It is a widespread misunderstanding that, when two slits are open but a detector is added to the experiment to determine which slit a photon has passed through, then the interference pattern no longer forms and the experimental apparatus yields two simple patterns, one from each slit, superposed without interference. Such a result would be obtained only if the results of two experiments were superposed in which either one or the other slit is closed. However, there are many other methods to determine whether a photon passed through a slit, for instance by placing an atom at the position of each slit and monitoring whether one of these atoms is influenced by a photon passing it. In general in such experiments the interference pattern will be changed but not be completely wiped out. Interesting experiments of this latter kind have been performed with photons[7] and with neutrons.[8]

Restriction to the two experiments in which either both slits are open or one slit is closed has given rise to the idea of wave-particle complementarity according to which a microscopic object (photon, electron, etc.) would manifest itself as a particle in the which-way experiment but as a wave in the interference experiment. This idea has been felt to be counterintuitive by those not being content with an instrumentalist interpretation of quantum mechanics in which that theory is accepted as just describing phenomena without providing explanations.

The most baffling part of this experiment comes when only one photon at a time is fired at the barrier with both slits open. The pattern of interference remains the same, as can be seen if many photons are emitted one at a time and recorded on the same sheet of photographic film. The clear implication is that something with a wavelike nature passes simultaneously through both slits and interferes with itself — even though there is only one photon present. (The experiment works with electrons, atoms, and even some molecules too.)

Richard Feynman was fond of saying that all of quantum mechanics can be gleaned from carefully thinking through the implications of this single experiment.[9]

The underpinning of this experiment

Circles of equal sizes are drawn along an existing wavefront and their tangents are smoothed to form the next predicted wavefront location.

Christiaan Huygens understood the basic idea of how light propagates and how to predict its path through a physical apparatus. He understood that a light source emits a series of waves comparable to the way that water waves spread out from something like a fishing float that is jiggled up and down and bobs on the water surface. He said that the way to predict where the next wave front will be found is to generate a series of concentric circles on a sufficiently large number of points on a known wave front and then draw a curve that will pass tangent to all the resulting circles out in front of the known wave front. The diagram given here shows what happens when a flat wave front is extended in this manner, and what happens when a curved wave front is extended in the same way. Augustin Fresnel (1788-1827) based his proof that the wave nature of light does not contradict the observed fact that light propagates in a straight line in homogeneous media on Huygens' work, and also based himself on Huygens' ideas to give a complete account of diffraction and interference phenomena known at his time.[10] See the article Huygens–Fresnel principle for more information.

Note that there are four gaps between crests hitting the screen -- places that will be darker in the resultant diffraction pattern visible to an observer

The second drawing shows what happens when a flat wave front encounters a slit in a wall. Following the same principle elucidated above, it is clear that the new wave front will "bulge out" from the slit and light will be experienced as having diverged around the edges of the slit. The result is called a diffraction pattern.

The third drawing shows the explanation for interference based on the classical idea of a single wave front that represents all the light energy emitted by a source at one moment. Since photons diverge beyond the barrier wall, the distance between parts of any pattern they form on the target wall increases as the distance they have to travel increases, a fact that is well known from everyday experience with things like automobile headlights whose beams are not parallel. But decreasing the distance between slits will also increase the distance between fringes (colored bands such as the sixteen shown in the second photograph above). Increasing the wavelength will also increase the distance between fringes as long as the slits are wide enough to permit the passage of light of that wavelength. Slits that are very wide in comparison to the frequency of the photons involved (e.g., two ordinary windows in a single wall) will permit light to appear to go "straight through."

J is the distance between fringes. J = Dλ/B "D" = dist. S2 to F, λ = wavelength, B = dist. b to c[11]

When light came to be understood as the result of electrons falling from higher energy orbits to lower energy orbits, the light that is delivered to some surface in any short interval of time came to be understood as ordinarily representing the arrival of very many photons, each with its own wave front. In understanding what actually happens in the two-slit experiment it became important to find out what happens when photons are emitted one by one.[10]

When it became possible to perform that experiment, it became apparent that a single photon has its own wave front that passes through both slits, and that the single photon will show up on the detector screen according to the net probability values resulting from the co-incidence of the two probability waves coming by way of the two slits. When a great number of photons are sent through the apparatus one by one and recorded on photographic film, the same interference pattern emerges that had been seen before when many photons were being emitted at the same time. The low intensity double-slit experiment was first performed by Taylor in 1909,[12] by reducing the level of incident light until on average only one photon was being transmitted at a time.[10] Note that it is the probabilities of photons appearing at various points along the detection screen that add or cancel. So if there is a cancellation of waves at some point that does not mean that a photon disappears; it means that the probability of a photon's appearing at that point will disappear, and the probability that it will appear somewhere else increases.

Importance to physics

Sketch of the layout of a classical optical double-slit experiment. Note that lasers are commonly used today and replace the incoherent source of light and the top pinhole.

Although the double-slit experiment is now often referred to in the context of quantum mechanics, it is generally thought to have been first performed by the English scientist Thomas Young in the year 1801 in an attempt to resolve the question of whether light was composed of particles (Newton's "corpuscular" theory), or rather consisted of waves traveling through some ether, just as sound waves travel in air. The interference patterns observed in the experiment seemed to discredit the corpuscular theory, and the wave theory of light remained well accepted until the early 20th century, when evidence began to accumulate which seemed instead to confirm the particle theory of light.[13]

The double-slit experiment, and its variations, then became a classic thought experiment for its clarity in expressing the central puzzles of quantum mechanics.

It was shown experimentally in 1972 that in a Young 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.[14] The experimental conditions were such that the photon density in the system was much less than unity.

A Young double slit experiment was not performed with anything other than light until 1961, when Clauss Jönsson of the University of Tübingen performed it with electrons[15][16], and not until 1974 in the form of "one electron at a time", in a laboratory at the University of Milan, by researchers led by Pier Giorgio Merli, of LAMEL-CNR Bologna.

The results of the 1974 experiment were published and even made into a short film, but did not receive wide attention. The experiment was repeated in 1989 by Tonomura et al. at Hitachi in Japan. Their equipment was better, reflecting 15 years of advances in electronics and a dedicated development effort by the Hitachi team. Their methodology was more precise and elegant, and their results agreed with the results of Merli's team. Although Tonomura asserted that the Italian experiment had not detected electrons one at a time—a key to demonstrating the wave-particle paradox—single electron detection is clearly visible in the photos and film taken by Merli and his group.[17]

In September 2002, the double-slit experiment of Claus Jönsson was voted "the most beautiful experiment" by readers of Physics World.[18]

Importance to philosophy

The double-slit experiment has been of great interest to philosophers, because the quantum mechanical behavior that it reveals has forced them to reevaluate their ideas about classical concepts such as "particles",[11] "waves", "location", and "movement from one place to another".

In contrast to the way of conceptualizing the macroscopic world of everyday experience, attempting to describe the motion of a single photon is problematic. As Philipp Frank observes, investigating the motion of single particles through a single slit can obtain a description of the pattern of photon strikes on a target screen. However, "the pattern of fringes for two slits is not the superposition of the two patterns for single slits. Hence, there is no law of motion that would determine the trajectory of a single photon and allow us to derive the observed facts that occur when photons pass two slits."[11] Experience in the micro world of sub-atomic particles forces us to reconceptualize some of our most commonplace ideas.

One of the most striking consequences of the new science is that it is not in agreement with the belief of Laplace that an omniscient entity, knowing the initial positions and velocities of all particles in the universe at one time, could predict their positions at any future time. (To paraphrase Laplace's idea, the positions and velocities of all things at any given time depend absolutely on their previous positions and velocities and the absolute laws that govern physical interactions.) Laplace believed that such particles would follow the laws of motion discovered by Newton, but twentieth century physics made it clear that the motions of sub-atomic particles and even some small atoms cannot be predicted by using the laws of Newtonian physics.[11] For instance, most of the orbits for electrons moving around atomic nuclei that are permitted by Newtonian physics are excluded by the new physics. And it is not even clear what the "movement" of a particle such as a photon may be when it is not clear that it "goes through" either one slit or the other, but it is clear that the probability of its arrival at various points on the target screen is a function of its wavelength and of the distance between the slits. Whereas Laplace would expect an omniscient mind to be able to predict with absolute confidence the arrival of a photon at some specific point on the target screen, it turns out that the particle may arrive at one of a great number of points, but that the percentage of particles that arrive at each of such points is determined by the laws of the new physics.

Results observed

Thomas Young's sketch of two-slit interference, based on his observations of water waves.[19]

The bright bands observed on the screen happen when the light has interfered constructively—where a crest of a wave meets a crest from another wave. The dark regions show destructive interference—a crest meets a trough. Constructive interference occurs when

\frac{n\lambda}{d} = \frac{x}{L} \quad\Leftrightarrow\quad{n}{\lambda}=\frac{xd}{L}\;,


λ is the wavelength of the light,
d is the separation of the slits, the distance between A and B in the diagram to the right
n is the order of maximum observed (central maximum is n = 0),
x is the distance between the bands of light and the central maximum (also called fringe distance), and
L is the distance from the slits to the screen centerpoint.

This is only an approximation and depends on certain conditions.[20]

It is possible to work out the wavelength of light using this equation and the above apparatus. If d and L are known and x is observed, then λ can be easily calculated.

A detailed treatment of the mathematics of double-slit interference in the context of quantum mechanics is given in the article on Englert-Greenberger duality.

Shape of interference fringes

The theoretical shapes of the interference fringes observed in a Young double slit experiment are straight lines, which is easily proved.

In case two pinholes are used instead of slits, as in Young's original experiment, hyperbolic fringes are observed.

If the two sources are placed on a line perpendicular to the screen, the shape of the interference fringes is circular as the individual paths travelled by light from the two sources are always equal for a given fringe. This can be done in simpler way by placing a mirror parallel to a screen at a distance and a source of light just above the mirror. (Note the extra phase difference of π due to reflection at the interface of a denser medium).

Quantum version of experiment

The wavefronts resulting from two pinholes.

By the 1920s, various other experiments (such as the photoelectric effect) had demonstrated that light interacts with matter only in discrete, "quantum"-sized packets called photons.[21]

If sunlight is replaced with a light source that is capable of producing just one photon at a time, or if the beam of light is attenuated to the point that only one photon at a time can get through, as G. I. Taylor did as early as 1909, one photon at a time can pass through the apparatus with the identical result of interference fringes.[22] It should be noted, however, that strictly speaking, no experiment with a true single-photon (Fock state) source was done until 1986[2] .

If either slit is covered, the individual photons hitting the screen, over time, create an ordinary diffraction pattern. But if both slits are left open, the pattern of photons hitting the screen, over time, again becomes a series of light and dark fringes. This result seems to both confirm and contradict the wave theory. If light were not to behave like a wave, there would be no interference pattern. On the other hand, if light were actually a wave then light energy would not arrive in discrete quantities (quanta) and would be spread over more space the farther the detector screen was placed from the screen with the slits in it.

There is a variation of the double-slit experiment in which detectors are placed in either or both of the two slits in an attempt to determine which slit the photon passes through on its way to the screen. Placing a detector even in just one of the slits will result in the disappearance of the interference pattern. The detection of a photon involves a physical interaction between the photon and the detector of the sort that physically changes the detector. (If nothing changed in the detector, it would not detect anything.) If two photons of the same frequency were emitted at the same time they would be coherent. If they went through two unobstructed slits then they would remain coherent and arriving at the screen at the same time but laterally displaced from each other they would exhibit interference. However, if one or both of them were to encounter a detector, time could be required for each to interact with its detector and they would most likely fall out of step with each other—that is, they would decohere. They would then arrive at the screen at slightly different times and could not interfere because the first to arrive would have already interacted with the screen before the second got there. If only one photon is involved, it must be detected at one or the other detector, and its continued path goes forward only from the slit where it was detected.[9]

Copenhagen interpretation

The Copenhagen interpretation is a consensus among some of the pioneers in the field of quantum mechanics that it is undesirable to posit anything that goes beyond the mathematical formulae and the kinds of physical apparatus and reactions that enable us to gain some knowledge of what goes on at the atomic scale. One of the mathematical constructs that enables experimenters to predict very accurately certain experimental results is sometimes called a probability wave. In its mathematical form it is analogous to the description of a physical wave, but its "crests" and "troughs" indicate levels of probability for the occurrence of certain phenomena (e.g., a spark of light at a certain point on a detector screen) that can be observed in the macro world of ordinary human experience.

The probability "wave" can be said to "pass through space" because the probability values that one can compute from its mathematical representation are dependent on time. One cannot speak of the location of any particle such as photon between the time it is emitted and the time it is detected simply because in order to say that something is located somewhere at a certain time one has to detect it (of course, since photons travel at a known speed (the speed of light) at any given time (stated to Planck accuracy) you can calculate (to within Planck distance) where the 'probability' field is 'centered', but until the particle is detected, you can not be certain 'exactly' where it is). The requirement for the eventual appearance of an interference pattern is that particles be emitted, and that there be a screen with at least two slits between the emitter and the detection screen. Experiments observe nothing whatsoever between the time of emission of the particle and its arrival at the detection screen. However, it is essential that both slits be an equal distance from the center line, and that they be within a certain maximum distance of each other that is related to the wavelength of the particle being emitted. If a ray tracing is then made as if a light wave as understood in classical physics is wide enough to encounter both slits and passes through both of them, then that ray tracing will accurately predict the appearance of maxima and minima on the detector screen when many particles pass through the apparatus and gradually "paint" the expected interference pattern.

Note that the existence of any such particle is known only at the point of emission and the point of detection. If by "object A exists" is meant "object A is detected at point x,y,z,t," then this object "exists" only at the point of emission and the point of detection. In between times it is completely out of sensible interaction with the things of our universe, out of sensible interaction with the macro world. What is going on in the apparatus is something that is not known.

It is perhaps not so astounding that one knows nothing about what a light particle is doing between the time it is emitted from the sun and the time it triggers a reaction in one's body, but the remarkable consequence discovered by this experiment is that anything that one does to try to locate a photon between the emitter and the detection screen will change the results of the experiment in a way that everyday experience would not lead one to expect. If, for instance, any device is used in any way that can determine whether a particle has passed through one slit or the other, the interference pattern formerly produced will then disappear.[citation needed]

Reason, as applied to the events of our ordinary macro experience, tells us that a particle must pass through one slit or the other. The experiment tells us that there must be at least two slits to produce an interference pattern, and that anything that locates the particle before it hits the screen will destroy the interference pattern. Recent experiments have tried to identify which of the two slits a particle is coming out of on its way to the detection screen. Doing so will also prevent interference. Even less in line with the expectations of human scale interactions with nature, if the information about which slit a given particle came through is "erased" before a photon has time to interact with the detector screen, interference will be restored. (See Quantum eraser experiment.)

Path Integral Formulation

The Copenhagen interpretation is similar to the path integral formulation of quantum mechanics provided by Feynman. Feynman stressed that his formulation is merely a mathematical description, not an attempt to describe a real process that we cannot measure. In the path integral formulation, the probability distribution of the outcome is the superposition of every possible classical path through space-time to get from point A to point B. In the double-slit experiment, point A might be the emitter, and point B the screen upon which the interference pattern appears, and the interference fringes are calculated as the sum of every possible path, including paths through both slits at once, to get from A to B. When a detector is placed at one of the slits, the situation changes, and we now have a different point B. Point B is now at the detector, and a new path proceeds from the detector to the screen. In this eventuality there is only empty space between (B =) A' and the new terminus B', no double slit in the way, and so an interference pattern no longer appears.

Relational interpretation

According to the relational interpretation of quantum mechanics, first proposed by Carlo Rovelli[23], observations such as those in the double-slit experiment result specifically from the interaction between the observer and the object being observed, 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 interaction includes information about the electron's position. This partially constrains the particle's eventual location at the screen. If it is observed not at a particular slit but rather at the screen, then there is no "which slit" 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 no different than if each individual electron had passed through both slits. It has also been suggested that space and distance themselves are relational, and that an electron can appear to be in "two places at once" — e.g., at both slits — because its spatial relations to particular points on the screen remain identical from both slit locations.[24]

When observed emission by emission

Electron buildup over time

Regardless of whether it is an electron, a proton, or something else existing on what is considered a "quantum" scale, where it will arrive at the screen is highly determinate (in that quantum mechanics predicts accurately the probability that it will arrive at any point on the screen). However, in what sequence members of a series of singly emitted things (e.g., electrons) build up the final distribution pattern is completely unpredictable. The experimental facts are so highly reproducible that there is virtually no argument about them, but the appearance of there being an uncaused event (because of the unpredictability of the sequencing) has aroused a great deal of cognitive dissonance and attempts to account for the sequencing by reference to supposed "additional variables."

For example, when electrons are fired at the target screen in bursts, it is easy to account for the interference pattern that results by assuming that electrons that travel in pairs are interfering with each other because they arrive at the screen at the same time, but when a laboratory apparatus was developed that could reliably fire single electrons at the screen[25], the emergence of an interference pattern suggested that each electron was interfering with itself; and, therefore, in some sense the electron had to be going through both slits.[26] For something that most people continue to imagine to be an unimaginably small particle to be able to interfere with itself would suggest that this "sub-atomic particle" was in two places at once, but that idea is strongly at odds with the truism, "You cannot be in two places at the same time," (see principle of contradiction). It was easier to conceptualize the electron as a wave than to accept another, more disturbing implication (from the point-of-view of our everyday notions of reality): that quantum objects are able to exist and behave in ways that defy classical interpretation.

However, when one electron (proton, photon, or whatever) is fired at a time, it also becomes possible to detect the point on the screen at which it arrives—and another result was demonstrated that could not easily be squared with experience of the macro world, the world of everyday experience.

In everyday experience we are accustomed to a seemingly analogous result. If one tests a firearm by locking it in a gun mount and firing several rounds at a target, a scatter pattern of bullet holes will appear in the target. We know from long experience that a poorly made gun firing poorly made ammunition will scatter shots fairly widely. We can learn and understand how flight path deviations are caused; more exacting construction of both firearms and ammunition leads to tighter and tighter patterns of bullet holes. But that is not what happens in the new double-slit experiment.

Returning again to electrons, when electrons are fired one at a time through a double-slit apparatus they do not cluster around two single points directly on lines between the emitter and the two slits, but instead one by one they fill in the same old interference pattern with which we have now become quite familiar. However, they do not arrive at the screen in any predictable order. In other words, knowing where all the previous electrons appeared on the screen and in what order tells us nothing about where the next electron will hit (although we can plainly work out the gross probability of where it will hit).[27]

The electrons (and the same applies to photons and to anything of atomic dimensions used) arrive at the screen in an unpredictable and arguably causeless random sequence, and the appearance of a causeless selection event in a highly orderly and predictable formulation of the by now familiar interference pattern has caused many people to try to find additional determinants in the system which, were they to become known, would account for why each impact with the target appears.[28]

Recent studies have revealed that interference is not restricted solely to elementary particles such as protons, neutrons, and electrons. Specifically, it has been shown that large molecular structures like fullerene (C60) also produce interference patterns.[29]

See also


  1. ^ Feynman, Richard P. (1965). The Feynman Lectures on Physics, Vol. 3. USA: Addison-Wesley. p. 1–8. ISBN 0201021188P. 
  2. ^ Darling, David (2007). "Wave - Particle Duality". The Internet Encyclopedia of Science. The Worlds of David Darling. Retrieved 2008-10-18. 
  3. ^ a b Feynman, Richard P.; Robert Leighton, Matthew Sands (1965). The Feynman Lectures on Physics. Massachusetts, USA: Addison-Wesley. pp. 1–1 to 1–9. ISBN 0201021188P. 
  4. ^ Harrison, David (2002). "Complementarity and the Copenhagen Interpretation of Quantum Mechanics". UPSCALE. Dept. of Physics, U. of Toronto. Retrieved 2008-06-21. 
  5. ^ Cassidy, David (2008). "Quantum Mechanics 1925-1927: Triumph of the Copenhagen Interpretation". Werner Heisenberg. American Institute of Physics. Retrieved 2008-06-21. 
  6. ^ Boscá Díaz-Pintado, María C. (29-31 March 2007). "Updating the wave-particle duality". 15th UK and European Meeting on the Foundations of Physics. Leeds, UK. Retrieved 2008-06-21. 
  7. ^ a b P. Mittelstaedt; A. Prieur, R. Schieder (1987). "Unsharp particle-wave duality in a photon split-beam experiment". Foundations of Physics 17: 891=903. doi:10.1007/BF00734319. 
  8. ^ J. Summhammer; H. Rauch, D. Tuppinger (1987). Phys. Rev. A 36: 4447. doi:10.1103/PhysRevA.36.4447. 
  9. ^ a b Greene, Brian (1999). The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. New York: W.W. Norton. pp. 97–109. ISBN 0393046885. 
  10. ^ a b c de Broglie, Louis (1953). The revolution in physics; a non-mathematical survey of quanta. Translated by Ralph W. Niemeyer. New York: Noonday Press. pp. 47, 117, 178–186. 
  11. ^ a b c d Philipp Frank, Philosophy of Science, p. 200f.
  12. ^ Sir Geoffrey Ingram Taylor, "Interference Fringes with Feeble Light", Proc. Cam. phil. Soc. 15, 114 (1909).
  13. ^ Albert Einstein, Essays in Science, Philosophical Library (1934), p. 100
  14. ^ Sillitto R and Wykes, C. 1972, Phys. Lett., An interference experiment with light beams modulated in anti-phase by an electro-optic shutter, 39A, 333-4
  15. ^ Jönsson C,(1961) Zeitschrift für Physik, 161:454-474
  16. ^ Jönsson C (1974). Electron diffraction at multiple slits. American Journal of Physics, 4:4-11.
  17. ^ See for more information and photographs (at the bottom of the article). See also Tonomura's Video clip 1 [1].
  18. ^ "The most beautiful experiment". Physics World 2002.
  19. ^ Rothman, Tony (2003). Everything's Relative and Other Fables in Science and Technology. New Jersey: Wiley. ISBN 0471202576. 
  20. ^ For a more complete discussion, with diagrams and photographs, see Arnold L Reimann, Physics, chapter 38.
  21. ^ Brian Greene, The Elegant Universe, pp. 94-97, recapitulates the history of these experiments.
  22. ^ A.P. French and Edwin F. Taylor, An Introduction to Quantum Mechanics, p. 91f.
  23. ^ Rovelli, Carlo (1996). arXiv:quant-ph/9609002 "Relational Quantum Mechanics". International Journal of Theoretical Physics 35: 1637-1678. arXiv:quant-ph/9609002. 
  24. ^ Filk, Thomas (2006). [ "Relational Interpretation of the Wave Function and a Possible Way Around Bell’s Theorem"]. International Journal of Theoretical Physics 45: 1205-1219. 
  25. ^ O Donati G F Missiroli G Pozzi May 1973 An Experiment on Electron Interference American Journal of Physics 41 639-644
  26. ^ Brian Greene, The Elegant Universe, p. 110
  27. ^ Brian Greene, The Elegant Universe, p. 104, pp. 109-114
  28. ^ Greene, Brian (2004). The Fabric of the Cosmos: Space, Time, and the Texture of Reality. Knopf. pp. 204–213. ISBN 0375412883. 
  29. ^ Nairz O, Arndt M, and Zeilinger A. Quantum interference experiments with large molecules. American Journal of Physics, 2003; 71:319-325.

Further reading

  • Feynman, Richard P. (1988). QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 0-691-02417-0. 
  • Frank, Philipp (1957). Philosophy of Science. Prentice-Hall. 
  • Gribbin, John (1999). Q is for Quantum: Particle Physics from A to Z. Weidenfeld & Nicolson. ISBN 0-7538-0685-1. 
  • French, A.F. and Edwin F. Taylor (1978). An Introduction to Quantum Physics. Norton. ISBN 0-393-09106-6. 
  • Greene, Brian (2000). The Elegant Universe. Vintage. ISBN 0-375-70811-1. 
  • Greene, Brian (2005). The Fabric of the Cosmos. Vintage. ISBN 0-375-72720-5. 
  • Hey, Tony (2003). The New Quantum Universe. Cambridge University Press. ISBN 0-5215-6457-3. 
  • Sears, Francis Weston (1949). Optics. Addison Wesley. 
  • Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed. ed.). W. H. Freeman. ISBN 0-7167-0810-8. 
  • Al-Khalili, Jim (2003). Quantum: A Guide for the Perplexed. Weidenfeld and Nicholson, London. ISBN 0-297-84305-2. 

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