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| {{Redirect|Slit experiment||Diffraction}}
| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you want to check out more info regarding [http://www.youtube.com/watch?v=90z1mmiwNS8 Best Dentists in DC] review the website. |
| {{Use dmy dates|date=July 2013}}
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| {{Quantum mechanics|cTopic=Experiments}}
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| The '''double-slit experiment''' is a demonstration that matter and energy can [[Introduction to quantum mechanics#Wave–particle duality|display characteristics]] of both [[classical physics|classically]] defined [[wave]]s and [[particle]]s; moreover, it displays the fundamentally probabilistic nature of [[quantum mechanics|quantum mechanical]] phenomena. The experiment belongs to a general class of "double path" experiments, in which a wave is split into two separate waves that later combine back into a single wave. Changes in the path lengths of both waves result in a phase shift, creating an [[interference pattern]]. Another version is the [[Mach–Zehnder interferometer]], which splits the beam with a mirror.
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| This experiment is often referred to as [[Young's interference experiment|'''Young's experiment''']] and 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.<ref name=Robinson2006>{{cite book|last=Robinson|first=Andrew|title=The Last Man Who Knew Everything|year=2006|publisher=Pi Press|location=New York, NY|isbn=0-13-134304-1|pages=123–124}}</ref>
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| In the basic version of this experiment, a [[Coherence (physics)|coherent light source]] such as a [[laser]] beam illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate.<ref name="Lederman">{{cite book | |
| | last = Lederman
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| | first = Leon M.
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| | coauthors = Christopher T. Hill
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| | title = Quantum Physics for Poets
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| | publisher = Prometheus Books
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| | year = 2011
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| | location = US
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| | pages = 102–111
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| | url = http://books.google.com/books?id=qY_yOwHg_WYC&pg=PA102
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| | doi =
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| | isbn = 1616142812}}</ref><ref name="Feynman">{{cite book
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| | last = Feynman
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| | first = Richard P.
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| | authorlink =
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| | coauthors = Robert B. Leighton, Matthew Sands
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| | title = The Feynman Lectures on Physics, Vol. 3
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| | publisher = Addison-Wesley
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| | year = 1965
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| | location = US
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| | pages = 1.1-1.8
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| | isbn = 0201021188}}</ref> The wave nature of light causes the light waves passing through the two slits to [[interference (wave propagation)|interfere]], producing bright and dark bands on the screen—a result that would not be expected if light consisted of classical particles (i.e., small chunks of matter).<ref name="Lederman" /><ref>Feynman, 1965, p. 1.5</ref> However, the light is always found to be absorbed at the screen at discrete points, as individual particles (not waves), the interference pattern appearing via the varying density of these particle hits on the screen.<ref>{{cite web
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| | last = Darling
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| | first = David
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| | authorlink = David Darling (astronomer)
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| | title = Wave–Particle Duality
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| | work = The Internet Encyclopedia of Science
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| | publisher = The Worlds of David Darling
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| | year = 2007
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| | url = http://www.daviddarling.info/encyclopedia/W/wave-particle_duality.html
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| | doi =
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| | accessdate = 2008-10-18}}</ref> Furthermore, versions of the experiment that include particle detectors at the slits find that each detected photon of light passes through one slit (as would a classical particle), but not through both slits (as would a wave).<ref>Feynman, 1965, p. 1.7</ref><ref>[http://books.google.com/books?id=qY_yOwHg_WYC&pg=PA109 Lederman, 2011, p. 109]</ref><ref name=" Müller-Kirsten">"''...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...''" {{cite book
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| | last = Müller-Kirsten
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| | first = H. J. W.
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| | title = Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral
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| | publisher = World Scientific
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| | year = 2006
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| | location = US
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| | pages = 14
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| | url = http://books.google.com/books?id=p1_Z81Le58MC&pg=PA14
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| | doi =
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| | id =
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| | isbn = 9812566910}}</ref><ref name="Plotnitsky">{{cite book
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| | last = Plotnitsky
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| | first = Arkady
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| | title = Niels Bohr and Complementarity: An Introduction
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| | publisher = Springer
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| | year = 2012
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| | location = US
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| | pages = 75–76
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| | url = http://books.google.com/books?id=dmdUp97S4AYC&pg=PA75
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| | isbn = 1461445175}}</ref><ref name="Rae">"''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 experiment.''" {{cite book
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| | last = Rae
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| | first = Alastair I. M.
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| | title = Quantum Physics: Illusion Or Reality?
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| | publisher = Cambridge University Press
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| | year = 2004
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| | location = UK
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| | pages = 9–10
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| | url = http://books.google.com/books?id=FVtMqukQ6g4C&pg=PA9
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| | doi =
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| | isbn = 1139455273}}</ref> These results demonstrate the principle of [[wave–particle duality]].
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| Other atomic-scale entities such as [[electrons]] are found to exhibit the same behavior when fired toward a double slit.<ref name="Feynman" /> Additionally, the detection of individual discrete impacts is observed to be inherently probabilistic, which is inexplicable using [[classical mechanics]].<ref name="Feynman" />
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| In 1999, [[buckyball]] [[molecule]]s (each of which comprises 60 carbon atoms), were found to exhibit wave-like interference.<ref name="buckyballs">[http://www.newscientist.com/article/mg20627596.100-quantum-wonders-corpuscles-and-buckyballs.html New Scientist: Quantum wonders: Corpuscles and buckyballs, 2010] (Introduction, subscription needed for full text, quoted in full in [http://postbiota.org/pipermail/tt/2010-May/007336.html])</ref><ref>[http://www.nature.com/nature/journal/v401/n6754/abs/401680a0.html Nature: Wave–particle duality of C<sub>60</sub> molecules, 14 October 1999]. Abstract, subscription needed for full text</ref> A buckyball is large enough (diameter about 0.7 [[Nanometre|nm]], nearly half a million times larger than a proton) to be seen under an [[electron microscope]].
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| In 2013, the double-slit experiment was successfully performed with molecules that each comprised 810 atoms (whose total mass was over 10,000 [[atomic mass units]]).<ref>"[https://medium.com/the-physics-arxiv-blog/462c39db8e7b Physicists Smash Record For Wave-Particle Duality]"</ref><ref>{{cite journal
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| | last =Eibenberger
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| | first =Sandra
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| | coauthors =et al.
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| | title =Matter-wave interference with particles selected from a molecular library with masses exceeding 10000 amu
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| | journal =[[Physical Chemistry Chemical Physics]]
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| | volume =15
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| | pages =pp. 14696–14700
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| | year =2013
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| | url =http://arxiv.org/abs/1310.8343
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| | doi =10.1039/C3CP51500A
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| }}</ref>
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| ==Overview==
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| [[Image:Single slit and double slit2.jpg|right|350px|thumb|Same double-slit assembly (0.7mm between slits); in top image, one slit is closed. In the single-slit image, a [[diffraction pattern]] (the faint spots on either side of the main band) forms due to the nonzero width of the slit. A diffraction pattern is also seen in the double-slit image, but at twice the intensity and with the addition of many smaller interference fringes.]]
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| If light consisted strictly of ordinary or [[Classical mechanics|classical]] 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 is a [[diffraction pattern]] in which the light is spread out. The smaller the slit, the greater the angle of spread. The top portion of the image on the right 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 refined apparatus. See [[Diffraction]].
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| Similarly, if light consisted strictly of classical particles and we illuminated two parallel slits, the expected pattern on the screen would simply be the sum of the two single-slit patterns. In reality, however, the pattern changes to one with a series of light and dark bands (See the bottom photograph to the right.) When [[Thomas Young (scientist)|Thomas Young]] (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 [[wavefront]]s.<ref name="Feynman" >{{cite book | last = Feynman | first = Richard P. | authorlink = Richard Feynman | coauthors = Robert Leighton, Matthew Sands | title = [[The Feynman Lectures on Physics]], Volume III | publisher = Addison-Wesley | year = 1965 | location = Massachusetts, USA
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| | pages = 1–1 to 1–9 | isbn = 0-201-02118-8P}}</ref> Young's experiment, performed in the early 1800s, played a vital part in the acceptance of the wave theory of light, vanquishing the [[corpuscular theory of light]] proposed by [[Isaac Newton]], which had been the accepted model of light propagation in the 17th and 18th centuries. However, the later discovery of the [[photoelectric effect]] 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 the [[Quantum mechanics|quantum]] nature of light into account.
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| The double-slit experiment (and its variations), conducted with individual particles, has become a classic [[thought experiment]] 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, [[Richard Feynman]] called it "a phenomenon which is impossible […] to explain in any [[classical mechanics|classical way]], and which has in it the heart of quantum mechanics. In reality, it contains the ''only'' mystery [of quantum mechanics]."<ref name="Feynman" /> Feynman was fond of saying that all of quantum mechanics can be gleaned from carefully thinking through the implications of this single experiment.<ref name="Greene_1999">{{cite book|last =Greene|first =Brian|authorlink =Brian Greene|title =[[The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory]] |publisher =W.W. Norton|location =New York|year =1999|pages = 97–109|isbn =0-393-04688-5 }}</ref> [[Časlav Brukner]] and [[Anton Zeilinger]] have succinctly expressed this limitation as follows:
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| <blockquote>The observer can decide whether or not to put detectors into the interfering path. That way, by deciding whether or not to determine the path through the two-slit experiment, he can decide which property can become reality. If he chooses not to put the detectors there, then the interference pattern will become reality; if he does put the detectors there, then the beam path will become reality. Yet, most importantly, the observer has no influence on the specific element of the world which becomes reality. Specifically, if he chooses to determine the path, he has no influence whatsoever which of the two paths, the left one or the right one, Nature will tell him is the one where the particle is found. Likewise, if he chooses to observe the interference pattern he has no influence whatsoever where in the observation plane he will observe a specific particle. Both outcomes are completely random.<ref name=Brukner-Zeilinger>{{cite journal | author = Brukner C, Zeilinger A | year = 2002 | title = Young’s experiment and the finiteness of information | journal = Philosophical Transactions of the Royal Society | volume = 360 | pages = 1061–1069 }}</ref></blockquote>
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| The [[Englert–Greenberger duality relation]] provides a detailed treatment of the mathematics of double-slit interference in the context of quantum mechanics.
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| A low-intensity double-slit experiment was first performed by [[Geoffrey Ingram Taylor|G. Taylor]] in 1909,<ref>[[Geoffrey Ingram Taylor|Sir Geoffrey Ingram Taylor]], "Interference Fringes with Feeble Light", ''Proc. Cam. Phil. Soc.'' 15, 114 (1909).</ref> by reducing the level of incident light until photon emission/absorption events were mostly nonoverlapping.
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| {{Anchor|Claus Jönsson}}
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| A double-slit experiment was not performed with anything other than light until 1961, when [[Claus Jönsson]] of the [[University of Tübingen]] performed it with electrons.<ref>Jönsson C,(1961) ''Zeitschrift für Physik'', '''161''':454–474 {{doi|10.1007/BF01342460}}</ref><ref>Jönsson C (1974). Electron diffraction at multiple slits. ''American Journal of Physics'', '''42''':4–11 {{doi|10.1119/1.1987592}}.</ref> In 2002, Jönsson's double-slit experiment was voted "the most beautiful experiment" by readers of ''[[Physics World]].''<ref>[http://physicsworld.com/cws/article/print/9746 "The most beautiful experiment"]. Physics World 2002.</ref>
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| The appearance of interference built up from individual photons could seemingly be explained by assuming that a single photon has its own associated wavefront 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.<ref>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.</ref> However, more complicated systems that involve two or more particles in superposition are not amenable to such a simple, classically intuitive explanation.<ref>Baggott, Jim (2011). ''The Quantum Story: A History in 40 Moments''. New York: Oxford University Press. pp. 76. ("The wavefunction of a system containing ''N'' particles depends on 3''N'' position coordinates and is a function in a 3''N''-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.")</ref>
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| ==Variations of the experiment==
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| ===Interference of individual particles===
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| [[Image:Double-slit experiment results Tanamura 2.jpg|thumb|right|200px|Electron buildup over time]]
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| An important version of this experiment involves single particles (or waves—for consistency, they are called particles here). Sending particles through a double-slit apparatus one at a time results in single particles appearing on the screen, as expected. Remarkably, however, an interference pattern emerges when these particles are allowed to build up one by one (see the image to the right). For example, when a laboratory apparatus was developed that could reliably fire one electron at a time through the double slit,<ref>Donati, O, Missiroli, G F, Pozzi, G (1973). An Experiment on Electron Interference. ''American Journal of Physics'' '''41''':639–644 {{doi|10.1119/1.1987321}}</ref> 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 at once''<ref>Brian Greene, ''The Elegant Universe,'' p. 110</ref>—an idea that contradicts our everyday experience of discrete objects. This phenomenon has also been shown to occur with atoms and even some molecules, including [[Buckminsterfullerene|buckyballs]].<ref name="buckyballs" /><ref>Olaf Nairz, Björn Brezger, Markus Arndt, and Anton Zeilinger, Abstract, "[http://prl.aps.org/abstract/PRL/v87/i16/e160401 Diffraction of Complex Molecules by Structures Made of Light]," ''Phys. Rev. Lett.'' 87, 160401 (2001)</ref><ref>Nairz O, Arndt M, and Zeilinger A. [http://hexagon.physics.wisc.edu/teaching/2010s%20ph531%20quantum%20mechanics/interesting%20papers/zeilinger%20large%20molecule%20interference%20ajp%202003.pdf Quantum interference experiments with large molecules]. American Journal of Physics, 2003; 71:319–325. {{doi|10.1119/1.1531580}}</ref> So experiments with electrons add confirmatory evidence to the view of Dirac that electrons, protons, neutrons, and even larger entities that are ordinarily called particles nevertheless have their own wave nature and even their own specific frequencies.
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| This experimental fact is highly reproducible, and the mathematics of quantum mechanics ([[#Classical wave-optics formulation|see below]]) allows us to predict the exact probability of an electron striking the screen at any particular point. However, the electrons 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 any future electron will hit, even though the probabilities at specific points can be calculated.<ref>Brian Greene, ''The Elegant Universe'', p. 104, pp. 109–114</ref> (Note that it is not the ''probabilities'' of photons appearing at various points along the detection screen that add or cancel, but the ''amplitudes''. Probabilities are the squares of amplitudes. Also note that if there is a cancellation of waves at some point, that does not mean that a photon disappears; it only means that the probability of a photon's appearing at that point will decrease, and the probability that it will appear somewhere else increases.) Thus, we have the appearance of a seemingly causeless selection event in a highly orderly and predictable formulation of the interference pattern. Ever since the origination of quantum mechanics, some theorists have searched for ways to incorporate additional determinants or "[[Hidden variable theory|hidden variables]]" that, were they to become known, would account for the location of each individual impact with the target.<ref name="Greene_2004">{{cite book|last =Greene|first =Brian|authorlink =Brian Greene|title =[[The Fabric of the Cosmos: Space, Time, and the Texture of Reality]] |publisher =Knopf|year =2004|pages = 204–213|isbn =0-375-41288-3}}</ref>
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| ===With particle detectors at the slits===
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| A famous ''gedanken'' experiment predicts that if particle detectors are positioned at the slits, showing through which slit a photon goes, the interference pattern will disappear.<ref name="Feynman" /> This ''which-way'' experiment illustrates the [[complementarity (physics)|complementarity]] principle that photons can behave as either particles or waves, but not both at the same time.<ref>{{cite web
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| | last = Harrison
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| | first = David
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| | coauthors =
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| | title = Complementarity and the Copenhagen Interpretation of Quantum Mechanics
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| | work = UPSCALE
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| | publisher = Dept. of Physics, U. of Toronto
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| | year = 2002
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| | url = http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Complementarity/CompCopen.html
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| | doi =
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| | accessdate = 2008-06-21}}</ref><ref>{{cite web
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| | last = Cassidy
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| | first = David
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| | coauthors =
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| | title = Quantum Mechanics 1925–1927: Triumph of the Copenhagen Interpretation
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| | work = Werner Heisenberg
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| | publisher = American Institute of Physics
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| | year = 2008
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| | url = http://www.aip.org/history/heisenberg/p09.htm
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| | doi =
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| | accessdate = 2008-06-21}}</ref><ref>{{cite conference
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| | first = María C.
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| | last = Boscá Díaz-Pintado
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| | coauthors =
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| | title = Updating the wave-particle duality
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| | booktitle = 15th UK and European Meeting on the Foundations of Physics
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| | pages =
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| | publisher =
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| | date = 29–31 March 2007
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| | location = Leeds, UK
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| | url = http://philsci-archive.pitt.edu/archive/00003568/
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| | doi =
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| | accessdate = 2008-06-21}}</ref>
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| Despite the importance of this ''gedanken'' in the history of quantum mechanics (for example, see the discussion on [[Einstein-Bohr_debates#Post-Revolution:_First_stage|Einstein's version of this experiment]]), technically feasible realizations of this experiment were not proposed until the 1970s.<ref name=Bartell1980>{{cite doi|10.1103/PhysRevD.21.1698|noedit}}</ref> (Naive implementations of the textbook ''gedanken'' are not possible because photons cannot be detected without absorbing the photon.) Currently, multiple experiments have been performed illustrating various aspects of complementarity,<ref name=Zeilinger1999>{{cite doi|10.1103/RevModPhys.71.S288}}</ref> including such experiments as the [[Delayed choice quantum eraser]].
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| However, an experiment performed in 1987 <ref name="Mittelstaedt">{{cite journal
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| | author=P. Mittelstaedt
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| | coauthors= A. Prieur, R. Schieder
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| | title=Unsharp particle-wave duality in a photon split-beam experiment
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| | journal=Foundations of Physics
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| | volume=17
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| | issue=9
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| | pages=891–903
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| | year=1987
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| | doi=10.1007/BF00734319
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| |bibcode = 1987FoPh...17..891M }}</ref><ref>D.M. Greenberger and A. Yasin, "Simultaneous wave and particle knowledge in a neutron interferometer", ''Physics Letters'' '''A 128''', 391–4 (1988).</ref> produced results that demonstrated that information could be obtained regarding which path a particle had taken without destroying the interference altogether. 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. 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.<ref>http://arstechnica.com/science/2012/05/disentangling-the-wave-particle-duality-in-the-double-slit-experiment/</ref> In order to do this, they used a setup such that particles coming to the screen were not from a point-like source, but from a source with two intensity maxima. It is debated whether this factor affects the validity of the experiment.<ref>{{cite web|last=Motl|first=Luboš|title=Pseudoscience hiding behind "weak measurements"|url=http://motls.blogspot.be/2012/09/pseudoscience-hiding-behind-weak.html#|accessdate=8 November 2013}}</ref>
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| ===By marking the wave===
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| Another way to show through which slit a photon goes is by marking wavefunctions emerging from each slit. The simplest version, [http://www.scientificamerican.com/slideshow.cfm?id=a-do-it-yourself-quantum-eraser given in an article in ''Scientific American'',] is to place polarisers before each slit with their axes orthogonal to each other. Then no interference pattern will be revealed, which is consistent with the fact that the polarizations performed offer experimenters which-path information. A polariser with an axis of 45° to the other polarisers in front of the detector "erases" this information, causing the interference pattern to reappear. The classical explantion are the [[Fresnel–Arago laws|Fresnel and Arago laws]], which state that orthogonal polarization does not interfere. The 45° polarisers force the two orthogonal polarizations to become parallel polarizations, which can interfere. A more complicated measurement uses two [[Waveplate|quarter wave plates]] at the slits, which transforms a linear to a rotating polarization.<ref name=Walborn>{{cite journal | author=S.P. Walborn | coauthors= M.O. Terra Cunha, S. Pádua and C.H. Monken. | title =Double-Slit Quantum Eraser
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| | journal=Physical Review A | volume =65 | issue =3 | pages =033818 | year =2002 | url =http://arxiv-web3.library.cornell.edu/abs/quant-ph/0106078 | doi =10.1103/PhysRevA.65.033818}}</ref> However, when two oppositely rotating waves are added (at the detector), the rotation disappears. Also, according the paper, the "not interference" pattern is actually the sum of two interference patterns which are shifted 180° to each other.
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| ===Delayed choice and quantum eraser variations===
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| The [[Wheeler's delayed choice experiment|delayed-choice experiment]] and the [[quantum eraser]] are sophisticated variations of the double-slit with particle detectors placed not at the slits but elsewhere in the apparatus. The first demonstrates that extracting "which path" information ''after'' a particle passes through the slits can seem to retroactively alter its previous behavior at the slits. The second demonstrates that wave behavior can be restored by erasing or otherwise making permanently unavailable the "which path" information.
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| ===Other variations===
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| [[Image:Double-slit wall sm.jpg|80px|thumb|right|A laboratory double-slit assembly; distance between top posts approximately 2.5 cm (one inch).]]
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| In 1967, Pfleegor and Mandel demonstrated two-source interference using two separate lasers as light sources.<ref>{{cite journal | journal=[[Physical Review]] | title=Interference of Independent Photon Beams | author=Pfleegor, R. L. and Mandel, L. | journal=Phys. Rev. |date=July 1967 | volume=159 | issue=5 | pages=1084–1088 | doi=10.1103/PhysRev.159.1084|bibcode = 1967PhRv..159.1084P }}</ref><ref>http://scienceblogs.com/principles/2010/11/interference_of_independent_ph.php></ref>
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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.<ref>{{cite journal | url=http://www.sciencedirect.com/science/article/pii/0375960172910158 | title=An interference experiment with light beams modulated in anti-phase by an electro-optic shutter | author=Sillitto, R.M. and Wykes, Catherine | journal=Physics Letters A | year=1972 | volume=39 | issue=4 | pages=333–334 | doi=10.1016/0375-9601(72)91015-8|bibcode = 1972PhLA...39..333S }}</ref><ref>[http://www.sillittopages.co.uk/80rms_35.html "To a light particle"]</ref> The experimental conditions were such that the photon density in the system was much less than unity.
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| The experiment has been performed with particles as large as [[Buckminsterfullerene|C60]] (Buckminsterfullerene).<ref>[http://www.quantum.at/research/molecule-interferometry-foundations/wave-particle-duality-of-c60.html Wave Particle Duality of C60]</ref>
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| In 2012, researchers at the [[University of Nebraska–Lincoln]] performed the double-slit experiment with electrons as described by [[Richard Feynman]], 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.<ref>{{cite journal
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| | last =Bach
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| | first =Roger
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| | coauthors =et al.
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| | title =Controlled double-slit electron diffraction
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| | journal =New Journal of Physics
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| | volume =15
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| | issue =3
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| | pages =033018
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| | date =March 2013
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| | url =http://arxiv.org/abs/1210.6243
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| | doi =10.1088/1367-2630/15/3/033018
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| }}</ref>
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| ==Classical wave-optics formulation==
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| [[File:Doubleslit3Dspectrum.gif|thumb|Two-slit diffraction pattern by a plane wave]]
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| [[Image:Doubleslit.svg|thumb|200px|right|Two slits are illuminated by a plane wave.]]
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| Much of the behaviour of light can be modelled using classical wave theory. The [[Huygens–Fresnel principle]] is one such model; it states that each point on a wavefront generates a secondary spherical wavelet, and that the disturbance at any subsequent point can be found by [[Superposition principle|summing]] the contributions of the individual wavelets at that point. This summation needs to take into account the [[Phase (waves)|phase]] as well as the [[amplitude]] of the individual wavelets. It should be noted that only the [[Intensity (physics)|intensity]] of a light field can be measured—this is proportional to the square of the amplitude.
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| In the double-slit experiment, the two slits are illuminated by a single laser beam. If the width of the slits is small enough (less than the wavelength of the laser light), the slits diffract the light into cylindrical waves. These two cylindrical wavefronts are 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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| If the viewing distance is large compared with the separation of the slits (the [[far field]]), 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 {{math|θ}} is given by:
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| :<math>d \sin \theta \approx d \theta</math>
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| 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 wavelength, one and a half wavelengths, etc., then the two waves cancel and the summed intensity is zero. This effect is known as [[Interference (optics)|interference]]. The interference fringe maxima occur at angles
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| :<math>~ d \theta_n = n \lambda,~ n=0,1,2,\ldots</math>
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| where λ is the [[wavelength]] of the light. The angular spacing of the fringes, {{math|θ<sub>''f''</sub>}}, is given by
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| :<math> \theta_f \approx \lambda / d </math>
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| The spacing of the fringes at a distance {{math|''z''}} from the slits is given by
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| :<math>~w=z \theta_f = z \lambda /d</math>
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| For example, if two slits are separated by 0.5mm ({{math|''d''}}), and are illuminated with a [[visible spectrum|0.6μm]] wavelength laser ({{math|λ}}), then at a distance of 1m ({{math|''z''}}), the spacing of the fringes will be 1.2mm.
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| If the width of the slits {{math|''b''}} is greater than the wavelength, the [[Fraunhofer diffraction]] equation gives the intensity of the diffracted light as:<ref>Jenkins FA and White HE, Fundamentals of Optics, 1967, McGraw Hill, New York</ref>
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| :<math>
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| \begin{align}
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| I(\theta)
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| &\propto \cos^2 \left [{\frac {\pi d \sin \theta}{\lambda}}\right]~\mathrm{sinc}^2 \left [ \frac {\pi b \sin \theta}{\lambda} \right]
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| \end{align}
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| </math>
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| Where the [[sinc function]] is defined as sinc(''x'') = sin(''x'')/(''x'') for ''x'' ≠ 0, and sinc(0) = 1.
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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 {{math|sinc}} function in this equation, and the second figure shows the combined intensity of the light diffracted from the two slits, where the {{math|cos}} function represent the fine structure, and the coarser structure represents diffraction by the individual slits as described by the {{math|sinc}} function.
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| Similar calculations for the [[near and far field|near field]] can be done using the [[Fresnel diffraction]] equation. 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 two diffracted patterns.<ref>Longhurst RS, Physical and Geometrical Optics, 1967, 2nd Edition, Longmans</ref>
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| ==Interpretations of the experiment==
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| Like the [[Schrödinger's cat]] thought experiment, the double-slit experiment is often used to highlight the differences and similarities between the various [[interpretations of quantum mechanics]].
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| ===Copenhagen interpretation===
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| {{unreferenced section|date=February 2012}}
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| 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.
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| 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 a 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. 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 distinct paths for the particle to take from the emitter to the detection screen. Experiments observe nothing whatsoever between the time of emission of the particle and its arrival at the detection screen. If a ray tracing is then made as if a light wave (as understood in classical physics) is wide enough to take both paths, 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.
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| ===Path-integral formulation===
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| [[Image:Wiener process 3d.png|thumb|200px|One of an infinite number of equally likely paths used in the Feynman path integral. (see also: [[Wiener process]].)]]
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| The Copenhagen interpretation is similar to the [[path integral formulation]] 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 [[functional integration]].
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| Each path is considered equally likely, and thus contributes the same amount. However, the [[phase (waves)|phase]] of this contribution at any given point along the path is determined by the [[action (physics)|action]] along the path:
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| ::<math>A_{\text{path}}(x,y,z,t) = e^{i S(x,y,z,t)}</math>
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| All these contributions are then added together, and the [[magnitude (mathematics)|magnitude]] of the final result is [[Square (algebra)|squared]], to get the probability distribution for the position of a particle:
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| ::<math>p(x,y,z,t) \propto \left\vert \int_{\text{all paths}} e^{i S(x,y,z,t)} \right\vert ^2 </math>
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| As is always the case when calculating [[probability]], the results must then be [[Normalizing constant|normalized]] by imposing:
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| ::<math>\iiint_{\text{all space}}p(x,y,z,t)\,\mathrm{d}V = 1</math>
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| To summarize, the probability distribution of the outcome is the normalized square of the norm of the [[superposition principle|superposition]], over all paths from the point of origin to the final point, of [[wave]]s [[wave propagation|propagating]] [[Proportionality (mathematics)|proportionally]] 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 [[Interference (wave propagation)|interference pattern]] 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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| ===Relational interpretation===
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| According to the [[Relational quantum mechanics|relational interpretation of quantum mechanics]], first proposed by [[Carlo Rovelli]],<ref>{{Cite journal|doi = 10.1007/BF02302261|last = Rovelli|first = Carlo|authorlink = Carlo Rovelli|title = Relational Quantum Mechanics|journal = International Journal of Theoretical Physics|volume = 35|issue = 8|pages = 1637–1678|year = 1996|arxiv = quant-ph/9609002 |bibcode = 1996IJTP...35.1637R }}</ref> observations such as those in the double-slit experiment result specifically from the interaction between the [[Observer (quantum physics)|observer]] (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. 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"—for example, at both slits—because its spatial relations to particular points on the screen remain identical from both slit locations.<ref>{{Cite journal|last = Filk|first = Thomas|title = Relational Interpretation of the Wave Function and a Possible Way Around Bell’s Theorem|journal = International Journal of Theoretical Physics|volume = 45|pages = 1205–1219|year = 2006|url = http://www.springerlink.com/content/v775765467462313/|doi = 10.1007/s10773-006-9125-0|arxiv = quant-ph/0602060 |bibcode = 2006IJTP...45.1166F }}</ref>
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| ==See also==
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| * [[Complementarity (physics)]]
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| * [[Delayed choice quantum eraser]]
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| * [[Dual Polarisation Interferometry]]
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| * [[Elitzur–Vaidman bomb tester]]
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| * [[Photon polarization]]
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| * [[Quantum coherence]]
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| * [[Schrödinger's cat]]
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| * [[Young's interference experiment]]
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| ==References==
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| {{reflist|colwidth=30em}}
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| ===Further reading===
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| *{{cite book | last = Al-Khalili | first = Jim |author-link=Jim Al-Khalili | title=Quantum: A Guide for the Perplexed | publisher=Weidenfeld and Nicholson |location=London | year=2003 | isbn=0-297-84305-2}}
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| *{{cite book | last = Feynman | first = Richard P. | title=QED: The Strange Theory of Light and Matter | publisher=Princeton University Press | year=1988 | isbn=0-691-02417-0}}
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| *{{cite book | last = Frank| first = Philipp | title=Philosophy of Science | publisher=Prentice-Hall | year=1957}}
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| *{{cite book | last = French | first = A.P. |first2=Edwin F. |last2=Taylor | title= An Introduction to Quantum Physics | publisher=Norton | year=1978 | isbn=0-393-09106-6}}
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| *{{cite book | last = Quznetsov | first = Gunn | title= Final Book on Fundamental Theoretical Physics | publisher=American Research Press | year=2011 | isbn=978-1-59973-172-8}}
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| *{{cite book | last = Greene | first = Brian | title= The Elegant Universe | publisher=Vintage | year=2000 | isbn=0-375-70811-1}}
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| *{{cite book | last = Greene | first = Brian | title= The Fabric of the Cosmos | publisher=Vintage | year=2005 | isbn=0-375-72720-5}}
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| *{{cite book | last = Gribbin | first = John |author-link=John Gribbin | title=Q is for Quantum: Particle Physics from A to Z | publisher=Weidenfeld & Nicolson | year=1999 | isbn=0-7538-0685-1}}
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| *{{cite book | last = Hey| first = Tony | title=The New Quantum Universe | publisher=Cambridge University Press | year=2003| isbn=0-521-56457-3}}
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| *{{cite book | last = Sears | first = Francis Weston | title=Optics | publisher=Addison Wesley | year=1949 }}
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| *{{cite book | last = Tipler | first = Paul | title=Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics | edition = 5th | publisher=W. H. Freeman | year=2004 | isbn=0-7167-0810-8}}
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| ==External links==
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| {{Commons category|Double-slit experiments}}
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| * [http://www.falstad.com/ripple/ex-2slit.html Java demonstration of double slit experiment, animated]
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| * [http://www.ianford.com/dslit Java demonstration of double slit experiment, point by point]
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| * [http://vsg.quasihome.com/interf.htm Java demonstration of Young's double slit interference]
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| * [http://homepage.univie.ac.at/Franz.Embacher/KinderUni2005/waves.gif Double-slit experiment animation]
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| * [http://video.google.com/videoplay?docid=5063999801799851614 Caltech: The Mechanical Universe, chapter 50 – Particles and Waves]
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| * [http://www.bo.imm.cnr.it/users/lulli/downintel/index.html Electron Interference movies from the Merli Experiment (Bologna-Italy, 1974)]
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| * [http://www.hitachi.com/rd/research/em/movie.html 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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| * [http://www.vega.org.uk/video/programme/66 Freeview video 'Electron Waves Unveil the Microcosmos' A Royal Institution Discourse by Akira Tonomura provided by the Vega Science Trust]
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| * [http://www.hitachi.com/rd/research/em/doubleslit.html Hitachi website that provides background on Tonomura video and link to the video]
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| * [http://schools.matter.org.uk/Content/Interference/formula.html Simple Derivation of Interference Conditions]
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| * [http://physdemo.phys.cmu.edu/newton_rings.htm Carnegie Mellon department of physics, photo images of Newton's rings]
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| * [http://www.physorg.com/news78650511.html "Single-particle interference observed for macroscopic objects"]
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| * [http://www.acoustics.salford.ac.uk/feschools/waves/diffract3.htm Huygens and interference]
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| * [http://www.strings.ph.qmw.ac.uk/~jmc/sefp/week9.pdf Huygens and interference]
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| * [http://demonstrations.wolfram.com/WaveParticleDualityInTheDoubleSlitExperiment/ 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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| * [http://www.stmary.ws/highschool/physics/home/notes/waves/WaveNatureOfLight.htm Wave Nature Of Light (High School Level) – Lots of graphics and simulations; double-slit equation with examples]
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| * [http://www.sillittopages.co.uk/80rms_35.html To a light particle]
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| *[[:File:DoubleSlitExperiment secondspace 2013-01-12.gif|A computer simulation of a Gaussian wave packet representing a single particle passing through two slits]]
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| {{DEFAULTSORT:Double-Slit Experiment}}
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| [[Category:Foundational quantum physics]]
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| [[Category:Physics experiments]]
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| [[Category:Quantum mechanics]]
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| [[Category:Wave mechanics]]
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| {{Link GA|zh}}
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