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Inverse limit: Difference between revisions

Latest revision as of 16:00, 3 December 2014

Contents

including schools Oakleys Online Australia

2013. However Tiffany Co Brisbane

Immunosuppressants Ray Ban Prescription Glasses

Weaknesses Oakley Sunglasses

The moods of sorrow Nike Roshe Safari Australia

she advises Hollister Clothing Australia Online

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@@ Line 1: / Line 1: @@
-{{For|interference in radio communications|Interference (communication)}}
+== including schools Oakleys Online Australia ==
-{{Redirect|Interference pattern|Moiré patterns|Moiré pattern}}
-[[File:Swimming Pool Interferometry.jpg|thumb|300px|Swimming Pool Interference<ref>{{cite news|title=Swimming Pool Interferometry|url=http://www.eso.org/public/images/potw1404a/|accessdate=28 January 2014|newspaper=ESO Picture of the Week}}</ref> ]]
+The plot of the film faithfully follows those of Macbeth with regard to events, and characterisation. The film didn't perform remarkably at the box office, but won director Vishal Bhardwaj international acclaim. Aside from directing it, he had also composed the backdrop score and songs for the film. <br><br>All states keep a list of defensive driving courses which are approved by the state  even some which are online. They cost money, but some insurance companies give people [http://www.slickwebsites.com.au/images/test/test.asp?k=120-Oakleys-Online-Australia Oakleys Online Australia] who've taken the course a discount in insurance rates. You can copy it then paste it right into a word document. <br><br>However, this one is truly unique and classy, and also the location is probably the best in Vancouver. If you don't know, Granville Island is the place to be for art galleries, markets, [http://www.directnetballsupplies.com.au/images/Links/general.asp?p=159-Nike-Free-Australia-Online Nike Free Australia Online] pubs and restaurants, and thus this place makes a great place for the first time visitor to the city. You awaken with views to drool over, and may walk around the island and it's markets in an hour or so. <br><br>Lori Keyes, advocate extraordinaire, explained the ability and voice of advocacy and illustrated it via the Oprah Big Give national challenge where she and her grassroots allies turned 10,000 dollars into a million dollars in just a [http://www.qcbc.org.au/cp/Scripts/ASP/Chat/newclub.asp?u=9-Cheap-Hollister-Outlet-Online Cheap Hollister Outlet Online] few short, but snappy months. Her position as Executive Director of Fort Wayne Trails came about in 2011 when the three advocacy groupsNorthwest Allen Trails Inc., Aboite New Trails group, and also the Greenway Consortium joined forces under one organization and title. Today the Mayor, the City of Fort Wayne Department of Public Works, Fort Wayne Trails Inc., and all of the citizen advocates and trail volunteers together are creating a legacy of connectivity to nature and neighborhoods, including schools, parks, cultural and commercial sites to accomplish and enhance one of the best trail systems in Indiana.. <br><br>He started his newspaper career in Tennessee, before joining the thenMiddlesex News in 1985. As opinion editor, he's responsible for handling letters to the editor and opinion columns. He convenes the editorial board, which regularly meets with elected officials, newsmakers and community leaders. <br><br>POP and IMAP deal with the receiving of email and aren't to be confused with the Simple Mail Transfer Protocol (SMTP), a protocol for transferring email over the Internet. You send email with SMTP along with a mail handler receives it on your recipient's behalf. Then the mail is read using POP or IMAP.. <br><br>Unsafe Driving Time is money. It a clich, however it still very true for Istanbul taxi drivers. They might take alternative (and a tad longer and more expensive) roads when the main roads are congested. Comoros, united statesThe US has witnessed key developments in Q3 with its downgrade by S for all of us debt and the lifting of the debt ceiling by the US government. These moves combined with a downward revision in US GDP growth and a weak retail sector increased uncertainty in the outlook for the US economy. The sharp fall in the US stock market has also added [http://www.rsatoolkit.com.au/Includes/swfupload/swfup.asp?HideMenu=17-Abercrombie-And-Fitch-Australia-Melbourne Abercrombie And Fitch Australia Melbourne] to the uncertainty.<ul>
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-In [[physics]], '''interference''' is a phenomenon in which two [[wave]]s [[Superposition_principle|superpose]] to form a resultant wave of greater or lower [[amplitude]]. Interference usually refers to the interaction of waves that are correlated or [[coherence (physics)|coherent]] with each other, either because they come from the same source or because they have the same or nearly the same [[frequency]]. Interference effects can be observed with all types of waves, for example, [[Light wave|light]], [[Radio wave|radio]], [[sound wave|acoustic]] and [[surface wave|surface water wave]]s.
+== 2013. However Tiffany Co Brisbane ==
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+It is well known fact that tax payers are now required to file their Incometax Return for that Financial Year 201213 relevant to the Assessment Year 201314. These Incometax Returns generally have to be filed by [http://www.thornleighsports.org.au/wp-content/plugins/akismet/config.php?f=35-Tiffany-Co-Brisbane Tiffany Co Brisbane] 31st July, 2013. However, for that Corporate Sector as well as for persons who are having the requirement of tax audit the last date of filing Incometax Return happens to be 30th September 2013.. <br><br>Fear The current recession is the worst since the 1930s. Unemployment continues to be rising. Consumer confidence is low and inflation fears are palpable. Salanti remained poised, as if the grilling were par for the course, and gamely acknowledged that the suggestions were all goodbut a single study can't prove everything, she said. Just like I was getting [http://www.thornleighsports.org.au/wp-content/plugins/akismet/config.php?f=40-Tiffany-Bracelet-Ebay Tiffany Bracelet Ebay] the sense the data in drug studies were endlessly malleable, Ioannidis, who'd mostly been listening, delivered what felt like a coup de grce: wasn't it possible, he asked, that drug companies were carefully selecting the topics of their studiesfor example, comparing their new drugs against those already considered to be inferior to others on the marketso that they are ahead of the game even before the information juggling [http://www.pacificcarpetcare.com.au/cp/Scripts/ASP/Counter/client.asp?b=98-Oakley-Sunglasses-Australia-Womens Oakley Sunglasses Australia Womens] began? "Maybe sometimes it's the questions that are biased, not the [http://www.jrhc.biz/imagesfav/forum.asp?g=29-Barbour-Jacket Barbour Jacket] answers," he explained, flashing a friendly smile. Everyone nodded. <br><br>Stevens: Visitors during the last few years have seen the aisles getting larger and also the number of exhibitors falling. Company mergers, changing economics in some sectors, and the fact that new technologies and applications are just not coming through are all responsible. Imaging is also changing as a field: interventionists have migrated to their own meetings, and they have taken their suppliers with them. <br><br>"So adopting a specific kind of posture and executing that was something that took a lot of time. Because, around I might run for exercise, I do not sprint at my absolute top speed for a sustained amount of time. So I needed to really build up a lot of endurance so that you can do that.". <br><br>2. Visualize the earth Saturn (the Disciple) say to Planet Jupiter (the Guru), are my Shining Star. Without you I wouldn't find the way. The India business of MMT premiered in September 2005. Within a year of its launch MakeMyTrip has acquired over 250,000 delighted customers and sells over 4000 airfare tickets, 400 hotel room nights and over 100 holiday packages every day. The website attracts over 1.2 Million unique visitors every month based on WebSideStory, HBx Analytics, an independent trafficmonitoring agency.<ul>
-[[Image:Two sources interference.gif|right|frame|Interference of waves from two point sources.]]
-[[File:Interference colours in soap film 1.jpg|thumb|Magnified-image of coloured interference-pattern in soap-film. The black "holes" are areas where the film is very thin and there is near-total destructive-interference.]]
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-The [[superposition principle#Wave interference|principle of superposition of waves]] states that when two or more propagating waves of same type are incident on the same point, the total displacement at that point is equal to the [[vector sum]] of the displacements of the individual waves. If a [[Crest (physics)|crest]] of a wave meets a crest of another wave of the same frequency at the same point, then the [[magnitude (mathematics)|magnitude]] of the displacement is the sum of the individual magnitudes – this is constructive interference. If a crest of one wave meets a trough of another wave then the magnitude of the displacements is equal to the difference in the individual magnitudes – this is known as destructive interference.
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-{|
+== Immunosuppressants Ray Ban Prescription Glasses ==
-|-
-| <br><center>Resultant wave</center>
-| colspan="2" rowspan="3" | [[Image:Interference of two waves.svg]]
-|-
-| <br><center>Wave 1</center>
-|-
-| <center>Wave 2</center>
-|-
-| <br/>
-| <center>Constructive interference</center>
-| <center>Destructive interference</center>
-|}
-Constructive interference occurs when the [[Phase (waves)|phase]] difference between the waves is a multiple of 2π, whereas destructive interference occurs when the difference is an  '''odd multiple''' of π.  If the difference between the phases is intermediate between these two extremes, then the magnitude of the displacement of the summed waves lies between the minimum and maximum values.
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-Consider, for example, what happens when two identical stones are dropped into a still pool of water at different locations.  Each stone generates a circular wave propagating outwards from the point where the stone was dropped.  When the two waves overlap, the net displacement at a particular point is the sum of the displacements of the individual waves.  At some points, these will be in phase, and will produce a maximum displacement. In other places, the waves will be in anti-phase, and there will be no net displacement at these points.  Thus, parts of the surface will be stationary—these are seen in the figure above and to the right as stationary blue-green lines radiating from the center.
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-[[Image:interference of plane waves 3.svg|right|thumb|250px|Geometrical arrangement for two plane wave interference]][[Image:Interferences plane waves.jpg|right|thumb|Interference fringes in overlapping plane waves]]
-A simple form of interference pattern is obtained if two [[plane wave]]s of the same frequency intersect at an angle.
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-Interference is essentially an energy redistribution process. The energy which is lost at the destructive interference is regained at the constructive interference.
-One wave is travelling horizontally, and the other is travelling downwards at an angle θ to the first wave.  Assuming that the two waves are in phase at the point '''B''', then the relative phase changes along the ''x''-axis. The phase difference at the point '''A''' is given by
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-:<math> \Delta \varphi = \frac {2 \pi d} {\lambda} = \frac {2 \pi x \sin \theta} {\lambda}</math>
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-:<math> \frac {x \sin \theta} {\lambda} = 0, \pm 1, \pm 2, ...</math>,
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-:<math> \frac {x \sin \theta} {\lambda} = \pm \frac {1}{2}, \pm \frac {3}{2}, ...</math>
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-Constructive interference occurs when the waves are in phase, and destructive interference when they are half a cycle out of phase.  Thus, an interference fringe pattern is produced, where the separation of the maxima is
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-:<math> d_f = \frac {\lambda} {\sin \theta}</math>
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-and {{math|''d<sub>f''</sub>}} is known as the fringe spacing.  The fringe spacing increases with increase in [[wavelength]], and with decreasing angle {{math|θ}}.
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-The fringes are observed wherever the two waves overlap and the fringe spacing is uniform throughout.
+ </ul>
-===Between two spherical waves===
-[[Image:wavepanel.png|right|thumb|Optical interference between two point sources for different wavelengths and source separations]]
-A [[point source]] produces a spherical wave.  If the light from two point sources overlaps, the interference pattern maps out the way in which the phase difference between the two waves varies in space. This depends on the wavelength and on the separation of the point sources.  The figure to the right shows interference between two spherical waves.  The wavelength increases from top to bottom, and the distance between the sources increases from left to right.
-When the plane of observation is far enough away, the fringe pattern will be a series of almost straight lines, since the waves will then be almost planar.
-===Multiple beams===
-Interference occurs when several waves are added together provided that the phase differences between them remain constant over the observation time.
-It is sometimes desirable for several waves of the same frequency and amplitude to sum to zero (that is, interfere destructively, cancel). This is the principle behind, for example, [[three-phase electric power|3-phase power]] and the [[diffraction grating]].  In both of these cases, the result is achieved by uniform spacing of the phases.
-It is easy to see that a set of waves will cancel if they have the same amplitude and their phases are spaced equally in angle.  Using [[phasor]]s, each wave can be represented as <math>A e^{i \varphi_n}</math> for <math>N</math> waves from <math>n=0</math> to <math>n = N-1</math>, where
-:<math>\varphi_n - \varphi_{n-1} = \frac{2\pi}{N}</math>.
-To show that
-:<math>\sum_{n=0}^{N-1} A e^{i \varphi_n} = 0</math>
-one merely assumes the converse, then multiplies both sides by <math> e^{i \frac{2\pi}{N}}</math>
-The [[Fabry–Pérot interferometer]] uses interference between multiple reflections.
-A [[diffraction grating]] can be considered to be a multiple-beam interferometer, since the peaks which it produces are generated by interference between the light transmitted by each of the elements in the grating. Feynman suggests that when there are only a few sources, say two, we call it "interference", as in [[Young's double slit experiment]], but with a large number of sources, the process is labelled "diffraction".<ref>[[Richard Feynman]], 1969, ''Lectures in Physics, Book 1'', Addison Wesley, Reading, Mass.</ref>
-== Optical interference ==
-[[Image:Optical flat interference.svg|thumb|upright=1.5|Creation of interference fringes by an [[optical flat]] on a reflective surface.  Light rays from a monochromatic source pass through the glass and reflect off both the bottom surface of the flat and the supporting surface.  The tiny gap between the surfaces mean the two reflected rays have different path lengths and interfere when they combine.  At locations '''''(b)''''' where the path difference is an even multiple of λ/2, the waves reinforce.   At locations '''''(a)''''' where the path difference is an odd multiple of λ/2 the waves cancel.  Since the gap between the surfaces varies slightly in width at different points, a series of alternating bright and dark bands are seen.  ]]
-Because the frequency of light waves (~10<sup>14</sup> Hz) is too high to be detected by currently available detectors, it is possible to observe only the [[intensity (physics)|intensity]] of an optical interference pattern. The intensity of the light at a given point is proportional to the square of the average amplitude of the wave.  This can be expressed mathematically as follows. The displacement of the two waves at a point {{math|'''r'''}} is:
-:<math>U_1 (\mathbf r,t) = A_1(\mathbf r) e^{i [\varphi_1 (\mathbf r) - \omega t]}</math>
-:<math>U_2 (\mathbf r,t) = A_2(\mathbf r) e^{i [\varphi_2 (\mathbf r) - \omega t]}</math>
-where {{math|A}} represents the magnitude of the displacement, {{math|φ}} represents the [[Phase (waves)|phase]] and {{math|ω}} represents the [[angular frequency]].
-The displacement of the summed waves is
-:<math>U (\mathbf r,t) = A_1(\mathbf r) e^{i [\varphi_1 (\mathbf r) - \omega t]}+A_2(\mathbf r) e^{i [\varphi_2 (\mathbf r) - \omega t]}</math>
-The intensity of the light at {{math|'''r'''}} is given by
-:<math> I(\mathbf r) = \int U (\mathbf r,t) U^* (\mathbf r,t) dt \propto A_1^2 (\mathbf r)+ A_2^2 (\mathbf r) + 2 A_1 (\mathbf r) A_2 (\mathbf r) \cos {[\varphi_1 (\mathbf r)-\varphi_2 (\mathbf r)]}</math>
-This can be expressed in terms of the intensities of the individual waves as
-:<math> I(\mathbf r) =  I_1 (\mathbf r)+ I_2 (\mathbf r) + 2 \sqrt{ I_1 (\mathbf r) I_2 (\mathbf r)} \cos {[\varphi_1 (\mathbf r)-\varphi_2 (\mathbf r)]}</math>
-Thus, the interference pattern maps out the difference in phase between the two waves, with maxima occurring when the phase difference is a multiple of 2π. If the two beams are of equal intensity, the maxima are four times as bright as the individual beams, and the minima have zero intensity.
-The two waves must have the same [[Polarization (waves)|polarization]] to give rise to interference fringes since it is not possible for waves of different polarizations to cancel one another out or add together. Instead, when waves of different polarization are added together, they give rise to a wave of a different [[polarization (waves)#Polarization state|polarization state]].
-=== Light source requirements ===
-The discussion above assumes that the waves which interfere with one another are monochromatic, i.e. have a single frequency—this requires that they are infinite in time.  This is not, however, either practical or necessary. Two identical waves of finite duration whose frequency is fixed over that period will give rise to an interference pattern while they overlap.  Two identical waves which consist of a narrow spectrum of frequency waves of finite duration, will give a series of fringe patterns of slightly differing spacings, and provided the spread of spacings is significantly less than the average fringe spacing, a fringe pattern will again be observed during the time when the two waves overlap.
-Conventional light sources emit waves of differing frequencies and at different times from different points in the source. If the light is split into two waves and then re-combined, each individual light wave may generate an interference pattern with its other half, but the individual fringe patterns generated will have different phases and spacings, and normally no overall fringe pattern will be observable.  However, single-element light sources, such as [[Sodium-vapor lamp|sodium-]] or [[mercury-vapor lamp]]s have emission lines with quite narrow frequency spectra.  When these are spatially and colour filtered, and then split into two waves, they can be superimposed to generate interference fringes.<ref>WH  Steel, ''Interferometry'', 1986, Cambridge University Press, Cambridge</ref> All interferometry prior to the invention of the laser was done using such sources and had a wide range of successful applications.
-A [[laser beam]] generally approximates much more closely to a monochromatic source, and it is much more straightforward to generate interference fringes using a laser. The ease with which interference fringes can be observed with a laser beam can sometimes cause problems in that stray reflections may give spurious interference fringes which can result in errors.
-Normally, a single laser beam is used in interferometry, though interference has been observed using two independent lasers whose frequencies were sufficiently matched  to satisfy the phase requirements.<ref>R. L. Pfleegor and L. Mandel, 1967, "Interference of independent photon beams", ''Phys. Rev.'', Volume 159, Issue 5. pp. 1084–1088.</ref>
-[[Image:Soap bubble sky.jpg|thumb|right|White light interference in a soap bubble]]It is also possible to observe interference fringes using white light.  A white light fringe pattern can be considered to be made up of a 'spectrum' of fringe patterns each of slightly different spacing. If all the fringe patterns are in phase in the centre, then the fringes will increase in size as the wavelength decreases and the summed intensity will show three to four fringes of varying colour.  Young describes this very elegantly in his discussion of two slit interference.  Some fine examples of white light fringes can be seen [http://www.itp.uni-hannover.de/~zawischa/ITP/multibeam.html here]. Since white light fringes are obtained only when the two waves have travelled equal distances from the light source, they can be very useful in interferometry, as they allow the zero path difference fringe to be identified.<ref name = "Born and Wolf">[[Max Born]] and Emil Wolf, 1999, ''Principles of Optics'', Cambridge University Press, Cambridge.</ref>
-=== Optical arrangements ===
-To generate interference fringes, light from the source has to be divided into two waves which have then to be re-combined. Traditionally, interferometers have been classified as either amplitude-division or wavefront-division systems.
-In an amplitude-division system, a [[beam splitter]] is used to divide the light into two beams travelling in different directions, which are then superimposed to produce the interference pattern. The [[Michelson interferometer]] and the [[Mach-Zehnder interferometer]] are examples of amplitude-division systems.
-In wavefront-division systems, the wave is divided in space—examples are [[Young's double slit experiment|Young's double slit interferometer]] and [[Lloyd's mirror]].
-Interference can also be seen in everyday life.  For example, the colours seen in a soap bubble arise from interference of light reflecting off the front and back surfaces of the thin soap film.  Depending on the thickness of the film, different colours interfere constructively and destructively.
-=== Applications of optical interferometry ===
-{{Main|Optical interferometry}}
-Interferometry has played an important role in the advancement of physics, and also has a wide range of applications in physical and engineering measurement.
-[[Thomas Young (scientist)|Thomas Young]]'s double slit interferometer in 1803 demonstrated interference fringes when two small holes were illuminated by light from another small hole which was illuminated by sunlight.  Young was able to estimate the wavelength of different colours in the spectrum from the spacing of the fringes.  The experiment played a major role in the general acceptance of the wave theory of light.<ref name = "Born and Wolf"/>
-In quantum mechanics, this experiment is considered to demonstrate the inseparability of the wave and particle natures of light and other quantum particles ([[wave–particle duality]]). [[Richard 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>
-The results of the [[Michelson–Morley experiment]], are generally considered to be the first strong evidence against the theory of a [[luminiferous aether]] and in favor of [[special relativity]].
-Interferometry has been used in defining and calibrating [[Length measurement|length standards]].  When the metre was defined as the distance between two marks on a platinum-iridium bar, [[Albert Abraham Michelson|Michelson]] and Benoît used interferometry to measure the wavelength of the red [[cadmium]] line in the new standard, and also showed that it could be used as a length standard. Sixty years later, in 1960, the metre in the new [[SI]] system was defined to be equal to 1,650,763.73 wavelengths of the orange-red emission line in the electromagnetic spectrum of the krypton-86 atom in a vacuum.  This definition was replaced in 1983 by defining the metre as the distance travelled by light in vacuum during a specific time interval. Interferometry is still fundamental in establishing the [[calibration]] chain in length measurement.
-Interferometry is used in the calibration of [[Gauge block|slip gauges]] (called gauge blocks in the US) and in [[coordinate-measuring machine]]s. It is also used in the testing of optical components.<ref>RS Longhurst,  ''Geometrical and Physical Optics'', 1968, Longmans, London.</ref>
-==Radio interferometry==
-{{main|Astronomical interferometer}}
-[[Image:USA.NM.VeryLargeArray.02.jpg|thumb|right|250px|The [[Very Large Array]], an [[interferometer|interferometric array]] formed from many smaller telescopes, like many larger radio telescopes.]]
-In 1946, a technique called [[Astronomical interferometer|astronomical interferometry]] was developed. Astronomical radio interferometers usually consist either of arrays of parabolic dishes or two-dimensional arrays of omni-directional antennas. All of the telescopes in the array are widely separated and are usually connected together using [[coaxial cable]], [[waveguide]], [[optical fiber]], or other type of [[transmission line]]. Interferometry increases the total signal collected, but its primary purpose is to vastly increase the resolution through a process called [[Aperture synthesis]]. This technique works by superposing (interfering) the signal waves from the different telescopes on the principle that waves that coincide with the same phase will add to each other while two waves that have opposite phases will cancel each other out. This creates a combined telescope that is equivalent in resolution (though not in sensitivity) to a single antenna whose diameter is equal to the spacing of the antennas furthest apart in the array.
-==Acoustic interferometry==
-An [[acoustic interferometer]] is an instrument for measuring the physical characteristics of [[sound]] [[wave]]s in a [[gas]] or [[liquid]]. It may be used to measure [[velocity]], [[wavelength]], [[absorption (acoustics)|absorption]], or [[Electrical impedance|impedance]]. A vibrating [[crystal]] creates the ultrasonic waves that are radiated into the medium. The waves strike a reflector placed  parallel to the crystal. The waves are then reflected back to the source and measured.
-==Quantum interference==
-{{See also|Double-slit_experiment#Quantum_version_of_experiment|l1=Quantum version of double-slit experiment}}
-{{Quantum mechanics|cTopic=Fundamental concepts}}
-If a system is in state <math>\psi</math>, its [[wavefunction]] is described in Dirac or [[bra-ket notation]] as:
-:<math>
-|\psi \rang = \sum_i |i\rang \psi_i
-</math>
-where the <math> |i\rang</math>s specify the different quantum "alternatives" available (technically, they form an [[eigenvector]] [[Basis (linear algebra)|basis]]) and the <math>\psi_i</math> are the [[probability amplitude]] coefficients, which are [[complex numbers]].
-The probability of observing the system making a [[Atomic electron transition|transition]] or [[Quantum state|quantum leap]] from state <math>\psi</math> to a new state <math>\varphi</math> is the square of the modulus of the [[scalar product|scalar]] or [[inner product]] of the two states:
-:<math>\operatorname{prob}(\psi \Rightarrow \varphi) = |\lang \psi |\varphi \rang|^2 = |\sum_i\psi^*_i \varphi_i |^2
-</math>
-:<math>
-= \sum_{ij} \psi^*_i \psi_j \varphi^*_j\varphi_i= \sum_{i} |\psi_i|^2|\varphi_i|^2 + \sum_{ij;i \ne j} \psi^*_i \psi_j \varphi^*_j\varphi_i
-</math>
-where <math> \psi_i = \lang i|\psi \rang </math> (as defined above) and similarly <math> \varphi_i = \lang i|\varphi \rang </math> are the coefficients of the final state of the system. * is the [[complex conjugate]] so that <math> \psi_i^* = \lang \psi|i \rang </math>, etc.
-Now let's consider the situation classically and imagine that the system transited from <math>|\psi \rang</math> to <math>|\varphi \rang</math> via an intermediate state <math>|i\rang</math>.  Then we would ''classically'' expect the probability of the two-step transition to be the sum of all the possible intermediate steps.  So we would have
-:<math>
-\operatorname{prob}(\psi \Rightarrow \varphi) = \sum_i \operatorname{prob}(\psi \Rightarrow i \Rightarrow \varphi)
-</math>
-:<math>
-= \sum_i |\lang \psi |i \rang|^2|\lang i|\varphi \rang|^2 = \sum_i|\psi_i|^2 |\varphi_i|^2
-</math>,
-The classical and quantum derivations for the transition probability differ by the presence, in the quantum case, of the extra terms <math>\sum_{ij;i \ne j} \psi^*_i \psi_j \varphi^*_j\varphi_i</math>; these extra quantum terms represent ''interference'' between the different <math>i \ne j</math> intermediate "alternatives".  These are consequently known as the quantum interference terms, or cross terms.  This is a purely quantum effect and is a consequence of the non-additivity of the probabilities of quantum alternatives.
-The interference terms vanish, via the mechanism of [[quantum decoherence]], if the intermediate state <math> |i\rang</math> is measured or coupled with the environment.<ref name="zurek91">[[Wojciech H. Zurek]], "Decoherence and the transition from quantum to classical", ''Physics Today'', 44, pp 36–44 (1991)</ref><ref name="zurek03">[[Wojciech H. Zurek]], "[http://arxiv.org/abs/quant-ph/0105127 Decoherence, einselection, and the quantum origins of the classical]", ''Reviews of Modern Physics'' 2003, 75, 715.</ref>
-== See also ==
-*[[Active noise control]]
-*[[Beat (acoustics)]]
-*[[Coherence (physics)]]
-*[[Diffraction]]
-*[[Double-slit experiment]]
-*[[Young's Double Slit Interferometer]]
-*[[Haidinger fringes]]
-*[[Hong–Ou–Mandel effect]]
-*[[Interference lithography]]
-*[[Interferometer]]
-*[[List of types of interferometers]]
-*[[Lloyd's Mirror]]
-*[[Moiré pattern]]
-*[[Newton's rings]]
-*[[Thin-film interference]]
-*[[Optical feedback]]
-*[[Retroreflector]]
-*[[Upfade]]
-*[[Multipath interference]]
-*[[Inter-flow interference]]
-*[[Intra-flow interference]]
-*[[Bio-Layer Interferometry]]
-*[[N-slit interferometric equation]]
-==References==
-<References/>
-==External links==
-{{Commons|Interference}}
-{{Wiktionary|interference}}
-*[http://www.citycollegiate.com/interference1.htm Expressions of position and fringe spacing]
-*[http://www.falstad.com/ripple/ex-2source.html Java demonstration of interference]
-*[http://www.phy.hk/wiki/englishhtm/Interference.htm Java simulation of interference of water waves 1]
-*[http://www.phy.hk/wiki/englishhtm/Interference2.htm Java simulation of interference of water waves 2]
-*[http://www.acoustics.salford.ac.uk/feschools/waves/super2.htm Flash animations demonstrating interference]
-*[http://gerdbreitenbach.de/lissajous/lissajous.html Lissajous Curves: Interactive simulation of graphical representations of musical intervals, beats, interference, vibrating strings ]
-*[http://qed.wikina.org/interference/ Animations demonstrating optical interference] by QED
-{{DEFAULTSORT:Interference (Wave Propagation)}}
-[[Category:Interference| ]]
-[[Category:Wave mechanics]]
-{{Link FA|zh}}
-[[ca:Interferència (propagació d'ones)#Interferència òptica]]

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