# Double-slit experiment The **double-slit experiment** sends [[Electromagnetic_radiation|light]], [[Electron|electrons]] or whole molecules through two closely spaced openings and records where they land. The screen shows bright and dark bands, the signature of [[Wave_interference|wave interference]], and yet each arrival is a single point, one quantum at a time. The classical half of the result is exact and old: the bright orders sit where the path difference is a whole number of wavelengths, d·sin(theta) = m·lambda.[^up3-ch3] In the microsim below the reader turns two slits into N slits, from 2 up to 50, and changes the ratio of slit spacing d to slit width a and the wavelength: the principal maxima do not move, they sharpen as 1/N, N − 2 weak secondary maxima appear between them, and whole orders vanish whenever d/a is a whole number. The intensity drawn is I = I0·(sin(N·phi/2)/(N·sin(phi/2)))²·(sin(b)/b)², phi = 2·pi·d·sin(theta)/lambda, b = pi·a·sin(theta)/lambda. On the Physics flagship this article serves Part II — Core theories at the section *Optics: interference and diffraction* (row P32). It is the hinge of the flagship's second half: read one way it closes classical wave optics, alongside [[Diffraction|diffraction]] and the [[Diffraction_grating|diffraction grating]]; read the other way it opens [[Quantum_mechanics|quantum mechanics]], the experiment the [[Schrödinger_equation|Schrödinger equation]] was written to describe and that no account of measurement can avoid. ## Overview Two conditions make fringes. The sources must be coherent, their phase relation steady long enough to average over,[^up3-ch3] and the path difference to a point on the screen must be comparable to a wavelength. In the far field that difference is d·sin(theta), so bright fringes fall at d·sin(theta) = m·lambda and dark ones at (m + 1/2)·lambda, m = 0, ±1, ±2, …[^up3-ch3] Near the axis the fringes are evenly spaced, y_m ≈ m·lambda·D/d for slit-to-screen distance D, but that fails at large angles, where the position is D·tan(theta).[^up3-ch3] Because sin(theta) cannot exceed 1, only orders up to m_max = d/lambda exist. For the textbook's pair, d = 0.0100 mm lit by a helium–neon [[Laser|laser]] at 632.8 nm, d/lambda = 15.80, so fifteen orders appear each side and the third sits at 10.94° (derived).[^up3-ch3][^up3-ch4] The quantum content is not in any of that. It appears when the source is turned down until quanta cross the apparatus one at a time. The arrivals are random; the accumulated pattern is the same. Nothing in the wave picture explains the individual points, and nothing in the particle picture explains the bands. The experiment is therefore the standard exhibit for the claim that the [[Wave_function|wave function]] gives probabilities rather than a physical wave of matter, and Richard Feynman treated it as the only mystery of quantum mechanics.[^feynman] ## History [[Thomas_Young_(scientist)|Thomas Young]] reported the two-source experiment to the Royal Society in his 1804 Bakerian Lecture, using it to argue that light is a wave against the prevailing corpuscular account.[^young1804] Augustin-Jean Fresnel's diffraction theory and Maxwell's electromagnetism made the wave reading unavoidable for a century. The tension arrived with the quantum. [[Albert_Einstein|Einstein]]'s 1905 account of the [[Photoelectric_effect|photoelectric effect]] required light to deliver energy in lumps,[^einstein1905] and in 1909 G. I. Taylor ran a diffraction experiment at light levels so low that, by his estimate, only one quantum was in the apparatus at a time, and still obtained fringes after a three-month exposure.[^taylor1909] Louis de Broglie's 1924 thesis extended lambda = h/p to matter, and Clinton Davisson and Lester Germer confirmed it in 1927 by scattering electrons from a nickel crystal.[^davisson1927] Claus Jönsson demonstrated true multi-slit [[Electron_diffraction|electron interference]] with fabricated slits in 1961.[^jonsson1961] Pier Giorgio Merli, Gian Franco Missiroli and Giulio Pozzi recorded the build-up dot by dot in 1976,[^merli1976] and Akira Tonomura's group repeated it in 1989 with an electron biprism and single-electron detection.[^tonomura1989] In 1999 Markus Arndt and co-workers at Vienna obtained fringes with [[Buckminsterfullerene|C₆₀ molecules]], objects of 60 carbon atoms with internal temperature.[^arndt1999] Readers of *Physics World* voted the single-electron version the most beautiful experiment in physics in 2002.[^crease2002] ## Variations of the experiment The two-slit geometry is a template rather than a single apparatus, and almost every conceptual question in quantum mechanics has been posed as a variation on it. They fall into three families: change what goes through the slits, from photons to electrons to molecules; change the apparatus while keeping the logic, as beam-splitter interferometers do; or add a measurement and watch the fringes pay for it. The last family is where the physics is contested. ### Interference from individual particles Reducing the flux does not change the pattern, only the time taken to build it. In the Merli and Tonomura films the first few hundred electrons look like noise; by tens of thousands the fringes are unmistakable.[^merli1976][^tonomura1989] That rules out any explanation in which the fringes come from particles interacting with each other, since at these rates the previous particle is absorbed before the next is emitted. The same holds for [[Neutron_diffraction|neutrons]], atoms and large molecules, the C₆₀ work showing that internal structure and a hot internal state do not by themselves destroy the fringes.[^arndt1999] What does destroy them is any process that leaves a record of the path. Mass matters only through the de Broglie wavelength: at a given speed lambda = h/p falls as the particle gets heavier, so the fringe spacing shrinks until it is finer than any detector. That, together with the difficulty of keeping a large object from interacting with anything on the way, is what makes the experiment hard for molecules and impossible for everyday objects; nothing in the theory forbids it. ### Mach–Zehnder interferometer The Mach–Zehnder interferometer replaces the slits with a beam splitter, two mirrors and a second beam splitter. Its arms are separated in space, often by metres, so a phase shift can be applied to one alone. The double-slit logic survives: with both paths open the output ports show complementary interference, and blocking one arm sends half the intensity to each port. Because the arms are macroscopically distinct it is the preferred apparatus for delayed-choice and which-path work, on the same instrument logic as the [[Michelson_interferometer|Michelson interferometer]]. ### "Which-way" experiments and the principle of complementarity Any measurement that determines which slit the particle used destroys the fringes, and the loss is gradual: partial path information gives partial contrast. Bohr's principle of complementarity states that the wave and particle descriptions are both needed and never simultaneously applicable, and the double slit is his standard illustration. Marking the path with a [[Photon|photon]] of a second field, or with the recoil of a movable slit screen, suffices — the marker need never be read. The disturbance is not necessarily mechanical: the correlation between the particle and the marker, not the momentum kick, is what removes the interference. This is the sharpest quantitative statement the experiment supports. Writing V for the fringe visibility and D for the distinguishability of the two paths, every measurement so far obeys V² + D² ≤ 1, with equality for a pure state: partial path knowledge costs contrast at a fixed exchange rate rather than switching the fringes off at a threshold. A double slit with a variable-efficiency path marker traces out that curve. ### Delayed choice and quantum eraser variations John Wheeler asked in 1978 what happens if the decision to measure path or interference is taken after the particle has passed the slits. The answer, confirmed in later experiments, is that the outcome always matches the measurement actually made; no account survives in which the particle "decided" at the slits what to be. The quantum eraser goes further: path information stored in a second particle can be discarded, and fringes reappear in the subset of data sorted by the eraser's outcome, never in the raw total.[^kim2000] The pattern is recovered by post-selection, so no signal travels backwards in time. ### Weak measurement Weak measurement extracts a little information from each of many identical runs, disturbing each only slightly, and averages. Applied to a two-slit apparatus by Sacha Kocsis and colleagues in 2011, it produced average photon trajectories running from the slits to the screen — curves resembling the streamlines of the probability current.[^kocsis2011] They are ensemble averages, not paths of individual photons, and the limit is stated here because the images are easily read as more than they are. ### Other variations Single-slit [[Diffraction|diffraction]] is the same experiment with one opening, and N-slit interference is the [[Diffraction_grating|diffraction grating]], where commercial rulings exceed 1,000 lines per millimetre.[^up3-ch4] A macroscopic analogue exists in fluid mechanics: a droplet bouncing on a vibrating bath is guided by the wave it makes and, passing a two-slit barrier, produces a statistical pattern resembling interference.[^couder2006] Whether the analogy runs deeper than the mathematics of a pilot wave is disputed, and no claim is made here that it does. ## Classical wave-optics formulation For N identical slits of width a, spaced d apart and illuminated at wavelength lambda, the far-field intensity is the product of an interference factor set by d and a single-slit envelope set by a: I = I0 · (sin(N·phi/2)/(N·sin(phi/2)))² · (sin(b)/b)², phi = 2·pi·d·sin(theta)/lambda, b = pi·a·sin(theta)/lambda. This is the equation the microsim computes, and it combines two results the source states separately: the positions from d·sin(theta) = m·lambda,[^up3-ch3] and the heights from the single-slit intensity I = I0·(sin(b)/b)², which the book derives by summing phasors and letting their number go to infinity.[^up3-ch4] The displayed right-hand sides were lost in the extraction of that text, the N-slit amplitude fraction among them, so both bracketed factors above are standard forms supplied against the surviving results rather than transcribed — an ILLUSTRATIVE reconstruction, and one of this page's open items.[^manual-gap] The reader has three controls: N from 2 to 50, the ratio d/a, and lambda. Three things follow. Adding slits does not move the principal maxima — they stay at d·sin(theta) = m·lambda — but sharpens them, because sin(N·phi/2) oscillates N times faster than its denominator; their width falls as 1/N, which is why a grating resolves what two slits cannot.[^up3-ch3] Exactly N − 2 secondary maxima appear between neighbouring principal ones, fainter as N grows. And an order goes missing whenever an interference maximum meets an envelope zero, which happens when d/a is a whole number.[^up3-ch4] In the textbook's worked case, a = 0.020 mm, d = 0.20 mm and lambda = 500 nm, the first envelope zero is at sin(theta) = lambda/a = 0.025, where the order d/a = 10 is suppressed, so nine orders survive each side and the central peak holds 19 bright fringes,[^up3-ch4] in general 2·ceil(d/a) − 1 (derived). The single-slit factor rewards a second look. Its minima are at a·sin(theta) = m·lambda, its side maxima fall slightly short of b = (m + 1/2)·pi, and the book's circle estimate puts the first side maximum at 1/(1.5·pi)² = 0.045 of the peak; the exact root is b = 1.4303·pi at a height of 0.0472 (derived).[^up3-ch4] Beside the curve the microsim draws a chain of 30 phasors at the pointer's angle: straight on axis, curling as the angle grows, closing into a full circle exactly at each minimum — the geometric reason the intensity is zero there rather than merely small. At N = 50 and d/lambda = 15.8 the pattern has roughly 1,600 zeros across the screen, so the sim either takes about 13,000 samples or restricts itself to |sin(theta)| ≤ 0.2.[^manual-gap] The same arithmetic runs backwards to measure something. In the 1887 Michelson–Morley experiment a shift of dN = 2·L·v²/(lambda·c²) fringes was expected on rotating an interferometer of arm length L through a supposed ether wind v; with L = 11 m, v = 3×10⁴ m/s and lambda = 589 nm that is 0.37 fringe, and the observed shift was a small fraction of it.[^michelson1887] The [[Speed_of_light|speed of light]] came out the same in every direction, and [[Special_relativity|special relativity]] followed. ## Path-integral formulation Feynman's formulation assigns to every path from source to detection point an amplitude exp(i·S/hbar), with S the classical action along that path, and adds them. The double slit is the simplest truncation of that sum: two paths instead of infinitely many, whose actions differ by the path length times the wavenumber, so the relative phase is 2·pi·d·sin(theta)/lambda and two unit phasors add to the cos² fringe pattern directly.[^feynman] The phasor chain drawn in the microsim is literally this sum for the continuum of paths through one open slit, which is why it curls: neighbouring paths differ in phase by a fixed increment. The formulation also explains why the classical path dominates for heavy objects — away from the stationary point of S the phases cancel, and the width of the surviving bundle shrinks as hbar/S, so the [[Correspondence_principle|correspondence principle]] appears here as a statement about [[Calculus_of_variations|stationary action]] rather than about large quantum numbers. ## Interpretations of the experiment Every interpretation of quantum mechanics reproduces the same numbers for this experiment; they differ over what is happening between emission and detection. None is distinguished by any measurement yet made here, so the sections below are not competing predictions but competing accounts of one uncontested calculation, each stated as its proponents state it rather than as its critics do. ### Standard quantum physics On the minimal account the two slits contribute amplitudes that add, the squared modulus of the sum gives the probability [[Probability_density_function|density]] on the screen, and the theory says nothing further about the route. The formalism is complete as a predictive instrument, and the question of the path is held to be outside its scope. ### Complementarity [[Niels_Bohr|Bohr]]'s reading makes the apparatus part of the phenomenon: a setup that reveals path is a different phenomenon from one that reveals fringes, and asking for both at once is asking about an experiment that cannot be built. The wave and particle pictures are complementary descriptions of one whole. ### Copenhagen interpretation In the broader Copenhagen tradition the wave function is a device for computing outcomes of measurements described in classical terms, and it collapses on detection. The fringes belong to the wave function; the single spot belongs to the irreversible act of registration. ### Relational interpretation The relational reading takes states to be relative to a physical system rather than absolute. Whether the particle has a definite path is not a property of the particle but of its relation to whatever it has interacted with, so the presence or absence of fringes is a fact about a correlation, not about a hidden trajectory. ### Many-worlds interpretation The many-worlds account keeps unitary evolution and drops collapse: both branches persist, detector and observer become entangled with them, and the apparent single outcome is what one branch records. Interference survives while the branches stay coherent and vanishes when the environment records the path — decoherence rather than collapse. ### De Broglie–Bohm theory De Broglie–Bohm theory restores definite trajectories, each particle passing through exactly one slit, guided by a wave that passes through both. The statistics match the standard theory because the initial positions are distributed as the squared wave function. The guiding wave is non-local, and the trajectories the theory predicts resemble the averaged curves recovered by weak measurement.[^kocsis2011] ## See also - [[Diffraction]] - [[Diffraction_grating]] - [[Wave_interference]] - [[Thin-film_interference]] - [[Michelson_interferometer]] - [[Electron_diffraction]] - [[Schrödinger_equation]] - [[Uncertainty_principle]] ## References [^up3-ch3]: Sanny, Jeff; Ling, Samuel; et al. (2016). *University Physics Volume 3*. OpenStax. Chapter 3, "Interference", pp. 119–144 (coherence at p. 121; constructive and destructive path differences at p. 122; d·sin θ = mλ at p. 123; y_m ≈ mλD/d at p. 124; the He-Ne example and m_max = 15 at pp. 124–125; N-slit principal and secondary maxima at p. 126; the He-Ne wavelength 632.8 nm at pp. 135–136). Portal Book 079. Open Textbook Library record: https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^up3-ch4]: Sanny, Ling et al. (2016). *University Physics Volume 3*. Chapter 4, "Diffraction", pp. 145–182 (far-field condition at p. 147; single-slit minima at pp. 148 and 153; β and the phasor construction at pp. 150–152; I = I0(sin β/β)² at pp. 152–153; the circle estimate 1/(1.5π)² at pp. 151–152; the product of interference and envelope and missing orders at pp. 155–157; the 19-fringe example at p. 157; commercial gratings above 1,000 lines/mm at p. 158). Portal Book 079. [^manual-gap]: Portal Books research note, sub-manual 04 "Atomic, Quantum, Statistical and Electromagnetic Physics", §8.1, §8.4, §8.5 and §A.1: in the extracted text of *University Physics Volume 3* every displayed right-hand side in Chapter 3 (pp. 122–134) and Chapter 4 (pp. 148–182) was lost, including the N-slit secondary-maximum amplitude fraction at p. 126; standard forms were supplied and are to be verified against the PDF pages (pages to pin). The same note records the sampling requirement of roughly 13,000 points at N = 50. [^young1804]: Young, Thomas (1804). "The Bakerian Lecture: Experiments and calculations relative to physical optics." *Philosophical Transactions of the Royal Society of London*, 94: 1–16. [^einstein1905]: Einstein, Albert (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt." *Annalen der Physik*, 322 (6): 132–148. [^taylor1909]: Taylor, Geoffrey Ingram (1909). "Interference fringes with feeble light." *Proceedings of the Cambridge Philosophical Society*, 15: 114–115. [^davisson1927]: Davisson, C.; Germer, L. H. (1927). "Diffraction of electrons by a crystal of nickel." *Physical Review*, 30 (6): 705–740. https://doi.org/10.1103/PhysRev.30.705 [^jonsson1961]: Jönsson, Claus (1961). "Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten." *Zeitschrift für Physik*, 161: 454–474. [^merli1976]: Merli, Pier Giorgio; Missiroli, Gian Franco; Pozzi, Giulio (1976). "On the statistical aspect of electron interference phenomena." *American Journal of Physics*, 44 (3): 306–307. [^tonomura1989]: Tonomura, Akira; Endo, Junji; Matsuda, Tsuyoshi; Kawasaki, Takeshi; Ezawa, Hiroshi (1989). "Demonstration of single-electron buildup of an interference pattern." *American Journal of Physics*, 57 (2): 117–120. [^arndt1999]: Arndt, Markus; Nairz, Olaf; Vos-Andreae, Julian; Keller, Claudia; van der Zouw, Gerbrand; Zeilinger, Anton (1999). "Wave–particle duality of C₆₀ molecules." *Nature*, 401: 680–682. [^crease2002]: Crease, Robert P. (2002). "The most beautiful experiment." *Physics World*, volume 15, September 2002 (reader poll result). [^kim2000]: Kim, Yoon-Ho; Yu, Rong; Kulik, Sergei P.; Shih, Yanhua; Scully, Marlan O. (2000). "Delayed 'choice' quantum eraser." *Physical Review Letters*, 84 (1): 1–5. https://doi.org/10.1103/PhysRevLett.84.1 [^kocsis2011]: Kocsis, Sacha; Braverman, Boris; Ravets, Sylvain; Stevens, Martin J.; Mirin, Richard P.; Shalm, L. Krister; Steinberg, Aephraim M. (2011). "Observing the average trajectories of single photons in a two-slit interferometer." *Science*, 332 (6034): 1170–1173. [^couder2006]: Couder, Yves; Fort, Emmanuel (2006). "Single-particle diffraction and interference at a macroscopic scale." *Physical Review Letters*, volume 97. [^michelson1887]: Michelson, Albert A.; Morley, Edward W. (1887). "On the relative motion of the Earth and the luminiferous ether." *American Journal of Science*, Series 3, 34: 333–345. [^feynman]: Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew. *The Feynman Lectures on Physics*, Volume III, Chapter 1 ("Quantum Behavior") and Chapter 3 ("Probability Amplitudes"); the path-integral rule is developed further in Volume II, Chapter 19. ### Further reading - Sanny, Ling et al. (2016). *University Physics Volume 3*. OpenStax. Chapters 3 and 4, pp. 119–182. Portal Book 079 — the page-cited source for every classical result on this page. - Likharev, Konstantin (2013). *Part QM: Quantum Mechanics*. Chapter 1, pp. 5–30 (the wave function and the probability interpretation the two-slit result forces). Portal Book 047. Open Textbook Library record: https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics - Feynman, Leighton and Sands, *The Feynman Lectures on Physics*, Volume III, Chapter 1. ## External links - [University Physics Volume 3](https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3), OpenStax, Open Textbook Library record — Portal Book 079 ### Interactive animations The Wikitube microsim above is this page's interactive treatment; the pair's *External links* section lists further animations, none of which Wikitube has checked and therefore none of which it links. ### Single particle experiments The single-particle work is cited in the body from the primary papers: Taylor (1909), Merli, Missiroli and Pozzi (1976), Tonomura and colleagues (1989) and Arndt and colleagues (1999). ### Hydrodynamic analog The bouncing-droplet analogue is cited in the body from Couder and Fort (2006); Wikitube links no secondary video of it. ### Computer simulations Numerical treatments belong to [[Computational_physics|computational physics]]; the sampling requirement for the N-slit pattern is recorded in this page's source note. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Double-slit_experiment.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Double-slit experiment* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Double-slit_experiment.html" data-title="Double-slit experiment"></div> *Built from `MICROSIM_GUIDE/specs/sims/Double-slit_experiment.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Double-slit_experiment) : [Wikitube](https://en.wikitube.io/wiki/Double-slit_experiment) · pinned revision [1373964616](https://en.wikipedia.org/w/index.php?oldid=1373964616) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. 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