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| :''Not to be confused with the [[Cauchy–Born rule]] in crystal mechanics.''
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| The '''Born rule''' (also called the '''Born law''', '''Born's rule''', or '''Born's law''') is a [[Physical law|law]] of [[quantum mechanics]] which gives the [[probability]] that a [[measurement]] on a [[quantum system]] will yield a given result. It is named after its originator, the physicist [[Max Born]]. The Born rule is one of the key principles of quantum mechanics. There have been many attempts to [[Formal proof|derive]] the Born rule from the other assumptions of [[quantum mechanics]], with inconclusive results; the [[Many Worlds Interpretation]] for example cannot derive the Born rule.<ref>N.P. Landsman, [http://www.math.ru.nl/~landsman/Born.pdf "The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle."], in ''Compendium of Quantum Physics'' (eds.) F.Weinert, K. Hentschel, D.Greenberger and B. Falkenburg (Springer, 2008), ISBN 3-540-70622-4</ref> However, within the [[Quantum Bayesianism]] interpretation of quantum theory, it has been shown to be an extension of the standard [[Law of Total Probability]], which takes into account the Hilbert space dimension of the physical system involved.<ref>[http://arxiv.org/pdf/1003.5209v1.pdf Fuchs, C. A. ''QBism: the Perimeter of Quantum Bayesianism'' 2010 ]</ref>
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| == The rule ==
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| The Born rule states that if an [[observable]] corresponding to a [[Hermitian operator]]
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| <math>A</math>
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| with discrete [[spectrum (functional analysis)|spectrum]] is measured in a system with normalized [[wave function]]
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| <math>\scriptstyle|\psi\rang</math>
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| (''see'' [[bra-ket notation]]), then
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| * the measured result will be one of the [[Eigenvalue, eigenvector and eigenspace|eigenvalues]] <math>\lambda</math> of <math>A</math>, and
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| * the probability of measuring a given eigenvalue <math>\lambda_i</math> will equal <math>\scriptstyle\lang\psi|P_i|\psi\rang</math>, where <math>P_i</math> is the projection onto the eigenspace of <math>A</math> corresponding to <math>\lambda_i</math>.
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| :(In the case where the eigenspace of <math>A</math> corresponding to <math>\lambda_i</math> is one-dimensional and spanned by the normalized eigenvector <math>\scriptstyle|\lambda_i\rang</math>, <math>P_i</math> is equal to <math>\scriptstyle|\lambda_i\rang\lang\lambda_i|</math>, so the probability <math>\scriptstyle\lang\psi|P_i|\psi\rang</math> is equal to <math>\scriptstyle\lang\psi|\lambda_i\rang\lang\lambda_i|\psi\rang</math>. Since the [[complex number]] <math>\scriptstyle\lang\lambda_i|\psi\rang</math> is known as the ''[[probability amplitude]]'' that the state vector <math>\scriptstyle|\psi\rang</math> assigns to the eigenvector <math>\scriptstyle|\lambda_i\rang</math>, it is common to describe the Born rule as telling us that probability is equal to the amplitude-squared (really the amplitude times its own [[complex conjugate]]). Equivalently, the probability can be written as <math>\scriptstyle|\lang\lambda_i|\psi\rang|^2</math>.)
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| In the case where the spectrum of <math>A</math> is not wholly discrete, the [[spectral theorem]] proves the existence of a certain [[projection-valued measure]] <math>Q</math>, the spectral measure of <math>A</math>. In this case,
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| * the probability that the result of the measurement lies in a measurable set <math>M</math> will be given by <math>\scriptstyle\lang\psi|Q(M)|\psi\rang</math>.
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| If we are given a wave function
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| <math>\scriptstyle\psi</math>
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| for a single structureless particle in position space,
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| this reduces to saying that the probability density function
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| <math>p(x,y,z)</math>
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| for a measurement of the position at time
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| <math>t_0</math>
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| will be given by
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| <math>p(x,y,z)=</math><math>\scriptstyle|\psi(x,y,z,t_0)|^2.</math>
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| == History ==
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| The Born rule was formulated by Born in a 1926 paper.<ref>''Zur Quantenmechanik der Stoßvorgänge'', Max Born, Zeitschrift für Physik, ''37'', #12 (Dec. 1926), pp. 863–867 (German); English translation, ''On the quantum mechanics of collisions'', in ''Quantum theory and measurement'', section I.2, J. A. Wheeler and W. H. Zurek, eds., Princeton, NJ: Princeton University Press, 1983, ISBN 0-691-08316-9.</ref>
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| In this paper, Born solves the [[Schrödinger equation]] for a scattering problem and, inspired by Einstein's work on the photoelectric effect,<ref>[http://nobelprize.org/physics/laureates/1954/born-lecture.pdf "Again an idea of Einstein’s gave me the lead. He had tried to make the duality of particles - light quanta or photons - and waves comprehensible by interpreting the square of the optical wave amplitudes as probability density for the occurrence of photons. This concept could at once be carried over to the psi-function: |psi|<sup>2</sup> ought to represent the probability density for electrons (or other particles)."] from Born's Nobel Lecture on the statistical interpretation of quantum mechanics</ref> concluded, in a footnote, that the Born rule gives the only possible interpretation of the solution. In 1954, together with [[Walter Bothe]], Born was awarded the Nobel Prize in Physics for this and other work.<ref>[http://nobelprize.org/physics/laureates/1954/born-lecture.pdf Born's Nobel Lecture on the statistical interpretation of quantum mechanics]</ref> [[John von Neumann]] discussed the application of [[spectral theory]] to Born's rule in his 1932 book.<ref>''Mathematische Grundlagen der Quantenmechanik'', John von Neumann, Berlin: Springer, 1932 (German); English translation ''Mathematical Foundations of Quantum Mechanics'', transl. Robert T. Beyer, Princeton, NJ: Princeton University Press, 1955.</ref>
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| ==References==
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| {{reflist}}
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| ==See also==
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| *[[Gleason's theorem]]
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| *[[Transactional interpretation]] of quantum mechanics
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| ==External links==
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| *[http://www.sciencedaily.com/releases/2010/07/100722142640.htm Quantum Mechanics Not in Jeopardy: Physicists Confirm a Decades-Old Key Principle Experimentally] ScienceDaily (July 23, 2010)
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| {{DEFAULTSORT:Born Rule}}
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| [[Category:Quantum measurement]]
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Wilber Berryhill is the name his mothers and fathers gave him and he totally digs that name. He functions as a bookkeeper. My spouse and I live in Mississippi and I love every day residing right here. To climb is some thing I truly appreciate performing.
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