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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Born's rule ： ウィキペディア英語版 Born rule The Born rule (also called the Born law, Born's rule, or Born's law) is a 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 derive the Born rule from the other assumptions of quantum mechanics, with inconclusive results; for example, while it has been claimed that Born's law can be derived from the Many Worlds Interpretation, the proofs have been criticized as circular.〔N.P. Landsman, ("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〕 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.〔(Fuchs, C. A. ''QBism: the Perimeter of Quantum Bayesianism'' 2010 )〕 The Born rule can however be derived in the ambit of so-called Hidden-Measurements Interpretation of quantum mechanics, by averaging over all possible measurement-interactions that can take place between the quantum entity and the measuring system.〔Aerts, D. (1986). A possible explanation for the probabilities of quantum mechanics, Journal of Mathematical Physics, 27, pp. 202-210.〕〔Aerts, D. and Sassoli de Bianchi, M. (2014). The extended Bloch representation of quantum mechanics and the hidden-measurement solution to the measurement problem. Annals of Physics 351, Pages 975–1025 ((Open Access )).〕 == The rule == The Born rule states that if an observable corresponding to a Hermitian operator $A$ with discrete spectrum is measured in a system with normalized wave function $\backslash scriptstyle|\backslash psi\backslash rang$ (''see'' bra–ket notation), then * the measured result will be one of the eigenvalues $\backslash lambda$ of $A$, and * the probability of measuring a given eigenvalue $\backslash lambda\_i$ will equal $\backslash scriptstyle\backslash lang\backslash psi|P\_i|\backslash psi\backslash rang$, where $P\_i$ is the projection onto the eigenspace of $A$ corresponding to $\backslash lambda\_i$. :(In the case where the eigenspace of $A$ corresponding to $\backslash lambda\_i$ is one-dimensional and spanned by the normalized eigenvector $\backslash scriptstyle|\backslash lambda\_i\backslash rang$, $P\_i$ is equal to $\backslash scriptstyle|\backslash lambda\_i\backslash rang\backslash lang\backslash lambda\_i|$, so the probability $\backslash scriptstyle\backslash lang\backslash psi|P\_i|\backslash psi\backslash rang$ is equal to $\backslash scriptstyle\backslash lang\backslash psi|\backslash lambda\_i\backslash rang\backslash lang\backslash lambda\_i|\backslash psi\backslash rang$. Since the complex number $\backslash scriptstyle\backslash lang\backslash lambda\_i|\backslash psi\backslash rang$ is known as the ''probability amplitude'' that the state vector $\backslash scriptstyle|\backslash psi\backslash rang$ assigns to the eigenvector $\backslash scriptstyle|\backslash lambda\_i\backslash rang$, 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 $\backslash scriptstyle|\backslash lang\backslash lambda\_i|\backslash psi\backslash rang|^2$.) In the case where the spectrum of $A$ is not wholly discrete, the spectral theorem proves the existence of a certain projection-valued measure $Q$, the spectral measure of $A$. In this case, * the probability that the result of the measurement lies in a measurable set $M$ will be given by $\backslash scriptstyle\backslash lang\backslash psi|Q(M)|\backslash psi\backslash rang$. If we are given a wave function $\backslash scriptstyle\backslash psi$ for a single structureless particle in position space, this reduces to saying that the probability density function $p(x,y,z)$ for a measurement of the position at time $t\_0$ will be given by $p(x,y,z)=$$\backslash scriptstyle|\backslash psi(x,y,z,t\_0)|^2.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Born rule」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.