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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 Englert–Greenberger–Yasin duality relation ： ウィキペディア英語版 Englert–Greenberger–Yasin duality relation The Englert–Jaeger–Shimony–Greenberger–Yasin duality relation relates the visibility, $V$, of interference fringes with the definiteness, or distinguishability, $D$, of the photons' paths in quantum optics. As an inequality: :$D^+\; V^\backslash le\; 1\; \backslash ,$ Although it is treated as a single relation, it actually involves two separate relations, which mathematically look very similar. The first relationship was first experimentally shown by Greenberger and Yasin in 1988. It was later theoretically derived by Jaeger, Shimony, and Vaidman in 1995. This relation involves correctly guessing which of the two paths the particle would have taken, based on the initial preparation. Here $D$ can be called the predictatbility, and is sometimes denoted by $P$. A year later Englert, in 1996, apparently unaware of this result, derived a related relation which dealt with knowledge of the two paths using an apparatus. Here $D$ is called the distinguishability. ==The mathematics of two-slit diffraction== This section reviews the mathematical formulation of the double-slit experiment. The formulation is in terms of the diffraction and interference of waves. The culmination of the development is a presentation of two numbers that characterizes the visibility of the interference fringes in the experiment, linked together as the Englert–Greenberger duality relation. The next section will discuss the orthodox quantum mechanical interpretation of the duality relation in terms of wave-particle duality. Of this experiment, Richard Feynman once said that it ''"has in it the heart of quantum mechanics. In reality it contains the only mystery."'' The wave function in the Young double-aperture experiment can be written as :$\backslash Psi\_(x)+\backslash Psi\_(x).$ The function :$\backslash Psi\_A(x)=C\_A\; \backslash Psi\_(x-x\_A)$ is the wave function associated with the pinhole at ''A'' centered on $x\_A$; a similar relation holds for pinhole ''B''. The variable $x$ is a position in space downstream of the slits. The constants $C\_A$ and $C\_B$ are proportionality factors for the corresponding wave amplitudes, and $\backslash Psi\_0(x)$ is the single hole wave function for an aperture centered on the origin. The single-hole wave-function is taken to be that of Fraunhofer diffraction; the pinhole shape is irrelevant, and the pinholes are considered to be idealized. The wave is taken to have a fixed incident momentum $p\_0=h/\backslash lambda$: :$\backslash Psi\_(x)\backslash propto\; \backslash frac$ where $|x|$ is the radial distance from the pinhole. To distinguish which pinhole a photon passed through, one needs some measure of the distinguishability between pinholes. Such a measure is given by〔Actually, what is called "distinguishability $D$" here is usually referred to as "predictability $P$".〕 :$D=|P\_A-P\_B|,\; \backslash ,$ where $P\_$ and $P\_$ are the probabilities of finding that the particle passed through aperture ''A'' and aperture ''B'' respectively. Since the Born probability measure is given by :$P\_=\backslash frac|^2\}$ and :$P\_=\backslash frac|^2\}$ then we get: :$D=\backslash left|\backslash ;\backslash frac\backslash ,\backslash right|$ We have in particular $D=0$ for two symmetric holes and $D=1$ for a single aperture (perfect distinguishability). In the far-field of the two pinholes the two waves interfere and produce fringes. The intensity of the interference pattern at a point ''y'' in the focal plane is given by :$I(y)\backslash propto\; 1+V\backslash cos$ where $p\_y=\; h/\backslash lambda\backslash cdot\; \backslash sin(\backslash alpha)$ is the momentum of the particle along the ''y'' direction, $\backslash phi=\backslash text(C\_A)-\backslash text(C\_B)$ is a fixed phase shift, and $d$ is the separation between the two pinholes. The angle α from the horizontal is given by $\backslash sin(\backslash alpha)\backslash simeq\; \backslash tan(\backslash alpha)=y/L$ where $L$ is the distance between the aperture screen and the far field analysis plane. If a lens is used to observe the fringes in the rear focal plane, the angle is given by $\backslash sin(\backslash alpha)\backslash simeq\; \backslash tan(\backslash alpha)=y/f$ where $f$ is the focal length of the lens. The visibility of the fringes is defined by :$V=\backslash frac$ where $I\_\backslash mathrm$ and $I\_\backslash mathrm$ denote the maximum and minimum intensity of the fringes respectively. By the rules of constructive and destructive interference we have :$I\_\backslash mathrm\; \backslash propto\; ||C\_A|+|C\_B||^2$ :$I\_\backslash mathrm\; \backslash propto\; ||C\_A|-|C\_B||^2$ Equivalently, this can be written as :$V=2\backslash frac.$ And hence we get, for a single photon in a pure quantum state, the duality relation :$\backslash beginV^2+D^2\; =\; 1\; \backslash end$ There are two extremal cases with a straightforward intuitive interpretation: In a single hole experiment, the fringe visibility is zero (as there are no fringes). That is, $V=0$ but $D=1$ since we know (by definition) which hole the photon passed through. On the other hand, for a two slit configuration, where the two slits are indistinguishable with $D=0$, one has perfect visibility with $I\_\backslash mathrm\; =\; 0$ and hence $V=1$. Hence in both these extremal cases we also have $V^+D^=1$. The above presentation was limited to a pure quantum state. More generally, for a mixture of quantum states, one will have :$V^+D^\backslash leq\; 1.\; \backslash ,$ For the remainder of the development, we assume the light source is a laser, so that we can assume $V^+D^=1$ holds, following from the coherence properties of laser light. 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「Englert–Greenberger–Yasin duality relation」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.