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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 Universality and quantum systems ： ウィキペディア英語版 Universality and quantum systems In quantum mechanics, universality is the observation that there are properties for a large class of systems that are independent of the exact structural details of the system. The notion of universality is familiar in the study and application of statistical mechanics to various physical systems since its introduction in a very precise fashion by Leo Kadanoff. Although not quite the same, it is closely related to universality as applied to quantum systems. This concept links to the essence of renormalization and scaling in many problems. Renormalization is based on the notion that a measurement device of wavelength $\backslash lambda$ is insensitive to details of structure at distances much smaller than $\backslash lambda$. An important consequence of universality is that one can mimic the ''real'' short-structure distance of the measurement device and the system to be measured by ''simple'' short-distance structure. Even though it is seen that scaling, universality and renormalization are closely related, they are not to be used interchangeably.〔Stanley E., ''Scaling, universality and renormalization: Three pillars of modern critical phenomena'' Reviews of Modern Physics, Vol. 71, No. 2, Centenary 1999〕 ==Universal Characteristics of Shapes of Potential Steps== For low energy particles, the exact detail of the potential step is irrelevant. This means that the intermediate change in the potential between $0$ and $V\_0$ is irrelevant for particles with extremely low energies. An important consequence of this is that theoretical calculations made for a Heaviside step function potential work perfectly well for potentials that have a finite spatial rate of change in potential: finite ∆$V$ over ∆$x$, which is the case with physical situations, for e.g. - Quantum confinement systems of nano dots, etc. For a general potential barrier: :$V(x)\; =$ \begin 0 & x < -a\\ V(x) & -a < x < a\\ V_0 & a < x \end, The time-dependent Schrödinger's equation in one-dimension is: :$i\backslash hbar\backslash frac\; \backslash Psi(x,t)\; =\; \backslash left\; (\backslash frac\backslash nabla^2\; +\; V(x,t)\backslash right)\; \backslash Psi(x,t)$ It is instructive to analyse this equation after expanding $\backslash Psi(x,t)$ in the plane wave basis $\backslash exp()$. $p$, then, is real for $E$ > $V$. For when $E$ < $V$, the relevant length scale in this problem is the range of the wavefunction under the barrier which is given by: $r\; =\; \backslash frac\{\backslash sqrt\{2mE\}\}$. For low-energy waves incident on this potential from the left: $\backslash lambda$ ≫ $a$. Essentially, for low energy, or large wavelength, particles, the exact detail of the potential barrier is irrelevant. This is a feature that is universal to different $V(x)$ for a well behaved function $V(x)$. 〔Avery J., ''Hyperspherical Harmonics and Generalized Sturmians'' 1989〕 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「Universality and quantum systems」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.