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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 Square potential ： ウィキペディア英語版 Rectangular potential barrier In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. The problem consists of solving the one-dimensional time-independent Schrödinger equation for a particle encountering a rectangular potential energy barrier. It is usually assumed, as here, that a free particle impinges on the barrier from the left. Although a particle hypothetically behaving as a point mass would be reflected, a particle actually behaving as a matter wave has a finite probability that it will penetrate the barrier and continue its travel as a wave on the other side. In classical wave-physics, this effect is known as evanescent wave coupling. The likelihood that the particle will pass through the barrier is given by the transmission coefficient, whereas the likelihood that it is reflected is given by the reflection coefficient. Schrödinger's wave-equation allows these coefficients to be calculated. ==Calculation== The time-independent Schrödinger equation for the wave function $\backslash psi(x)$ reads :$H\backslash psi(x)=\backslash left()\backslash psi(x)=E\backslash psi(x)$ where $H$ is the Hamiltonian, $\backslash hbar$ is the (reduced) Planck constant, $m$ is the mass, $E$ the energy of the particle and :$V(x)=V\_0()$ is the barrier potential with height $V\_0\; >$ 0 and width $a$. is the Heaviside step function. The barrier is positioned between $x=0$ and $x=a$. The barrier can be shifted to any $x$ position without changing the results. The first term in the Hamiltonian, $-\backslash frac\backslash frac\backslash psi$ is the kinetic energy. The barrier divides the space in three parts (). In any of these parts, the potential is constant, meaning that the particle is quasi-free, and the solution of the Schrödinger equation can be written as a superposition of left and right moving waves (see free particle). If $E>V\_0$ : : :$\backslash psi\_R(x)=\; C\_r\; e^\; +\; C\_l\; e^\backslash quad\; x>a$ where the wave numbers are related to the energy via : The index r/l on the coefficients A and B denotes the direction of the velocity vector. Note that, if the energy of the particle is below the barrier height, $k\_1$ becomes imaginary and the wave function is exponentially decaying within the barrier. Nevertheless, we keep the notation r/l even though the waves are not propagating anymore in this case. Here we assumed $E\backslash neq\; V\_0$. The case $E=V\_0$ is treated below. The coefficients $A,\; B,\; C$ have to be found from the boundary conditions of the wave function at $x=0$ and $x=a$. The wave function and its derivative have to be continuous everywhere, so. :$\backslash psi\_L(0)=\backslash psi\_C(0)$ :$\backslash frac\backslash psi\_L(0)\; =\; \backslash frac\backslash psi\_C(0)$ :$\backslash psi\_C(a)=\backslash psi\_R(a)$ :$\backslash frac\backslash psi\_C(a)\; =\; \backslash frac\backslash psi\_R(a)$. Inserting the wave functions, the boundary conditions give the following restrictions on the coefficients :$A\_r+A\_l=B\_r+B\_l$ :$ik\_0(A\_r-A\_l)=ik\_1(B\_r-B\_l)$ :$B\_re^+B\_le^=C\_re^+C\_le^$ :$ik\_1(B\_re^-B\_le^)=ik\_0(C\_re^-C\_le^)$. 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「Rectangular potential barrier」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.