2重障壁ポテンシャルに障壁よりも低いエネルギーをもつ台形波束を入射して共鳴トンネル現象が観測されるか調べた。時間を含む1次元シュレディンガー方程式を数値的に解き,パソコン画面上で波束の運動を観察した。共鳴は,2重障壁の間に波束が長時間にわたって多量にたまることによって識別できた。また,波束をフーリエ変換して得た波数分布から透過波と反射波の割合をみることによって,波束が2重障壁をトンネル透過する程度が判別できた。2重障壁の固有の量子エネルギー準位に対応して台形波束が2重障壁を共鳴的にトンネル透過することが,いろいろな条件の場合について,鮮明に観測された。その結果,2重障壁は特定のエネルギーを持つ波束すなわち粒子のみを選択的に透過させる優れたフィルターの役割をすることがわかった。|A resonance tunnel effect was investigated by leading a wave-packet of trapezoid shape into double potential barriers. The time-dependent Schrödinger equation was numerically solved, and the motion of a wave packet was observed on the screen of a personal computer. The state that a wave packet is resonant with double potential barriers showed a distinct phenomenon that a large part of a wave packet is stored between two barriers for a long time. Tunneling of a wave packet was clearly recognized by comparing an amount of the transmitted wave with that of the reflected one in the distribution of wave vectors that was obtained by the Fourier transformation of a wave packet. A resonance tunnel effect was demonstrated for various kinds of double potential barriers. It was showed that a couple of double potential barriers functions as an excellent filter that allows particles with specified energies to pass through.
2重障壁ポテンシャルに障壁よりも低いエネルギーをもつ台形波束を入射して共鳴トンネル現象が観測されるか調べた。時間を含む1次元シュレディンガー方程式を数値的に解き,パソコン画面上で波束の運動を観察した。共鳴は,2重障壁の間に波束が長時間にわたって多量にたまることによって識別できた。また,波束をフーリエ変換して得た波数分布から透過波と反射波の割合をみることによって,波束が2重障壁をトンネル透過する程度が判別できた。2重障壁の固有の量子エネルギー準位に対応して台形波束が2重障壁を共鳴的にトンネル透過することが,いろいろな条件の場合について,鮮明に観測された。その結果,2重障壁は特定のエネルギーを持つ波束すなわち粒子のみを選択的に透過させる優れたフィルターの役割をすることがわかった。|A resonance tunnel effect was investigated by leading a wave-packet of trapezoid shape into double potential barriers. The time-dependent Schrödinger equation was numerically solved, and the motion of a wave packet was observed on the screen of a personal computer. The state that a wave packet is resonant with double potential barriers showed a distinct phenomenon that a large part of a wave packet is stored between two barriers for a long time. Tunneling of a wave packet was clearly recognized by comparing an amount of the transmitted wave with that of the reflected one in the distribution of wave vectors that was obtained by the Fourier transformation of a wave packet. A resonance tunnel effect was demonstrated for various kinds of double potential barriers. It was showed that a couple of double potential barriers functions as an excellent filter that allows particles with specified energies to pass through.
1次元の周期ポテンシャル中を進む電子波のBragg反射とBloch振動の様子を, 時間に依存する1次元シュレディンガー方程式を数値的に解くことによって調べた。電子波をGauss型波束と台形波束とに近似して, 波束の実空間と波数空間における運動を, 計算機画面上で観察した。多色波に相当するGauss型波束と単色波に相当する台形波束がBragg反射する時の様子とその差異が, 鮮明に観測された。観測結果を, 電子のエネルギーバンド構造を用いて考察した。周期ポテンシャル中を進む台形波束に, 外部から一定の電圧をかけて, 波束がBloch振動する時の様子を観測した。電圧の大きさ, 格子間隔の長さ, 入射波数の大きさ, 周期ポテンシャルの高さを変化させた時の, 波束のBloch振動の特徴が鮮明に観測された。観測結果を, 外部電圧が加わった場合のエネルギーバンド構造を用いて考察した。|Behaviors of the Bragg reflection and the Bloch vibration of a wave packet of an electron which is propagating in one-dimensional periodic potentials were investigated by numerically solving the time-dependent Schrödinger equation of one dimension. Motions of a wave packet in real space and in wave-number space were observed on the screen of a personal computer, where we used a Gaussian wave-packet and a trapezoid one. We clearly observed the behaviors of the Bragg reflections of a Gaussian wave-packet which corresponds to a wave with a variety of colors and a trapezoid wave-packet which corresponds to a single-color wave. The results of the observation were understood by using the energy-band structure of an electron in a periodic potential. We loaded a constant electric-field on a trapezoid wave-packet which is propagating in a periodic potential, and observed the motion of the wave packet. Characteristics of the Bloch vibration of the wave packet were sharply demonstrated when we changed the magnitude of an electric field, the lattice distance, the magnitude of an incident wave-number and the height of the periodic potential.
In previous studies, by the methods of wave mechanics, of one-dimensional motion of particles in cases in which there are intervals in which the value of the potential energy function V(x) exceeds the value of the total energy E, attention has been confined to wave functions of the form $f(x, E)\mathrm{exp}(\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iEt}}{h})$. In the present note wave packets are considered, instead of these trains of waves.The function V(x) is taken as follows: V(x),=0forx<0andforx>a,=V0>0for0<x<a. A wave function is set up which initially represents a wave packet moving toward the point x=0 from the left. The separation of the incident packet into a reflected packet and a transmitted packet is studied. It is found that the transmitted packet appears at the point x=a at about the time at which the incident packet reaches the point x=0, so that there is no appreciable delay in the transmission of the packet through the barrier.
The resonant tunneling of a wave packet is studied, for a one-dimensional double barrier, by solving the time-dependent Schrödinger equation. The transmission is determined as a function of the spatial width of the incident wave packet. It is shown that if the double barrier acts as an energy filter, then the spatial width of the transmitted wave packet is increased, due to the trapping of the particle between the two barriers for a time, ℏ/Γ, where Γ is the width of the resonance. The reflected wave packet consists of two components: one corresponding to reflection from the first barrier and the other due to the particle being trapped between the barriers for a time ℏ/Γ.
Using the time parameter in the time-dependent Schr\"odinger equation, we study the time of flight for a particle tunneling through a square barrier potential. Comparing the mean and variance of the energy and the flight time for transmitted and reflected particles, using both density and flux distributions, we find that, when accounting for momentum filtering, the suitably normalized transmitted and reflected distributions are identical in both the density and flux cases. In contrast to previous studies, we demonstrate that these results do not imply a vanishing tunneling time, but rather that the time it takes to tunnel through a square barrier is precisely given by the reflected phase time. For wide barriers, this becomes independent of the barrier width, as predicted independently by MacColl and Hartman. We show that these conclusions can be reached using a variety of arguments, including purely quantum mechanical ones. Analysis of the shapes of the distributions under consideration reveals that wave-packet reshaping is not an explanation for the MacColl-Hartman effect. The results presented here have direct implications for understanding recent experimental results in the study of the barrier crossing of rubidium atoms. The finite width of an incident wave packet significantly ``masks'' the tunneling time, and induces substantial asymmetry between the flight times of transmitted and reflected atoms.
シュレディンガー方程式をパーソナルコンピュータを用いて数値計算することによって,典型的な数種類の1次元井戸型ポテンシャル中を運動する粒子の波動関数,エネルギー,運動エネルギーおよび位置エネルギーを求めた。求めた波動関数の形とポテンシャルの形とを,エネルギーと共に,同一の図に図示することによって,量子論の特徴が視覚的かつ直感的に理解できることを示した。量子論と古典論とのつながり,エネルギーの増減の原因,分子形成の有様,固体の電子のエネルギーバンド形成の原因などが視覚的に理解されている。また,解析的には予測することが不可能な,特徴的な量子状態や現象が数値計算をすることによって初めて発見できる例を2,3示した。|The wave function, the energy, the kinetic energy and the potential energy of a particle moving in one-dimensional potentials of several typical quantum wells were obtained by calculating the Schrödinger equation with a personal computer. It was shown that we can visually and intuitively understand the characteristics of quantum mechanic by comparing the shape of wave function and the energy with the shape of potential. A visual interpretation is made on a relation between quantum mechanics and classical one, an origin of increase and decrease of energy, a condition of formation of molecules, an origin of formation of energy band structure for electrons in solid, and so on. It was also indicated that some kinds of characteristic quantum states can only be found by the numerical calculation.