无限深方势阱本征值和本征态的三种求解方法

一维无限深方势阱模型是量子力学理想模型,经典教材中势阱的边界一般取得比较特殊.或关于坐标原点具有对称性,或势阱左边界位于坐标原点.本文首先展示了如何利用3种方法求解一维任意边界无限深方势阱能量本征值和对应的本征态,不同方法得到的结果彼此之间等价,讨论分析了这3种方法的推导结果,然后得到关于一维任意边界无限深方势阱能量本征值和本征态的通式,从中比较容易看出这两个物理量均与阱宽有关,并且本征波函数与边界值有关,最后将一维结果拓展到二维和三维任意边界无限深方势阱情况.     

一维无限深梯形势阱中微观粒子波函数和能级的半解析解

利用特殊函数研究了一个数学形式简单但物理内涵丰富的新势阱——一维无限深梯形势阱,并得到了处于该势阱中微观粒子的波函数和能级的半解析解,其结果可以用来从理论上分析梯形沟道势阱的二维电子气的物理机制.此外指明了它与熟知的一维无限深方势阱和三角势阱的联系与区别,并基于对应原理讨论了该模型下粒子波函数的特征.     

一维有限深方势阱的α0迭代求解法

介绍了用α０迭代方法求一维有限深方势阱的解析近似解，给出了解的通式和可求任意精度要求解的α０循环迭代计算程序流程。在新的基础上建立有限深方势阱的能级和波函数与无限深方势阱的能级和波函数的联系。     

有限深方势阱图解法的改进

对有限深方势阱中的能级方程做一个简单的数学变换,从而将常用的图解法中的函数图像简化,使得有限深方势阱问题的物理图像变得更清晰,更有利于大家理解这个简单而重要的模型.文中还利用改进后的图解法简单讨论了一些有限深方势阱的性质.     

关于半壁无限深势阱束缚态存在条件的讨论

一维无限深势阱问题是量子力学中的经典问题,也是国内外量子力学初学者必学内容之一。一维半壁无限深势阱作为一维无限深势阱的变形,也是量子力学教学中常见的物理模型。然而关于半壁无限深势阱这一模型,许多教学和科研工作者甚至教材之中,都存在不严谨之处;尤其是对于束缚态存在条件的求解,许多教材与习题册没有对临界条件进行讨论验证。本文借助图解法对半壁无限深势阱束缚态存在条件进行了讨论并给出了严格解,纠正了长久以来人们对该问题的错误认识,对量子力学领域本科教学与实际科研工作有一定帮助。     

(nc-Si/SiO2)/SiO2多层量子点结构的激子能级

采用球型量子点模型,应用有效质量近似理论,研究了(nc-Si/SiO2)/SiO2多层量子点结构的激子能级和波函数.结果表明,有限深势阱模型的引入更符合实际更加准确.无论在无限深或有限深势阱下,激子质心运动部分基态能量随最子点半径的减小而急剧增大.对于相同的量子点半径α,无限深势阱下的质心部分能量总比有限深势阱高,且二者的差距随α的减小小而增大.     

基于GUI的三维无限深势阱可视化研究

无限深势阱中粒子的量子力学图像是教师教学和学生学习的一个难点,本文利用Matlab的GUI功能对三维无限深势阱粒子的量子化行为进行可视化研究。通过对控件的选取与设置,分别给出了三维无限深势阱不同量子数下零平面、极值点所在平面的波函数切片分布图,以及垂直x方向任意截面处的波函数立体分布图,实现了三维无限深势阱任意量子数及任意截面处波函数的可视化。利用GUI实现三维无限深势阱粒子量子化行为的可视化,可以很好地实现人机互动,便于形象分析不同条件下的波函数分布情况,有助于教师教学和学生学习。     

量子力学中无限深势阱问题的教学研究

在一维问题的基础上探讨了二维无限深势阱的问题,发掘二维无限深势阱在不同边界约束情况下不同于一维问题的特征和应用.     

无限深势阱问题的可视化研究

基于计算软件对一些常见的无限深势阱进行了可视化研究,并设计了GUI界面实现对势阱的选择.在选定势阱后,设置好相应的势阱参数和量子数,便可依次绘制出低维势阱的波函数、概率密度函数和三维势阱中的电子云图像.文中分析了二维和三维势阱中能级的简并度问题,并重点讨论了不同势阱波函数和概率密度函数随量子数的变化规律.将势阱问题可视...     

一维有限深方势阱中的束缚态

主要对量子力学一维有限深方势阱中运动粒子的束缚态存在条件进行讨论.通过求解定态薛定谔方程,得到粒子运动满足的超越方程,借助于Mathematica软件求解该超越方程得到粒子的能级结构,对粒子束缚态存在条件进行分析.结果表明,对于在一维有限深对称方势阱中运动的粒子,总会有束缚态的存在,和势阱的宽度、深度及粒子的质量无关.而在半壁有限深势阱中运动的粒子,出现束缚态是有条件的.并且分别给出了在两种势阱中粒子存在多个束缚态需要满足的条件.     

Localised solutions of Hartree equations for narrow-band crystals

We consider the Hartree equations for a system of an infinite number of electrons in a periodic potential consisting of a periodic array of wells. The filling fraction is assumed to be of one electron per well. We prove that if the wells are deep enough to admit a bound state and if they are separated by a distance large enough, then the Hartree equations have a solution in which all single particle wave functions decay exponentially.     

A critical evaluation of closure methods via two simple dynamic systems

Two simple dynamic systems with cubic nonlinearity and additive Gaussian white noise are used to assess the performance and the usefulness of closure methods in nonlinear random vibration. One of the systems has a single potential well while the other has two potential wells. It is shown that the performance of closure methods is determined by the structure of the moment equations rather than the way in which these equations are closed. For the system with one potential well, any closure method is satisfactory. For the system with two potential wells, closure methods can be inaccurate irrespective of the closure level. It is also shown that moment equations can be augmented with moment inequalities to solve approximately the infinite hierarchy of moment equations.     

Reorientations of molecules between unequal potential wells in solid pentafluorochlorobenzene according to 19F NMR and 35Cl NQR data

This paper presents the results of an experimental and theoretical study of reorientational molecular motion between unequal potential wells in a three-well potential bounded by infinite walls. The temperature dependences of the nuclear spin-lattice relaxation times in crystalline pentafluorochlorobenzene, C₆F₅Cl, (magnetic for fluorine 19 and quadrupole for chlorine 35) have been analyzed on the basis of the theoretical expressions obtained here. The parameters of the orientational potential with unequal wells have been established as a result of this interpretation of the experimental data. Fiz. Tverd. Tela (St. Petersburg) 39, 877–879 (May 1997)     

Dynamics of a two-level system interacting with a random classical field

The dynamics of a particle interacting with a random classical field in a two-well potential is studied by the functional integration method. The probability of particle localization in either of the wells is studied in detail. Certain field-averaged correlation functions for quantum-mechanical probabilities and the distribution function for the probabilities of final states (which can be considered as random variables in the presence of a random field) are calculated. The calculated correlators are used to discuss the dependence of the final state on the initial state. One of the main results of this work is that, although the off-diagonal elements of the density matrix disappear with time, a particle in the system is localized incompletely (wave-packet reduction does not occur), and the distribution function for the probability of finding particle in one of the wells is a constant at infinite time.     

Energy levels of a particle in an infinite potential well with identical square-potential barriers

We have evaluated the energy levels of a particle in an infinite potential well containing identical square-potential barriers of equal width and separation, in a symmetric as well as an asymmetric arrangement, as functions of the number and height of the potential barriers. For this purpose we have derived analytical equations for the energy eigenvalues which can be solved with a personal computer. We show plots of the normalized energy levels of particles in potential wells containing up to three identical square-potential barriers with heights varying from zero to infinity. These can be applied to models of inversion and libration of molecules, electronic states in dye molecules, quantum well structures, etc.     

The diamagnetic susceptibility of hydrogenic donor in two-dimensional semiconductors with anisotropic effective mass of carriers

In this study, the effects of quantum confinement and effective mass anisotropy parameter on the diamagnetic susceptibility of a hydrogenic donor placed in GaAs, Si, and Ge quantum wells with infinite confinement potential are investigated in the effective mass and parabolic band approximations by using two and one parameter trial wave functions. It is observed that the diamagnetic susceptibility of a hydrogenic donor in anisotropic quantum wells is essentially equal to the transverse diamagnetic susceptibility part when well widths are larger than L > 100 Å, and the impurity is located at center. Moreover, a two parameter trial wave function model gives higher values of diamagnetic susceptibility, except for χ_z (GaAs).     

Potential wells for classical acoustic waves

The acceleration theorem of Bloch waves is utilized to construct random potential wells for classical acoustic waves in systems composed of alternating‘cavities’and‘couplers’.One prominent advantage of this method is these‘cavities’and‘couplers’are all monolayer structures.It allows forming more compact classical potential wells,which leads to the miniaturization of acoustic devices.We systematically investigate properties of harmonic,tangent,hyperbolic function,and square classical potential wells in quasi-periodic superlattices.Results show these classical potential wells are analogues of quantum potential wells.Thus some technologies and concepts in quantum potential well fields may be generalized to classical acoustic wave fields.In addition,some abnormal cases regarding forming classical potential wells are also found.     

抛物量子阱中束缚极化子的极化势和结合能

利用改进的Lee-Low-Pines(LLP)方法,用变分法计算了无限深抛物量子阱中同时考虑与体纵光学声子和界面纵光学声子相互作用的束缚极化子的极化势和结合能.数值计算得出:阱宽较大时极化势很小,阱宽较小时极化势较大,所以对于较窄的抛物阱必须考虑极化势.对于给定阱宽的抛物阱,随着远离阱中心极化势迅速减小,当到达阱的界面附近极化势又开始增大.阱宽较小时,束缚极化子的结合能随着阱宽L的增大而急剧减小;阱宽较大时,结合能减小的非常缓慢,最后接近体材料中的三维值.     

Kinetic Arrest Originating in Competition Between Attractive Interaction and Packing Force

We discuss the situation where attractive and repulsive portions of the interparticle potential both contribute significantly to glass formation. We introduce the square-well potential as prototypical model for this situation, and reject the Baxter model as a useful model for comparison to experiment on glasses, based on our treatment within mode coupling theory. We present explicit results for various well widths, and show that, for narrow wells, there is a useful analytical formula that would be suitable for experimentalists working in the field of colloidal science. We raise the question as to whether, in a more exact treatment, the sticky-sphere limit might have an infinite glass transition temperature or a high but finite one.     

Magnetic Kronig-Penney-type graphene superlattices: finite energy Dirac points with anisotropic velocity renormalization

We study the energy band structure of magnetic graphene superlattices with delta-function magnetic barriers and zero average magnetic field. The dispersion relation obtained using the T-matrix approach shows the emergence of an infinite number of Dirac-like points at finite energies, while the original Dirac point is still located at the same place as that for pristine graphene. The carrier group velocity at the original Dirac point is isotropically renormalized, but at finite energy Dirac points it is generally anisotropic. An asymmetry in the width between the wells and the barriers of the periodic potential induces a shift of the original Dirac point in the zero-energy plane, keeping the velocity renormalization isotropic.