Journal of Heze University

Existence Conditions of Particle's Bound State in One-dimensional Unfinite Deep Potential Well

Le Wang

Published Jan 1, 2014• 2 citations

Abstract

For a moving particle in an semi- infinite potential well of one dimension,a continuous and monotonous function is introduced by solving stationary Schr? dinger equation,then Existence conditions of one even more Particle's bound state are analyzed by means of the zero- point theorem of continuous function. The results show the number of bound state is connected with the width and depth of one – dimensional Infinite potential well. For a certain quality of particle,more bound states need much deeper and wider potential well.

Related papers

Graphical Approach to Eigenvalue Problems of The One-Dimensional Finite Deep Square Potential Well

Shu Long

The one-dimensional finite-depth potential well problem under certain boundary conditions discussed in this paper is mainly the eigenvalue problem in quantum mechanics. The basis of this article is the solution of Schrodinger equation and the treatment of transcendental equation. In many quantum mechanics textbooks, the one-dimensional infinite potential well problem is regarded as the foundation of quantum mechanics because it is easy to handle and understand. On this basis, this paper discusses the more challenging finite-depth potential well situation. In this paper, the transcendental equation is obtained by analogy between the wave function with time and the continuity of the boundary conditions. In order to solve the problem that transcendental equation is difficult to solve in this process, this paper adopts the method of solving numerical solutions by using images, which includes two different cases of odd and even functions. This paper reveals the possibility of particles appearing outside the well in this problem, and on this basis, some discussions and analysis are made on the one-dimensional asymmetric finite-depth potential well problem.

Highlights in Science Engineering and TechnologyAug 20, 20240 citations

Solving Numerically Schrödinger Equation for Confined Particle in the Finite Potential Well

B. Bounthong

The quantum model of finite potential well is a most subject in the book of quantum mechanics. The solutions maybe derive from one dimension Schrödinger’s equation for describe a bound state energy of confined particle in the finite potential well. Sometime, the problem of confined particle in the finite potential well does not have an exact solution yet, but there are in fact exact solution.In this work a numerical technique is adopted to find the exact solution for the quantum model of finite potential well. To achieve this numerical solution, I use Fortran code computer program by the Newton-Raphson method for calculating the Eigen state energy for a particle (electron) confined in the finite potential well. Five values of potential width and depth have studied, the results showed that large potential width and large potential depth led me to more bound states, while smaller potential width and depth led me to fewer bound states.A comparison between infinite potential and finite potential has been presented. The results bound state energy and penetration. The results of comparison showed that the wave function of particle confined in the infinite potential cannot penetrate in the well, while the wave function of particle in the finite potential was penetrate in the well; the energy levels of an infinite well are much higher than that the corresponding energy levels for finite potential well. For confined particle in 2 or 3-dimension, we can reduce from result of 1-dimension.

Vālasān vithanyāsāt mahāvithanyālai Suphānuvong.Jun 20, 20220 citations

Analysis of the Solution of Schrdinger Equation for Free Particle

Luo Guan

Based on the theory of Schrdinger equation for the free particle,the solution of that equation was discussed in detail. It is found that the equation only exist an insignificant solution. And the mediocre solution can be also obtained from the one-dimensional infinitely deep potential well in a potential well width tending to infinite and from one-dimensional square potential barrier in the barrier height tending to zero. Although the solution is insignificant,but it describes some ideal that is consistent with that of classical mechanics. It explains the connection between the classical mechanics and quantum mechanics. The mediocre solutions show that 1) For infinite free particles,it is followed by Newton' s first motion law,2) For infinite free particle,it does not exist matter wave characteristics. Only if there is some potential( i. e. interactions),the Wave-particle duality be embodied.

Chongqing Shifan Daxue xuebao. Ziran kexue banJan 1, 20140 citations

Further Discussions on One Dimensional Infinitely Deep Potential Well Model

Ce-Xiang Mei

One-dimensional infinitely deep potential well(ODIDPW) model is one kind of the fundamental problems in quantum mechanics.This paper firstly uses the coordinate-momentum uncertainty relation and the operator theory respectively to prove why the energy of particles in ODIDPW is always greater than zero,and then makes a further discussions on stationary state wavefunction,the quantization of physical quantity,energy level interval,zero-point energy,the coordinate-momentum uncertainty relation and the applicable condition of the ODIDPW model.

Journal of Xianyang Normal UniversityJan 1, 20100 citations

BOUND STATES IN 2D AND 3D FINITE DEEP SPHERICAL SQUARE POTENTIAL WELLS

Leping LIU, Jiaze SUN, Xibin LI +1 more

By solving the Schrodinger equations in three-dimensional (3D) and two-dimensional (2D) finite depth center spherically symmetric square potential well, the eigenequations of bounded energy levels and their distributions are obtained, and the analytical solution for any bound state wave function is given. Both the eigenequations of the energy levels as well as the stationary state wave functions are closely related to the (spherical) Bessel functions. It is found that for the 3D case, there exists a bound solution when 2μa2V0/ħ2>π2/4 is present. While for the 2D case, there exists a bound state if 2μa2V0/ħ2>0.1993. Finally, numerical methods are used to analyze the characteristics of probability density distribution of different quantum states. It is found that in both three-dimensional and two-dimensional situations, the maximum number of probability density distributions inside the well corresponds exactly to the quantum number n. The larger n or the smaller α is, the higher the probability of discovering particles outside the well. In addition, the angular momentum quantum number l or m can characterize the degree to which particles deviate from the center or the uniform distribution.

Wuli yu gongcheng.Apr 1, 20251 citation

On the Expansion Problem of the Wavefunctions of a Stationary State for a Particle in One-Dimensinal Potentional Well with Infinitely High Walls

Liu Deng

The expansion problem of the wavefunctions of a stationary state for a particle in one dimensional potential well with infinitely high walls in studied. We found, in the interval ( 0≤x≤a ), that the wavefunctions of a stationary state can be expressed as the superposition of two plane waves, which traveling in the opposite directions, when the energy quantum number n is even. However, it can not be done when n is odd, because they are not the period functions in this interval. If only if the width of the well tends to infinity, or n is very large, and the expanding coefficients on the base of continuous spectrum become to the superposition of two symmetrical delta functions, all of the stationary wavefunctions can be expanded as the superposition of two plane waves.

Journal of Shanxi Teachers UniversityJan 1, 20000 citations