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Journal of Xianyang Normal University

Further Discussions on One Dimensional Infinitely Deep Potential Well Model

Ce-Xiang Mei

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

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.

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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.
First, this article explains the formula of one - dimensional infinitely deep square potential well's energy levels from stationary state schrodinger equation, and then deduces the same formula from one - dimensional half infinitely high square potential well' s energy levels diagram of graphic. solution, and proves that one - dimensional leveles are limit of one - dimensional half infinitely high square potential well' s energy levels. Finally it discusses the energy levels' number of one - dimensional half infinitely high square potential well and gives out the differentiateating formula about it.
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.
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.
Abstract The problem of a particle moving in an infinite well or one-dimensional box, defined by the potentialis one of the most familiar and easiest to solve of all problems in introductory quantum mechanics. It is the simplest case in which to study the phenomenon of quantized energy levels in bound states, one of the “smoking guns “ of wave mech- anics. In this chapter and the next, we use this problem (and some related ones) as a tool in understanding the formalism used in solving the time-independent Schrödinger equation for bound states in a potential, understanding their phys- ical interpretation, as well as some of the formal properties of its solutions. In Chapter 7, we also use the infinite well as a model system in which to examine the role that spin and the Pauli exclusion principle play in multiparticle quantum systems.