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Journal of Jinzhong University

The Wave Function of the One-Dimensional Infinite Potential Well

Jia Xu Jin

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

In this article,via numerical method,we have shown the wave function diagrams corresponding to the different analytical solutions of the infinitely deep square potential,and the reason which causes the different phase of the analytic solution has been discussed.The probability density distribution curve is given,through which we found that the probability density distribution of the analytical solutions are identical even though the phase of them are different.This conclusion reflects in one side that a phase factor difference of the wave functions can exist in quantum mechanics.

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In this article,via numerical method,we have shown the wave function diagrams corresponding to the different analytical solutions of the infinitely deep square potential,and the reason which causes the different phase of the analytic solution has been discussed.The probability density distribution curve is given,through which we found that the probability density distribution of the analytical solutions are identical even though the phase of them are different.This conclusion reflects in one side that a phase factor difference of the wave functions can exist in quantum mechanics.
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.
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.
As Wigner quasi-probability distribution function,the Wigner function contains the entire information in the evolution of the quantum states in the phase space.The function also has great significance in physics.This paper introduces the definition and nature of the Wigner function,as well as the general expression.Then,it used the integral method given Wigner function of a one-dimensional infinitely deep square potential well.Finally,we discuss the characteristics of the distribution of the function and point out that it is a real function of the phase space.
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.