From the point view of teaching research,discuss the Eigen function solution and the effects of width of potential well about the energy and the wave function,to simplify the solution of Eigen function process,and to highlight the physical significance of the one-dimensional infinitely deep potential well.
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.
This chapter solves the eigen-problems for a few simple one-dimensional systems to acquire experience in solving the Schrodinger eigen-equation. One-dimensional problems are physically interesting not only because they are simple, but also because a number of more complicated higher-dimensional problems can be reduced to the solutions of equations in one dimension. For a symmetric potential, an eigenfunction of the Schrödinger equation can be always chosen to have a definite parity. The probability density of the plane wave solution is a constant, independent of the spatial coordinate of the particle. The eigensolution of a particle in an infinite square well consists of an infinite number of discrete energy levels. In classical mechanics, a particle cannot penetrate a potential barrier whose height is higher than the energy of the particle.
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.
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.