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American Journal of Physics

Position Expectation Values for an Electron in an Infinite Tilted Well

Noble M. Johnson, John N. Churchill

Published Apr 1, 1970• 1 citation
Publisher

Abstract

A numerical technique is used to determine the manner in which the expectation values of position for an electron in an infinite tilted well depend on the potential gradient and quantum state of the system. The results are compared to the classical case and to the exact analytic expression for the special case in which the total energy of the electron is equal to the maximum value of the linear potential.

DOI
10.1119/1.1976371

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A numerical technique is used to determine the manner in which the expectation values of position for an electron in an infinite tilted well depend on the potential gradient and quantum state of the system. The results are compared to the classical case and to the exact analytic expression for the special case in which the total energy of the electron is equal to the maximum value of the linear potential.
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In this paper we have solved analytically the Schrodinger equation for a microscopic particle in a one-dimensional box with two infinite walls, which the potential function inside it, has a linear form. Based on the solutions of this special quantum mechanical system, we have shown that as the quantum number approaches infinity the expectation values of microscopic particle position and square of the linear momentum are equal to the classical time average of particle position and its square of the linear momentum, respectively.
In this paper it is considered a particle in an infinite potential well of special smooth form. To obtain eigenfunctions and eigenvalues of the problem, a method of Darboux transformations is applied. An orthonormal set of the eigenfunctions is presented. Some important properties of the eigenfunctions are shown. Principal characteristics of the particle in the well are discussed. A function of distribution of the probability density is analyzed. Conditions for determination of points of extrema of the probability density are formulated and corresponding transcendental equations are presented. An approximated method for determination of points of extrema is also proposed. Expected values of position and momentum of the particle are calculated explicitly. For some lower energy states, distributions of probability density are shown graphically in comparison with distributions for the infinite square well. As an additional useful result, some integrals of special combinations of trigonometric functions are presented.
The objective of this study is to numerically solve and apply two approximate methods to investigate the Time Independent Schrödinger Equation (TISE) in one dimension for an infinite potential square well. These two numerical methods are the Numerov Method (NM) and Matrix Method (MM). As a simulation tool, MATLAB, a high-level programming language and an efficient simulation tool, is used for modeling and solving TISE in one dimension. Exact analytical solutions for these potential functions are obtained and compared with numerical solutions and computational techniques. The energy eigenvalues and Eigen functions of a particle (such as an electron) restricted to move inside this potential are discussed as an illustration. The numerically calculated energies of several states with increasing numbers of points were obtained from both methods and compared with the simulation results of the exact solution. As an exemplary case, the first five wave functions are accurately determined numerically where the discreteness is found since the wave function vanishes at the boundary. The obtained results show very good agreement and the similarity is clearly confirmed between the three cases. This agreement confirms that this approach was highly accurate and efficient. The accuracy and the convergence of the numerical obtained results were easily checked. The stability of these methods is due to the fact that there are no restrictions on the time steps to be taken. The merits of these numerical methods are to avoid a huge expense in time when solving Schrödinger equation.
Abstract Resonant tunneling is a quantum phenomenon that has been extensively studied and applied in electronic devices. In this paper, the transmission coefficient and resonant energies of a particle passing through a linear potential barrier are derived and investigated. We propose that approximated values of resonant energies can be achieved by considering the energy levels of a particle confined in an infinite tilted potential well. It is found that approximation of high resonant energies is more accurate than that of low resonant energies.