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Blucher Physics Proceedings

A Equação de Schroedinger num Cenário deComprimento Minimo

A. O. G. Oakes

Published Mar 1, 2021• 0 citations
Publisher

Abstract

In this work, we intend to investigate the application of the Schroedinger equation in a minimum length scenario for a potential already known in nature.In this perspective, we will apply to the potential barrier, whose ordinary case is already widely known in the literature.We intend to show the solutions of the Schroedinger equation in this scenario, the stationary solutions and, finally, discuss the transmission coefficient of the wave.

DOI
10.5151/xiecfa-oakes_r1

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In this work, we intend to investigate the application of the Schroedinger equation in a minimum length scenario for a potential already known in nature.In this perspective, we will apply to the potential barrier, whose ordinary case is already widely known in the literature.We intend to show the solutions of the Schroedinger equation in this scenario, the stationary solutions and, finally, discuss the transmission coefficient of the wave.
Recent work shows that the Schroedinger equation can be solved exactly based only on classical least action. The computation is based on solving a HamiltonJacobi equation for the action computing the classical density accordingly along all stationary action paths and finally constructing the exact wave function based on these classical quantities alone. The method requires that the action Laplacian or more generally the Laplacian of the propagated density be purely time-varying along stationary action paths. In the case of arbitrary nonlinear potentials this condition can still be verified without loss of generality by using a time rescaling. As the complexity of the wave computation is thus shifted to that of the action and the possible time rescaling this paper proposes a new analytical approach both to compute the action itself for a general nonlinear Hamilton-Jacobi pde and to concurrently construct a time rescaling as needed In contrast to solving the Schroedinger equation directly this computation of action and density extends naturally to systems with nonlinear potentials or position dependent inertia tensors. Hence in principle it can replace the approximations of quantum perturbation theory. For general nonlinear potentials, extending a result of Duru and Kleinert the time rescaling in a given metric is shown to correspond to a change of variables in the computed action density and wave with each eigenwave computed using that change of variables. We show that the approach makes it straightforward to construct the quantum wave for basic cases where no exact solution has been yet derived with the time rescaling unifying the computation. The approach first illustrated for the known three dimensional hyperbolic potential waves of the hydrogen atom is then used to compute the quantum waves for a quartic oscillator for which so far no direct exact solutions are known.
Abstract The reflection and transmission coefficients describe the behavior of the matter wave incident on a potential barrier. They can be expressed in terms of the probability with which the matter wave can be reflected or transmitted. The central equation accounting for the behavior of the matter wave is the Schrödinger equation. The Schrödinger equation is the second order partial differential equation. However, in a stationary state, the Schrödinger equation is reduced to the time independent Schrödinger equation. This time independent Schrödinger equation is the second order linear ordinary differential equation. Since the time independent Schrödinger equation is linear, superposition of any of the two solutions to the time independent Schrödinger equation is also a solution. In this paper, we focus on the superposition of various potentials. The reflection and transmission coefficients from the superposition of various potentials are obtained. A comparison between the exact coefficients and those obtained by the 2 × 2 transfer matrix is made. The relationship between the transmission coefficient of the superposed potential and that of each individual potential is found. The results show that the transmission coefficient obtained from the 2 × 2 transfer matrix is of a lower bound on the exact transmission coefficient.
A method, based on the use of Milne's function www, for the numerical determination of quantum-mechanical reflection coefficients in one-dimensional problems to any preassigned degree of accuracy is outlined. The case in which a "barrier" is both preceded and followed by a field-free space is considered in detail, sections 2-4, and a numerical example is worked out, section 5. The procedure in the case in which a "barrier" is preceded by a field-free space and is followed by a potential for which Schroedinger's equation can be solved analytically is outlined, sections 6-7, and the special case in which a "barrier" is followed by a uniform field is considered in some detail, section 8. The case in which a "barrier" is both preceded and followed by potentials for which Schroedinger's equation can be solved analytically is mentioned, section 8. Formulas are given for the evaluation in terms of $\ensuremath{\psi}$ and its first derivative, of the respective densities of the dextral and of the sinistral current flowing past any point xxx in regions where the total energy is greater than the potential energy, section 6. A procedure is given for finding solutions representing unidirectional beams at infinity, section 6.