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College Physics

General boundary conditions and the continuity of probability current density

Liu Deng

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

For a particle inside a one-dimensional finite-deep square well potential, we try to substitute the continuity condition of the first-order derivative of logarithm of wave functions by the general boundary conditions. It seems more reasonable than that replaceing the usual boundary condition forcing wave functions to vanish at two walls for a free particle confined in a one-dimensional box. It is found that the continuity condition of probability current density might be damaged, if the values of parameters ρ and θ are arbitrary. In addition, the constrained conditions for the parameters ρ and θ are given and a new class of analytical solutions is obtained.\;

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arXiv:1705.05517•May 16, 2017•0 citations
For a particle inside a one-dimensional finite-deep square well potential, we try to substitute the continuity condition of the first-order derivative of logarithm of wave functions by the general boundary conditions. It seems more reasonable than that replaceing the usual boundary condition forcing wave functions to vanish at two walls for a free particle confined in a one-dimensional box. It is found that the continuity condition of probability current density might be damaged, if the values of parameters ρ and θ are arbitrary. In addition, the constrained conditions for the parameters ρ and θ are given and a new class of analytical solutions is obtained.;
One-dimensional particle states are constructed according to orthogonality conditions, without requiring boundary conditions. Free particle states are constructed using Dirac's delta function orthogonality conditions. The states (doublets) depend on two quantum numbers: energy and parity. With the aid of projection operators the particles are confined to a constrained region, in a way similar to the action of an infinite well potential. From the resulting overcomplete basis only the mutually orthogonal states are selected. Four solutions are found, corresponding to different non-commuting Hamiltonians. Their energy eigenstates are labeled with the main quantum number n and parity "+" or "-". The energy eigenvalues are functions of n only. The four cases correspond to different boundary conditions: (I) the wave function vanishes on the boundary, (II) the derivative of the wavefunction vanishes on the boundary,(III) periodic (symmetric) boundary conditions, (IV) periodic (antisymmetric)boundary conditions . Among the four cases, only solution (III) forms a complete basis in the sense that any function in the constrained region, can be expanded with it. By extending the boundaries of the constrained region to infinity, only solution (III) converges uniformly to the free particle states. Orthogonality seems to be a more basic requirement than boundary conditions. By using projection operators, confinement of the particle to a definite region can be achieved in a conceptually simple and unambiguous way, and physical operators can be written so that they act only in the confined region.
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.
Analytical solutions to the time-dependent Shrödinger equation in one dimension are developed for time-independent potentials, one consisting of an infinite wall and a repulsive delta function. An exact solution is obtained by means of a convolution of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions. Square-integrability and the boundary conditions are satisfied. The probability for the particle to be found inside the potential well is calculated and shown to exhibit non-exponential decay decreasing at large times as t−3t^{-3}t−3. The result is generalized for all square-integrable solutions to this problem.
For a quantum confinement model, the wave function of a particle is zero outside the confined region. Due to this, the negative energy states are, in fact, square integrable. As negative energy states are not physical, we need to impose some boundary conditions in order to avoid these states. For the case of the infinite square well model, we show the possible boundary conditions that avoid the existence of the negative energy states. The well-known boundary condition requiring the wave function be continuous on both sides of the well is one of the boundary conditions that avoids negative energy states. However, there are other types of boundary condition that can also avoid the negative energy states. This shows that there are the different branch of physical systems that can be established on the infinite square well model.