Simple one-dimensional potentials

J. Ronald Farias

A simple exact solution of a one-dimensional Schr\"odinger equation which describes the motion of a particle in a continuous potential $V(x)$ and scattering from a finite set of $\ensuremath{\delta}$-function potentials of arbitrary positions and strengths is found.

Tapan Kumar Das

D. Baye, M. Dufour, Benjamin Fuks E

Yueyang Wu

One-dimensional quantum problems have always been an important issue in various branches of quantum mechanics fields, and many quantum models can be idealized as one-dimensional potential profiles. Therefore, it is necessary to investigates the way to deal with and calculate the problems. This paper proposes a new and simple method for simulation and calculation of one-dimensional quantum problems. To be specific, by representing continuous X values by a series of discrete lattice points, the Hamiltonian matrix is constructed for the system in the way of dealing with monomer and many-body problems, so as to simply calculate the energy level distribution and draw the wave function image. In terms of simulating one-dimensional infinite deep potential well, one-dimensional finite deep potential well, one-dimensional multi-potential well and other one-dimensional quantum systems with this method, this paper shows that the method is accurate and practical. Compared with other methods for one-dimensional quantum problems, this paper also presents the superiority of this method. To deal with the problem based on such a method can save the computation cost and time cost, which is more convenient to study the one-dimensional quantum problem in the future. These results shed light on studying complex one-dimensional quantum problems conveniently.

Tao Xiang

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.

Jochen Pade