Cambridge University Press eBooks

One-Dimensional Potential Steps, Barriers, and Wells

Ana Marı́a Cetto, Luis de la Peña

Published Dec 18, 2025• 0 citations

Abstract

Chapter 5 familiarizes the student with some of the most basic features of quantum systems, by applying the Schrödinger equation to simple, one-dimensional problems. It shows that it is possible to obtain qualitatively correct conclusions about the behavior of quantum particles in the presence of more or less arbitrary potentials by replacing them with piecewise-constant potentials, which greatly simplifies the mathematics. Replacing the real potential with a simpler one that retains its basic characteristics has the great advantage of allowing us to study the essence of the physical situation, as illustrated by the examples of the one-dimensional potential step, the square well, the square potential barrier and the symmetric double well. These examples are used to discuss specific features such as quantum tunneling, and applications such as masers and molecular clocks.

DOI: 10.1017/9781009679633.006

Related papers

Applications of Schrödinger Equation: Potential (Quantum) Wells

Z. Ficek

The wave nature of particles plays an important role in their physical properties that, for example, particles confined into a small bounded area can have only particular discrete energies. This chapter explains how to solve the time-independent Schrodinger equation. It considers four cases of a one-dimensional motion of particles confined in potentials rapidly changing with x : infinite potential quantum well; a potential step; square-well potential; and tunneling through a potential barrier. The chapter helps to learn the characteristic properties of solutions to this equation. It shows the differences between the predictions of quantum mechanics and classical physics. Quantum tunneling is important in the understanding of a number of physical phenomena such as thermonuclear reactions and conduction in metals and semiconductors. The most advanced application of quantum tunneling is the scanning tunneling microscope. The details of the calculations are left for the readers as a tutorial problem.

Mar 3, 20170 citations

Wave Mechanics and Its Simple Applications

Ajay Sharma

The Schrödinger equation assumes that a particle behaves as a wave and yields a solution in terms of wave function and the energy of the particle under consideration. Once the wave function is known, then everything about the particle can be deduced from the wave function. This chapter covers the development of the Schrödinger wave equation and its applications to one-dimensional systems. Time-dependent and time-independent Schrödinger equations, bound and scattering states, probability density, probability current and expectation values, the solution of the time-independent 1D-Schrödinger equation for a free particle, infinite as well as finite potential wells, potential steps, and the finite potential barrier are discussed in detail in the chapter. The phenomenon of tunneling, relevance and the importance of the free particle, potential wells and potential barriers to contemporary fields of physics and chemistry, periodic solids and their band structures, the Kronig-Penney model, confined states in quantum wells, and wires and dots are other topics presented in innovative manners in the chapter. The importance of the topics presented here is demonstrated through carefully chosen solved examples and unsolved exercises from contemporary fields of sciences.

CRC Press eBooksJun 18, 20210 citations

One-dimensional Eigen-problem

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.

CRC Press eBooksMar 31, 20220 citations

First Steps in Quantum Physics: Basics – 2. Model Examples (Square Wells)

M. Avramov, D. I. Marvakov

Cases of a particle in a one-dimensional square well – infinitely deep and with finite depth – are also analyzed in detail. As an example, the adsorption of a hydrogen atom on a metal surface by a qualitative and accurate solution of the problem is considered.

Natural Science and Advanced Technology EducationMay 1, 20210 citations

Solutions of Schrödinger’s equation in one dimension

A.P. French, Edwin F. Taylor

This chapter examines some explicitly one-dimensional systems, including the finite square well, the harmonic oscillator, and the application of harmonic oscillator theory to molecular vibrations. It analyzes the computer solutions of the Erwin Schrodinger equation. Although there are no real one-dimensional square wells in nature, many real systems have potentials that resemble the square well. Near the center of force the different wave functions must differ from one another or else they would not satisfy the Schrodinger equation for different values of energy. In contemporary research almost all the manipulation of the Schrodinger equation is done not analytically but rather by computer using numerical methods. A major advantage of the dimensionless form of the Schrodinger equation is that a single solution for one kind of system is trivially easy to extend to all similar systems. The dimensionless form of the Schrödinger equation can be solved numerically if it is approximated by a so-called difference equation.

Routledge eBooksMay 11, 20180 citations

Applications of the Formalism-I

Shabnam Siddiqui

This chapter discusses several applications of the formalism of theory of quantum mechanics. The time-independent Schrodinger equation is applied to understand simple quantum systems such as a free particle, a particle in an infinite square well, step potentials, potential barrier and finite square well. The chapter deals with the simplest case, a free particle. It is the most elementary application of the formalism. A free particle means that no forces are acting on the particle, and hence, there is no interaction between the particle and its environment. According to classical mechanics, a free particle moves with a constant velocity along a straight line. The energy of the wave associated with the free quantum particle is same as that of the classical free particle. The chapter describes a quantum free particle as a wave packet traveling in space and time. It discusses a quantum particle under the influence of a force.

CRC Press eBooksOct 10, 20180 citations