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Other One-Dimensional Potentials

Published Apr 13, 2006• 0 citations
Publisher

Abstract

Abstract Our study of the infinite well family of potentials has provided us with an array of insights into the physical meaning and mathematical structure of wave mechan- ics. There are other quantum mechanical properties which do not appear in such systems, and in this chapter we study several model potentials which illustrate many new aspects, both formal and intuitive, which have wide applications.

DOI
10.1093/oso/9780198530978.003.0008

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Abstract Our study of the infinite well family of potentials has provided us with an array of insights into the physical meaning and mathematical structure of wave mechan- ics. There are other quantum mechanical properties which do not appear in such systems, and in this chapter we study several model potentials which illustrate many new aspects, both formal and intuitive, which have wide applications.
Abstract The problem of a particle moving in an infinite well or one-dimensional box, defined by the potentialis one of the most familiar and easiest to solve of all problems in introductory quantum mechanics. It is the simplest case in which to study the phenomenon of quantized energy levels in bound states, one of the “smoking guns “ of wave mech- anics. In this chapter and the next, we use this problem (and some related ones) as a tool in understanding the formalism used in solving the time-independent Schrödinger equation for bound states in a potential, understanding their phys- ical interpretation, as well as some of the formal properties of its solutions. In Chapter 7, we also use the infinite well as a model system in which to examine the role that spin and the Pauli exclusion principle play in multiparticle quantum systems.
The mathematical formalism of quantum theory in terms of vectors and operators in infinite-dimensional complex vector spaces is very abstract. The definitions of many mathematical quantities used do not seem to have an intuitive meaning, which makes it difficult to appreciate the mathematical formalism and understand quantum mechanics. This book provides intuition and motivation to the mathematics of quantum theory, introducing the mathematics in its simplest and familiar form, for instance, with three-dimensional vectors and operators, which can be readily understood. Feeling confident about and comfortable with the mathematics used helps readers appreciate and understand the concepts and formalism of quantum mechanics. This book is divided into four parts. Part I is a brief review of the general properties of classical and quantum systems. A general discussion of probability theory is also included which aims to help in understanding the probability theories relevant to quantum mechanics. Part II is a detailed study of the mathematics for quantum mechanics. Part III presents quantum mechanics in a series of postulates. Six groups of postulates are presented to describe orthodox quantum systems. Each statement of a postulate is supplemented with a detailed discussion. To make them easier to understand, the postulates for discrete observables are presented before those for continuous observables. Part IV presents several illustrative applications, which include harmonic and isotropic oscillators, charged particle in external magnetic fields and the Aharonov–Bohm effect. For easy reference, definitions, theorems, examples, comments, properties and results are labelled with section numbers. Various symbols and notations are adopted to distinguish different quantities explicitly and to avoid misrepresentation. Self-contained both mathematically and physically, the book is accessible to a wide readership, including astrophysicists, mathematicians and philosophers of science who are interested in the foundations of quantum mechanics.
We provide an introduction to a study of applications of noncommutative calculus to quantum statistical physics. Centered on noncommutative calculus, we describe the physical concepts and mathematical structures appearing in the analysis of large quantum systems and their consequences. These include the emergence of algebraic approach and the necessity of employment of infinite-dimensional structures. As an illustration, a quantization of stochastic processes, new formalism for statistical mechanics, quantum field theory, and quantum correlations are discussed.