Applied Quantum Mechanics, Second Edition
Using real-world engineering examples to engage the reader, the author of this detailed text makes quantum mechanics accessible and relevant to the engineering student.
TABLE OF CONTENTS Applied Quantum Mechanics, Second Edition
Chapter 5: Eigenstates and Operators
5.1 Introduction
Quantum mechanics is a very successful description of atomic scale systems. The simplicity of the mathematical description in terms of noncommuting linear operators is truly remarkable. The elegant, embedded symmetries are, in themselves, aesthetically pleasing and have been a source of inspiration for some studying this subject. Of course, this mathematical description uses postulates to provide a logical framework with which to make contact with the results of experimental measurements.
5.1.1 The postulates of quantum mechanics
From the material developed in the previous chapters, we may write down four assumptions or postulates for quantum mechanics.
5.1.1.1 Postulate 1
Associated with every physical observable is a corresponding operator from which results of measurement of the observable may be deduced.
We assume that each operator is linear and satisfies an eigenvalue equation of the form ? n = a n ? n, in which the eigenvalues a n are real numbers and the eigenfunctions ? n form a complete orthogonal set in state-function space. The eigenvalues, which may take on discrete values or exist for a continuous range of values, are guaranteed to be real (and hence measurable) if the corresponding operator is Hermitian. We also note that, in general, the eigenfunctions themselves are complex and hence not directly measurable.
5.1.1.2 Postulate 2
The only possible result of a measurement on a single system of a physical observable associated with the operator is an eigenvalue of the operator .
In this...
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