vol 4: Glossary
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Operator
[Borowski, Operation, Operator]
An operator is any symbol used to represent an operation.
An operation is any procedure such as addition, subtraction, multiplication, differentiation etc that transforms some given input into a unique output according to certain mechanistic rules.
More particularly, an operator is a mapping of one set into another which has a certain structure (defined by algebraic operations, a topology or an order relation). This general definition of an operator is very similar to a mapping or function.
Im computer terms, operators are called code, microcode, programs, routines, subroutines, procedures, functions and so on. In communication theory, the encoding and decoding of messages may be considered the work of an operator.
There are many more specific definitions of operator, and the different operators thus arising have taxonomic relationships similar to the functions they operate on.
The notion of operator includes recursion, so that operators may operate on operators to transform them into different operators. Our main practical application of operators is in the exposition of quantum mechanics.
Books
| Davis, Harry F, Fourier Series and Orthogonal Functions, Dover 1963 Jacket: This incisive text deftly combines both theory and practical examples to introduce and explore Fourier series and orthogonal functions and applications of the Fourier method to the solutiuon of boundary-value problems. Directed to advanced undergraduate and graduate studients in mathematics as well as in physics and engineering, the book requires no prior knowledge of partial differential equations or advanced vector anbalysis. Students familiar with partial derivatives, multiple integrals, vecotrs and elementary differential equations will find the text both accessible and challenging.' Amazon back |
| Dirac, P A M, The Principles of Quantum Mechanics (4th ed), Oxford UP/Clarendon 1983 Jacket: '[this] is the standard work in the fundamental principles of quantum mechaincs, indispensible both to the advanced student and the mature research worker, who will always find it a fresh source of knowledge and stimulation.' (Nature) Amazon back |
| Feynman, Richard P et al, The Feynman Lectures on Physics (volume 3) : Quantum Mechanics, Addison Wesley 1970 Foreword: 'This set of lectures tries to elucidate from the beginning those features of quantum mechanics which are the most basic and the most general. ... In each instance the ideas are introduced together with a detailed discussion of some specific examples - to try to make the physical ideas as real as possible.' Matthew Sands Amazon back |
| Kreyszig , Erwin, Introductory Functional Analysis with Applications, John Wiley and Sons 1989 Amazon: 'Kreyszig's "Introductory Functional Analysis with Applications", provides a great introduction to topics in real and functional analysis. This book is part of the Wiley Classics Library and is extremely well written, with plenty of examples to illustrate important concepts. It can provide you with a solid base in these subjects, before one takes on the likes of Rudin and Royden. I had purchased a copy of this book, when I was taking a graduate course on real analysis and can only strongly recommend it to anyone else.' Krishnan S. Kartik Amazon back |
| von Neumann, John, (translated from the German edition by Robert T Beyer), Mathematical Foundations of Quantum Mechanics, Princeton University Press 1983 Jacket: '... a revolutionary book that caused a sea change in theoretical physics. ... JvN begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which vN regards as the definitive form of quantum mechanics. ... Regarded as a tour de force at the time of its publication, this book is still indispensible for those interested in the fundamental issues of quantum mechanics.' Amazon back |
| Weyl, Hermann, and translated by H P Robertson, The Theory of Groups and Quantum Mechanics, Dover 1950 Jacket: 'This book is devoted to the consistent and systematic application of group theory to quantummechanics. Beginning with a detailed introduction to the classical theory of groups, Dr Weyl continues with an account of the fundamental results of quantum physics. There follows a rigorous investigation of the relations holding between the mathematical and physical theories.' Amazon back |
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