Inevitably, in order to remain within the confines of a small volume, it has proved necessary to omit much interesting material from the present work owing to the large number of different applications of variational principles that have been carried out recently. An effort has been made to keep the treatment of variational principles at a reasonably elementary level, the general aim being to provide a fairly broad view of the way in which variational principles have been applied to various problems in theoretical physics, although a certain amount of emphasis has been given to their role in the quantum theory of scattering.
This edition is the same as the original edition of my book apart from the correction of a few minor errors and misprints. The search for unifying concepts and general principles has been one of the most important aims of mathematicians from early times and, as before, I would like to emphasize that some of the most aesthetically satisfying and illuminating general principles are the variational principles that are the subject of this book.
Variational principles are concerned with the maximum and minimum properties, or more generally the stationary properties, of an extensive range of quantities of mathematical and physical interest spanning a wide field of applications. Not only are these variational principles characterized by an elegant mathematical structure, but also they often possess the greatest practical utility in the solution of important problems in physics and chemistry and provide the most reliable method of accurately determining the values of many physical quantities of fundamental importance such as energy eigenvalues.
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Variational principles have an ancient history and arose in antiquity in connection with the solutions of the isoperimetric and the geodesic problems. The isoperimetric problem, proposed by Greek mathematicians of the second century B.
Variational Principles and Methods in Theoretical Physics and Chemistry
These two problems have many generalizations. For example, the triangle with a given perimeter enclosing the greatest area is equilateral and, in general, the polygon having a given perimeter that encloses the greatest area is a regular polygon. Also, the closed surface possessing a given area that surrounds the greatest volume is a sphere. Further, the shortest path connecting two points on a circular cylinder is a helix, a result that is related to the shape of the DNA molecule. Another ancient variational problem, first solved by the Greek mathematician and scientist Heron of Alexandria in the first century A.
This is an example of the principle of least time that was later named after the seventeenth century mathematician Fermat who examined the refraction of a light ray at an interface separating two different media and showed that it takes the route of the shortest possible time. However, this variational principle is not necessarily a minimum principle since a light ray that passes from one focal point to the other focal point of an ellipse by reflection at a mirror within the ellipse, which is tangential to the ellipse at its minor axis, takes the trajectory with the maximum possible total distance between the foci connected by two straight-line segments with a vertex at the mirror.
Thus we have here an example of a principle of maximum time.
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Product Details Table of Contents. Pages: Product dimensions: 6. Table of Contents Preface; Part I. Classical Mathematics and Physics: 1.
Moiseiwitsch Variational Principles
History of variational theory; 2. Classical mechanics; 3.
Applied mathematics; Part II. Bound States in Quantum Mechanics: 4. Time-independent quantum mechanics; 5.
ISBN 13: 9780521803915
Independent-electron models; 6. Time-dependent theory and linear response; Part III. This book brings together the essential ideas and methods behind applications of variational theory in theoretical physics and chemistry. The emphasis is on understanding physical and computational applications of variational methodology rather than on rigorous mathematical formalism.
The text begins with an historical survey of familiar variational principles in classical mechanics and optimization theory, then proceeds to develop the variational principles and formalism behind current computational methodology for bound and continuum quantum states of interacting electrons in atoms, molecules, and condensed matter. It covers multiple-scattering theory, including a detailed presentation of contemporary methodology for electron-impact rotational and vibrational excitation of molecules.
The book ends with an introduction to the variational theory of relativistic fields.
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Ideal for graduate students and researchers in any field that uses variational methodology, this book is particularly suitable as a backup reference for lecture courses in mathematical methods in physics and theoretical chemistry. Seller Inventory CBL More information about this seller Contact this seller. Book Description Cambridge University Press, Never used!
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