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After a quarter century of discoveries that rattled the foundations of classical mechanics and electrodynamics, the year 1926 saw the publication of two works intended to provide a theoretical structure to support new quantum explanations of the subatomic world. Heisenberg's matrix mechanics and Schrodinger's wave mechanics provided compatible but mathematically disparate ways of unifying the discoveries of Planck, Einstein, Bohr and many others. Efforts began immediately to prove the equivalence of these two structures, culminated successfully by John von Neumann's 1932 volume Mathematical Foundations of Quantum Mechanics. This forms the springboard for the current effort. We begin with a presentation of a minimal set of von Neumann postulates while introducing language and notation to facilitate subsequent discussion of quantum calculations based in finite dimensional Hilbert spaces. Chapters which follow address two-state quantum systems (with spin one-half as the primary example), entanglement of multiple two-state systems, quantum angular momentum theory and quantum approaches to statistical mechanics. A concluding chapter gives an overview of issues associated with quantum mechanics in continuous infinite-dimensional Hilbert spaces.
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Table of Contents

PostulatesTwo-state SystemsEntanglementQuantum Angular MomentumQuantum Many-body ProblemInfinity, and BeyondAppendix: Relevant results from linear algebraAppendix: Directory of definitions and notation

About the Author

Professor Williams joined the faculty at Washington and Lee in 1974, and retired from full-time teaching in the summer of 2011. He attended the University of Virginia, receiving a B. S. in Physics in 1963 and a Ph.D. in Physics (under the direction of M. E. Rose) in 1967.Upon completion of the Ph.D. he was awarded a National Research Council / National Science Foundation Postdoctoral Research Fellowship which funded two years of theoretical research (with M. Danos) at the National Bureau of Standards, in Washington, DC. He then spent three semesters as Gastdozent at the Universitaet Erlangen-Nuernberg in Germany, followed by a one term teaching appointment at the Virginia Military Institute in Lexington, Virginia.From the fall of 1971 through the end of 1973, he served as a staff scientist at Kaman Sciences, Colorado Springs, Colorado, studying electromagnetic shielding, field propagation, transmission and reception.In January 1974, Dr. Williams was appointed Assistant Professor of Physics at Washington and Lee. He was promoted to Associate Professor in 1979, and to Professor in 1984. For the three academic years spanning 1986-89 he served as Associate Dean of the College of Arts and Sciences, and served as Head of Physics and Engineering from 1989 until June, 2000. During the academic year 2002-03, he served as Acting Dean of the College. From 2003-07 he was University Provost. In the fall of 2007 he returned to the Department of Physics and Engineering and served as Department Head in 2010 and 2011. Professor Williams has provided consulting services to the National Bureau of Standards and the Los Alamos National Laboratory. He has received research support from the National Bureau of Standards, Los Alamos National Laboratory, the Research Corporation and the National Science Foundation.

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