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CRC Press eBooks

Miscellaneous Results from Elementary Quantum Mechanics

Hans Albrecht Bethe

Published Nov 23, 2018• 1 citation
Publisher

Abstract

This chapter will serve to summarize the results of elementary quantum mechanics and to set the notation that will be employed in this book. For a more complete treatment of these topics, the reader is referred to any textbook on quantum mechanics.
DOI
10.1201/9780429493645-2

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Quantum theory, in its conventional formulation, is built on the theory of Hilbert spaces and operators. In this chapter we go through this basic material, which is central for the rest of the book. Our treatment is mainly intended as a refresher and a summary of useful results. It is assumed that the reader is already familiar with some of these concepts and elementary results, at least in the case of finite-dimensional inner product spaces. We present proofs for propositions and theorems only if the proof itself is considered to be instructive and illustrative. This gives us the freedom to present the material in a topical order rather than in the strict order of mathematical implication. Good references for this chapter are the functional analysis textbooks by Conway [45], Pedersen [113] and Reed and Simon [121]. These books also contain the proofs that we skip here. Hilbert spaces As an introduction, before a formal definition is given, one may think of a Hilbert space as the closest possible generalization of the inner product spaces c d to infinite dimensions. Actually, there are no finite-dimensional Hilbert spaces (up to isomorphisms) other than c d spaces. The crucial requirement of completeness becomes relevant only in infinite-dimensional spaces. This defining property of Hilbert spaces guarantees that they are well-behaved mathematical objects, and many calculations can be done almost as easily as in c d .
This book offers a complete revision for its introduction to the quantum theory of light, including notable developments as well as improvements in presentation of basic theory and concepts, with continued emphasis on experimental aspects. The author provides a thorough overview on basic methods of classical and quantum mechanical measurements in quantum optics, enabling readers to analyze, summarize, and resolve quantum optical problems. The broad coverage of concepts and tools and its practical, experimental emphasis set it apart from other available resources. New discussions of timely topics such as the concept of the photon and distinguishability bring the entire contents up to date. Key Features: Provides a complete update of a classic textbook for the field. Features many new topics, including optical coherence, coherent and incoherent imaging, turbulence-free interferometry. Includes new chapters for intensity fluctuation correlation and thermal light ghost imaging, and biphoton imaging. Offers a complete overhaul of the introductory theory to give a more coherent and thorough treatment. Expands on discussions of optical tests of quantum theory, Popper’s experiment, Einstein’s locality questions, and the delayed choice quantum eraser.