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Birkhäuser Basel eBooks

A Glance at Quantum Mechanics

Published Jan 1, 2009• 0 citations
Publisher

Abstract

Since this book is closely connected to quantum mechanics, these introductory words will briefly and informally recall some postulates of this theory. Interested readers are urged to consult traditional books on quantum mechanics to complement the discussion ahead and for descriptions of experimental evidence that give rise to the postulates. In physics there are several “equivalent formulations” of quantum mechanics, and a catalogue of the most important appears in [StyO2]. The discussion here is restricted to the nonrelativistic case.

DOI
10.1007/978-3-7643-8795-2_1

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Since this book is closely connected to quantum mechanics, these introductory words will briefly and informally recall some postulates of this theory. Interested readers are urged to consult traditional books on quantum mechanics to complement the discussion ahead and for descriptions of experimental evidence that give rise to the postulates. In physics there are several “equivalent formulations” of quantum mechanics, and a catalogue of the most important appears in [StyO2]. The discussion here is restricted to the nonrelativistic case.
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Although there are many textbooks that deal with the formal apparatus of quantum mechanics (QM) and its application to standard problems, none take into account the developments in the foundations of the subject which have taken place in the last few decades. There are specialized treatises on various aspects of the foundations of QM, but none that integrate those topics with the standard material. This book aims to remove that unfortunate dichotomy, which has divorced the practical aspects of the subject from the interpretation and broader implications of the theory.The book is intended primarily as a graduate level textbook, but it will also be of interest to physicists and philosophers who study the foundations of QM. Parts of it could be used by senior undergraduates too.
In this chapter we shall present the basic postulates of quantum physics, generalizing the results obtained in the preceding chapter for the two special cases of photon polarization and spin 1/2. In general, the space of states will a priori have any dimension N, which may even be infinite, rather than only two dimensions. The postulates which we present in this chapter fix the general conceptual framework of quantum mechanics and do not directly provide the tools necessary for solving specific problems. The solution of a specific physical problem always involves a modeling stage, where the system to be studied is simplified, the approximations to be used are defined, and so on, and this modeling stage inevitably rests on more or less heuristic arguments which cannot be derived within the general framework of quantum physics. In Section 3.2.5 we gave an example of a heuristic procedure leading to the solution of a specific problem, that of the motion of a spin 1/2 in a magnetic field. Other sets of postulates can be used. For example, another approach is to state the postulates of quantum mechanics in terms of path integrals. As is often the case, the same physical theory can be dressed in various different mathematical clothes. Finally, it should be emphasized that the postulates of quantum physics give rise to some difficult epistemological problems which are still largely under debate and which we do not discuss in this book. The interested reader may consult, for example, the book by Isham [1995].
Introduction to the Maths and Physics of Quantum Mechanics details the mathematics and physics that are needed to learn the principles of quantum mechanics. It provides an accessible treatment of how to use quantum mechanics and why it is so successful in explaining natural phenomena. This book clarifies various aspects of quantum physics such as ‘why quantum mechanics equations contain “I”, the imaginary number?’, ‘Is it possible to make a transition from classical mechanics to quantum physics without using postulates?’ and ‘What is the origin of the uncertainty principle?’. A significant proportion of discussion is dedicated to the issue of why the wave function must be complex to properly describe our “real” world. The book also addresses the different formulations of quantum mechanics. A relatively simple introductory treatment is given for the “standard” Heisenberg matrix formulation and Schrodinger wave-function formulation and Feynman path integrals and second quantization are then discussed. This book will appeal to first- and second-year university students in physics, mathematics, engineering and other sciences studying quantum mechanics who will find material and clarifications not easily found in other textbooks. It will also appeal to self-taught readers with a genuine interest in modern physics who are willing to examine the mathematics and physics in a simple but rigorous way. Key Features: Written in an engaging and approachable manner, with fully explained mathematics and physics concepts Suitable as a companion to all introductory quantum mechanics textbooks Accessible to a general audience
This monograph is written within the framework of the quantum mechanical paradigm. It is modest in scope in that it is restricted to some obser vations and solved illustrative problems not readily available in any of the many standard (and several excellent) texts or books with solved problems that have been written on this subject. Additionally a few more or less standard problems are included for continuity and purposes of comparison. The hope is that the points made and problems solved will give the student some additional insights and a better grasp of this fascinating but mathematically somewhat involved branch of physics. The hundred and fourteen problems discussed have intentionally been chosen to involve a minimum of technical complexity while still illus trating the consequences of the quantum-mechanical formalism. Concerning notation, useful expressions are displayed in rectangular boxes while calculational details which one may wish to skip are included in square brackets.
I start by briefly overviewing quantum mechanics as it is standardly formulated and by discussing the question of whether this standard formulation needs to be supplemented by an interpretation. The overview is based mainly on Von Neumann (1932, 1955) and its aim is not to give the reader a crash course in quantum mechanics, but to present those parts of the standard formalism which I use in this book. For a more complete treatment, the reader may consult the standard textbooks on quantum mechanics or, for instance, Sudbery (1986) or Redhead (1987). The standard formulation The standard formulation of quantum mechanics can be introduced in four steps. The first step is that in quantum mechanics one describesthe physics of a system by means of a Hilbert space ℋ. This Hilbert space is a complex linear vector space on which an inner product is defined. Let's adopt the convention (please consult the Glossary at the end of the book for notational conventions) that α refers to a system and that is the Hilbert space that is associated with this system. Let ∣ℋ α 〉 denote a vector in ℋ α and let 〈ψ α ∣φ α 〉 be the inner product between the vectors ∣ψ α 〉 and ∣φ α 〉.