com The Theoretical Foundations of Quantum Mechanics www. Baaquie The Theoretical Foundations of Quantum Mechanics 123 www. Baaquie Department of Physics National University of Singapore Singapore ISBN 978-1-4614-6223-1 ISBN 978-1-4614-6224-8 (eBook) DOI 10.1007/978-1-4614-6224-8 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012954422 © Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.
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Printed on acid-free paper Springer is part of Springer Science+Business Media (www.com Preface Quantum theory introduces a fundamentally new framework for thinking about Nature and entails a radical break with the paradigm of classical physics. In spite of the fact that the shift of paradigm from classical to quantum mechanics has been going on for more than a century, a conceptual grasp of quantum mechanics has till today proved elusive. According to leading quantum theorist Richard Feynman, “It is safe to say that no one understands quantum mechanics” [13]. The foundations of quantum mechanics have been studied by many authors, and most of their books have been written for specialists working on the foundations of quantum mechanics and quantum measurement [1, 4, 16]—requiring an advanced knowledge of mathematics and of quantum mechanics [23, 25, 36].
An exception is the book by Isham [19], which is very clearly written and discusses the principles of quantum mechanics for a wider audience. Given the ubiquitous presence of quantum mechanics in almost all branches of science and of engineering, there is a need for a book on the enigmatic workings of quantum mechanics to be accessible to a wider audience. This book on the foundations of quantum mechanics is for the nonspecialists and written at a level accessible to undergraduates, both from science and engineering, who have taken an introductory course on quantum mechanics. The mathematical formalism has been kept to a minimum and requires only a familiarity with calculus and linear algebra.
The emphasis in all the topics is on analyzing the concepts and ideas that are expressed in the symbols of quantum mechanics. Linear vector spaces and operators form the mathematical bedrock of quantum mechanics, and a few derivations have been done to clarify these structures. In this book the Schrödinger equation is never solved; instead, the focus is on the paradoxes and theoretical conundrums of quantum mechanics as well as on the conceptual basis required for addressing these. In particular, this book concentrates on issues such as the inherent (quantum) indeterminateness of Nature and the essential role of quantum measurement in defining a consistent interpretation of quantum mechanics.
The unusual properties of many widely used technologies are due to quantum phenomena. Indeed, most of what goes under the name of high technology is a direct v www.com vi Preface result of the workings of quantum mechanics, and many modern conveniences that we take for granted today would be impossible without it.1 Although quantum mechanics has qualitatively changed our view of Nature, a satisfactory understanding of it is still far from complete, and one can be sure there are a lot of surprises still awaiting us in the future. The main focus of this book is to address the reasons why quantum mechanics is so enigmatic and extraordinary. A theoretical framework for quantum mechanics is proposed in an attempt to clarify the underpinnings of quantum mechanics, namely the transempirical quantum principle, which states the following: A physical entity has two forms of existence, an indeterminate transempirical form when it is not observed and a determinate empirical form when it is observed.
The transempirical and empirical forms have completely different behavior. The empirical form is intuitive and is the (experimentally) observed determinate state of the entity, whereas the indeterminateness of the transempirical form of the entity leads to all the paradoxes of quantum mechanics. 1 For example, electronic devices, from computers, television, to mobile phones, are all based on semiconductors, and airplanes, ships, and cars all use semiconductors in an essential manner. More complex technologies such as superconductors, scanning electron microscope, magnetic resonance imaging (MRI), and lasers; fabrication of new drugs; modern materials science; and the study of nanoscale phenomenon all draw upon quantum mechanics.com Acknowledgments I would like to acknowledge and express my heartfelt thanks to many outstanding teachers who inspired me to study quantum mechanics and marvel at its mysteries.
As an undergraduate, my formative views on quantum mechanics were greatly influenced by Khodadad Khan, A. Rafiqullah, George Zweig, Gerald “Gerry” Neugebauer, Clifford M. Will, and Jeffrey E. Mandula and by The Feynman Lectures on Physics [24].
As a graduate student, I was a tutor for a course taught by Kurt Gottfried and learned of his views on quantum mechanics; his book on the subject [15] continues to be, in my view, one of the best. I had the good fortune of conversing with Richard P. Feynman on many occasions, and at times I had the pleasure of even debating with him. His profound observations still ring in my ears.
I had the privilege of doing my Ph. thesis under the guidance of Kenneth G. Wilson; his visionary conception of quantum mechanics and of quantum field theory greatly enlightened and inspired me and continues to do so till today. I thank Kenneth Hong, Thomas Osiopowicz, Setiawan, Pan Tang, Duxin, Kuldip Singh, Rafi Rashid, Oh Choo Hiap, N.
Hari Das and Cao Yang for helpful discussions. I want to specially thank Dagomir Kaszlikowski and Ravishankar Ramanathan for generously sharing their valuable insights on quantum mechanics. I owe a special vote of thanks to Frederick H. Willeboordse for a careful reading of the manuscript that clarified many concepts and helped me to make a more coherent presentation of the subtleties of quantum mechanics.
I am particularly indebted to Zahur Ahmed for his advice on the book and for his invaluable observations on its draft.com Contents 1 Synopsis. 1 2 The Quantum Entity and Quantum Mechanics .1 What Is a Classical Entity? .2 The Entity in Quantum Mechanics .3 Describing an Indeterminate Quantum Entity .4 The Copenhagen Quantum Postulate .5 Five Pillars of Quantum Mechanics .6 Degree of Freedom Space F .9 The Schrödinger Equation for State ψ (t, F ) .10 Indeterminate Quantum Paths .11 The Process of Measurement .12 Summary: Quantum Entity. 22 3 Quantum Mechanics: Empirical and Trans-empirical .1 Real Versus Exist.2 Empirical, Trans-empirical, and Indeterminate .3 Quantum Mechanics and the Trans-empirical .4 Quantum Degree of Freedom F Is Trans-empirical.5 The Quantum State ψ : Transition .6 Trans-empirical Domain and Laws of Physics .7 Quantum Superposition: Trans-empirical Paths .8 Trans-empirical Interpretation of Two-Slit Experiment .9 The Trans-empirical Quantum Principle .10 Does the Quantum State ψ (t, F ) “Exist”?. 46 4 Degree of Freedom F ; State Space V .1 Dirac’s Formulation of the Quantum State .2 State Space and Experiment .3 Quantum Degree of Freedom F .4 Binary Degree of Freedom and State Space .5 Degree of Freedom F(2N+1) : State Space V(2N+1) .6 Continuous Degree of Freedom .7 Basis States for State Space .8 Unitary Transformation: Momentum Basis.1 Operators: Trans-empirical to Empirical.3 Eigenstates: Projection Operators .4 Operators and Quantum Numbers.5 Periodic Degree of Freedom.6 Position and Momentum Operators x̂ and p̂ .7 Heisenberg Commutation Equation .8 Expectation Value of Operators .9 The Schrödinger Equation .10 Heisenberg Operator Formulation.
90 6 Density Matrix: Entangled States.2 The Outer Product .3 Partial Trace for Outer Products.5 The Schmidt Decomposition .6 Reduced Density Matrix .7 Separable Quantum Systems .8 Entangled Quantum States .9 A Pair of Entangled Spins .11 Pure and Mixed Density Matrix .1 The EPR Paradox .2 The Bell-CHSH Operator.3 Classical Probability: Objective Reality .4 The Bell Inequality .5 The Bell Inequality Non-violation .6 Bell Inequality Violation: Entangled States .7 The Bell–Kochen–Specker Inequality .8 Commuting and Non-commuting Operators .com Contents xi 8 Quantum Superposition .1 Superposing State Vectors .2 Probability and Probability Amplitudes .3 Empirical and Trans-Empirical Paths .5 The Mach–Zehnder Interferometer .6 Determinate Empirical Paths: No Interference .7 Indeterminate Trans-Empirical Paths: Interference .11 Partial Quantum Eraser. 169 9 Quantum Theory of Measurement .1 Measurement: Trans-Empirical to Empirical .2 Position Projection Operator .3 Repeated Observations in Quantum Mechanics .4 Expectation Value of Projection Operators .5 The Experimental Device.6 The Process of Measurement .7 Mixed Density Matrix ρM .8 Reduced Density Matrix ρR .9 Preparation of a Quantum State .10 The Heisenberg Uncertainty Principle .11 Theories of Quantum Measurement. 202 10 The Stern–Gerlach Experiment .2 Classical and Quantum Predictions .3 The Stern–Gerlach Hamiltonian .4 Electron’s Time Evolution .5 Entanglement of Spin and Device .6 Summary of Spin Measurement .7 Irreversibility and Collapse of State Vector .8 Interpretation of Spin Measurement. 219 11 The Feynman Path Integral .1 Probability Amplitude and Time Evolution .3 Superposition of Trans-Empirical Paths .4 The Dirac–Feynman Formula .6 The Feynman Path Integral .com xii Contents 11.7 Path Integral for Evolution Kernel .8 Composition Rule for Probability Amplitudes .9 Trans-Empirical Paths and Path Integral .10 State Vector and Trans-Empirical Paths .11 Path Integral Quantization: Action .12 Hamiltonian from Lagrangian .1 Three Formulations of Quantum Mechanics .2 Interpretations of Quantum Mechanics.
253 Glossary of Terms. 257 List of Symbols .com Synopsis 1 The epoch-making idea of the quantum as a fundamental property of Nature was introduced by Max Planck in 1900. Quantum mechanics is undoubtedly one of the most important and experimentally accurate scientific theory in the history of science.1 Its range of applications and mathematical depth are unmatched, and quantum mechanics continues to yield novel and unexpected results—in technology as well as in all scientific fields, including physics and mathematics. Paradoxically enough, in spite of all its empirical and mathematical success quantum mechanics— due to its strange and enigmatic conceptual framework—has, until now, defied all attempts to reach a satisfactory understanding of its inner workings.
The human being’s five physical senses are based on natural processes that can perceive only a finite range of physical phenomena. In the case of electromagnetic radiation, only a tiny and limited range of its wavelengths are visible to the human eye, with radiation of much longer and much shorter wavelengths being invisible. Since the smallest allowed quantum of energy for light (and for atoms) is truly minuscule when compared to the energies we encounter in daily life, there are only a few physical process, most of them being man-made, where one can directly observe quantum phenomena using one’s five senses. When we extend our five senses with experimental devices and instruments, we can probe more deeply into Nature’s secrets, and the quantum aspect of Nature becomes more apparent.