com Gennaro Auletta Shang-Yung Wang PAN STANFORD irrr PUBLISHING www.com Published by Pan Stanford Publishing Pte. Penthouse Level, Suntec Tower 3 8 Temasek Boulevard Singapore 038988 Email: editorial@panstanford.com Web: www.com British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Quantum Mechanics for Thinkers Copyright© 2014 Pan Stanford Publishing Pte. All rights reserved.
This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-4411-71-4 (Hardcover) ISBN 978-981-4411-72-1 (eBook) Printed in the USA www.com To our families www.com Contents Foreword xi Introduction 1 PART I BASIC ISSUES: STATES 1 Classical Mechanics 9 1.1 Classical-Mechanical Description 9 1.2 Basic Principle of Classical Mechanics 12 1.3 Summary 15 2 Superposition Principle 17 2.1 Origin and Foundations of Quantum Mechanics 17 2.2 Classical and Quantum Superposition 18 2.3 A Photon in an Interferometer 20 2.5 Formulation of the Superposition Principle 26 2.6 Transmission, Reflection, and Phase Shift 27 2.7 Action of the Second Beam Splitter 33 2.8 Computing the Detection Probabilities 35 2.9 Summary 37 3 Quantum States as Vectors 39 3.2 Action of the Polarization Filter 40 3.3 Vector Spaces and Bases 42 3.4 Scalar Products and Brackets 44 3.5 Polarization Filters as Projectors 49 3.6 Projectors as Matrices 51 www.com viii I Contents 3.7 Action and Properties of Projectors 55 3.8 Summary 59 4 Bases and Operations 61 4.1 Corpuscular Nature of Light 61 4.2 Further Experimental Evidences 65 4.4 Quantum Observables in General 69 4.5 Different Bases and Superposition 72 4.6 Change of Basis as a Unitary Transformation 75 4.7 Not all Operations Commute 80 4.8 Features vs Properties 84 4.9 Summary 85 5 Complementarity Principle 87 5.1 Undulatory Nature of Matter 87 5.2 Interferometry with a Blocked Path 88 5.3 Classical and Quantum Probability 90 5.4 Double Slit Experiment 93 5.5 Path Predictability and Interference Visibility 96 5.6 Delayed Choice Experiment 101 5.7 Summary 103 PART II FORMAL Issues: OBSERVABLES 6 Position and Momentum 107 6.1 Position Operator: Discrete Case 107 6.2 From Summation to Integration 110 6.3 Position Operator: Continuous Case 117 6.4 Derivatives: From Finite to Infinitesimal Quantities 123 6.5 Partial and Total Derivatives 132 6.6 Momentum as Generator of Space Translations 136 6.8 Commutation and Uncertainty Relations 147 6.9 Conceptual Aspects of the Uncertainty Relations 153 6.10 Summary 156 7 Energy and Quantum Dynamics 157 7.1 Hamiltonian and Classical Dynamics 157 www.com Contents I iK 7.3 Time Evolution as a Unitary Transformation 162 7.4 Active and Passive Transformations 166 7.5 Schrodinger and Heisenberg Pictures 171 7.10 Summary 198 8 Angular Momentum and Spin 201 8.1 Angular Momentum as Generator of Rotations 202 8.2 Angular Momentum Operator 207 8.3 Quantization of Angular Momentum 210 8.4 Angular Momentum Eigenfunctions 216 8.5 Central Potential and the Hydrogen Atom 223 8.6 Spin Angular Momentum 235 8.7 Addition of Angular Momenta 244 8.8 Identical Particles and Spin 248 8.9 Summary 253 PART Ill ONTOLOGICAL ISSUES: PROPERTIES 9 Measurement Problem 257 9.1 Statement of the Problem 257 9.2 Density Matrix and Projectors 260 9.5 Role of the Environment 269 9.6 Entropy and Information 271 9.7 Reversibility and Irreversibility 280 9.9 Summary 292 10 Non-Locality and Non-Separability 293 10.2 Bohr's and Schrodinger's Criticism of EPR 300 10.3 EPR-Bohm Experiment 303 www.7 Kochen-Specker Theorem 325 10.8 Summary 331 11 Quantum Information 333 11.1 Nature of Information 333 11.7 Mutual Information and Entanglement 374 11.8 Information and Non-Separability 386 11.2 Bounds on Information Acquisition 400 12.5 Fundamental Information Triad 413 12.6 Summary 416 Bibliography 417 Author Index 433 Subject Index 437 Solutions to Selected Problems 445 www.com Foreword The discovery of quantum mechanics and its comprehension are at the basis of the foundations of modern technology.
This fact is not widely recognized. I believe that if one asks the layman which are the most important technological applications of quantum mechanics, he would mostly select nuclear power. After some reflections he could mention lasers, but he would not think of the most important one, i., the transistor that is at the basis not only of computers but of practically any device we commonly use (with some notable exceptions like bicycles, wind surfs, and skis). People who are not trained in quantum mechanics can use a transistor without difficulties, and with some minor technical training they can understand the specifications and use transistors to build simple devices like a wireless radio: transistors behave in a way that is not very different from the old thermionic tubes.
However, quantum mechanics has been crucial in the design of transistors, which, when finally constructed, worked exactly as predicted by quantum mechanics. In spite of the ubiquitousness of quantum mechanics appli- cations, quantum mechanics remains some kind of mystery not only for learned people with a humanistic background, but also for most of the scientists, with the exception of physicists and chemists. The intrinsic difficulty in understanding the principles of quantum mechanics certainly contributes to this deplorable situation. However, this situation is worsened by an aura of incomprehensibility that derives from most of the presentations of quantum mechanics that one find in the literature.
Indeed, books that describe quantum mechanics may be divided into two main categories: www.com xii I Foreword • Those that require an advanced knowledge of mathematical analysis (differential and integral calculus), thus casting away most of the people. Such books are perfect for people interested in getting a working knowledge of quantum mechanics, but are of no use for those interested in knowing only what quantum mechanics is and in understanding its implications. • Those that are directed toward the general public. Although some of these books are excellent, their presentation is limited to a qualitative description.
By the time one gets ready to see how all extraordinary properties of quantum mechanics could be implemented in a quantitative description of the system, the presentation, in most cases, stops, usually adding something like "More details would be too technical; they need too much mathematics and therefore cannot be described here." At the end, quantum mechanics seems to be something like magic that can be understood only by fifth-level wizards. On the contrary, this book makes a strong effort to arrive to a quantitative formulation of quantum mechanical for very simple systems, a formulation that is constructed using minimal mathema- tical requirements. In this way the reader can easily arrive at the conceptual core of quantum mechanics in its precise mathematical formulation without having to know analysis and calculus. This can be done only if the authors are very careful in choosing the model systems that one uses for the presentation: the choice made in this book is very appropriate so that the reader becomes acquainted with the formalism of quantum mechanics in the simplest possible way.
Only in the second part of the book, after a minimal description of the analytic mathematical tools needed, the reader finds the extension (to a generic system) of the formalism that he or she has learned in the first part. In this way the reader arrives at an understanding of the usual formalism of quantum mechanics separating the conceptual steps, described in the first part, from the technical issues, described in the second part. In the last part of the book, Ontological Issues, the authors discuss the general implications of quantum mechanics that have www.com Foreword I xiii been discussed in many places, including popularization articles: the measurement problem, non-locality and non-separability, quantum information, and finally the interpretation of quantum mechanics. The authors' viewpoint on these highly debated subjects is deep and original: the presentation is quite concise, although it does not shy away from giving technical details where needed.
The book is well written and is very readable. It fulfills at its best the premise of the title Quantum Mechanics for Thinkers. Giorgio Parisi www.com Introduction Reasons for Studying Quantum Mechanics Quantum mechanics represents one of the great conceptual revolu- tions of the 20th century. It has raised a huge number offundamental questions of both physical and philosophical kind.
• What does matter mean at all? • What are the main properties or characteristics of matter? • Can matter be reduced to information? • Is our universe probabilistic at the most fundamental level? • Are there non-local correlations in nature? • Are non-causal interconnections between physical systems possible? • Is the bound on the speed of information propagation set by the theory of relativity violated? • What do terms like state, observable, and property mean at all? • Can physical reality exist without observers? • Are observers necessary for having a macroscopic world? • What are the general features of information processing and exchange in our universe? These questions (and there are also many others) give a first feeling about the depth of the conceptual turn represented by quantum mechanics. Even those classical hypotheses or laws that have passed the quantum mechanical check have somehow been transformed or at least been corrected. It is important for people who desire to deal with fundamental problems in science, especially in quantum theory or in those fields (like chemistry, mathematics, and informatics) that are closely related to quantum theory, to have a deep and clear understanding of this kind of problems. This book provides such an www.com 2 I1ntroduction opportunity.
We think that undergraduate students in physics could also take advantage of this book, and then transition to more difficult stuff. This book could also be of some use in the last years of the high school. Indeed, one of the major problems we find for these classes is that most of our students go out of the school without having ever heard a single word about quantum mechanics, that is, about the basic physical theory that we have, and it is likely that most of them will never have the opportunity to come back to these issues. The book is also addressed to people interested in the philosophy of science or in problems at the interface between science and philosophy.
As a matter of fact, one of the biggest problems of modern thought is a fracture between science and philosophy causing severe alienation to both fields. Indeed, science without philosophy can become a pure technique, where finally ad hoc solutions and pure simulations dominate, whereas philosophy without science can shift toward esotericism and aestheticism. As a matter of fact, the issues that have been raised within natural sciences, and especially in physics, have always implied a deep shift of the philosophical paradigms. The affirmation of Galilean and Newtonian classical mechanics, which is an important part of the first scientific revolution, has led to a radical rearrangement of the theory of knowledge, first making of the physical science a privileged reference and then, with Kant's doctrine of the a priori synthetic judgments, as the unique and authentic form of knowledge.
Quantum mechanics implies, or should imply, even a more radical change of the philosophical modules. However, this has happened in an incomplete and partial form. This is because the discussion on the foundations of this theory is not yet accomplished and so far has not even been dealt with at a sufficiently deep level. Theoretically deal- ing with the foundations of quantum mechanics is an urgent task', es- pecially considering its huge predictive power and the wide domain of applicability.
Its practical consequences already determine many aspects of our modern society (atomic bombs and atomic energy, semiconductors, transistors, and photovoltaic cells, lasers and light- emitting diodes, applications to technology of new states of matter like Bose-Einstein condensates, etc.) and many other may be deter- mined in the near future (quantum cryptography, quantum telepor- tation, quantum computation, photography without light, etc.