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Auletta | Wang
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Quantum Mechanics for Thinkers provides a quick access to quantum mechanics without dealing with a true textbook that demands proper specialized studies in physics and related mathematics.
The book consists of three parts: Basic Issues, in which the basic notions of quantum mechanics are introduced; Formal Issues, in which more advanced topics are discussed; and Ontological Issues, in which the conceptual and interpretational problems are dealt with. The book deals with the most recent developments in quantum information and non-locality. It comprises 70 figures, which are a crucial instrument to acquaint the readers with abstract problems in a “representative” way, and 30 in-section boxes, which assist the readers to solve even the most difficult mathematical problems. The book has about 130 problems (most of them solved) to help the readers test themselves and verify how well they were able to understand the topics.
One of its kind, this is the only book available in the market that introduces undergraduate students in physics and scholars of adjacent fields (chemistry, mathematics, engineering, information science, biology, and philosophy) to the study of such a difficult field in an easy-to-understand language. It also offers a considerable number of clear and analytical treatments for what are considered the most difficult conceptual problems of the theory. Although a textbook, and therefore not a popular book, it can also prove useful for a very bright spectrum of potential readers.
Gennaro Auletta is a senior researcher at the University of Cassino, Italy. Prof. Auletta has authored and coauthored 17 books, including Quantum Mechanics (with G. Parisi and M. Fortunato; Cambridge University Press, 2009, 2013) and Cognitive Biology: Dealing with Information from Bacteria to Minds (Oxford University Press, 2011), and more than 70 papers. His areas of interest are metaphysics, philosophy of nature, logic, foundations and interpretation of quantum mechanics, quantum information, system biology, cognitive biology,
top-down causation in biology and neurosciences, and mathematical definition of complexity.
Shang-Yung Wang is an associate professor of physics at Tamkang University, Taiwan. His current research interests include elementary particles, cosmology, quantum information, and foundations of quantum mechanics.
Quantum Mechanics for Thinkers
ISBN 978-981-4411-71-4V367
QUANTUM MECHANICS for THINKERS
Gennaro AulettaShang-Yung Wang
QUANTUM MECHANICS for THINKERS
for the WorldWind PowerThe Rise of Modern Wind Energy
Preben MaegaardAnna KrenzWolfgang Palz
editors
Pan Stanford Series on Renewable Energy — Volume 2
QUANTUM MECHANICS for THINKERS
Gennaro AulettaShang-Yung Wang
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Published by
Pan Stanford Publishing Pte. Ltd.
Penthouse Level, Suntec Tower 3
8 Temasek Boulevard
Singapore 038988
Email: [email protected]
Web: www.panstanford.com
British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.
Quantum Mechanics for Thinkers
Copyright c© 2014 Pan Stanford Publishing Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying,recording or any information storage and retrieval system now known or tobe 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
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To our families
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Contents
Foreword xi
Introduction 1
PART I BASIC ISSUES: STATES
1 Classical Mechanics 91.1 Classical-Mechanical Description 9
1.2 Basic Principle of Classical Mechanics 12
1.3 Summary 15
2 Superposition Principle 172.1 Origin and Foundations of Quantum Mechanics 17
2.2 Classical and Quantum Superposition 18
2.3 A Photon in an Interferometer 20
2.4 Probability Amplitudes 24
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 393.1 Photon Polarization 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
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viii Contents
3.7 Action and Properties of Projectors 55
3.8 Summary 59
4 Bases and Operations 614.1 Corpuscular Nature of Light 61
4.2 Further Experimental Evidences 65
4.3 Quantization Principle 67
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 875.1 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 1076.1 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
6.7 Momentum Representation 143
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 1577.1 Hamiltonian and Classical Dynamics 157
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Contents ix
7.2 Schrodinger Equation 160
7.3 Time Evolution as a Unitary Transformation 162
7.4 Active and Passive Transformations 166
7.5 Schrodinger and Heisenberg Pictures 171
7.6 Free Particle 174
7.7 Harmonic Oscillator 178
7.8 Density Matrix 187
7.9 Composite Systems 192
7.10 Summary 198
8 Angular Momentum and Spin 2018.1 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 III ONTOLOGICAL ISSUES: PROPERTIES
9 Measurement Problem 2579.1 Statement of the Problem 257
9.2 Density Matrix and Projectors 260
9.3 Projection Postulate 262
9.4 Basis Ambiguity 266
9.5 Role of the Environment 269
9.6 Entropy and Information 271
9.7 Reversibility and Irreversibility 280
9.8 Schrodinger’s Cat 289
9.9 Summary 292
10 Non-Locality and Non-Separability 29310.1 EPR Paper 293
10.2 Bohr’s and Schrodinger’s Criticism of EPR 300
10.3 EPR–Bohm Experiment 303
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x Contents
10.4 Bell Theorem 306
10.5 Entanglement Swapping 316
10.6 Eberhard Theorem 319
10.7 Kochen–Specker Theorem 325
10.8 Summary 331
11 Quantum Information 33311.1 Nature of Information 333
11.2 Information Accessibility 336
11.3 Potential Information 344
11.4 Quantum Computation 348
11.5 Quantum Teleportation 364
11.6 Quantum Cryptography 369
11.7 Mutual Information and Entanglement 374
11.8 Information and Non-Separability 386
11.9 Summary 390
12 Interpretation 39312.1 Information Acquisition 393
12.2 Bounds on Information Acquisition 400
12.3 Operations 408
12.4 Theoretical Entities 410
12.5 Fundamental Information Triad 413
12.6 Summary 416
Bibliography 417
Author Index 433
Subject Index 437
Solutions to Selected Problems 445
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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.e., 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:
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xii 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
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Foreword 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
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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 of fundamental
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
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2 Introduction
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.).
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Introduction 3
Aim of the Book
We find that most of the problems that we have stated above are
often treated by public opinion and even by cultivated laypersons
with superficiality and without a true understanding of the physical
and conceptual foundations of quantum theory. It is very often heard
or read that quantum mechanics allows telepathy or that reality
does not exist. Statements like these show a deep misunderstanding
about the true meaning of quantum theory. Therefore, the main aim
of the book is to allow the students, the scholars, the philosophers,
and even the laypersons interested in these issues to have a quick
access to quantum mechanics without dealing with a true textbook
that demands proper specialized studies in physics (and related
mathematics) for about a couple of years. The phrase “quick access”
does not mean that this is a popular science book. It is in fact a
scientific book, but addressed to people who do not already posses
the prerequisite for dealing with such a sophisticated scientific stuff.
In order to understand the theoretic and philosophical problems
in quantum mechanics, it is indeed necessary to master certain
formal instruments. In other words, this book does contain quite
a few equations. However, we have tried to reduce the formalism
to the minimum extent required for understanding the basis of the
theory. Moreover, we have also explained from scratch mathematical
tools like vector and matrix algebra, probability (in the first part of
the book), as well as integration and differentiation (in the second
part of the book). This is the reason why the first part is confined
to an algebraic approach. In this way, the book is somehow self-
contained and only presupposes some high-school background in
mathematics.
What Is Required of the Reader?
Although we shall try to do things as simply as possible, this does
not mean that the reader shall not meet some difficulties and should
not make some efforts to understand the mathematics and the
underlying physics. However, our basic assumption is that the study
of this book is in the range of university students and scholars of
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4 Introduction
any faculty, or of any cultivated layperson, who are motivated and
interested in deepening their knowledge of the subject. Where the
reader should meet some particular difficulties in mathematics, we
recommend to make use of some online resources where many
mathematical concepts are explained with different degrees of
difficulty. In particular, we suggest the online mathematics reference
MATHWORLD.a As an alternative, the reader can also take into
account the Mathematics Portal of WIKIPEDIAb and the ENCYCLOPEDIA
OF MATHEMATICS.c Finally, for a first introduction to this type of
mathematics we strongly recommend the textbook by Heller.d
To minimize the mathematics and to emphasize the underlying
physics, we have chosen to present many of the technical details in
the form of in-section boxes and end-of-section problems. There are
30 boxes and 130 problems altogether, with the solutions to most of
the problems provided at the end of the book. However, the reader is
encouraged to try to work out the problems by him- or herself before
resorting to the solutions provided. There are also many resuming
tables that help the reader quickly find the information that he or
she desires. Moreover, we have included 70 figures which not only
provide a kind of graphical help but often can even be understood as
an integral part of the explanation. In order to help the reader better
organize the concepts developed in the book, we have composed a
summary of the main concepts at the end of each chapter. Finally, the
book contains an extensive bibliography of about 150 entries, and
two full, accurate, and comprehensive subject and author indexes for
assisting the reader’s quick search.
While this book could be an excellent starting point for self-
study of quantum mechanics, it is obviously better if the reader
is helped by someone with a physics background in dealing with
this study. This could happen through an introductory course to
quantum mechanics but also through a tutorial. A word of caution
is also necessary. The present book does not substitute a complete
course in quantum mechanics as taught in any physics department
and taking advantage of more advanced textbooks.e With the help of
ahttp://mathworld.wolfram.combhttp://en.wikipedia.org/wiki/Portal:Mathematicschttp://www.encyclopediaofmath.orgd(Heller, 2006).e(Le Bellac, 2006), (Auletta et al., 2009).
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Introduction 5
this book, a careful reader can understand what quantum mechanics
is (and this is the aim of the book) but cannot learn to make use
of it. In other words, this book helps the reader understand what
quantum mechanics is, what are its conceptual foundations, and how
its basic formalism works, but does not make of him or her an expert
in quantum mechanics.
Outline of the Book
The book is divided into three major parts and is organized as
follows:
I. Basic Issues
The first part contains five fundamental chapters. Chapter 1
provides a short review of classical mechanical concepts.
Chapters 2–5 represent the foundation block that deals with the
basic notions of quantum theory. However, already Chapter 5
raises many conceptual problems that may give a taste of what
follows. Some readers may be satisfied to study this part. It is
relatively easy but also needs some time, especially if the reader
has never been engaged with mathematics or he or she was,
but many years ago. Our suggestion in this case is to read each
chapter repeatedly before going further in order to become fully
familiar with this language.
II. Formal Issues
The second part consists of three technical chapters, Chapters 6–
8. In this part we introduce some of the most important quantum
mechanical observables: position and momentum, energy, and
angular momentum and spin. Arguably, this is probably the
most difficult part for people not acquainted with physics or
mathematics. It is, however, necessary if one really wants to
understand the deep meaning of the philosophical conclusions
drawn in the subsequent chapters. We stress that the main
difficulty is not in the equations themselves, since each step
is explained and we presume that a patient reader who will
follow those steps shall also be able to consistently progress.
The main problem is rather in the large quantity of information
packed together. Then we suggest to proceed by taking time
in order to assimilate each step and eventually read each
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6 Introduction
section again and again. If, having tried several times, the
reader does not understand the meaning of certain steps, we
suggest to take them as facts since it is plausible that some
developments can become clearer afterwards. If the reader
encounters insurmountable difficulties, he or she may initially
skip the latter sections of these chapters and try to come back
to specific aspects when the third part of the book demands the
knowledge of some previous notions. Sooner or later, however,
if the reader wants a deeper understanding of the theory, it
becomes necessary to study the whole of it. Indeed, only to
have assimilated the Schrodinger equation or the model of the
hydrogen atom can bring the reader even to really appreciate
not only the usefulness but even the beauty of this theory.
III. Ontological Issues
The third part is composed of four advanced chapters. The
measurement problem is dealt with in Chapter 9, the issue of
quantum non-locality in Chapter 10, and quantum information
in Chapter 11. Finally, the interpretation of the theory that puts
together the previous three subjects (and all the main issues
raised in the book) is dealt with in Chapter 12. As a matter
of fact, Chapters 9–12 represent the block that will turn out
to be the most satisfactory one for people searching for a true
understanding of quantum mechanics. Here, the reader can
appreciate how worthwhile was the previous study for arriving
at such a point!
It is likely that the first part could be very useful for students in
the last years of the high school or for laypersons who intend to
understand the very basic notions of quantum mechanics. Scholars
in mathematics and chemistry will especially like the second part,
while scholars in informatics and philosophy will perhaps find
the third part more interesting. Undergraduate students in physics
should study the whole book carefully as a kind of fore-preparation
for the more technical studies.
Last but not least, the reader can visit the book’s websitea for
communications about the book and errata of the book.
ahttp://www.gennaroauletta.net/qmftbook