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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


for the WorldWind PowerThe Rise of Modern Wind Energy

Preben MaegaardAnna KrenzWolfgang Palz

editors

Pan Stanford Series on Renewable Energy — Volume 2


Gennaro AulettaShang-Yung Wang

April 28, 2014 11:0 PSP Book - 9in x 6in 00-Auletta–prelims

Published by

Pan Stanford Publishing Pte. Ltd.

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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.

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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


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


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


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


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:


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


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


April 28, 2014 10:57 PSP Book - 9in x 6in auletta

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


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.).


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


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).


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


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