Foundations of Quantum Mechanics

Undergraduate Lecture Notes in Physics

Foundations

of Quantum

Mechanics

Travis Norsen

An Exploration of the Physical Meaning

of Quantum Theory

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Travis Norsen

Foundations of Quantum

Mechanics

An Exploration of the Physical Meaning

of Quantum Theory

123

Travis Norsen

Department of Physics

Smith College

Northampton, MA

USA

ISSN 2192-4791 ISSN 2192-4805 (electronic)

Undergraduate Lecture Notes in Physics

ISBN 978-3-319-65866-7 ISBN 978-3-319-65867-4 (eBook)

DOI 10.1007/978-3-319-65867-4

Library of Congress Control Number: 2017949150

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Bohr always would go in for this remark,

‘You cannot really explain it in the

framework of space and time.’ By God, I was

determined I was going to explain it in the

framework of space and time.

—John Slater

Preface

This textbook is intended as a lifeline to physics students (of either the traditional or

the autodidactic variety) who have had some preliminary exposure to quantum

mechanics but who want to actually try to make physical and conceptual sense

of the theory in the same way that they have been trained and expected to do when

learning about other areas of physics. Its main goals are (i) to help students

appreciate and understand the concerns that people like Einstein, Schrödinger, and

Bell have had with traditional formulations of the theory and (ii) to introduce

students to the several extant formulations of quantum theory which purport to

address at least some of the concerns and provide candidate accounts of what

quantum theory might actually imply about how the micro-physical world works.

The book grew out of, and its structure in many ways reflects, the “special topics

in physics” course on the Foundations of Quantum Mechanics that I taught at Smith

College in the Spring of 2016. In this seminar-style course, students would read

through each new chapter (and attempt a few of the end-of-chapter Projects that

I recommended as appropriate pre-class exercises) prior to our weekly three-hour

meeting. During our time together in class, we would discuss the more difficult

concepts and derivations from the text, students would share their (sometimes only

partial) solutions to the assigned pre-class projects (and we would discuss and

complete those as needed), and then we would tackle some additional projects.

Not surprisingly, then, I envision the book being most straightforwardly useful

for a similarly structured elective course in a physics department (or perhaps for a

philosophy-of-physics course focused on the Foundations of Quantum Mechanics

in a philosophy department). But the fact that the chapters were created as pre-class

readings (as opposed to transcripts of “lecture notes”) perhaps makes this, com￾pared to most physics textbooks, unusually readable and accessible to individuals

for whom it is not the textbook for any official course—e.g., interested physics

students who are not lucky enough to find themselves in a department that offers an

elective course on the Foundations of Quantum Mechanics, or just anyone with an

interest in the puzzling and fascinating history, philosophy, and, really, physics of

quantum physics.

vii

The book begins with two introductory chapters. Chapter 1 (“Pre-Quantum

Theories”) introduces a number of important concepts and ideas in the context of

classical physics theories such as Newtonian gravity and Maxwellian electrody￾namics. Chapter 2 (“Quantum Examples”) then provides a lightning overview of

some quantum mechanical formalism and examples that serve as a foundation for

later discussions. The level of these two chapters (as well as the rest of the book)

reflects the background preparation I was able to expect for the students in my

course at Smith College: they had taken a sophomore-level “modern physics”

course including exposure to Schrödinger’s equation and 1-D wave mechanics

(but had not yet taken, or were in some cases taking concurrently, a junior-level

“quantum mechanics” course using, for example, the text by Griffiths); similarly,

they had seen Maxwell’s equations before (in a 100-level introductory course and

perhaps also a 300-level E&M course) and had a fairly strong prior exposure to

vector calculus and differential equations. Still, the students found some of the

material in the two introductory chapters quite fresh and challenging.

Readers who are missing one or more of the prerequisites I just mentioned (or

readers who are pursuing physics outside of, or perhaps decades beyond, any

organized undergraduate physics curriculum) should thus anticipate some struggle

with some mathematical details in the first two chapters. However, I want to

reassure people in this category that it will be OK, and that they should get what

they can out of the first two chapters and press forward into the rest of the book. Let

me explain my attitude here with an example. I don’t think you can fully appreciate

Bell’s Theorem (the subject of Chap. 8) without digesting, in Chap. 1, the rea￾sonableness of Bell’s formulation of “locality” as a generalization of the specifically

deterministic sort of local causality exhibited by Maxwellian electrodynamics (in

contrast to Newtonian gravitation). But for readers for whom understanding the

mathematical details is too big a stretch, it will suffice to merely accept that Bell’s

formulation purports to be a natural generalization of the important relativistic

locality of classical E&M.

After the two introductory chapters, the book turns toward the first goal men￾tioned above: Helping students appreciate and understand the concerns that people

like Einstein, Schrödinger, and Bell had with traditional formulations of quantum

theory. We begin in Chap. 3 by studying “The Measurement Problem” which was

most famously illustrated by Schrödinger’s infamous cat and then emphasized and

significantly clarified by Bell. Chapter 4 tackles “The Locality Problem” which was

most famously brought out in the 1935 paper of Einstein, Podolsky, and Rosen—

although, as we will discuss in detail, this canonical presentation does not perfectly

capture Einstein’s fundamental objection to the orthodox interpretation. Finally,

Chap. 5 introduces “The Ontology Problem”—a concern that was intensely wor￾rying to Schrödinger, Einstein, and others in the early days of quantum mechanics,

but which has, unfortunately, been largely forgotten in the instrumentalist and

anti-realist wake of the Copenhagen orthodoxy (and which, again unfortunately,

remains under-appreciated even by certain schools of anti-Copenhagen quantum

realism). One of the things I like best about this book is that it gives the ontology

problem pride of place alongside the (more widely recognized) measurement and

viii Preface

locality problems as one of the “big three” concerns that clearheaded physicists

should have in mind when they are evaluating and developing candidate theories.

Having thus surveyed the central problems that one would hope to see resolved,

the book turns to reviewing and assessing the menu of available resolutions. We

cover, in particular, what I consider the four most important perspectives on

quantum mechanics that curious and intelligent physics students should understand.

These include: in Chap. 6, “The Copenhagen Interpretation” (which is a

self-confessed non-candidate for genuinely explaining micro-physics in an ordi￾nary, realist way, but is of historical and sociological interest nevertheless since it

has been the official, if only superficially understood and half-heartedly accepted,

orthodoxy of the physics community for nearly a century); in Chap. 7,

“The Pilot-Wave Theory” of de Broglie and Bohm; in Chap. 9, “The Spontaneous

Collapse Theory” of Ghirardi, Rimini, Weber, and Pearle; and, finally, in Chap. 10,

“The Many-Worlds Theory” of Everett. Chapter 8, on “Bell’s Theorem,” is a kind

of sequel to Chap. 4 which explains the Earth-shattering advance that Bell was led

to from his study of the pilot-wave theory.

The material in this second half of the book is, to a large but not perfect extent,

organized historically. Thus, the Copenhagen Interpretation (largely developed in

the 1930s) comes first, the pilot-wave theory (originally proposed by de Broglie in

1927 but then largely forgotten until Bohm resurrected the idea in 1952) comes

second, then we turn to Bell’s theorem of 1964 (which, as mentioned, was directly

stimulated by Bell’s contemplation of a seemingly troubling feature of the

pilot-wave theory), and then the spontaneous collapse theories (which only began to

be developed in the 1980s). Everett’s many-worlds theory is presented last, despite

the fact that Everett proposed it in 1957 (between Bohm’s resurrection of the

pilot-wave idea and Bell’s presentation of his important theorem), both because the

theory was not widely recognized as a serious candidate account of quantum

phenomena until much more recently, and also because I think it is hard to recover

from studying something rather surreal and focus on something rather more

mundane!

Note that it might be slightly puzzling that the Copenhagen Interpretation is only

covered (in Chap. 6) after we have reviewed the measurement, locality, and

ontology problems (in Chaps. 3, 4, and 5, respectively)—this despite the fact that

these “problems” were largely raised in response to the interpretive pronounce￾ments of Bohr and Heisenberg and their colleagues. I structured things this way in

part because I assume that students will already have been exposed, as part of a

“modern physics” type course, to the basic Copenhagen philosophy of insisting on

the completeness of the description in terms of wave functions alone (but also,

paradoxically, denying the reality of wave functions) and then foreclosing further

discussion as somehow scientifically inappropriate. So I thought students would be

able to appreciate the somewhat-reactionary concerns of, for example, Einstein and

Schrödinger, without any explicit prior discussion of the Copenhagen philosophy.

In addition, I think having a clear sense of the critics’ concerns can help motivate

students to actually care about what, exactly, Bohr and Heisenberg said: Did they

really assert what the critics reacted against, and did they have viable answers to the

Preface ix

criticisms? Finally, I thought that giving Bohr and Heisenberg the last word (after

hearing from the critics) was a good way to try to maintain the neutrality that I have

aimed at throughout the book—despite, perhaps obviously, not thinking very highly

of the Copenhagen philosophy.

In the Smith College course, we went through these topics at a pace of one

chapter per week. That left a couple of weeks at the end of the semester, during

which the students each picked a topic they were individually interested in

exploring further, did some independent reading and research, and then gave a

presentation back to the class summarizing what they had learned and uncovered.

This structure is reflected in the present book, which closes with an “Afterword”

that tries to bring an (admittedly limited) element of closure to the covered topics by

summarizing where things stand and then provides an informal laundry list of

recommended topics for further exploration, including pointers to some more

contemporary literature.

I attempt, though, even in the ten chapters of the book, to build bridges to the

primary literature. There is, for example, extensive quoting from the published

papers (as well as the private correspondence) of Einstein, Lorentz, Schrödinger,

Bell, Bohr, Heisenberg, etc., and many of the end-of-chapter Projects invite stu￾dents to read some accessible piece of primary literature and report on things they

find interesting or surprising. Indeed, one of my goals with this book is to help

students appreciate the extent to which their own confusions and concerns about

quantum mechanics are not something to feel ashamed of (a feeling that is too-often

the result of the “shut up and calculate” attitude that quantum physics professors

frequently take toward the subjects we cover). Instead, students should feel proud

that they can understand, and indeed in many cases will have anticipated without

realizing it, concerns that were shared by some of the giants of twentieth-century

physics—concerns that have unfortunately been suppressed and forgotten rather

than adequately addressed. To capture the intended spirit of the book in this respect,

I can do no better than quote from an email from my friend Kenny Felder who read

drafts of most of the chapters:

Reading [this], I have I think exactly the sense that you want me to have—or perhaps the

meta-sense that you want me to have—in any case it’s a wonderful sense that I really have

never had before. I have the sense of a group of men who are very smart but perfectly

human, right at the dawn of the quantum revolution, desperately trying to figure out what

the experimental evidence is actually telling them. I see them throwing ideas around, trying

and rejecting theories, alone and in correspondence with each other. And I get the sense that

somewhere between them and us, that search for a coherent theory more or less evapo￾rated—not because the questions were answered, but more because people kind of forgot

about them—and you’re trying to revitalize that quest. It’s exciting!

Let me finally say something about the end-of-chapter “Projects” which I consider

to be an essential component of the book, just as they were an essential component

of the course it grew out of. Some of these are rather like traditional end-of-chapter

exercises, which ask students, for example, to fill in gaps in derivations from the

text or apply concepts introduced in the text to simple concrete examples. But many

of the Projects are considerably more challenging and open-ended. For example, as

x Preface

mentioned above, some invite students to read an article or essay that has been

discussed in the text and report back on things they find interesting, surprising, or

novel. Some projects invite students to use Mathematica or another programming

language to create helpful visualizations or numerical solutions of difficult prob￾lems. There are even a few Projects (perhaps most suitable for students using the

text in the context of a traditional course) asking students to interview a few

physicists to get a sense of how real people think about some issue. It is hoped that

the diversity and open-endedness of the Projects will allow students with many

different backgrounds, technical abilities, and interests to stay actively engaged with

the material (before, during, after, and/or without classroom time, as appropriate in

each individual case).

Let me close by thanking Darby Bates, Jean Bricmont, Kira Chase, Kenny

Felder, and Trevor Wright for reading, and providing significant helpful feedback

on, earlier versions of at least some of the chapters. I also owe a more generalized

debt of gratitude to my tireless and inspiring wife Sarah, and my kids Finn and Tate,

for helping keep me grounded and happy—as well as to my wonderful parents,

Steve and Carol, for believing and investing in me (especially by supporting my

own undergraduate education at Harvey Mudd College, where my interest in the

subject matter of this book began).

Northampton, USA Travis Norsen

Preface xi

Contents

1 Pre-Quantum Theories .................................... 1

1.1 Newtonian Mechanics.................................. 1

1.2 Maxwellian Electrodynamics ............................ 5

1.3 Locality ............................................ 8

1.4 Bell’s Formulation of “Locality” ......................... 13

1.5 Ontology............................................ 18

1.6 Measurement ........................................ 22

1.7 Abstract Spaces ...................................... 25

References............................................... 31

2 Quantum Examples....................................... 33

2.1 Overview ........................................... 33

2.2 Particle-in-a-Box...................................... 36

2.3 Free Particle Gaussian Wave Packets ...................... 38

2.4 Diffraction and Interference ............................. 44

2.5 Spin ............................................... 47

2.6 Several Particles ...................................... 51

References............................................... 57

3 The Measurement Problem................................. 59

3.1 The Quantum Description of Measurement.................. 59

3.2 Formal Treatment ..................................... 64

3.3 Schrödinger’s Cat and Einstein’s Bomb .................... 69

3.4 Hidden Variables and the Ignorance Interpretation ............ 74

3.5 Wrap-Up............................................ 79

References............................................... 85

4 The Locality Problem ..................................... 87

4.1 Einstein’s Boxes...................................... 87

4.2 EPR ............................................... 96

4.3 Einstein’s Discussions of EPR ........................... 100

xiii

4.4 Bohm’s Reformulation ................................. 104

4.5 Bell’s Re-Telling ..................................... 107

References............................................... 113

5 The Ontology Problem .................................... 115

5.1 Complexity and Reality ................................ 115

5.2 Configuration Space ................................... 118

5.3 Ontology, Measurement, and Locality ..................... 122

5.4 Schrödinger’s Suggestion for a Density in 3-Space ........... 129

5.5 So Then What?....................................... 133

References............................................... 139

6 The Copenhagen Interpretation ............................. 141

6.1 Bohr’s Como Lecture .................................. 142

6.2 Heisenberg .......................................... 148

6.3 Bohr on Einstein’s Diffraction Example .................... 154

6.4 The Photon Box Thought Experiment ..................... 160

6.5 Bohr’s Reply to EPR .................................. 166

6.6 Contemporary Perspectives.............................. 169

References............................................... 174

7 The Pilot-Wave Theory.................................... 177

7.1 Overview ........................................... 178

7.2 Particle in a Box...................................... 182

7.3 Other Single Particle Examples........................... 185

7.4 Measurement ........................................ 188

7.5 Contextuality ........................................ 194

7.6 The Many-Particle Theory and Nonlocality ................. 199

7.7 Reactions ........................................... 205

References............................................... 212

8 Bell’s Theorem........................................... 215

8.1 EPRB Revisited ...................................... 215

8.2 A Preliminary Bell Inequality ............................ 218

8.3 The Real Bell (and the CHSH) Inequality .................. 222

8.4 Experiments ......................................... 227

8.5 What Does It Mean?................................... 231

8.6 (Bell’s) Locality Inequality Theorem ...................... 236

References............................................... 243

9 The Spontaneous Collapse Theory ........................... 245

9.1 Ghirardi, Rimini, and Weber ............................ 246

9.2 Multiple Particle Systems and Measurement................. 254

9.3 Ontology, Locality, and Relativity ........................ 259

9.4 Empirical Tests of GRW ............................... 265

References............................................... 271

xiv Contents

10 The Many-Worlds Theory ................................. 273

10.1 The Basic Idea ....................................... 274

10.2 Probability .......................................... 280

10.3 Ontology............................................ 286

10.4 Locality ............................................ 291

References............................................... 302

Afterword .................................................. 303

Contents xv

Chapter 1

Pre-Quantum Theories

In this introductory chapter we review two theories from classical physics –

Newtonian mechanics and Maxwellian electrodynamics – and use them to intro￾duce a number of concepts (such as determinism, locality, ontology, measurement,

and configuration space) that we will explore in the context of quantum mechanics

in subsequent chapters.

1.1 Newtonian Mechanics

As a first example of a “pre-quantum theory” let’s consider the picture of the universe

formulated by Isaac Newton. The theory, in a nutshell, says that the physical world

consists of particles interacting by means of forces which the particles exert on one

another and which influence the particles’ motions. About the particles, Newton

wrote:

...it seems probable to me, that God in the Beginning form’d Matter in solid, massy, hard,

impenetrable, moveable Particles, of such Sizes and Figures, and with such other Properties,

and in such Proportion to Space, as most conduced to the End for which he form’d them;

and that these primitive Particles being Solids, are incomparably harder than any porous

Bodies compounded of them; even so very hard, as never to wear or break in pieces.... [A]ll

material Things seem to have been composed of the hard and solid Particles above-mention’d,

variously associated.... [1, pp. 400–2]

Newton’s endorsement of the idea that observable macroscopic objects are composed

of invisibly small, indestructible particles is a kind of bridge between the speculative

notion of atomism that had been introduced by Ancient Greek philosophers such

as Democritus, and the more scientific atomic theory of matter that grew out of

chemistry and physics in the centuries following Newton.

Regarding the forces that these particles exert on one another, Newton wrote that

Bodies act one upon another by the Attractions of Gravity, Magnetism, and Electricity; and

these Instances shew the Tenor and Course of Nature, and make it not improbable but that

© Springer International Publishing AG 2017

T. Norsen, Foundations of Quantum Mechanics, Undergraduate Lecture

Notes in Physics, DOI 10.1007/978-3-319-65867-4_1

1