61,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
QUANTUM MECHANICS From classical analytical mechanics to quantum mechanics, simulation, foundations & engineering Quantum mechanics is a fundamental and conceptually challenging area of physics. It is usually assumed that students are unfamiliar with Lagrangian and Hamiltonian formulations of classical mechanics and the role played by probability. As a result, quantum physics is typically introduced using heuristic arguments, obscuring synergies with classical mechanics. This book takes an alternative approach by leveraging classical analytical mechanics to facilitate a natural transition to quantum physics. By doing so, a solid foundation for understanding quantum phenomena is provided. Key features of this textbook include: * Mathematics and Classical Analytical Mechanics: The necessary mathematical background and classical analytical mechanics are introduced gradually, allowing readers to focus on one conceptual challenge at a time. * Deductive Approach: Quantum mechanics is presented on the firm foundation of classical analytical mechanics, ensuring a logical progression of concepts. * Pedagogical Features: This book includes helpful notes, worked examples, problems, computational challenges, and problem-solving approaches to enhance understanding. * Comprehensive Coverage: Including advanced topics such as open quantum systems, phase-space methods, and computational methods for quantum physics including good programming practice and code design. Much of the code needed to reproduce figures throughout this book is included. * Consideration of Foundations: The measurement problem and correspondence principle are addressed, including an open and critical discussion of their interpretation and consequences. * Introduction to Quantum Systems Engineering: This is the first book to introduce Quantum Systems Engineering approaches for applied quantum technologies development. This textbook is suitable for undergraduate students in physics and graduate students in mathematics, chemistry, engineering, and materials science.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 749
Veröffentlichungsjahr: 2023
Ähnliche
Table of Contents
Cover
Table of Contents
Title Page
Copyright
Acronyms
About the Authors
Preface
Note
Acknowledgements
About the Companion Website
Introduction
Notes
1 Mathematical Preliminaries
1.1 Introduction
1.2 Generalising Vectors
1.3 Linear Operators
1.4 Representing Kets as Vectors, and Operators as Matrices and Traces
1.5 Tensor Product
1.6 The Heisenberg Uncertainty Relation
1.7 Concluding Remarks
Notes
2 Notes on Classical Mechanics
2.1 Introduction
2.2 A Brief Revision of Classical Mechanics
2.3 On Probability in Classical Mechanics
2.4 Damping
2.5 Koopman–von Neumann (KvN) Classical Mechanics
2.6 Some Big Problems with Classical Physics
Notes
3 The Schrödinger View/Picture
3.1 Introduction
3.2 Motivating the Schrödinger Equation
3.3 Measurement
3.4 Representation of Quantum Systems
3.5 Closing Remarks and the Axioms of Quantum Mechanics
Notes
4 Other Formulations of Quantum Mechanics
4.1 Introduction
4.2 The Heisenberg Picture
4.3 Wigner's Phase-space Quantum Mechanics
4.4 The Path Integral Formulation of Quantum Mechanics
4.5 Closing Remarks
Notes
5 Vectors and Angular Momentum
5.1 Introduction
5.2 On Curvilinear Coordinates (Using Spherical Coordinates as an Example)
5.3 The Theory of Orbital and General Angular Momentum
5.4 Addition of Angular Momentum
Notes
6 Some Analytic and Semi-analytic Methods
6.1 Introduction
6.2 Problems of the Form
Ĥ
0
+ Ŵ
6.3 The Variational Method
6.4 Instantaneous Energy Eigenbasis
6.5 Moving Basis
6.6 Time periodic Systems and Floquet Theory
6.7 Two-level Systems
Note
7 Applications and Examples
7.1 Introduction
7.2 Position Representation Examples of Particles and Potentials
7.3 The Harmonic Oscillator
7.4 The Hydrogen Atom
7.5 The Dihydrogen Ion
7.6 The Jaynes–Cummings Model
7.7 The Stern–Gerlach Experiment
Notes
8 Computational Simulation of Quantum Systems
8.1 Introduction
8.2 General Points for Consideration
8.3 Some Overarching Coding Principles
8.4 A Small Generic Quantum Library
8.5 Concluding Remarks
8.6 Appendix I – Some Useful Calculated Quantities
8.7 Appendix II – Wigner Function Code
Notes
9 Open Quantum Systems
9.1 Introduction
9.2 Classical Brownian Motion
9.3 Master Equations
9.4 Master Equation Approximations and Their Implications
9.5 A Master Equation Derivation Example
9.6 Unravelling the Master Equation
Notes
10 Foundations: Measurement and the Quantum-to-Classical Transition
10.1 Introduction
10.2 The Measurement Problem
10.3 Refining the Idea of Measurement
10.4 My First Foray into Model-based Measurement
10.5 Two Other Measurement Devices and Their Classical Limit
10.6 The Quantum-to-Classical Transition
10.7 A Model-based Approach to Quantum Measurement
10.8 Questions for the Reader to Ponder
Notes
11 Quantum Systems Engineering
11.1 Introduction
11.2 What Is Systems Engineering?
11.3 Quantum Systems Engineering
11.4 Concluding Remarks
Notes
Bibliography
Index
End User License Agreement
List of Illustrations
Chapter 2
Figure 2.1 Illustration of the principle of least action determining a traje...
Figure 2.2 An exaggerated illustration of how Poisson bracket and Hamiltonia...
Figure 2.3 Evolution of a phase space distribution for the damped driven Duf...
Figure 2.4 Set-up of the Stern–Gerlach experiment. Hot silver atoms leave th...
Figure 2.5 Stylised set-up of the Stern–Gerlach experiment.
Chapter 4
Figure 4.1 Examples based on the displaced vacuum ‘coherent state’. (c,d) sh...
Figure 4.2 Examples showing how the phase space of the Wigner function chang...
Chapter 5
Figure 5.1 An example basis state showing the magnitude of by radius for
Chapter 6
Figure 6.1 Time evolution plot of the probabilities associated to measuring ...
Figure 6.2 Time evolution plot of the probabilities associated to measuring ...
Chapter 7
Figure 7.1 Potential energy for a particle in an infinite well.
Figure 7.2 Energy eigenfunctions displaced vertically by their corresponding...
Figure 7.3 Probability densities for the first three energy eigenfunctions o...
Figure 7.4 Schematic of a potential barrier of magnitude and width with ...
Figure 7.5 Particle in an infinite well.
Figure 7.6 Examples of a ground and first excited state wave function for a ...
Figure 7.7 The energy eigenbasis for the harmonic oscillator (a) with their ...
Figure 7.8 The dense dotted parabola shows the potential of the harmonic osc...
Figure 7.9 Diagram showing the ion system comprising two protons and one e...
Figure 7.10 Diagram showing the two possible pictures used to describe the d...
Figure 7.11 Diagram showing the bonded and antibonded states of the dihydrog...
Figure 7.12 A schematic of the physical set-up that the Jaynes–Cumming model...
Figure 7.13 Diagram showing the energy level structures, and , of the fie...
Figure 7.14 Diagram showing the breaking of degeneracy when . Note the new ...
Figure 7.15 To help understand the later superposition examples, we show the...
Figure 7.16 The atomic inversion for an atom initially prepared in its excit...
Figure 7.17 Diagram showing the approximate atomic inversion of the Jaynes–C...
Figure 7.18 The set-up of the Stern–Gerlach experiment: a beam of Spin-1/2 p...
Chapter 8
Figure 8.1 An illustration of the diamond structure that can arise if multip...
Figure 8.2 Top: Plot of data outputted by our code for the initial state s...
Chapter 9
Figure 9.1 Diagram showing the difference between the spectral density (dark...
Figure 9.2 Plot comparing (effective) spectral densities with and without di...
Figure 9.3 Plot comparing sub-Ohmic (dashed), Ohmic (solid), and super-Ohmic...
Figure 9.4 The first three energy eigenvalues for a SQUID ring with circuit ...
Chapter 10
Figure 10.1 Schematic of a SQUID ring coupled to an oscillator (tank circu...
Figure 10.2 An approximate, and incomplete, schematic of a photomultiplier t...
Figure 10.3 (a) The experimental set-up for Young's double-slit experiment, ...
Figure 10.4 Example Poincaré sections for the quantum state diffusion evolut...
Figure 10.5 Schematic of the set-up of a minimal ‘fully quantum mechanical’ ...
Guide
Cover
Title Page
Copyright
Acronyms
About the Authors
Preface
Acknowledgements
About the Companion Website
Introduction
Table of Contents
Begin Reading
Index
End User License Agreement
Pages
iii
iv
xiii
xiv
xv
xvii
xviii
xix
xx
xxi
xxiii
xxiv
xxv
xxvi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
395
396
397
398
399
400
401
402
403
Quantum Mechanics
From Analytical Mechanics to Quantum Mechanics, Simulation, Foundations, and Engineering
Mark Julian Everitt
Department of PhysicsLoughborough University
Kieran Niels Bjergstrom
Quantum TechnologiesLoughborough University Science and Enterprise Park
Stephen Neil Alexander Duffus
Department of PhysicsLoughborough University
This edition first published 2024© 2024 John Wiley & Sons Ltd.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of Mark Julian Everitt, Kieran Niels Bjergstrom and Stephen Neil Alexander Duffus to be identified as the author(s) of this work has been asserted in accordance with law.
Registered OfficesJohn Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USAJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK
For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com.
Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats.
Trademarks: Wiley and the Wiley logo are trademarks or registered trademarks of John Wiley & Sons, Inc. and/or its affiliates in the United States and other countries and may not be used without written permission. All other trademarks are the property of their respective owners. John Wiley & Sons, Inc. is not associated with any product or vendor mentioned in this book.
Limit of Liability/Disclaimer of WarrantyWhile the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
Library of Congress Cataloging-in-Publication Data applied for:
Paperback ISBN: 9781119829874
Cover design by WileyCover image: © M.J. Everitt
Acronyms
BCH
Baker Campbell Hausdorff
BSIM
Berkeley short-channel IGFET model (IGFET – insulated gate field-effect transistor)
CL
Caldeira–Leggett
CLE
Caldeira–Leggett equation
CSCO
complete set of commuting observables
DC
direct current
DRY
don't repeat yourself
DY
dynodes
FORTRAN
formula translation (language)
GPS
global positioning system
IDE
integrated development environment
ISO
International Standards Organisation
KVN
Koopman–von Neumann
I/O
input–output
JJ
Josephson junction
LC
electrical oscillator circuits with for inductance and for capacitance
LCR
damped electrical oscillator circuits with for inductance, for capacitance and
R
for resistance
MA
measurement apparatus
MASER
microwave amplification by stimulation emission of radiation
MBSE
model-based systems engineering
MOT
magneto-optical trap
probability density function
PMT
photo-multiplier tubes
PNT
positioning, navigation, and timing
RSJ
resistivity shunted junction (model)
SOLID
the principles of:
S
ingle-responsibility;
O
pen–closed;
L
iskov substitution;
I
nterface segregation;
D
ependency inversion
TRL
technology-readiness level
QAA
Quality Assurance Agency
QBM
quantum Brownian motion
QHO
quantum harmonic oscillator
QO
quantum object
QSD
quantum state diffusion
R&D
research and development
RF
radio frequency
SHO
simple harmonic oscillator
SOI
system of interest
SQUID
superconducting quantum interference device
SW
Stratonovich–Weyl
TDD
test-driven development
TDSE
time-dependent Schrödinger equation
TISE
time-independent Schrödinger equation
UTF
unicode transformation format
V&V
verification and validation
About the Authors
MARK JULIAN EVERITT
is a senior lecturer and director of studies in the Department of Physics at Loughborough University. His research interests are the foundations of quantum mechanics (such as measurement and the quantum to classical transition, currently with a focus on phase-space methods), open quantum systems, modelling and simulation of quantum systems and devices. Most recently he is leading an effort to establish the disciplines of quantum and low-TRL systems engineering.
KIERAN NIELS BJERGSTROM
is director of a technology and strategy consultancy focused on innovative technology translation and commercialisation. He advises organisations globally on the strategic impact of quantum technologies, paths to implementation, and investment priorities. His academic background began in theoretical physics looking at realistically modelling open quantum systems. His PhD in Quantum Systems Engineering was the first on the topic.
STEPHEN NEIL ALEXANDER DUFFUS
is a university teacher within the Physics Department at Loughborough University. He has an established reputation of communicating complex ideas in an engaging and accessible fashion. During his PhD, his main area of research was in open quantum systems.
Preface
This book has arisen from my having to write a new set of lecture notes for a second-year module on quantum mechanics at Loughborough University. Considering that very many textbooks on quantum physics already exist, it seems only right to provide some justification as to why there should be yet another. This lecture series is not simply an evolution of my previous teaching of quantum physics, it has been for me a fresh look at how to introduce the subject. The approach I wanted to use is one for which I was surprised not to find a suitable text for the module's reading list. Many who have taught quantum mechanics complain that the first time students typically encounter the Hamiltonian is in this subject. Frustratingly, it makes the subject seem more removed from classical physics than it really is. It also means that students have to wrestle with more new concepts than they should need to. Maybe as a corollary of this historical imperative, most current textbooks are written assuming an audience with no familiarity with Hamiltonian or Lagrangian mechanics. Further, students at this stage of their education tend not to have been provided an appreciation of the role of probability in classical mechanics, such as embodied in Liouville's theorem. As such, they are ill equipped to understand the importance of, and the subtleties in, the probabilistic interpretation of quantum mechanics. This, in turn, makes it difficult to have an open and honest discussion about the metaphysics of the subject. Due to the importance of each of these considerations, I wanted to be able to address all the above and more.
The possibility of making substantial improvements to our undergraduate teaching arose after I became involved in revisions of the regulatory framework for UK physics degrees1 and at about the same time I became Director of Studies for Physics at Loughborough University. I took this opportunity to lead a major review of the Loughborough physics degree. Those of us involved in the review sought to put our provision on the firmest ground possible, leading to the first-year curriculum developing a very strong classical physics foundation. Our first year now covers the analytical mechanics topics referred to above and culminates in presenting the Lagrangian for the electromagnetic field in its covariant form (as well as much-expanded mathematics, laboratory, and computational content). As a result of these changes, we had achieved my ambition to engineer an opportunity to teach quantum physics in a way that makes a strong connection to the framework of classical analytical classical mechanics. Much of this work is the outcome of exploring some of the various paths that this background enabled. In this respect, the Schrödinger picture, and later the Heisenberg picture, Feynman path integrals, and the Wigner phase-space approach are all introduced and motivated from formal classical mechanics. This is done because each represents an independent pathway into the subject and is best understood in its own right, rather than being derived from each other (although that interlinking is presented subsequently).
Later chapters focus on the consequences of the formalism, some difficulties of quantum theory (such as the measurement problem, interpretations, and the quantum-to-classical transition), applications and problem-solving, including computational methods (the tone of this chapter was set from my experience teaching a first-year computational physics module). We close with a chapter on quantum systems engineering. I started to develop the idea for this subject in 2014, based on a coffee-shop conversation with Vincent Dwyer on reliability engineering (for which I am eternally grateful). With him and many others (such as Kieran Bjergstrom and Michael Henshaw), we have been developing this subject for a number of years. Even though it is still in its infancy, we believe the ideas are mature and valuable enough to deserve a chapter in a book. As with the computational methods chapter, many of the ideas we present here are far more wide-ranging than quantum mechanics, it is just that quantum mechanics brings with it some unique demands that make the subject even more fascinating. This last chapter contains some rather important ideas, even for the theorist. I am becoming increasingly of the view that systems thinking and systems methods can lead to a better way of doing science, not just quantum physics. Making clear the requirements of a model and/or experiment, the acceptance, verification, and validation criteria not only embody the scientific methods but also formalise and systematise the approach, making it less likely that we will overlook assumptions and draw incorrect conclusions from our work.
There are several target audiences for this book:
To support a first or second course on quantum mechanics following or co-teaching with a course on Hamiltonian and Lagrangian mechanics (
Chapters 1
–
7
).
To support advanced courses on quantum mechanics where coding, open systems, philosophy, or engineering is included.
To introduce the subject to mathematics majors, who often study the required classical mechanics in courses on dynamical systems or analytical mechanics.
At master's level (potentially as part of integrated PhD programmes) which often contains advanced conversion course elements (for those switching disciplines) in countries like the UK and US.
For faculty designing new (or redesigning existing) quantum mechanics courses, where the existence of a text such as this might better facilitate their preferred teaching of the subject over existing texts.
As a ‘reader’ in quantum mechanics for the self-motivated advanced student.
Mark Julian Everitt
Loughborough University
2023
Note
1
Participating in a review of the QAA subject benchmarks statements and Institute of Physics Accreditation.
Acknowledgements
I first and foremost need to thank my co-authors. I was very pleased to be joined in writing this book by two of my former PhD students, Steve and Kieran. It is a source of immense pleasure when one's students go on to be better than one self in subjects that you taught them. The decoherence chapter was led by Steve and based on much of the PhD work of himself and Kieran (my contribution was limited to the narrative and unravellings content). Steve has enjoyed immense success teaching quantum physics to our third-year students over the past few years. We combined our experience to develop the methods and applications chapters. Kieran and I (and others) have been working on developing the subject of quantum systems engineering for some time now. He has a rare talent in his ability to encapsulate and articulate the complexities of this subtle discipline. Working on the last chapter together developed some existing, and stimulated a number of new, insights from us both that we hope the reader will enjoy and will find pleasantly surprising. I also appreciate both of their substantial efforts in editing and proofreading which provided a level of coherence that was well needed in this work. In this last and most challenging of tasks we were very pleased to receive the help of Niels Bjergstrom, whose gargantuan effort resulted in a well-constructed text. For their general input and advice on the first few chapters, I am very grateful to Sasha Balanov and John Samson. Sasha, together with Adam Thompson, Mark Greenaway, and Thomas Steffen, provided very valuable feedback on the Computational Simulation of Quantum Systems chapter (with Adam's professional programmers' eyes being especially useful in ensuring that key topics such as logging were included). We are also very grateful to Finlay Potter for his careful student perspective, careful reading, and valuable observations on the open-systems chapter.
In addition to this, I am deeply grateful to a number of people who have helped deepen my understanding of, and perspectives on, quantum physics in different ways over many years. This includes, but is not limited to: Terry Clark for being my mentor for many years and showing me what commitment to science means; Jason Ralph for many a long conversation on the foundations of quantum mechanics; Todd Tilma, Russell Rundle, John Samson (again), and Ben Davies for adventures in phase space; Tim Spiller, Bill Munro, Kae Nemoto, Ray Bishop, and Alex Zagoskin for always asking good, hard, interesting questions; Vincent Dwyer, Michael Henshaw, Jack Lemon, Laura Justham, Simon Devitt, and Susannah Jones for exploring new ways of engineering quantum systems (as well as my father, who as a professional systems engineer, influenced my world view to naturally accommodate a holistic view); to Pieter Kok, Dan Browne, Derek Raine, and Antje Kohnle for many interesting conversations on how to teach quantum mechanics (especially Pieter's well-voiced issues with unphysical beams in diagrams). Last, but not least, I need to acknowledge the support of my family, Lennie, Alex, and Sophie, who have allowed me the time to work on this text (and who will be seeing much more of me in the future).
Mark Julian Everitt
About the Companion Website
This book is accompanied by the companion website:
http://www.wiley.com/go/everitt/quantum
The website includes: Repository to be discussed in Chapter 8.
Introduction
This book introduces quantum mechanics from the perspective of classical analytical mechanics. The idea of this approach is to highlight more clearly the similarities of quantum and classical mechanics, as well as the differences. As a consequence, we believe that this makes the theoretical framework of the theory more intuitive where this is possible. Our intent is that the actual peculiarities of quantum mechanics are more clearly identified than in treatments where, e.g. the relationship between the dynamics of classical and quantum probability densities is considered.
Chapters 1–4 were written to be studied in order, subject to the various caveats listed below. There is more flexibility in the order in which later material can be studied. The content up to and including Chapter 7 forms the basis of what we consider a first course in quantum mechanics, with the remaining chapters constituting either additional reading or supporting more advanced courses.
In Chapter 1 we present most of the mathematics used in the text, introducing core concepts and notations (such as that of Dirac). We have found that some students can struggle to become comfortable with such notation. Separating the concerns of mathematics from the physics we find improves the understanding of both the mathematics and the physics. Presumably this is because it reduces the number of concepts that students need to think about at any one time. Our approach has the added advantage that we can introduce concepts such as the Heisenberg uncertainty principle in their general mathematical form. Such results have utility beyond quantum mechanics, which this presentation makes clear. In the specific case of the uncertainty principle, the separation of concerns enables us to introduce the result without incorrectly confusing it with ideas of measurement disturbance. This chapter might either be studied entirely on its own before engaging with the rest of the text, or it may be interleaved with the content of the rest of the book. Especially in early chapters, we cross-reference sections back to the prerequisite mathematical material contained within this chapter to enable either approach to be taken.
As some knowledge of Hamiltonian and Lagrangian mechanics is required, Chapter 2 provides a self-contained introduction to the subject. Even if you are familiar with most of this material, it is worth reading, as it contains important material on a formulation of classical mechanics due to Koopman and von Neumann that greatly helps in motivating the Schrödinger equation. Historically, the idea was to make classical mechanics look like quantum mechanics. We have chosen to present the ideas ahistorically, as it is an odd but not conceptually difficult jump to move from the Liouville equation to Koopman–von Neumann classical mechanics. Once we have Koopman–von Neumann classical mechanics, the Schrödinger equation does not seem anywhere near as surprising as it might do without this context. This chapter finishes with a discussion of the breakdown of classical mechanics and the correspondence principle (some understanding of which is important at this stage, as it allows us to understand the flexibility we have in formulating new theories consistently with existing ones).
In Chapter 3 we introduce the Schrödinger formulation of quantum mechanics. We try to do this in a way that makes as few assumptions as possible. This leads to a somewhat lengthy discussion, but it is one that draws out the key assumptions and issues of the Schrödinger picture (such as that the initial state is all that is needed to determine a system's evolution, just like in the Liouville equation). We motivate the Schrödinger equation as being similar in form to the Koopman–von Neumann equation of motion, but where we move to an operator formalism that replaces some Poisson brackets with commutation relations. We make only a few assumptions about the state (i.e. it is a vector in a vector space), and our discussion progresses through measurement axioms before discussing that the representation of a quantum system is a matter of choice and that, e.g. the wave function is just the position representation of a quantum state. This discussion allows us to make clear the axioms associated with dynamics and measurement, and separate these clearly from the ideas of representation.
We then, in Chapter 4, look at some alternative paths into quantum mechanics, the Heisenberg, Wigner phase space, and (a very brief introduction to) Feynman path integral formulations. We do not derive these from the Schrödinger formulation, but instead re-argue from first principles. We do this for two reasons: (i) to show that they are not subordinate to the Schrödinger view, and (ii) because they can be more naturally argued for in a way that helps develop our discussion of the similarities as well as the differences between quantum and classical physics. In fact, there is a strong case for arguing that either of these pictures might be a more natural starting point for developing the subject of quantum physics. We chose to start with the Schrödinger picture, simply because this is the dominant one in most of the textbook and research literature. We think it would be relatively straightforward to mix the Heisenberg picture discussion of Chapter 4 with elements of Chapter 3 to form an alternative opening to the subject.
Chapter 5 introduces vectors and angular momentum and is somewhat unusual in so far as we include an extensive discussion of curvilinear coordinates in quantum mechanics. Our reason for doing this is that many classical mechanics problems become easier to treat by using curvilinear coordinates, if that suits the symmetry of the problem. Such simplification is not seen in quantum mechanics. One of our key aims of this text is to highlight as clearly as possible the similarities and differences between quantum and classical physics. For problems with spherical symmetry, simplification is actually found through an analysis of angular momentum, and we wanted to explain why this is in fact the case. One advantage of this introduction to the subject is that the expression of the kinetic energy operator in terms of radial and angular momentum components does arise naturally (in the usual treatment this is discovered through analysis of the three-dimensional kinetic and angular momentum operators in the coordinate representation which, while effective, lacks elegance). Those not interested in that level of detail can happily skim-read most of Section 5.2. The remainder of the chapter contains the theory needed to understand really important applications, such as the quantum physics of hydrogen. Note that it is often the case that the harmonic oscillator is introduced before angular momentum. We have chosen not to do this, as angular momentum is part of the general theory and the harmonic oscillator is simply a very important example.
The harmonic oscillator discussion in Section 7.3 is not predicated on the content in Chapter 5 and can be studied beforehand.
Chapters 6 and 7 contain methods and applications and important examples such as hydrogen, molecules, and the Jaynes–Cummings model. The order of study can be somewhat flexible.
For the modern physicist, computation has become as important as mathematics for many tasks, for example enabling the solution of problems with no analytic solution. Many texts cover technical aspects of algorithm design pertinent to scientific computing. While there is some literature [93] on good practice in scientific computing, there is a limited amount of textbook resources for physics. In Chapter 8 our focus is on good practice and design principles using quantum physics as an example. In a world where artificial intelligence is getting better at writing routine code, it is these higher-level skills that the physicist will require more and more. This chapter can equally well be used to support either a quantum mechanics module with coding elements, or a coding course where quantum mechanics would provide valuable example applications1.
In Chapter 9 we provide an introduction to open quantum systems. While there is already substantial literature on this subject, it is a challenging one. Based on our undergraduate teaching and project supervision, our intention is to make this material as accessible as possible, expanding on those areas where our experience has found that students need assistance with the material presented in the existing literature.
Many treatments of quantum mechanics set down a philosophical interpretation of the subject early on. It is one of the main discussion points of the theory that there are multiple interpretations of quantum mechanics and that the subject contains unresolved metaphysical issues. In Chapter 10 we take advantage of much of the preceding content of the book to have an in-depth, open, and honest discussion on some aspects of the foundations of quantum mechanics. Our intent is to stimulate thought and discussion rather than present a single perspective. Interestingly, a discussion of the measurement problem is pertinent to the verification and validation issues that are presented in the final chapter. This provides an interesting link between the very philosophical foundations of quantum mechanics and the very applied goal of engineering quantum systems.
Finally, in Chapter 11 we turn to a general discussion of some challenges associated with the engineering of quantum technologies. We introduce an approach to quantum systems engineering that we believe has value, not just for quantum applications, but to physics as a whole – from the scientific method to technology development2. While it has a place in experimental research physics and the scientific method, we avoid extensive discussion of established or modified-for-laboratory systems engineering tools that also apply to non-quantum technologies. Our discussion centres on the different engineering needs of quantum from non-quantum systems, and why this results in challenges that a quantum physicist might enjoy. This is an emerging field at the forefront of innovation, and we caveat our writings as something of a personal, evolving, perspective
Throughout this document, the following boxes denote different types of materials, as follows:
Normal notes will look like this
Really important notes and observations will look like this
Prerequisite Material: References to needed material or cross-referencing will look like this.
Notes
1
Much of the conceptual content of this chapter appears in the Loughborough first-year course on computational physics but with classical physics rather than quantum mechanical applications.
2
For this reason, some of this subject matter, as well as some conventional systems engineering, has been strategically introduced into the Loughborough undergraduate curriculum. We find the latter is especially useful in support of laboratory and group project work.
1Mathematical Preliminaries
“Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.”
Bertrand Russell
“The mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which nature has chosen.”
Paul Dirac
1.1 Introduction
Different people learn in different ways. Some like to study the mathematics they need in some depth before looking at the physics; others prefer to mix and learn both subjects together, a bit at a time. People will also begin their journey into quantum mechanics with different levels of mathematical knowledge. For this reason, the core mathematics needed for the study of quantum mechanics has been organised here in a single chapter, with cross-referencing in later chapters to sections in this chapter. You may wish to study this chapter in depth before proceeding to the Physics that follows. Alternatively, you may wish to skip this chapter and refer back to it as you discover new and unfamiliar mathematics (or skim-read it, just to see what is here). Our aim is to facilitate your study of this material in a way that best suits you.
Quantum mechanics makes use of several branches of mathematics. It does, however, focus on one branch above most others, which centres on a generalisation of vectors, dot products, and linear operators on those vectors. In mathematics, we use the term space to refer to a set (collection of mathematical objects) together with some additional properties. Hence we will study vectors in a vector space where the additional properties are vector addition and multiplication by a scalar. We will also study vectors in an inner product space which is a vector space but with the additional property of a generalisation of the dot product called the inner product. We intend the mathematics we introduce in this chapter to be sufficiently detailed to enable an understanding of the physics that follows. The treatment is only at this level and, for the sake of brevity, is not a complete treatment of the subject. The language of quantum mechanics is framed in terms of vectors in an extension of inner product space called a Hilbert space. This constrains the space used to describe quantum systems by additional specific mathematical properties1.
To be consistent with the wider literature, we will often use the term Hilbert space when we only need the properties of an inner product space2. Most of our focus will be on the mathematics to do with Hilbert spaces, but some other topics, such as expectation values of random variables, will be briefly introduced. This means that the flow of the text in this chapter will not be as seamless as in more comprehensive treatments. As our approach removes the need to interleave mathematics and physics, better emphasising physics arguments, we hope that you will find this a worthwhile compromise. It is worth noting that while our focus is quantum mechanics, the mathematics introduced in this chapter is applicable to many other subjects. Hilbert spaces are very useful for the study of signals and for gaining a deep understanding of Fourier analysis. A specific benefit of using this approach is that we can introduce the Heisenberg uncertainty principle as a purely mathematical proposition and demonstrate that it has much more general applicability than its usual quantum physics application would imply. The important consequence of doing this is that it extracts from the argument the often incorrectly made assertions about measurement disturbance (we will properly understand what Heisenberg's uncertainty principle actually means for quantum physics when we discuss its realisation in the phase space methods section of Chapter 4). Even if you already have knowledge of vector spaces and functional3 analysis, this chapter will be of value because it provides some physics context and notation.
1.2 Generalising Vectors
1.2.1 Vector Spaces
Powerful mathematical methods and principles are often developed by identifying shared features of different mathematical objects and exploring the consequences of a generalisation. The advantage of this approach is that we can start with something that is easy to understand (in this case arrows on a plane) and learn the consequences of combined properties of that system (lemmas, propositions, and theorems), which we can then apply to other systems that share those same properties. For instance, the Cartesian co-ordinates , phase space coordinates , functions (including probability density functions – which shall be introduced later, but are essentially what you would think from the name), sequences, and many other things share a certain set of properties in common with vectors. This means that the theorems that hold for arrows-on-a-plane also hold for all these other mathematical objects. We will later see that we use vectors and functions to hold the information that characterises a system. Therefore, generalising vectors will prove to be of central importance to quantum mechanics, and a good understanding now will prepare us for the discussions that follow.
We start by identifying those features and abstracting them into a list of properties (called axioms) that describe the generalised mathematical concept of a vector. Note that, where possible, we have provided a diagrammatic example of each axiom to emphasise the reductionist nature of the approach. In general, vectors , and with scalars (complex or real numbers) and have the following properties:
Adding two vectors together results in another vector, i.e. if and are vectors, then is also a vector. In terms of arrows, this is the notion of putting the tail of the arrow on the head of arrow – the resultant vector connects the tail of to the head of . The mathematical term for this is that the vector space is closed under addition.
Multiplying a vector by a scalar gives a new vector (closed under scalar multiplication).
If the scalar is real, then this is just a scale-factor that stretches or shrinks the vector by that amount (i.e. multiplication of the length of the vector).
Note that a negative scaling will flip the arrow into the opposite direction.
If the scalar is complex, then think of this as scaling the length by and adding the phase . This can be thought of as a scale and a twist in the complex plan orthogonal to the vector as the direction of the vector does not change.
Commutativity of vector addition, which states that the resultant vector from adding two vectors together does not change if you change the order of the sum:
Associativity of vector addition, , which states that it does not matter which order three vectors are added.
There is a vector of zero length, commonly notated as , such that for any . This is referred to as the null vector or zero vector. Note that we choose not to use , as we will want to use 0 later for another purpose.
For any vector , there exists another vector, denoted (inverse element), such that the sum . Note that as only addition is defined (tail of one arrow on the head of another arrow) subtraction does not formally need to exist – but we do use the shorthand .
Compatibility is satisfied:
Multiplication by unity, commonly the number one, leaves a vector unchanged (multiplicative identity):
Distributivity across vector sum:
Distributivity across scalar sum:
Any example of mathematical objects that satisfies all of the above conditions can be considered a kind of vector. Furthermore, a set of objects satisfying the conditions together with the rules of addition and multiplication by a scalar is called a vector space. Examples of vector spaces include: complex numbers, vectors (obviously), matrices, co-ordinates [can be seen as and ], sequences (with elementwise addition), series, linear equations, and last, but by no means least, functions [seen by considering and ]. To provide a concrete example: if and , then the sum of these functions is also a function, as is a number times the function. It is this analogy to vectors that explains how functions can act as a basis (with this example pertaining to Fourier analysis).
More importantly, this means that any theorem we prove for vectors, as defined above, will hold for all other things that satisfy these conditions too!
The co-ordinates of a physical system in phase space are vectors in a vector space.
1.2.2 Inner Product
We are familiar with the idea of the dot product . Having now observed that we can generalise the notion of a vector to other things, we can also generalise the notion of the dot product to any vector space.
The dot product has the following important properties:
(where indicates complex conjugate)
, and if and only if
Exercise 1.1 Show that these relations hold for three-element column vectors where (where superscript means transpose).
The third condition is needed to ensure is real. As such, we can interpret as a length in a way that corresponds to the normal notion of Euclidean distance.
The first and third conditions imply:
which is sometimes, unnecessarily, listed separately.
From the second and third conditions, .
The generalisation of the dot product is termed an inner product or scalar product and is denoted by . Other notations for the inner product do exist in the literature, which all look different (and may alter the order of and ) but mean the same thing. The notation is historically reserved for standard vectors and Cartesian coordinates. We use , as it is the historical origin for a notation developed by Dirac, which we are soon to discuss – and which is needed to be able to properly engage with the literature of quantum mechanics.
As with the dot product, the inner product takes the form of a mapping (strictly speaking a sesquilinear form4) of any two vectors (of the same kind but no longer restricted to traditional vectors) to a number, and satisfies the same properties as the dot product:
As with the dot product, the third condition is needed as it ensures is real. As such, we can interpret as a length, which means we can do interesting things such as ask: what is the distance between two functions? We call the norm of . The norm generalises the notion of the modulus of a vector.
The first and third conditions imply that:
The second and third conditions imply that , just as for the dot product.
Two important examples of the inner product are:
The usual dot product for standard vectors
(so we retain all the good maths that we know to work).
And, for complex valued functions of a real parameter:
which means we can apply many ideas that we understand from the geometry of vectors to functions. The lemma (a small theorem) we consider next is an excellent and important example of this, and this will become especially important when we begin to use functions to describe physical systems.
When integrating, our usual notation is to put the differential of the variable before the integrand. This is a standard notation, but not one you may have seen before. The reason we choose to do this is that it enables us to see which variable we are integrating against from the outset. This can be very useful when things get complicated.
Introducing an inner product allows us to make further analogies to spatial vectors such as the angle between any two vectors and by . We will say the two vectors orthogonal if .
The Cauchy–Schwarz Inequality states that:
We will later see that the proof of Heisenberg's uncertainty relation relies on this inequality in its derivation, so it is rather important.
For vectors with a dot product, we know that the inner product, , is just the amount of in the direction of , and . So this is a generalisation of the geometrical idea that the (square or the length of vector ) times the (square or the length of vector ) is greater than, or equal to, the (square of the amount of in the direction of ). Specifically, for the dot product, we have (where is the angle between and ). So the result is perhaps not surprising for traditional vectors as – it is its generalization that makes it interesting.
As a specific example for functions, using the definition of the inner product provided earlier, the result
is perhaps less obvious.
Exercise 1.2 Prove the Cauchy–Schwarz Inequality just using the features (axioms) of the generalised vector and inner product. Hints: where is the inequality in the definition of the inner product? - and consider .
1.2.3 Dirac Notation
The way we denoted the inner product introduces the possibility of a new notation for signifying vectors in an inner product space. The approach has the immediate advantage of indicating that such vectors have an inner product associated with them, so we can make use of theorems such as the Cauchy–Schwarz inequality. It is called Dirac notation and comes from noticing that the inner product has a shape. If we consider the inner product , we can split it up into and consider the two sides as entities in their own right: a bra and a ket . Note that we have dropped the use of bold to denote ‘vectors’ as this is also convention. Where boldface is used in the following discussion, you can assume it refers to a conventional column vector.
Here, we consider that the ket is just the ‘vector’ itself. In Dirac notation the axioms (the features that we identified as common in our generalisation of the arithmetic of arrows on a plane) for a vector space now read:
Commutativity of addition:
Associativity of addition:
There is a vector of zero length commonly denoted such that for any .
For any there exists another vector (inverse element) such that (we use the shorthand ).
Compatibility is satisfied:
Multiplicative identity:
Distributivity across a vector sum:
Distributivity across a scalar sum:
We also can write so long as it is clear that labels the vector and is a scalar. Please compare these directly with the examples for the properties of ‘normal’ vectors explained earlier. Convince yourself that we followed exactly the same reasoning here, but added a bracket around the vector instead of using a bold typeface (or putting an arrow over the top) – as this is all we have done to reach Dirac notation.
While getting used to Dirac notation it may help to informally think of the bra and ket using the following analogy, if
In this way we can see that the bra can be thought of as a vector with a dot product waiting to happen. Because of the fact the inner/dot-product is linear the bra behaves just like a vector, so can be thought of as being a vector in its own right (just as behaves like a vector). So, the if bra version of one ket vector acts on another ket it will transform that ket into a scalar in just the same way as one can multiply row and column vectors. In other words we make the analogy,
Alternatively, if we are dealing with a function space, the bra is just like
Exercise 1.3 Show that the set of bras also satisfy all the axioms of a vector space and can therefore also be considered vectors in their own right. Hint: The argument for commutativity of addition is to take and post multiply by some arbitrary ket and by inner product rules
as we made no assumptions about so .
From the preceding exercise we see that to every ket in a vector space we can associate a bra vector through the inner product so that
and we can treat bras just like vectors. In some cases, there may even be more bra than ket vectors. For instance, you may recall that the Dirac delta function not really a function (it's the limit of a distribution) and it is only well-defined under an integral – so there is strictly speaking no ket for . But there is a bra vector as the following integral is well-defined:
The set of bras is called the dual space and it may be bigger than the vector space itself.
1.2.4 Basis and Dimension
In this section we generalise the idea of basis vectors such as , , and . Just as with conventional vectors, any vector can be represented in terms of a basis set of vectors as a linear combination
where the are suitable coefficients. A basis is said to be complete if all possible vectors (in the vector space) can be represented by that basis. It is said to be orthogonal if each basis vector is perpendicular to every other basis vector, and orthonormal if each basis vector is also of length one. Most bases you will encounter in this book are complete and orthonormal. Note that continuous bases are also possible, and in such cases the series is replaced by an integral.
The minimum number of basis vectors needed to represent any vector in a space is the dimension of the space, which may in some cases be infinite. An example of an infinite basis is the sines and cosines in the Fourier expansion of a function.
As with Cartesian systems, the situation is greatly simplified if the vectors in the basis are orthogonal (just like , , and ), and each basis vector is of length one. For orthonormal bases, we often use the standard notation:
(in the continuous case, the Kronecker delta is replaced by a Dirac delta function). The vector with elements is a representation of in that basis (we will consider this point in much more detail in Section 1.4).
While we can make an analogy of vector space basis vectors to , spatial basis vectors in quantum mechanics are a different kind of vector and the two should not be confused. When we later consider angular momentum in quantum mechanics, we will discuss Cartesian and spherical polar coordinates as examples of spatial bases. For non-Cartesian coordinate systems, quantum-classical analogies are not as simple as one might expect. For example, defining an (unnormalised) spatial basis vector for the coordinate as
has some subtle difficulties if its application is attempted in quantum mechanics (if you have not seen this notation before it might help to note, e.g. that as expected).
1.3 Linear Operators
1.3.1 Definition and Some Key Properties of Linear Operators
Recall that if a matrix acts on a vector it creates another vector,
In this way, a matrix encodes the operations of rotating and scaling a vector. In the same way we generalised vectors and the dot product, we may also generalise the notion of a matrix. Such a generalisation is termed a (linear) operator, and its definition is motivated by the arithmetic of matrices as follows (the ‘hat’ denotes that an object is an operator):
Operators transform vectors to other vectors (of the same kind),
Linearity means that for vectors and and a scalar , and
The operators and are equivalent operators if, for all , the following holds:
This is true even if and look different – operators are defined by their effect on vectors.
As with matrices, operators are
commutative under addition:
have associativity of addition:
have associativity under multiplication (e.g. composition of rotation matrices):
The null operator maps all vectors to the zero vector
The identity operator (or ) maps all vectors to themselves
The notation does not look like the matrix-vector notation . In physics, to emphasise the vector nature of the ket, we often use the alternative notation,
where it is understood that acts to the right, just as a matrix acts on a vector. Most of the time this is not confusing, but sometimes it is. In these cases it may help to return to the notation.
When operators appear in an inner product between two vectors
is said to be a matrix element of the operator as has two indices just like the element of a matrix. Recall that if is a basis, a vector with elements is a representation of in that basis. In the same way, the matrix with elements represents as an actual matrix in that basis. The action of will be the same as (again, we will consider this more deeply in Section 1.4).
For the case where the same vector appears on both sides,
we term this an expectation value of the operator (the reason for this will be explained later). This is sometimes given the shorthand , or if the actual vector is not relevant to the discussion.
1.3.2 Expectation Value of Random Variables
We briefly digress from our discussion of operators to that of random variables, specifically to justify the term expectation value introduced above. Consider some random variable (traditionally labelled using a capital letter) , representing the outcome of an experiment. First, let us assume that has a finite number, or countably infinite number, of outcomes , and the probability of each outcome occurring is . If we performed many experiments and averaged the results, we would get the expected value of the experiment:
Some care needs to be taken when dealing with expectation values – they don't necessarily tell you what outcome you would get (the expected outcome of a random string of zeros and ones with equal weight would be half – not an outcome that is ever measured). Since many distributions in physics are Gaussian, or in some other way localised, the expectation value can be a very useful concept (especially when coupled with higher-order moment or cumulant analysis – a subject not covered here).
Generalisation to a continuous case is, as one would expect,
and further generalisations to sets of outcomes that depend on multiple outcomes, such as the position on the plane, is again a natural extension of this idea
Simply, this adds up all the values of the random variable times their outcome probabilities to get the mean of all possible outcomes.
If, as in the above example, we don't actually know the values of or of – if we have some function of the random variables and , say – but we do know the probability density , we can invoke the law of the unconscious statistician, which states:
(the two-dimensional example is given, for reference, in Section 2.3).
With this definition of expectation in mind, let us lay some groundwork that links random variables to Dirac notation, and therein quantum mechanics. This may seem purely notational, but it also puts down the foundations for an interesting representation of classical mechanics by Koopman and von Neumann, which looks very much like quantum mechanics (and is briefly discussed at the end of Chapter 2).
Consider some arbitrary probability density function over some set of random variables . Then there will exist some functions such that (which, for reasons that will become apparent when we study quantum mechanics, we allow to be complex). We restrict in the space of square integrable functions (the integral of the modules squared exists and is finite) and so it is also a vector in that space. The space has the inner product given in Eq. (1.6), so we can write the function as a ket and define:
By doing this, the expectation value of any function of random variables can be written as:
This is why we consider Eq. (1.8) the expectation value of an operator. We will give a more physical argument for this terminology in Section 2.5.
1.3.3 Inverse of Operators
Not all operators have an inverse, but if one exists it is defined as follows: for an operator its inverse is denoted and must obey:
1.3.4 Hermitian Adjoint Operators
Consider the matrix-vector equation
where is some scalar. The Hermitian adjoint (or Hermitian conjugate) of is a matrix denoted that satisfies:
or, more succinctly, is the matrix associated with that satisfies
Exercise 1.4 Use , written in the form of a column vector, along with this equation
to show that .
As an example to help you with the above exercise, let us consider the matrix
then,
Further,
You can now show that
which equals the same thing.
Hermitian adjoint and adjoint mean two different things – here we are talking about the former, also known as the conjugate transpose or Hermitian transpose
