Fundamental Quantum Mechanics for EngineersLeon van Dommele

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Fundamental Quantum Mechanics for Engineers

Leon van Dommelen

12/20/07 Version 3 beta 3.4

Copyright 2004 and on, Leon van Dommelen You are allowed to copy or print out this work inunmodified work for your personal use You are allowed to attach additional notes, corrections,and additions, as long as they are clearly identified as not being part of the original documentnor written by its author

Distribution of this document for pay or for any other economic gain to a general audiencewithout permission is strictly prohibited As an exception, its unmodified web pages may belinked to freely, and may be displayed within your own frames, even for gain Conversions tohtml of the pdf version of this document are stupid, since there is a much better native htmlversion already available, so try not to do it

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To my parents

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Why Another Book on Quantum Mechanics?

With the current emphasis on nanotechnology, quantum mechanics is becoming increasinglyessential to engineering students Yet, the typical quantum mechanics texts for physics stu-dents are not written in a style that most engineering students would likely feel comfortablewith Furthermore, an engineering education provides very little real exposure to modernphysics, and introductory quantum mechanics books do little to fill in the gaps The empha-sis tends to be on the computation of specific examples, rather than on discussion of the broadpicture Undergraduate physics students may have the luxury of years of further courses topick up a wide physics background, engineering graduate students not really In addition, thecoverage of typical introductory quantum mechanics books does not emphasize understanding

of the larger-scale quantum system that a density functional computation, say, would be usedfor

Hence this book, written by an engineer for engineers As an engineering professor with anengineering background, this is the book I wish I would have had when I started learning realquantum mechanics a few years ago The reason I like this book is not because I wrote it; thereason I wrote this book is because I like it

This book is not a popular exposition: quantum mechanics can only be described properly inthe terms of mathematics; suggesting anything else is crazy But the assumed background inthis book is just basic undergraduate calculus and physics as taken by all engineering under-graduates There is no intention to teach students proficiency in the clever manipulation ofthe mathematical machinery of quantum mechanics For those engineering graduate studentswho may have forgotten some of their undergraduate calculus by now, there are some quickand dirty reminders in the notations For those students who may have forgotten some ofthe details of their undergraduate physics, frankly, I am not sure whether it makes much of adifference The ideas of quantum mechanics are that different from conventional physics Butthe general ideas of classical physics are assumed to be known I see no reason why a brightundergraduate student, having finished calculus and physics, should not be able to understandthis book A certain maturity might help, though There are a lot of ideas to absorb

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a reader would not be able to put down until she had finished it Obviously, this goal wasunrealistic I am far from a professional writer, and this is quantum mechanics, after all, not

a murder mystery But I have been told that this book is very well written, so maybe there

is something to be said for aiming high

To prevent the reader from getting bogged down in mathematical details, I mostly avoidnontrivial derivations in the text Instead I have put the outlines of these derivations in notes

at the end of this document: personally, I enjoy checking the correctness of the mathematicalexposition, and I would not want to rob my students of the opportunity to do so too

While typical physics texts jump back and forward from issue to issue, I thought that wouldjust be distracting for my audience Instead, I try to follow a consistent approach, with ascentral theme the method of separation-of-variables, a method that most mechanical graduatestudents have seen before already It is explained in detail anyway To cut down on the issues

to be mentally absorbed at any given time, I purposely avoid bringing up new issues until Ireally need them Such a just-in-time learning approach also immediately answers the questionwhy the new issue is relevant, and how it fits into the grand scheme of things

The desire to keep it straightforward is the main reason that topics such as Clebsch-Gordancoefficients (except for the unavoidable introduction of singlet and triplet states) and Paulispin matrices have been shoved out of the way to a final chapter My feeling is, if I can give

my students a solid understanding of the basics of quantum mechanics, they should be in agood position to learn more about individual issues by themselves when they need them Onthe other hand, if they feel completely lost in all the different details, they are not likely tolearn the basics either

I also try to go slow on the more abstract vector notation permeating quantum mechanics,usually phrasing such issues in terms of a specific basis Abstract notation may seem to becompletely general and beautiful to a mathematician, but I do not think it is going to beintuitive to a typical engineer

Knowledgeable readers may also note that I try to stay clear of abstract mathematical models

if I can For example, the discussion of solids avoids the usual Kronig-Penney or Dirac combssort of models in favor of a physical discussion of realistic one-dimensional crystals

I try to be as consistent as possible Electrons are grey tones at the initial introduction ofparticles, and so they stay through the rest of the book Nuclei are red dots Occupiedquantum states are red, empty ones grey That of course required all figures to be custommade They are not intended to be fancy but consistent and clear

When I derive the first quantum eigenfunctions, for a pipe and for the harmonic oscillator, Imake sure to emphasize that they are not supposed to look like anything that we told thembefore It is only natural for students to want to relate what we told them before about themotion to the completely different story we are telling them now So it should be clarified

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that (1) no, they are not going crazy, and (2) yes, we will eventually explain how what theylearned before fits into the grand scheme of things.

Another difference of approach in this book is the way it treats classical physics conceptsthat the students are likely unaware about, such as canonical momentum, magnetic dipolemoments, Larmor precession, and Maxwell’s equations They are largely “derived“ in quantumterms, with no appeal to classical physics I see no need to rub in the student’s lack ofknowledge of specialized areas of classical physics if a satisfactory quantum derivation isreadily given

This book is not intended to be an exercise in mathematical skills Review questions aretargeted towards understanding the ideas, with the mathematics as simple as possible I alsotry to keep the mathematics in successive questions uniform, to reduce the algebraic effortrequired I know

Finally, this document faces the very real conceptual problems of quantum mechanics

head-on, including the collapse of the wave functihead-on, the indeterminacy, the nonlocality, and thesymmetrization requirements The usual approach, and the way I was taught quantum me-chanics, is to shove all these problems under the table in favor of a good sounding, but uponexamination self-contradictory and superficial story Such superficiality put me off solidlywhen they taught me quantum mechanics, culminating in the unforgettable moment whenthe professor told us, seriously, that the wave function had to be symmetric with respect toexchange of bosons because they are all truly the same, and then, when I was popping myeyes back in, continued to tell us that the wave function is not symmetric when fermions areexchanged, which are all truly the same I would not do the same to my own students And Ireally do not see this professor as an exception Other introductions to the ideas of quantummechanics that I have seen left me similarly unhappy on this point One thing that reallybugs me, none had a solid discussion of the many worlds interpretation This is obviously notbecause the results would be incorrect, (they have not been contradicted for half a century,)but simply because the teachers just do not like these results I do not like the results myself,but basing teaching on what the teacher would like to be true rather on what the evidenceindicates is true remains absolutely unacceptable in my book

And I do hope this book will manage to convince you that quantum mechanics is a very cinating subject, whatever you think of it Our ancestors have ferreted out impressive detailsabout how nature works, mainly out of plain curiosity Its benefits gave us the technologicalworld we live in as well as the leisure time to appreciate what they did Going the next step

fas-is what being an engineer fas-is all about

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This book is for a large part based on my reading of the excellent book by Griffiths, [3].

It includes a concise summary of the material of Griffiths’ chapters 1-5 (about 250 pages),written by an engineer who was learning the material himself at the time

Somewhat to my surprise, I find that my coverage actually tends to be closer to Yariv’s book,[10] I still think Griffiths is more readable for an engineer, though Yariv has some itemsGriffiths does not

The discussions on two-state systems are mainly based on Feynman’s notes, [2, chapters 11] Since it is hard to determine the precise statements being made, much of that has beenaugmented by data from web sources, mainly those referenced

8-The nanomaterials lectures of colleague Anter El-Azab that I audited inspired me to add abit on simple quantum confinement to the first system studied, the particle in the box Thatdoes add a bit to a section that I wanted to keep as simple as possible, but then I figure it alsoadds a sense that this is really relevant stuff for future engineers I also added a discussion ofthe effects of confinement on the density of states to the section on the free electron gas

I thank Swapnil Jain for pointing out that the initial subsection on quantum confinement inthe pipe was definitely unclear and is hopefully better now

I thank Johann Joss for pointing out a mistake in the formula for the averaged energy oftwo-state systems

The section on solids is mainly be based on Sproull, [8], a good source for practical knowledgeabout application of the concepts It is surprisingly up to date, considering it was written half

a century ago Various items, however, come from Kittel [4] The discussion of ionic solidsreally comes straight from hyperphysics [3] I prefer hyperphysics’ example of NaCl, instead

of Sproull’s equivalent discussion of KCl

The section on the Born-Oppenheimer approximation comes from Wikipedia, [8}, with ifications including the inclusion of spin

mod-The section on the Hartree-Fock method is mainly based on Szabo and Ostlund [9], a written book, with some Parr and Yang [6] thrown in

well-The many-worlds discussion is based on Everett’s exposition, [1] It is brilliant but quiteimpenetrable

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Comments and Feedback

If you find an error, please let me know The same if you find points that are unclear tothe intended readership, ME graduate students with a typical exposure to mathematics andphysics, or equivalent General editorial comments are also welcome I’ll skip the philosophicaldiscussions I am an engineer

Feedback can be e-mailed to me at quantum@dommelen.net

This is a living document I am still adding some things here and there, and fixing variousmistakes and doubtful phrasing Even before every comma is perfect, I think the documentcan be of value to people looking for an easy to read introduction to quantum mechanics at

a calculus level So I am treating it as software, with version numbers indicating the level ofconfidence I have in it all

History

• The first version of this manuscript was posted Oct 24, 2004

• A revised version was posted Nov 27, 2004, fixing a major blunder related to a nastyproblem in using classical spring potentials for more than a single particle The fix re-quired extensive changes This version also added descriptions of how the wave function

of larger systems is formed

• A revised version was posted on May 4, 2005 I finally read the paper by Everett, III onthe many worlds interpretation, and realized that I had to take the crap out of prettymuch all my discussions I also rewrote everything to try to make it easier to follow Iadded the motion of wave packets to the discussion and expanded the one on Newtonianmotion

• May 11 2005 I got cold feet on immediately jumping into separation of variables, so Iadded a section on a particle in a pipe

• Mid Feb, 2006 A new version was posted Main differences are correction of a number

of errors and improved descriptions of the free electron and band spectra There is also

a rewrite of the many worlds interpretation to be clearer and less preachy

• Mid April, 2006 Various minor fixes Also I changed the format from the “article” tothe “book” style

• Mid Jan, 2007 Added sections on confinement and density of states, a commutatorreference, a section on unsteady perturbed two state systems, and an advanced chapter

on angular momentum, the Dirac equation, the electromagnetic field, and NMR Fixed

a dubious phrasing about the Dirac equation and other minor changes

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Answers are in the new solution manual.

• 4/2 2007 There are now lists of key points and review questions for chapter 2 Thatmakes it the 3 beta 2 version So I guess the final beta version will be 3 beta 6 Variousother fixes I also added, probably unwisely, a note about zero point energy

• 5/5 2007 There are now lists of key points and review questions for chapter 3 Thatmakes it the 3 beta 3 version Various other fixes, like spectral line broadening, Helium’srefusal to take on electrons, and countless other less than ideal phrasings And fullsolutions of the harmonic oscillator, spherical harmonics, and hydrogen wave functionODEs, Mandelshtam-Tamm energy-time uncertainty, (all in the notes.) A dice is now adie, though it sounds horrible to me Zero point energy went out again as too speculative

• 5/21 2007 An updated version 3 beta 3.1 to correct a poorly written subsection onquantum confinement for the particle in a pipe Thanks to Swapnil Jain for pointingout the problem I do not want people to get lost so early in the game, so I made it apriority correction In general, I do think that the sections added later to the documentare not of the same quality as the original with regard to writing style The reason issimple When I wrote the original, I was on a sabbatical and had plenty of time tothink and rethink how it would be clearest The later sections are written during thefew spare hours I can dig up I write them and put them in I would need a year off to

do this as it really should be done

• 7/19 2007 Version 3 beta 3.2 adds a section on Hartree-Fock It took forever Mymain regret is that most of them who wasted my time in this major way are probably

no longer around to be properly blasted Writing a book on quantum mechanics by anengineer for engineers is a minefield of having to see through countless poor definitionsand dubious explanations It takes forever In view of the fact that many of thosephysicist were probably supported by tax payers much of the time, it should not be such

an absolute mess!

There are some additions on Born-Oppenheimer and the variational formulation thatwere in the Hartree-Fock section, but that I took out, since they seemed to be too general

to be shoved away inside an application Also rewrote section 4.9 and subsection 4.11.2

to be consistent, and in particular in order to have a single consistent notation Zeropoint energy (the vacuum kind) is back What the heck

• 9/9 2007 Version 3 beta 3.3 mainly adds sections on solids, that have been combinedwith rewritten free and nearly free electron gas sections into a full chapter on solids Therest of the old chapter on examples of multiple particle systems has been pushed backinto the basic multiple particle systems chapter A completely nonsensical discussion in

a paragraph of the free electron gas section was corrected; I cannot believe I have readover that several times I probably was reading what I wanted to say instead of what Isaid The alternative name “twilight terms” has been substituted for “exchange terms.”Many minor changes

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• 12/20 2007 Version 3 beta 3.4 cleans up the format of the “notes.” No more need forloading an interminable web of 64 notes all at the same time over your phone line to read

20 words It also corrects a few errors, one pointed out by Johann Joss, thanks (It isgood to know some people are actually looking at any of this.) It also also extends somefurther griping about correlation energy to all three web locations You may surmisefrom the lack of progress that I have been installing linux on my home PC You areright

Wish List

Donald Knuth, in his versions of TeX, was approaching π I seem to be approaching π + εwith ε < 0.26 Geez, I hope not!

I would like to add key points and review questions to all basic sections I am inching up to

it Very slowly If I ever get chapter 4 done, it will be version 3.4, no kidding Up to version3.6 (3.7 will be left without for now, as being all advanced, elective material.)

After that, the idea is to run all this text through a style checker to eliminate the dead wood.Also, ispell seems to be missing misspelled words Probably thinks they are TeX

It would be nice to put frames around all key formulae Many are already there

There is supposed to be a second volume or additional chapter on computational methods,

in particular density-functional theory Actually, at the time of this writing, I would already

be in heaven if I managed to get around to writing a small section on DFT In short, don’thold your breath But something may be there eventually How old are you, and how is yourhealth?

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Why another book on quantum mechanics? vii

Acknowledgments x

Comments and Feedback xi

History xi

Wish list xiii

List of Figures xxiii List of Tables xxvii 1 Mathematical Prerequisites 1 1.1 Complex Numbers 1

1.2 Functions as Vectors 4

1.3 The Dot, oops, INNER Product 6

1.4 Operators 9

1.5 Eigenvalue Problems 10

1.6 Hermitian Operators 11

1.7 Additional Points 13

1.7.1 Dirac notation 14

1.7.2 Additional independent variables 14

2 Basic Ideas of Quantum Mechanics 15 2.1 The Revised Picture of Nature 15

2.2 The Heisenberg Uncertainty Principle 17

2.3 The Operators of Quantum Mechanics 19

2.4 The Orthodox Statistical Interpretation 21

2.4.1 Only eigenvalues 21

2.4.2 Statistical selection 22

2.5 Schr¨odinger’s Cat [Background] 24

2.6 A Particle Confined Inside a Pipe 25

2.6.1 The physical system 25

2.6.2 Mathematical notations 26

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2.6.4 The Hamiltonian eigenvalue problem 27

2.6.5 All solutions of the eigenvalue problem 28

2.6.6 Discussion of the energy values 32

2.6.7 Discussion of the eigenfunctions 33

2.6.8 Three-dimensional solution 35

2.6.9 Quantum confinement 39

2.7 The Harmonic Oscillator 41

2.7.1 The Hamiltonian 42

2.7.2 Solution using separation of variables 42

2.7.3 Discussion of the eigenvalues 46

2.7.4 Discussion of the eigenfunctions 48

2.7.5 Degeneracy 51

2.7.6 Non-eigenstates 53

3 Single-Particle Systems 57 3.1 Angular Momentum 57

3.1.1 Definition of angular momentum 57

3.1.2 Angular momentum in an arbitrary direction 58

3.1.3 Square angular momentum 60

3.1.4 Angular momentum uncertainty 63

3.2 The Hydrogen Atom 64

3.2.1 The Hamiltonian 64

3.2.2 Solution using separation of variables 65

3.2.3 Discussion of the eigenvalues 69

3.2.4 Discussion of the eigenfunctions 72

3.3 Expectation Value and Standard Deviation 76

3.3.1 Statistics of a die 77

3.3.2 Statistics of quantum operators 79

3.3.3 Simplified expressions 80

3.3.4 Some examples 81

3.4 The Commutator 84

3.4.1 Commuting operators 84

3.4.2 Noncommuting operators and their commutator 86

3.4.3 The Heisenberg uncertainty relationship 86

3.4.4 Commutator reference [Reference] 88

3.5 The Hydrogen Molecular Ion 90

3.5.1 The Hamiltonian 90

3.5.2 Energy when fully dissociated 91

3.5.3 Energy when closer together 92

3.5.4 States that share the electron 93

3.5.5 Comparative energies of the states 94

3.5.6 Variational approximation of the ground state 95

3.5.7 Comparison with the exact ground state 97

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4 Multiple-Particle Systems 99

4.1 Wave Function for Multiple Particles 99

4.2 The Hydrogen Molecule 100

4.2.1 The Hamiltonian 101

4.2.2 Initial approximation to the lowest energy state 101

4.2.3 The probability density 102

4.2.4 States that share the electrons 103

4.2.5 Variational approximation of the ground state 106

4.2.6 Comparison with the exact ground state 107

4.3 Two-State Systems 107

4.4 Spin 110

4.5 Instantaneous Interactions [Background] 111

4.6 Multiple-Particle Systems Including Spin 116

4.6.1 Wave function for a single particle with spin 116

4.6.2 Inner products including spin 118

4.6.3 Wave function for multiple particles with spin 119

4.6.4 Example: the hydrogen molecule 119

4.6.5 Triplet and singlet states 120

4.7 Identical Particles 121

4.8 Global Symmetrization [Background] 122

4.9 Ways to Symmetrize the Wave Function 123

4.10 Matrix Formulation [Advanced] 126

4.11 Atoms Heavier Than Hydrogen 128

4.11.1 The Hamiltonian eigenvalue problem 128

4.11.2 Approximate solution using separation of variables 128

4.11.3 Hydrogen and helium 129

4.11.4 Lithium to neon 131

4.11.5 Sodium to argon 134

4.11.6 Kalium to krypton 135

4.12 Exclusion-Principle Repulsion 135

4.13 Chemical Bonds 136

4.13.1 Covalent sigma bonds 137

4.13.2 Covalent pi bonds 137

4.13.3 Polar covalent bonds and hydrogen bonds 138

4.13.4 Promotion and hybridization 139

4.13.5 Ionic bonds 141

4.13.6 Limitations of valence bond theory 142

5 Solids 143 5.1 Molecular solids 143

5.2 Ionic solids 148

5.3 Introduction to band theory 152

5.4 Metals 153

5.4.1 Lithium 154

5.4.2 One-dimensional crystals 155

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5.4.4 Analysis of the wave functions 158

5.4.5 Floquet (Bloch) theory 159

5.4.6 Fourier analysis 161

5.4.7 The reciprocal lattice 161

5.4.8 The energy levels 162

5.4.9 Electrical conduction 163

5.4.10 Merging and splitting bands 164

5.4.11 Three-dimensional metals 166

5.5 Covalent Materials 169

5.6 Confined Free Electrons 172

5.6.1 The Hamiltonian eigenvalue problem 173

5.6.2 Solution by separation of variables 173

5.6.3 Discussion of the solution 175

5.6.4 A numerical example 177

5.6.5 The density of states and confinement [Advanced] 178

5.6.6 Relation to Bloch functions 182

5.7 Nearly-Free Electrons [Advanced] 184

5.7.1 The lattice structure 185

5.7.2 The small perturbation approach 188

5.7.3 Zeroth order solution 189

5.7.4 First order solution 194

5.7.5 Second order solution 197

5.7.6 Discussion of the energy changes 198

5.8 Quantum Statistical Mechanics 203

5.9 Additional Points [Advanced] 206

5.9.1 Thermal properties 206

5.9.2 Ferromagnetism 211

5.9.3 X-ray diffraction 213

6 Time Evolution 221 6.1 The Schr¨odinger Equation 221

6.1.1 Energy conservation 222

6.1.2 Stationary states 223

6.1.3 Time variations of symmetric two-state systems 224

6.1.4 Time variation of expectation values 225

6.1.5 Newtonian motion 226

6.2 Unsteady perturbations of two-state systems 227

6.2.1 Schr¨odinger equation for a two-state system 227

6.2.2 Stimulated and spontaneous emission 229

6.2.3 Absorption of radiation 230

6.3 Conservation Laws and Symmetries [Background] 233

6.4 The Position and Linear Momentum Eigenfunctions 237

6.4.1 The position eigenfunction 237

6.4.2 The linear momentum eigenfunction 239

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6.5 Wave Packets in Free Space 241

6.5.1 Solution of the Schr¨odinger equation 241

6.5.2 Component wave solutions 242

6.5.3 Wave packets 243

6.5.4 The group velocity 245

6.6 Motion near the Classical Limit 246

6.6.1 General procedures 247

6.6.2 Motion through free space 248

6.6.3 Accelerated motion 249

6.6.4 Decelerated motion 249

6.6.5 The harmonic oscillator 250

6.7 Scattering 251

6.7.1 Partial reflection 251

6.7.2 Tunneling 252

7 Some Additional Topics 255 7.1 All About Angular Momentum [Advanced] 255

7.1.1 The fundamental commutation relations 256

7.1.2 Ladders 257

7.1.3 Possible values of angular momentum 260

7.1.4 A warning about angular momentum 262

7.1.5 Triplet and singlet states 263

7.1.6 Clebsch-Gordan coefficients 265

7.1.7 Pauli spin matrices 269

7.2 The Relativistic Dirac Equation [Advanced] 271

7.2.1 The Dirac idea 271

7.2.2 Emergence of spin from relativity 274

7.3 The Electromagnetic Field [Advanced] 276

7.3.1 The Hamiltonian 276

7.3.2 Maxwell’s equations 278

7.3.3 Electrons in magnetic fields 285

7.4 Nuclear Magnetic Resonance [Advanced] 287

7.4.1 Description of the method 287

7.4.2 The Hamiltonian 288

7.4.3 The unperturbed system 290

7.4.4 Effect of the perturbation 292

7.5 The Variational Method [Advanced] 294

7.5.1 Basic variational statement 294

7.5.2 Differential form of the statement 295

7.5.3 Example application using Lagrangian multipliers 296

7.6 The Born-Oppenheimer Approximation [Advanced] 298

7.6.1 The Hamiltonian 298

7.6.2 The basic Born-Oppenheimer approximation 300

7.6.3 Going one better 302

7.7 The Hartree-Fock Approximation [Advanced] 305

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7.7.2 The Hamiltonian 310

7.7.3 The expectation value of energy 311

7.7.4 The canonical Hartree-Fock equations 314

7.7.5 Additional points 316

7.8 Some Topics Not Covered [Advanced] 322

7.9 The Meaning of Quantum Mechanics [Background] 325

7.9.1 Failure of the Schr¨odinger Equation? 325

7.9.2 The Many-Worlds Interpretation 327

A Notes 335 A.1 Notes on the Mathematical Prerequisites 335

A.1.1 Derivation of the Euler identity 335

A.1.2 Nature and real eigenvalues 335

A.1.3 Are Hermitian operators really like that? 336

A.2 Notes on the Basic Ideas of Quantum Mechanics 336

A.2.1 Why the linear momentum operators are Hermitian 336

A.2.2 Why boundary conditions are tricky 336

A.2.3 Three-dimensional solutions from one-dimensional ones 337

A.2.4 Derivation of the harmonic oscillator solution 339

A.2.5 More on the harmonic oscillator and uncertainty 342

A.3 Notes on the Single-Particle Systems 342

A.3.1 Derivation of a vector identity 342

A.3.2 Derivation of the spherical harmonics 343

A.3.3 Derivation of the hydrogen radial wave functions 345

A.3.4 Definitions of the Laguerre polynomials 346

A.3.5 Justification of the expression for the expectation value 347

A.3.6 Why commuting operators have a common set of eigenvectors 347

A.3.7 Derivation of the generalized uncertainty relationship 348

A.3.8 Derivation of the commutator rules 349

A.3.9 How the hydrogen molecular ion integrals are done 349

A.3.10 In what sense the variational approximation is best 350

A.3.11 More on the accuracy of variational approximation 350

A.3.12 Why the hydrogen molecular ion wave function is positive 351

A.3.13 Symmetries of the hydrogen molecular ion wave function 352

A.4 Notes on the Multiple-Particle Systems 352

A.4.1 Bell’s actual analysis 352

A.4.2 Why spin does not change the hydrogen molecule ground state 353

A.4.3 Limitations of the shielding approximation 353

A.4.4 Why the s states get the closest to the origin 354

A.5 Notes on Solids 354

A.5.1 Ambiguities in the definition of electron affinity 354

A.5.2 Why Floquet theory should be called so 356

A.6 Notes on Time Evolution 357

A.6.1 Why energy eigenstates are physically stationary 357

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A.6.2 More precise description of two-state systems 357

A.6.3 Derivation of the evolution of expectation values 357

A.6.4 The virial theorem 358

A.6.5 The energy-time uncertainty relationship 358

A.6.6 Justification of the two-state approximation in atom radiation 359

A.6.7 About spectral broadening 360

A.6.8 Why symmetry eigenvalues are preserved 360

A.6.9 Remark on the edges of the wave packet 361

A.7 Notes on the Additional Topics 361

A.7.1 Physical justification of the fundamental commutation relations 361

A.7.2 Angular momentum components have only zero in common 362

A.7.3 Components of vectors are less than the total vector 362

A.7.4 Finding the spherical harmonics using ladder operators 362

A.7.5 Why angular momenta components can be added 363

A.7.6 Why the Clebsch-Gordan tables can be read either way 363

A.7.7 How to make your very own Clebsch-Gordan tables 363

A.7.8 Machine language version of the Clebsch-Gordan tables 364

A.7.9 The triangle inequality in quantum mechanics 364

A.7.10 Awkward questions about spin 365

A.7.11 More awkwardness about spin 366

A.7.12 Derivation of a vectorial triple product property 367

A.7.13 More on Maxwell’s third law 367

A.7.14 Setting the record straight on alignment 367

A.7.15 Solving the NMR equations 368

A.7.16 A basic description of Lagrangian multipliers 368

A.7.17 The generalized variational principle 370

A.7.18 Born-Oppenheimer approximation and spin-degeneracy 371

A.7.19 Derivation of the Born-Oppenheimer approximation 372

A.7.20 Why a single Slater determinant does not work 376

A.7.21 Simplification of the Hartree-Fock energy 377

A.7.22 Constraints on the Coulomb and exchange integrals 380

A.7.23 Generalized orbitals 382

A.7.24 Derivation of the Hartree-Fock equations 383

A.7.25 Why the Fock operator is Hermitian 389

A.7.26 Basic science (BS) behind “correlation energy” 390

A.7.27 Everett’s theory and vacuum energy 393

A.7.28 A tenth of a googol in universes 394

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List of Figures

1.1 The classical picture of a vector 41.2 Spike diagram of a vector 41.3 More dimensions 41.4 Infinite dimensions 51.5 The classical picture of a function 51.6 Forming the dot product of two vectors 61.7 Forming the inner product of two functions 72.1 A visualization of an arbitrary wave function 162.2 Combined plot of position and momentum components 182.3 The uncertainty principle illustrated 182.4 Classical picture of a particle in a closed pipe 252.5 Quantum mechanics picture of a particle in a closed pipe 252.6 Definitions 262.7 One-dimensional energy spectrum for a particle in a pipe 322.8 One-dimensional ground state of a particle in a pipe 342.9 Second and third lowest one-dimensional energy states 342.10 Definition of all variables 362.11 True ground state of a particle in a pipe 372.12 True second and third lowest energy states 382.13 A combination of ψ111 and ψ211 seen at some typical times 402.14 The harmonic oscillator 412.15 The energy spectrum of the harmonic oscillator 472.16 Ground state ψ000 of the harmonic oscillator 492.17 Wave functions ψ100 and ψ010 502.18 Energy eigenfunction ψ213 502.19 Arbitrary wave function (not an energy eigenfunction) 543.1 Spherical coordinates of an arbitrary point P 583.2 Spectrum of the hydrogen atom 703.3 Ground state wave function ψ100 of the hydrogen atom 723.4 Eigenfunction ψ200 733.5 Eigenfunction ψ210, or 2pz 743.6 Eigenfunction ψ211 (and ψ21−1) 743.7 Eigenfunctions 2px, left, and 2py, right 753.8 Hydrogen atom plus free proton far apart 91

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3.10 The electron being anti-symmetrically shared 933.11 The electron being symmetrically shared 944.1 State with two neutral atoms 1034.2 Symmetric state 1044.3 Antisymmetric state 1054.4 Separating the hydrogen ion 1124.5 The Bohm experiment 1134.6 The Bohm experiment, after the Venus measurement 1134.7 Spin measurement directions 1144.8 Earth’s view of events 1154.9 A moving observer’s view of events 1164.10 Approximate solutions for hydrogen (left) and helium (right) 1304.11 Approximate solutions for lithium (left) and beryllium (right) 1324.12 Example approximate solution for boron 1344.13 Covalent sigma bond consisting of two 2pz states 1374.14 Covalent pi bond consisting of two 2px states 1384.15 Covalent sigma bond consisting of a 2pz and a 1s state 1384.16 Shape of an sp3 hybrid state 1404.17 Shapes of the sp2 (left) and sp (right) hybrids 1415.1 Possible polarizations of a pair of hydrogen atoms 1455.2 Billiard-ball model of the salt molecule 1495.3 Billiard-ball model of a salt crystal 1505.4 The salt crystal disassembled to show its structure 1515.5 Sketch of electron energy spectra in solids 1525.6 The lithium atom, scaled more correctly than in chapter 4.11 1545.7 Body-centered-cubic (bcc) structure of lithium 1555.8 Fully periodic wave function of a two-atom lithium “crystal.” 1565.9 Flip-flop wave function of a two-atom lithium “crystal.” 1575.10 Wave functions of a four-atom lithium “crystal.” The actual picture is that ofthe fully periodic mode 1585.11 Reciprocal lattice of a one-dimensional crystal 1615.12 Schematic of energy bands 1625.13 Energy versus linear momentum 1645.14 Schematic of merging bands 1655.15 A primitive cell and primitive translation vectors of lithium 1665.16 Wigner-Seitz cell of the bcc lattice 1675.17 Schematic of crossing bands 1705.18 Ball and stick schematic of the diamond crystal 1715.19 Allowed wave number vectors 1755.20 Schematic energy spectrum of the free electron gas 1765.21 Occupied wave number states and Fermi surface in the ground state 1775.22 Density of states for the free electron gas 179

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5.23 Energy states, top, and density of states, bottom, when there is severe ment in the y-direction, as in a quantum well 1805.24 Energy states, top, and density of states, bottom, when there is severe confine-ment in both the y- and z-directions, as in a quantum wire 1815.25 Energy states, top, and density of states, bottom, when there is severe confine-ment in all three directions, as in a quantum dot or artificial atom 1835.26 Wave number vectors seen in a cross section of constant kz Top: sinusoidalsolutions Bottom: exponential solutions 1865.27 Assumed simple cubic reciprocal lattice, shown as black dots, in cross-section.The boundaries of the surrounding primitive cells are shown as thin red lines 1875.28 Occupied states for one, two, and three free electrons per physical lattice cell 1905.29 Redefinition of the occupied wave number vectors into Brillouin zones 1925.30 Second, third, and fourth Brillouin zones seen in the periodic zone scheme 1935.31 The typical exponential free electron wave function to be corrected for thelattice potential is shown as a red dot 1935.32 The ~k-lattice and k-sphere in wave number space 1955.33 Tearing apart of the wave number space energies 1995.34 Effect of a lattice potential on the energy The energy is represented by thesquare distance from the origin, and is relative to the energy at the origin 1995.35 Bragg planes seen in wave number space cross section 2005.36 Occupied states for the energies of figure 5.34 if there are two valence electronsper lattice cell Left: energy Right: wave numbers 2015.37 Smaller lattice potential From top to bottom shows one, two and three valenceelectrons per lattice cell Left: energy Right: wave numbers 2025.38 Sketch of electron energy spectra in solids 2035.39 Specific heat at constant volume of gases Temperatures from absolute zero to

confine-1200 K Data from NIST-JANAF and AIP 2075.40 Specific heat at constant pressure of solids Temperatures from absolute zero

to 1200 K Carbon is diamond; graphite is similar Water is ice and liquid.Data from NIST-JANAF, CRC, AIP, Rohsenow et al 2095.41 Depiction of an electromagnetic ray 2145.42 Law of reflection in elastic scattering from a plane 2145.43 Scattering from multiple “planes of atoms” 2155.44 Difference in travel distance when scattered from P rather than O 2166.1 Emission and absorption of radiation by an atom 2296.2 Approximate Dirac delta function δε(x−ξ) is shown left The true delta functionδ(x− ξ) is the limit when ε becomes zero, and is an infinitely high, infinitelythin spike, shown right It is the eigenfunction corresponding to a position ξ 2386.3 The real part (red) and envelope (black) of an example wave 2426.4 The wave moves with the phase speed 2436.5 The real part (red) and magnitude or envelope (black) of a typical wave packet 2446.6 The velocities of wave and envelope are not equal 2446.7 A particle in free space 2486.8 An accelerating particle 249

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6.10 Unsteady solution for the harmonic oscillator The third picture shows themaximum distance from the nominal position that the wave packet reaches 2506.11 A partial reflection 2516.12 An tunneling particle 2526.13 Penetration of an infinitely high potential energy barrier 2527.1 Example bosonic ladders 2597.2 Example fermionic ladders 2597.3 Triplet and singlet states in terms of ladders 2657.4 Clebsch-Gordan coefficients of two spin one half particles 2667.5 Clebsch-Gordan coefficients for lb equal to one half 2677.6 Clebsch-Gordan coefficients for lb equal to one 2687.7 Relationship of Maxwell’s first equation to Coulomb’s law 2797.8 Maxwell’s first equation for a more arbitrary region The figure to the rightincludes the field lines through the selected points 2807.9 The net number of field lines leaving a region is a measure for the net chargeinside that region 2807.10 Since magnetic monopoles do not exist, the net number of magnetic field linesleaving a region is always zero 2817.11 Electric power generation 2827.12 Two ways to generate a magnetic field: using a current (left) or using a varyingelectric field (right) 2837.13 Larmor precession of the expectation spin (or magnetic moment) vector aroundthe magnetic field 2917.14 Probability of being able to find the nuclei at elevated energy versus time for agiven perturbation frequency ω 2937.15 Maximum probability of finding the nuclei at elevated energy 2937.16 A perturbing magnetic field, rotating at precisely the Larmor frequency, causesthe expectation spin vector to come cascading down out of the ground state 2947.17 Bohm’s version of the Einstein, Podolski, Rosen Paradox 3287.18 Non entangled positron and electron spins; up and down 3287.19 Non entangled positron and electron spins; down and up 3287.20 The wave functions of two universes combined 3297.21 The Bohm experiment repeated 3317.22 Repeated experiments on the same electron 332

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List of Tables

2.1 One-dimensional eigenfunctions of the harmonic oscillator, [3, p 56] 453.1 The first few spherical harmonics, from [3, p 139] 613.2 The first few radial wave functions for hydrogen, from [3, p 154] 684.1 Abbreviated periodic table of the elements, showing element symbol, atomicnumber, ionization energy, and electronegativity 132

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−1 is not an ordinary, “real”, number, since there is no real number whose square

is −1; the square of a real number is always positive This section summarizes the mostimportant properties of complex numbers

First, any complex number, call it c, can by definition always be written in the form

where both cr and ci are ordinary real numbers, not involving √

−1 The number cr is calledthe real part of c and ci the imaginary part

You can think of the real and imaginary parts of a complex number as the components of atwo-dimensional vector:

The length of that vector is called the “magnitude,” or “absolute value” |c| of the complexnumber It equals

|c| =qc2

r + c2

i

Complex numbers can be manipulated pretty much in the same way as ordinary numbers can

A relation to remember is:

HH

HHc∗H

You can get the magnitude of a complex number c by multiplying c with its complex conjugate

c∗ and taking a square root:

which is indeed the same as before

From the above graph of the vector representing a complex number c, the real part is cr =

|c| cos α where α is the angle that the vector makes with the horizontal axis, and the imaginarypart is ci =|c| sin α So you can write any complex number in the form

c =|c| (cos α + i sin α)The critically important Euler identity says that:

cos α + i sin α = eiα (1.5)

1.1 COMPLEX NUMBERS 3

So, any complex number can be written in “polar form” as

where both the magnitude |c| and the angle α are real numbers

Any complex number of magnitude one can therefor be written as eiα Note that the only tworeal numbers of magnitude one, 1 and −1, are included for α = 0, respectively α = π Thenumber i is obtained for α = π/2 and −i for α = −π/2

(See note {A.1.1} if you want to know where the Euler identity comes from.)

Key Points

¦ Complex numbers include the square root of minus one, i, as a valid number

¦ All complex numbers can be written as a real part plus i times an imaginary part,where both parts are normal real numbers

¦ The complex conjugate of a complex number is obtained by replacing i everywhere by

1 Multiply out (2 + 3i)2 and then find its real and imaginary part

2 Show more directly that 1/i =−i

3 Multiply out (2 + 3i)(2− 3i) and then find its real and imaginary part

4 Find the magnitude or absolute value of 2 + 3i

5 Verify that (2− 3i)2 is still the complex conjugate of (2 + 3i)2 if both are multipliedout

6 Verify that e−2i is still the complex conjugate of e2i after both are rewritten usingthe Euler identity

Figure 1.1: The classical picture of a vector.

However, the same vector may instead be represented as a spike diagram, by plotting thevalue of the components versus the component index:

Figure 1.2: Spike diagram of a vector

(The symbol i for the component index is not to be confused with i =√

1.2 FUNCTIONS AS VECTORS 5For a large number of dimensions, and in particular in the limit of infinitely many dimensions,the large values of i can be rescaled into a continuous coordinate, call it x For example, xmight be defined as i divided by the number of dimensions In any case, the spike diagrambecomes a function f (x):

Figure 1.4: Infinite dimensions

The spikes are usually not shown:

Figure 1.5: The classical picture of a function

In this way, a function is just a vector in infinitely many dimensions

Key Points

¦ Functions can be thought of as vectors with infinitely many components

¦ This allows quantum mechanics do the same things with functions as you can do withvectors

1.3 The Dot, oops, INNER Product

The dot product of vectors is an important tool It makes it possible to find the length of avector, by multiplying the vector by itself and taking the square root It is also used to check

if two vectors are orthogonal: if their dot product is zero, they are In this subsection, thedot product is defined for complex vectors and functions

The usual dot product of two vectors ~f and ~g can be found by multiplying components withthe same index i together and summing that:

~· ~g ≡ f1g1+ f2g2+ f3g3

(The emphatic equal, ≡, is commonly used to indicate “is by definition equal” or “is alwaysequal.”) Figure 1.6 shows multiplied components using equal colors

Figure 1.6: Forming the dot product of two vectors

Note the use of numeric subscripts, f1, f2, and f3 rather than fx, fy, and fz; it means the samething Numeric subscripts allow the three term sum above to be written more compactly as:

~· ~g ≡X

all i

figi

The Σ is called the “summation symbol.”

The length of a vector ~f , indicated by | ~f| or simply by f, is normally computed as

However, this does not work correctly for complex vectors The difficulty is that terms of theform f2

i are no longer necessarily positive numbers For example, i2 =−1

Therefore, it is necessary to use a generalized “inner product” for complex vectors, which puts

a complex conjugate on the first vector:

h ~f|~gi ≡X

all i

1.3 THE DOT, OOPS, INNER PRODUCT 7

If vector ~f is real, the complex conjugate does nothing, and the inner producth ~f|~gi is the same

as the dot product ~f·~g Otherwise, in the inner product ~f and ~g are no longer interchangeable;the conjugates are only on the first factor, ~f Interchanging ~f and ~g changes the inner productvalue into its complex conjugate

The length of a nonzero vector is now always a positive number:

The inner product of functions is defined in exactly the same way as for vectors, by multiplyingvalues at the same x position together and summing But since there are infinitely many x-values, the sum becomes an integral:

hf|gi =

Z

all xf∗(x)g(x) dx (1.9)

as illustrated in figure 1.7

Figure 1.7: Forming the inner product of two functions

The equivalent of the length of a vector is in case of a function called its “norm:”

||f|| ≡ qhf|fi =

sZ

all x|f(x)|2dx (1.10)The double bars are used to avoid confusion with the absolute value of the function

A vector or function is called “normalized” if its length or norm is one:

hf|fi = 1 iff f is normalized (1.11)

(“iff” should really be read as “if and only if.”)

Two vectors, or two functions, f and g are by definition orthogonal if their inner product iszero:

hf|gi = 0 iff f and g are orthogonal (1.12)Sets of vectors or functions that are all

• mutually orthogonal, and

1 = hf1|f1i = hf2|f2i = hf3|f3i =

Key Points

¦ For complex vectors and functions, the normal dot product becomes the inner product

¦ To take an inner product of vectors, (1) take complex conjugates of the components

of the first vector; (2) multiply corresponding components of the two vectors together;and (3) sum these products

¦ To take an inner product of functions, (1) take the complex conjugate of the firstfunction; (2) multiply the two functions; and (3) integrate the product function Thereal difference from vectors is integration instead of summation

¦ To find the length of a vector, take the inner product of the vector with itself, andthen a square root

¦ To find the norm of a function, take the inner product of the function with itself, andthen a square root

¦ A pair of functions, or a pair of vectors, are orthogonal if their inner product is zero

¦ A set of functions, or a set of vectors, form an orthonormal set if every one is orthogonal

to all the rest, and every one is of unit norm or length

!+

1.4 OPERATORS 9

2 Find the length of the vector Ã

1 + i3

!

3 Find the inner product of the functions sin(x) and cos(x) on the interval 0≤ x ≤ 1

4 Show that the functions sin(x) and cos(x) are orthogonal on the interval 0≤ x ≤ 2π

5 Verify that sin(x) is not a normalized function on the interval 0 ≤ x ≤ 2π, andnormalize it by dividing by its norm

6 Verify that the most general multiple of sin(x) that is normalized on the interval

operator that corresponds to multiplying by x If it is clear that something is an operator,such as d/dx, no hat will be used

It should really be noted that the operators that you are interested in in quantum mechanicsare “linear” operators: if you increase f by a number, Af increases by that same number;also, if you sum f and g, A(f + g) will be Af plus Ag

Key Points

¦ Matrices turn vectors into other vectors

¦ Operators turn functions into other functions

1.4 Review Questions

1 So what is the result if the operator d/dx is applied to the function sin(x)?

2 If, say,xc 2sin(x) is simply the function x2sin(x), then what is the difference between

However, eigenfunctions like ex are not very common in quantum mechanics since they come very large at large x, and that typically does not describe physical situations Theeigenfunctions of d/dx that do appear a lot are of the form eikx, where i = √

be-−1 and k is an

1.6 HERMITIAN OPERATORS 11arbitrary real number The eigenvalue is ik:

¦ If an operator turns a nonzero function into a multiple of that function, that function

is an eigenfunction of the operator, and the multiple is the eigenvalue

1.5 Review Questions

1 Show that eikx, above, is also an eigenfunction of d2/dx2, but with eigenvalue−k2

In fact, it is easy to see that the square of any operator has the same eigenfunctions,but with the square eigenvalues (Since the operators of quantum mechanics arelinear.)

2 Show that any function of the form sin(kx) and any function of the form cos(kx),where k is a constant called the wave number, is an eigenfunction of the operator

d2/dx2, though they are not eigenfunctions of d/dx

3 Show that sin(kx) and cos(kx), with k a constant, are eigenfunctions of the inversionoperator Inv, which turns any function f (x) into f (−x), and find the eigenvalues

• They always have real eigenvalues, not involving i = √−1 (But the eigenfunctions,

or eigenvectors if the operator is a matrix, might be complex.) Physical values such asposition, momentum, and energy are ordinary real numbers since they are eigenvalues

of Hermitian operators {A.1.2}

• Their eigenfunctions can always be chosen so that they are normalized and mutuallyorthogonal, in other words, an orthonormal set This tends to simplify the variousmathematics a lot

• Their eigenfunctions form a “complete” set This means that any function can be written

as some linear combination of the eigenfunctions In practical terms, that means thatyou only need to look at the eigenfunctions to completely understand what the operatordoes {A.1.3}

In the linear algebra of real matrices, Hermitian operators are simply symmetric matrices Abasic example is the inertia matrix of a solid body in Newtonian dynamics The orthonormaleigenvectors of the inertia matrix give the directions of the principal axes of inertia of thebody

The following properties of inner products involving Hermitian operators are often needed, sothey are listed here:

If A is Hermitian: hg|Afi = hf|Agi∗, hf|Afi is real (1.16)The first says that you can swap f and g if you take complex conjugate (It is simply areflection of the fact that if you change the sides in an inner product, you turn it into itscomplex conjugate Normally, that puts the operator at the other side, but for a Hermitianoperator, it does not make a difference.) The second is important because ordinary realnumbers typically occupy a special place in the grand scheme of things (The fact that theinner product is real merely reflects the fact that if a number is equal to its complex conjugate,

it must be real; if there was an i in it, the number would change by a complex conjugate.)

Key Points

¦ Hermitian operators can be flipped over to the other side in inner products

¦ Hermitian operators have only real eigenvalues

¦ Hermitian operators have a complete set of orthonormal eigenfunctions (or tors)

eigenvec-1.6 Review Questions

1 Show that the operator b2 is a Hermitian operator, butbi is not.