Table Of ContentESSENTIAL QUANTUM MECHANICS
This page intentionally left blank
Essential Quantum
Mechanics
GARY E. BOWMAN
Department of Physics and Astronomy
Northern Arizona University
1
3
GreatClarendonStreet,OxfordOX26DP
OxfordUniversityPressisadepartmentoftheUniversityofOxford.
ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship,
andeducationbypublishingworldwidein
Oxford NewYork
Auckland CapeTown DaresSalaam HongKong Karachi
KualaLumpur Madrid Melbourne MexicoCity Nairobi
NewDelhi Shanghai Taipei Toronto
Withofficesin
Argentina Austria Brazil Chile CzechRepublic France Greece
Guatemala Hungary Italy Japan Poland Portugal Singapore
SouthKorea Switzerland Thailand Turkey Ukraine Vietnam
OxfordisaregisteredtrademarkofOxfordUniversityPress
intheUKandincertainothercountries
PublishedintheUnitedStates
byOxfordUniversityPressInc.,NewYork
(cid:1)c GaryE.Bowman2008
Themoralrightsoftheauthorhavebeenasserted
DatabaserightOxfordUniversityPress(maker)
Firstpublished2008
Allrightsreserved.Nopartofthispublicationmaybereproduced,
storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,
withoutthepriorpermissioninwritingofOxfordUniversityPress,
orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriate
reprographicsrightsorganization.Enquiriesconcerningreproduction
outsidethescopeoftheaboveshouldbesenttotheRightsDepartment,
OxfordUniversityPress,attheaddressabove
Youmustnotcirculatethisbookinanyotherbindingorcover
andyoumustimposethesameconditiononanyacquirer
BritishLibraryCataloguinginPublicationData
Dataavailable
LibraryofCongressCataloginginPublicationData
Dataavailable
TypesetbyNewgenImagingSystems(P)Ltd.,Chennai,India
PrintedinGreatBritain
onacid-freepaperby
BiddlesLtd.,King’sLynn
ISBN978–0–19–922892–8(Hbk.)
ISBN978–0–19–922893–5(Pbk.)
1 3 5 7 9 10 8 6 4 2
Contents
Preface ix
1 Introduction: Three Worlds 1
1.1 Worlds 1 and 2 1
1.2 World 3 3
1.3 Problems 4
2 The Quantum Postulates 5
2.1 Postulate 1: The Quantum State 6
2.2 Postulate 2: Observables, Operators, and Eigenstates 8
2.3 Postulate 3: Quantum Superpositions 10
2.3.1 Discrete Eigenvalues 11
2.3.2 Continuous Eigenvalues 12
2.4 Closing Comments 15
2.5 Problems 16
3 What Is a Quantum State? 19
3.1 Probabilities, Averages, and Uncertainties 19
3.1.1 Probabilities 19
3.1.2 Averages 22
3.1.3 Uncertainties 24
3.2 The Statistical Interpretation 26
3.3 Bohr, Einstein, and Hidden Variables 28
3.3.1 Background 28
3.3.2 Fundamental Issues 30
3.3.3 Einstein Revisited 32
3.4 Problems 33
4 The Structure of Quantum States 36
4.1 Mathematical Preliminaries 36
4.1.1 Vector Spaces 36
4.1.2 Function Spaces 39
4.2 Dirac’s Bra-ket Notation 41
4.2.1 Bras and Kets 41
4.2.2 Labeling States 42
4.3 The Scalar Product 43
4.3.1 Quantum Scalar Products 43
4.3.2 Discussion 45
vi Contents
4.4 Representations 47
4.4.1 Basics 47
4.4.2 Superpositions and Representations 48
4.4.3 Representational Freedom 50
4.5 Problems 52
5 Operators 53
5.1 Introductory Comments 54
5.2 Hermitian Operators 56
5.2.1 Adjoint Operators 56
5.2.2 Hermitian Operators: Definition and Properties 57
5.2.3 Wavefunctions and Hermitian Operators 59
5.3 Projection and Identity Operators 61
5.3.1 Projection Operators 61
5.3.2 The Identity Operator 62
5.4 Unitary Operators 62
5.5 Problems 64
6 Matrix Mechanics 68
6.1 Elementary Matrix Operations 68
6.1.1 Vectors and Scalar Products 68
6.1.2 Matrices and Matrix Multiplication 69
6.1.3 Vector Transformations 70
6.2 States as Vectors 71
6.3 Operators as Matrices 72
6.3.1 An Operator in Its Eigenbasis 72
6.3.2 Matrix Elements and Alternative Bases 73
6.3.3 Change of Basis 75
6.3.4 Adjoint, Hermitian, and Unitary Operators 75
6.4 Eigenvalue Equations 77
6.5 Problems 78
7 Commutators and Uncertainty Relations 82
7.1 The Commutator 83
7.1.1 Definition and Characteristics 83
7.1.2 Commutators in Matrix Mechanics 85
7.2 The Uncertainty Relations 86
7.2.1 Uncertainty Products 86
7.2.2 General Form of the Uncertainty Relations 87
7.2.3 Interpretations 88
7.2.4 Reflections 91
7.3 Problems 93
8 Angular Momentum 95
8.1 Angular Momentum in Classical Mechanics 95
8.2 Basics of Quantum Angular Momentum 97
8.2.1 Operators and Commutation Relations 97
Contents vii
8.2.2 Eigenstates and Eigenvalues 99
8.2.3 Raising and Lowering Operators 100
8.3 Physical Interpretation 101
8.3.1 Measurements 101
8.3.2 Relating L2 and L 104
z
8.4 Orbital and Spin Angular Momentum 106
8.4.1 Orbital Angular Momentum 106
8.4.2 Spin Angular Momentum 107
8.5 Review 107
8.6 Problems 108
9 The Time-Independent Schr¨odinger Equation 111
9.1 An Eigenvalue Equation for Energy 112
9.2 Using the Schro¨dinger Equation 114
9.2.1 Conditions on Wavefunctions 114
9.2.2 An Example: the Infinite Potential Well 115
9.3 Interpretation 117
9.3.1 Energy Eigenstates in Position Space 117
9.3.2 Overall and Relative Phases 118
9.4 Potential Barriers and Tunneling 120
9.4.1 The Step Potential 120
9.4.2 The Step Potential and Scattering 122
9.4.3 Tunneling 124
9.5 What’s Wrong with This Picture? 125
9.6 Problems 126
10 Why Is the State Complex? 128
10.1 Complex Numbers 129
10.1.1 Basics 129
10.1.2 Polar Form 130
10.1.3 Argand Diagrams and the Role of the Phase 131
10.2 The Phase in Quantum Mechanics 133
10.2.1 Phases and the Description of States 133
10.2.2 Phase Changes and Probabilities 135
10.2.3 Unitary Operators Revisited 136
10.2.4 Unitary Operators, Phases, and Probabilities 137
10.2.5 Example: A Spin 1 System 139
2
10.3 Wavefunctions 141
10.4 Reflections 142
10.5 Problems 143
11 Time Evolution 145
11.1 The Time-Dependent Schro¨dinger Equation 145
11.2 How Time Evolution Works 146
11.2.1 Time Evolving a Quantum State 146
11.2.2 Unitarity and Phases Revisited 148
viii Contents
11.3 Expectation Values 149
11.3.1 Time Derivatives 149
11.3.2 Constants of the Motion 150
11.4 Energy-Time Uncertainty Relations 151
11.4.1 Conceptual Basis 151
11.4.2 Spin 1: An Example 153
2
11.5 Problems 154
12 Wavefunctions 157
12.1 What is a Wavefunction? 158
12.1.1 Eigenstates and Coefficients 158
12.1.2 Representations and Operators 159
12.2 Changing Representations 161
12.2.1 Change of Basis Revisited 161
12.2.2 From x to p and Back Again 161
12.2.3 Gaussians and Beyond 163
12.3 Phases and Time Evolution 165
12.3.1 Free Particle Evolution 165
12.3.2 Wavepackets 167
12.4 Bra-ket Notation 168
12.4.1 Quantum States 168
12.4.2 Eigenstates and Transformations 170
12.5 Epilogue 171
12.6 Problems 172
A Mathematical Concepts 175
A.1 Complex Numbers and Functions 175
A.2 Differentiation 176
A.3 Integration 178
A.4 Differential Equations 180
B Quantum Measurement 183
C The Harmonic Oscillator 186
C.1 Energy Eigenstates and Eigenvalues 186
C.2 The Number Operator and its Cousins 188
C.3 Photons as Oscillators 189
D Unitary Transformations 192
D.1 Unitary Operators 192
D.2 Finite Transformations and Generators 195
D.3 Continuous Symmetries 197
D.3.1 Symmetry Transformations 197
D.3.2 Symmetries of Physical Law 197
D.3.3 System Symmetries 199
Bibliography 201
Index 205
Preface
While still a relatively new graduate student, I once remarked to my advi-
sor,JimCushing,thatIstilldidn’tunderstandquantummechanics.Tothis
hepromptlyreplied:“You’llspendtherestofyourlifetryingtounderstand
quantummechanics!”Despitecountlessbooksthatthesubjecthasspawned
since it first assumed a coherent form in the 1920s, quantum mechanics
remains notoriously, even legendarily, difficult. Some may believe students
should be told that physics really isn’t that hard, presumably so as not to
intimidatethem.Idisagree:whatcanbemoredemoralizingthanstruggling
mightily with a subject, only to be told that it’s really not that difficult?
Let me say it outright, then: quantum mechanics is hard. In writing
this book, I have not found any “magic bullet” by which I can render
the subject easily digestible. I have, however, tried to write a book that is
neitherapopularizationnora“standard”text;abookthattakesamodern
approach, rather than one grounded in pedagogical precedent; a book that
focuses on elucidating the structure and meaning of quantum mechanics,
leaving comprehensive treatments to the standard texts.
Above all, I have tried to write with the student in mind. The pri-
marytargetaudienceisundergraduatesabouttotake,ortaking,theirfirst
quantum course. But my hope is that the book will also serve biologists,
philosophers,engineers,andotherthoughtfulpeople—peoplewhoarefasci-
natedbyquantumphysics,butfindthepopularizationstoosimplistic,and
the textbooks too advanced and comprehensive—by providing a foothold
on “real” quantum mechanics, as used by working scientists.
Popularizations of quantum mechanics are intended not to expound
the subject as used by working scientists, but rather to discuss “quantum
weirdness,”suchasBell’stheoremandthemeasurementproblem,interms
palatable to interested non-scientists. As such, the mathematical level of
such books ranges from very low to essentially nonexistent.
In contrast, the comprehensive texts used in advanced courses often
make daunting conceptual and mathematical demands on the reader.
Preparation for such courses typically consists of a modern physics course,
but these tend to be rather conceptual. Modern physics texts generally
take a semi-historical approach, discussing topics such as the Bohr atom
and the Compton effect. Formalism is minimized and description empha-
sized; the highly abstract mathematical and physical concepts of quantum
mechanicsremainlargelyuntouched.Thereisthusaratherlargegaptobe
bridged, and students in advanced courses may find that they must solve
