MOLECULARQUANTUM

MECHANICS,FOURTH EDITION

Peter AtkinsRonald Friedman

OXFORD UNIVERSITY PRESS

M O L E C U L A R Q U A N T U M M E C H A N I C S

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F O U R T H E D I T I O N

Peter Atkins

University of Oxford

Ronald Friedman

Indiana Purdue Fort Wayne

AC

MOLECULAR

QUANTUM

MECHANICS

ACGreat Clarendon Street, Oxford OX2 6DP

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Table of contents

Preface xiii

Introduction and orientation 1

1 The foundations of quantum mechanics 9

2 Linear motion and the harmonic oscillator 43

3 Rotational motion and the hydrogen atom 71

4 Angular momentum 98

5 Group theory 122

6 Techniques of approximation 168

7 Atomic spectra and atomic structure 207

8 An introduction to molecular structure 249

9 The calculation of electronic structure 287

10 Molecular rotations and vibrations 342

11 Molecular electronic transitions 382

12 The electric properties of molecules 407

13 The magnetic properties of molecules 436

14 Scattering theory 473

Further information 513

Further reading 553

Appendix 1 Character tables and direct products 557

Appendix 2 Vector coupling coefficients 562

Answers to selected problems 563

Index 565

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Detailed Contents

Introduction and orientation 1

0.1 Black-body radiation 1

0.2 Heat capacities 3

0.3 The photoelectric and Compton effects 4

0.4 Atomic spectra 5

0.5 The duality of matter 6

P R O B L E M S 8

1 The foundations of quantum mechanics 9

Operators in quantum mechanics 9

1.1 Linear operators 10

1.2 Eigenfunctions and eigenvalues 10

1.3 Representations 12

1.4 Commutation and non-commutation 13

1.5 The construction of operators 14

1.6 Integrals over operators 15

1.7 Dirac bracket notation 16

1.8 Hermitian operators 17

The postulates of quantum mechanics 19

1.9 States and wavefunctions 19

1.10 The fundamental prescription 20

1.11 The outcome of measurements 20

1.12 The interpretation of the wavefunction 22

1.13 The equation for the wavefunction 23

1.14 The separation of the Schrödinger equation 23

The specification and evolution of states 25

1.15 Simultaneous observables 25

1.16 The uncertainty principle 27

1.17 Consequences of the uncertainty principle 29

1.18 The uncertainty in energy and time 30

1.19 Time-evolution and conservation laws 30

Matrices in quantum mechanics 32

1.20 Matrix elements 32

1.21 The diagonalization of the hamiltonian 34

The plausibility of the Schrödinger equation 36

1.22 The propagation of light 36

1.23 The propagation of particles 38

1.24 The transition to quantum mechanics 39

P R O B L E M S 40

2 Linear motion and the harmonicoscillator 43

The characteristics of acceptable wavefunctions 43

Some general remarks on the Schrödinger equation 44

2.1 The curvature of the wavefunction 45

2.2 Qualitative solutions 45

2.3 The emergence of quantization 46

2.4 Penetration into non-classical regions 46

Translational motion 47

2.5 Energy and momentum 48

2.6 The significance of the coefficients 48

2.7 The flux density 49

2.8 Wavepackets 50

Penetration into and through barriers 51

2.9 An infinitely thick potential wall 51

2.10 A barrier of finite width 52

2.11 The Eckart potential barrier 54

Particle in a box 55

2.12 The solutions 56

2.13 Features of the solutions 57

2.14 The two-dimensional square well 58

2.15 Degeneracy 59

The harmonic oscillator 60

2.16 The solutions 61

2.17 Properties of the solutions 63

2.18 The classical limit 65

Translation revisited: The scattering matrix 66

P R O B L E M S 68

3 Rotational motion and the hydrogen atom 71

Particle on a ring 71

3.1 The hamiltonian and the Schrödinger

equation 71

3.2 The angular momentum 73

3.3 The shapes of the wavefunctions 74

3.4 The classical limit 76

Particle on a sphere 76

3.5 The Schrödinger equation and

its solution 76

3.6 The angular momentum of the particle 79

3.7 Properties of the solutions 81

3.8 The rigid rotor 82

Motion in a Coulombic field 84

3.9 The Schrödinger equation for

hydrogenic atoms 84

3.10 The separation of the relative coordinates 85

3.11 The radial Schrödinger equation 85

3.12 Probabilities and the radial

distribution function 90

3.13 Atomic orbitals 91

3.14 The degeneracy of hydrogenic atoms 94

P R O B L E M S 96

4 Angular momentum 98

The angular momentum operators 98

4.1 The operators and their commutation

relations 99

4.2 Angular momentum observables 101

4.3 The shift operators 101

The definition of the states 102

4.4 The effect of the shift operators 102

4.5 The eigenvalues of the angular momentum 104

4.6 The matrix elements of the angular

momentum 106

4.7 The angular momentum eigenfunctions 108

4.8 Spin 110

The angular momenta of composite systems 112

4.9 The specification of coupled states 112

4.10 The permitted values of the total angular

momentum 113

4.11 The vector model of coupled angular

momenta 115

4.12 The relation between schemes 117

4.13 The coupling of several angular momenta 119

P R O B L E M S 120

5 Group theory 122

The symmetries of objects 122

5.1 Symmetry operations and elements 123

5.2 The classification of molecules 124

The calculus of symmetry 129

5.3 The definition of a group 129

5.4 Group multiplication tables 130

5.5 Matrix representations 131

5.6 The properties of matrix representations 135

5.7 The characters of representations 137

5.8 Characters and classes 138

5.9 Irreducible representations 139

5.10 The great and little orthogonality theorems 142

Reduced representations 145

5.11 The reduction of representations 146

5.12 Symmetry-adapted bases 147

The symmetry properties of functions 151

5.13 The transformation of p-orbitals 151

5.14 The decomposition of direct-product bases 152

5.15 Direct-product groups 155

5.16 Vanishing integrals 157

5.17 Symmetry and degeneracy 159

The full rotation group 161

5.18 The generators of rotations 161

5.19 The representation of the full rotation group 162

5.20 Coupled angular momenta 164

Applications 165

P R O B L E M S 166

viii j CONTENTS

6 Techniques of approximation 168

Time-independent perturbation theory 168

6.1 Perturbation of a two-level system 169

6.2 Many-level systems 171

6.3 The first-order correction to the energy 172

6.4 The first-order correction to the wavefunction 174

6.5 The second-order correction to the energy 175

6.6 Comments on the perturbation expressions 176

6.7 The closure approximation 178

6.8 Perturbation theory for degenerate states 180

Variation theory 183

6.9 The Rayleigh ratio 183

6.10 The Rayleigh–Ritz method 185

The Hellmann–Feynman theorem 187

Time-dependent perturbation theory 189

6.11 The time-dependent behaviour of a

two-level system 189

6.12 The Rabi formula 192

6.13 Many-level systems: the variation of constants 193

6.14 The effect of a slowly switched constant

perturbation 195

6.15 The effect of an oscillating perturbation 197

6.16 Transition rates to continuum states 199

6.17 The Einstein transition probabilities 200

6.18 Lifetime and energy uncertainty 203

P R O B L E M S 204

7 Atomic spectra and atomic structure 207

The spectrum of atomic hydrogen 207

7.1 The energies of the transitions 208

7.2 Selection rules 209

7.3 Orbital and spin magnetic moments 212

7.4 Spin–orbit coupling 214

7.5 The fine-structure of spectra 216

7.6 Term symbols and spectral details 217

7.7 The detailed spectrum of hydrogen 218

The structure of helium 219

7.8 The helium atom 219

7.9 Excited states of helium 222

7.10 The spectrum of helium 224

7.11 The Pauli principle 225

Many-electron atoms 229

7.12 Penetration and shielding 229

7.13 Periodicity 231

7.14 Slater atomic orbitals 233

7.15 Self-consistent fields 234

7.16 Term symbols and transitions of

many-electron atoms 236

7.17 Hund’s rules and the relative energies of terms 239

7.18 Alternative coupling schemes 240

Atoms in external fields 242

7.19 The normal Zeeman effect 242

7.20 The anomalous Zeeman effect 243

7.21 The Stark effect 245

P R O B L E M S 246

8 An introduction to molecular structure 249

The Born–Oppenheimer approximation 249

8.1 The formulation of the approximation 250

8.2 An application: the hydrogen molecule–ion 251

Molecular orbital theory 253

8.3 Linear combinations of atomic orbitals 253

8.4 The hydrogen molecule 258

8.5 Configuration interaction 259

8.6 Diatomic molecules 261

8.7 Heteronuclear diatomic molecules 265

Molecular orbital theory of polyatomic

molecules 266

8.8 Symmetry-adapted linear combinations 266

8.9 Conjugated p-systems 269

8.10 Ligand field theory 274

8.11 Further aspects of ligand field theory 276

The band theory of solids 278

8.12 The tight-binding approximation 279

8.13 The Kronig–Penney model 281

8.14 Brillouin zones 284

P R O B L E M S 285

CONTENTS j ix

9 The calculation of electronic structure 287

The Hartree–Fock self-consistent field method 288

9.1 The formulation of the approach 288

9.2 The Hartree–Fock approach 289

9.3 Restricted and unrestricted Hartree–Fock

calculations 291

9.4 The Roothaan equations 293

9.5 The selection of basis sets 296

9.6 Calculational accuracy and the basis set 301

Electron correlation 302

9.7 Configuration state functions 303

9.8 Configuration interaction 303

9.9 CI calculations 305

9.10 Multiconfiguration and multireference methods 308

9.11 Møller–Plesset many-body perturbation theory 310

9.12 The coupled-cluster method 313

Density functional theory 316

9.13 Kohn–Sham orbitals and equations 317

9.14 Exchange–correlation functionals 319

Gradient methods and molecular properties 321

9.15 Energy derivatives and the Hessian matrix 321

9.16 Analytical derivatives and the coupled

perturbed equations 322

Semiempirical methods 325

9.17 Conjugated p-electron systems 326

9.18 Neglect of differential overlap 329

Molecular mechanics 332

9.19 Force fields 333

9.20 Quantum mechanics–molecular mechanics 334

Software packages for

electronic structure calculations 336

P R O B L E M S 339

10 Molecular rotations and vibrations 342

Spectroscopic transitions 342

10.1 Absorption and emission 342

10.2 Raman processes 344

Molecular rotation 344

10.3 Rotational energy levels 345

10.4 Centrifugal distortion 349

10.5 Pure rotational selection rules 349

10.6 Rotational Raman selection rules 351

10.7 Nuclear statistics 353

The vibrations of diatomic molecules 357

10.8 The vibrational energy levels of diatomic

molecules 357

10.9 Anharmonic oscillation 359

10.10 Vibrational selection rules 360

10.11 Vibration–rotation spectra of diatomic molecules 362

10.12 Vibrational Raman transitions of diatomic

molecules 364

The vibrations of polyatomic molecules 365

10.13 Normal modes 365

10.14 Vibrational selection rules for polyatomic

molecules 368

10.15 Group theory and molecular vibrations 369

10.16 The effects of anharmonicity 373

10.17 Coriolis forces 376

10.18 Inversion doubling 377

Appendix 10.1 Centrifugal distortion 379

P R O B L E M S 380

11 Molecular electronic transitions 382

The states of diatomic molecules 382

11.1 The Hund coupling cases 382

11.2 Decoupling and L-doubling 384

11.3 Selection rules 386

Vibronic transitions 386

11.4 The Franck–Condon principle 386

11.5 The rotational structure of vibronic transitions 389

The electronic spectra of polyatomic molecules 390

11.6 Symmetry considerations 391

11.7 Chromophores 391

11.8 Vibronically allowed transitions 393

11.9 Singlet–triplet transitions 395

The fate of excited species 396

11.10 Non-radiative decay 396

x j CONTENTS

11.11 Radiative decay 397

11.12 The conservation of orbital symmetry 399

11.13 Electrocyclic reactions 399

11.14 Cycloaddition reactions 401

11.15 Photochemically induced electrocyclic reactions 403

11.16 Photochemically induced cycloaddition reactions 404

P R O B L E M S 406

12 The electric properties of molecules 407

The response to electric fields 407

12.1 Molecular response parameters 407

12.2 The static electric polarizability 409

12.3 Polarizability and molecular properties 411

12.4 Polarizabilities and molecular spectroscopy 413

12.5 Polarizabilities and dispersion forces 414

12.6 Retardation effects 418

Bulk electrical properties 418

12.7 The relative permittivity and the electric

susceptibility 418

12.8 Polar molecules 420

12.9 Refractive index 422

Optical activity 427

12.10 Circular birefringence and optical rotation 427

12.11 Magnetically induced polarization 429

12.12 Rotational strength 431

P R O B L E M S 434

13 The magnetic properties of molecules 436

The descriptions of magnetic fields 436

13.1 The magnetic susceptibility 436

13.2 Paramagnetism 437

13.3 Vector functions 439

13.4 Derivatives of vector functions 440

13.5 The vector potential 441

Magnetic perturbations 442

13.6 The perturbation hamiltonian 442

13.7 The magnetic susceptibility 444

13.8 The current density 447

13.9 The diamagnetic current density 450

13.10 The paramagnetic current density 451

Magnetic resonance parameters 452

13.11 Shielding constants 452

13.12 The diamagnetic contribution to shielding 456

13.13 The paramagnetic contribution to shielding 458

13.14 The g-value 459

13.15 Spin–spin coupling 462

13.16 Hyperfine interactions 463

13.17 Nuclear spin–spin coupling 467

P R O B L E M S 471

14 Scattering theory 473

The formulation of scattering events 473

14.1 The scattering cross-section 473

14.2 Stationary scattering states 475

Partial-wave stationary scattering states 479

14.3 Partial waves 479

14.4 The partial-wave equation 480

14.5 Free-particle radial wavefunctions and the

scattering phase shift 481

14.6 The JWKB approximation and phase shifts 484

14.7 Phase shifts and the scattering matrix element 486

14.8 Phase shifts and scattering cross-sections 488

14.9 Scattering by a spherical square well 490

14.10 Background and resonance phase shifts 492

14.11 The Breit–Wigner formula 494

14.12 Resonance contributions to the scattering

matrix element 495

Multichannel scattering 497

14.13 Channels for scattering 497

14.14 Multichannel stationary scattering states 498

14.15 Inelastic collisions 498

14.16 The S matrix and multichannel resonances 501

The Green’s function 502

14.17 The integral scattering equation and Green’s

functions 502

14.18 The Born approximation 504

Appendix 14.1 The derivation of the Breit–Wigner

formula 508

Appendix 14.2 The rate constant for reactive

scattering 509

P R O B L E M S 510

CONTENTS j xi

Further information 513

Classical mechanics 513

1 Action 513

2 The canonical momentum 515

3 The virial theorem 516

4 Reduced mass 518

Solutions of the Schrödinger equation 519

5 The motion of wavepackets 519

6 The harmonic oscillator: solution by

factorization 521

7 The harmonic oscillator: the standard solution 523

8 The radial wave equation 525

9 The angular wavefunction 526

10 Molecular integrals 527

11 The Hartree–Fock equations 528

12 Green’s functions 532

13 The unitarity of the S matrix 533

Group theory and angular momentum 534

14 The orthogonality of basis functions 534

15 Vector coupling coefficients 535

Spectroscopic properties 537

16 Electric dipole transitions 537

17 Oscillator strength 538

18 Sum rules 540

19 Normal modes: an example 541

The electromagnetic field 543

20 The Maxwell equations 543

21 The dipolar vector potential 546

Mathematical relations 547

22 Vector properties 547

23 Matrices 549

Further reading 553

Appendix 1 557

Appendix 2 562

Answers to selected problems 563

Index 565

xii j CONTENTS

PREFACE

Many changes have occurred over the editions of this text but we have

retained its essence throughout. Quantum mechanics is filled with abstract

material that is both conceptually demanding and mathematically challen-

ging: we try, wherever possible, to provide interpretations and visualizations

alongside mathematical presentations.

One major change since the third edition has been our response to concerns

about the mathematical complexity of the material. We have not sacrificed

the mathematical rigour of the previous edition but we have tried in

numerous ways to make the mathematics more accessible. We have intro-

duced short commentaries into the text to remind the reader of the mathe-

matical fundamentals useful in derivations. We have included more worked

examples to provide the reader with further opportunities to see formulae in

action. We have added new problems for each chapter. We have expanded the

discussion on numerous occasions within the body of the text to provide

further clarification for or insight into mathematical results. We have set aside

Proofs and Illustrations (brief examples) from the main body of the text so

that readers may find key results more readily. Where the depth of pre-

sentation started to seem too great in our judgement, we have sent material to

the back of the chapter in the form of an Appendix or to the back of the book

as a Further information section. Numerous equations are tabbed with wwwto signify that on the Website to accompany the text [www.oup.com/uk/

booksites/chemistry/] there are opportunities to explore the equations by

substituting numerical values for variables.

We have added new material to a number of chapters, most notably the

chapter on electronic structure techniques (Chapter 9) and the chapter on

scattering theory (Chapter 14). These two chapters present material that is at

the forefront of modern molecular quantum mechanics; significant advances

have occurred in these two fields in the past decade and we have tried to

capture their essence. Both chapters present topics where comprehension

could be readily washed away by a deluge of algebra; therefore, we con-

centrate on the highlights and provide interpretations and visualizations

wherever possible.

There are many organizational changes in the text, including the layout of

chapters and the choice of words. As was the case for the third edition, the

present edition is a rewrite of its predecessor. In the rewriting, we have aimed

for clarity and precision.

We have a deep sense of appreciation for many people who assisted us in

this endeavour. We also wish to thank the numerous reviewers of the text-

book at various stages of its development. In particular, we would like to

thank

Charles Trapp, University of Louisville, USA

Ronald Duchovic, Indiana Purdue Fort Wayne, USA

Karl Jalkanen, Technical University of Denmark, Denmark

Mark Child, University of Oxford, UK

Ian Mills, University of Reading, UK

David Clary, University of Oxford, UK

Stephan Sauer, University of Copenhagen, Denmark

Temer Ahmadi, Villanova University, USA

Lutz Hecht, University of Glasgow, UK

Scott Kirby, University of Missouri-Rolla, USA

All these colleagues have made valuable suggestions about the content and

organization of the book as well as pointing out errors best spotted in private.

Many individuals (too numerous to name here) have offered advice over the

years and we value and appreciate all their insights and advice. As always, our

publishers have been very helpful and understanding.

PWA, Oxford

RSF, Indiana University Purdue University Fort Wayne

June 2004

xiv j PREFACE

There are two approaches to quantum mechanics. One is to follow the

historical development of the theory from the first indications that the

whole fabric of classical mechanics and electrodynamics should be held

in doubt to the resolution of the problem in the work of Planck, Einstein,

Heisenberg, Schrödinger, and Dirac. The other is to stand back at a point

late in the development of the theory and to see its underlying theore-

tical structure. The first is interesting and compelling because the theory

is seen gradually emerging from confusion and dilemma. We see experi-

ment and intuition jointly determining the form of the theory and, above

all, we come to appreciate the need for a new theory of matter. The second,

more formal approach is exciting and compelling in a different sense: there is

logic and elegance in a scheme that starts from only a few postulates, yet

reveals as their implications are unfolded, a rich, experimentally verifiable

structure.

This book takes that latter route through the subject. However, to set the

scene we shall take a few moments to review the steps that led to the revo-

lutions of the early twentieth century, when some of the most fundamental

concepts of the nature of matter and its behaviour were overthrown and

replaced by a puzzling but powerful new description.

0.1 Black-body radiation

In retrospect—and as will become clear—we can now see that theoretical

physics hovered on the edge of formulating a quantum mechanical descrip-

tion of matter as it was developed during the nineteenth century. However, it

was a series of experimental observations that motivated the revolution. Of

these observations, the most important historically was the study of black-

body radiation, the radiation in thermal equilibrium with a body that absorbs

and emits without favouring particular frequencies. A pinhole in an otherwise

sealed container is a good approximation (Fig. 0.1).

Two characteristics of the radiation had been identified by the end of the

century and summarized in two laws. According to the Stefan–Boltzmann

law, the excitance, M, the power emitted divided by the area of the emitting

region, is proportional to the fourth power of the temperature:

M ¼ sT4 ð0:1Þ

Introduction and orientation

0.1 Black-body radiation

0.2 Heat capacities

0.3 The photoelectric and

Compton effects

0.4 Atomic spectra

0.5 The duality of matter

The Stefan–Boltzmann constant, s, is independent of the material from whichthe body is composed, and its modern value is 56.7 nW m�2 K�4. So, a region

of area 1 cm2 of a black body at 1000 K radiates about 6 W if all frequencies

are taken into account. Not all frequencies (or wavelengths, with l¼ c/n),though, are equally represented in the radiation, and the observed peak moves

to shorter wavelengths as the temperature is raised. According to Wien’s

displacement law,

lmaxT ¼ constant ð0:2Þ

with the constant equal to 2.9 mm K.

One of the most challenging problems in physics at the end of the nine-

teenth century was to explain these two laws. Lord Rayleigh, with minor help

from James Jeans,1 brought his formidable experience of classical physics to

bear on the problem, and formulated the theoretical Rayleigh–Jeans law for

the energy density e(l), the energy divided by the volume, in the wavelengthrange l to lþ dl:

deðlÞ ¼ rðlÞ dl rðlÞ ¼ 8pkTl4

ð0:3Þ

where k is Boltzmann’s constant (k¼ 1.381 � 10�23 J K�1). This formulasummarizes the failure of classical physics. It suggests that regardless of

the temperature, there should be an infinite energy density at very short

wavelengths. This absurd result was termed by Ehrenfest the ultraviolet

catastrophe.

At this point, Planck made his historic contribution. His suggestion was

equivalent to proposing that an oscillation of the electromagnetic field of

frequency n could be excited only in steps of energy of magnitude hn, whereh is a new fundamental constant of nature now known as Planck’s constant.

According to this quantization of energy, the supposition that energy can be

transferred only in discrete amounts, the oscillator can have the energies 0,

hn, 2hn, . . . , and no other energy. Classical physics allowed a continuousvariation in energy, so even a very high frequency oscillator could be excited

with a very small energy: that was the root of the ultraviolet catastrophe.

Quantum theory is characterized by discreteness in energies (and, as we shall

see, of certain other properties), and the need for a minimum excitation

energy effectively switches off oscillators of very high frequency, and hence

eliminates the ultraviolet catastrophe.

When Planck implemented his suggestion, he derived what is now called

the Planck distribution for the energy density of a black-body radiator:

rðlÞ ¼ 8phcl5

e�hc=lkT

1� e�hc=lkT ð0:4Þ

This expression, which is plotted in Fig. 0.2, avoids the ultraviolet cata-

strophe, and fits the observed energy distribution extraordinarily well if we

take h¼6.626� 10�34 J s. Just as the Rayleigh–Jeans law epitomizes thefailure of classical physics, the Planck distribution epitomizes the inception of

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. ‘It seems to me,’ said Jeans, ‘that Lord Rayleigh has introduced an unnecessary factor 8 by

counting negative as well as positive values of his integers.’ (Phil. Mag., 91, 10 (1905).)

25

20

15

10

5

00 0.5 1.0 1.5 2.0

�kT hc/

�/(8

π(kT

)5 /

(hc)

4 )

Fig. 0.2 The Planck distribution.

Pinhole

Detectedradiation

Containerat a temperature T

Fig. 0.1 A black-body emitter can be

simulated by a heated container witha pinhole in the wall. The

electromagnetic radiation is reflected

many times inside the container andreaches thermal equilibrium with the

walls.

2 j INTRODUCTION AND ORIENTATION

quantum theory. It began the new century as well as a new era, for it was

published in 1900.

0.2 Heat capacities

In 1819, science had a deceptive simplicity. Dulong and Petit, for example,

were able to propose their law that ‘the atoms of all simple bodies have

exactly the same heat capacity’ of about 25 J K�1 mol�1 (in modern units).

Dulong and Petit’s rather primitive observations, though, were done at room

temperature, and it was unfortunate for them and for classical physics when

measurements were extended to lower temperatures and to a wider range of

materials. It was found that all elements had heat capacities lower than

predicted by Dulong and Petit’s law and that the values tended towards zero

as T ! 0.Dulong and Petit’s law was easy to explain in terms of classical physics by

assuming that each atom acts as a classical oscillator in three dimensions. The

calculation predicted that the molar isochoric (constant volume) heat capa-

city, CV,m, of a monatomic solid should be equal to 3R¼24.94 J K�1 mol�1,where R is the gas constant (R¼NAk, with NA Avogadro’s constant). Thatthe heat capacities were smaller than predicted was a serious embarrassment.

Einstein recognized the similarity between this problem and black-body

radiation, for if each atomic oscillator required a certain minimum energy

before it would actively oscillate and hence contribute to the heat capacity,

then at low temperatures some would be inactive and the heat capacity would

be smaller than expected. He applied Planck’s suggestion for electromagnetic

oscillators to the material, atomic oscillators of the solid, and deduced the

following expression:

CV;mðTÞ ¼ 3RfEðTÞ fEðTÞ ¼yET e

yE=2T

1� eyE=T

� �2ð0:5aÞ

where the Einstein temperature, yE, is related to the frequency of atomicoscillators by yE¼hn/k. The function CV,m(T)/R is plotted in Fig. 0.3, andclosely reproduces the experimental curve. In fact, the fit is not particularly

good at very low temperatures, but that can be traced to Einstein’s

assumption that all the atoms oscillated with the same frequency. When this

restriction was removed by Debye, he obtained

CV;mðTÞ ¼ 3RfDðTÞ fDðTÞ ¼ 3T

yD

� �3Z yD=T0

x4ex

ðex � 1Þ2dx ð0:5bÞ

where the Debye temperature, yD, is related to the maximum frequency of theoscillations that can be supported by the solid. This expression gives a very

good fit with observation.

The importance of Einstein’s contribution is that it complemented

Planck’s. Planck had shown that the energy of radiation is quantized;

3

2

1

0

Einstein

Debye

0 0.5 1 1.5 2T /�

CV

,m/R

Fig. 0.3 The Einstein and Debye

molar heat capacities. The

symbol y denotes the Einsteinand Debye temperatures,respectively. Close to T¼0 theDebye heat capacity is

proportional to T3.

0.2 HEAT CAPACITIES j 3

Einstein showed that matter is quantized too. Quantization appears to be

universal. Neither was able to justify the form that quantization took (with

oscillators excitable in steps of hn), but that is a problem we shall solve laterin the text.

0.3 The photoelectric and Compton effects

In those enormously productive months of 1905–6, when Einstein formu-

lated not only his theory of heat capacities but also the special theory

of relativity, he found time to make another fundamental contribution

to modern physics. His achievement was to relate Planck’s quantum

hypothesis to the phenomenon of the photoelectric effect, the emission of

electrons from metals when they are exposed to ultraviolet radiation. The

puzzling features of the effect were that the emission was instantaneous when

the radiation was applied however low its intensity, but there was no emis-

sion, whatever the intensity of the radiation, unless its frequency exceeded a

threshold value typical of each element. It was also known that the kinetic

energy of the ejected electrons varied linearly with the frequency of the

incident radiation.

Einstein pointed out that all the observations fell into place if the elec-

tromagnetic field was quantized, and that it consisted of bundles of energy

of magnitude hn. These bundles were later named photons by G.N. Lewis,and we shall use that term from now on. Einstein viewed the photoelectric

effect as the outcome of a collision between an incoming projectile, a

photon of energy hn, and an electron buried in the metal. This pictureaccounts for the instantaneous character of the effect, because even one

photon can participate in one collision. It also accounted for the frequency

threshold, because a minimum energy (which is normally denoted F andcalled the ‘work function’ for the metal, the analogue of the ionization

energy of an atom) must be supplied in a collision before photoejection can

occur; hence, only radiation for which hn>F can be successful. The lineardependence of the kinetic energy, EK, of the photoelectron on the frequency

of the radiation is a simple consequence of the conservation of energy,

which implies that

EK ¼ hn� F ð0:6Þ

If photons do have a particle-like character, then they should possess a

linear momentum, p. The relativistic expression relating a particle’s energy to

its mass and momentum is

E2 ¼ m2c4 þ p2c2 ð0:7Þ

where c is the speed of light. In the case of a photon, E¼ hn and m¼ 0, so

p ¼ hnc¼ h

lð0:8Þ

4 j INTRODUCTION AND ORIENTATION

This linear momentum should be detectable if radiation falls on an electron,

for a partial transfer of momentum during the collision should appear as a

change in wavelength of the photons. In 1923, A.H. Compton performed the

experiment with X-rays scattered from the electrons in a graphite target, and

found the results fitted the following formula for the shift in wavelength,

dl¼ lf� li, when the radiation was scattered through an angle y:

dl ¼ 2lC sin2 12 y ð0:9Þ

where lC¼ h/mec is called the Compton wavelength of the electron(lC¼2.426 pm). This formula is derived on the supposition that a photondoes indeed have a linear momentum h/l and that the scattering event is like acollision between two particles. There seems little doubt, therefore, that

electromagnetic radiation has properties that classically would have been

characteristic of particles.

The photon hypothesis seems to be a denial of the extensive accumulation

of data that apparently provided unequivocal support for the view that

electromagnetic radiation is wave-like. By following the implications of

experiments and quantum concepts, we have accounted quantitatively for

observations for which classical physics could not supply even a qualitative

explanation.

0.4 Atomic spectra

There was yet another body of data that classical physics could not elucidate

before the introduction of quantum theory. This puzzle was the observation

that the radiation emitted by atoms was not continuous but consisted of

discrete frequencies, or spectral lines. The spectrum of atomic hydrogen had a

very simple appearance, and by 1885 J. Balmer had already noticed that their

wavenumbers, ~nn, where ~nn¼ n/c, fitted the expression

~nn ¼ RH1

22� 1

n2

� �ð0:10Þ

where RH has come to be known as the Rydberg constant for hydrogen

(RH¼ 1.097�105 cm�1) and n¼3, 4, . . . . Rydberg’s name is commemoratedbecause he generalized this expression to accommodate all the transitions in

atomic hydrogen. Even more generally, the Ritz combination principle states

that the frequency of any spectral line could be expressed as the difference

between two quantities, or terms:

~nn ¼ T1 � T2 ð0:11Þ

This expression strongly suggests that the energy levels of atoms are confined

to discrete values, because a transition from one term of energy hcT1 to

another of energy hcT2 can be expected to release a photon of energy hc~nn, orhn, equal to the difference in energy between the two terms: this argument

0.4 ATOMIC SPECTRA j 5

leads directly to the expression for the wavenumber of the spectroscopic

transitions.

But why should the energy of an atom be confined to discrete values? In

classical physics, all energies are permissible. The first attempt to weld

together Planck’s quantization hypothesis and a mechanical model of an atom

was made by Niels Bohr in 1913. By arbitrarily assuming that the angular

momentum of an electron around a central nucleus (the picture of an atom

that had emerged from Rutherford’s experiments in 1910) was confined to

certain values, he was able to deduce the following expression for the per-

mitted energy levels of an electron in a hydrogen atom:

En ¼ �me4

8h2e20 1

n2n ¼ 1, 2, . . . ð0:12Þ

where 1/m¼1/meþ 1/mp and e0 is the vacuum permittivity, a fundamentalconstant. This formula marked the first appearance in quantum mechanics of

a quantum number, n, which identifies the state of the system and is used to

calculate its energy. Equation 0.12 is consistent with Balmer’s formula and

accounted with high precision for all the transitions of hydrogen that were

then known.

Bohr’s achievement was the union of theories of radiation and models of

mechanics. However, it was an arbitrary union, and we now know that it is

conceptually untenable (for instance, it is based on the view that an electron

travels in a circular path around the nucleus). Nevertheless, the fact that he

was able to account quantitatively for the appearance of the spectrum of

hydrogen indicated that quantum mechanics was central to any description of

atomic phenomena and properties.

0.5 The duality of matter

The grand synthesis of these ideas and the demonstration of the deep links

that exist between electromagnetic radiation and matter began with Louis de

Broglie, who proposed on the basis of relativistic considerations that with any

moving body there is ‘associated a wave’, and that the momentum of the body

and the wavelength are related by the de Broglie relation:

l ¼ hp

ð0:13Þ

We have seen this formula already (eqn 0.8), in connection with the prop-

erties of photons. De Broglie proposed that it is universally applicable.

The significance of the de Broglie relation is that it summarizes a fusion

of opposites: the momentum is a property of particles; the wavelength is

a property of waves. This duality, the possession of properties that in classical

physics are characteristic of both particles and waves, is a persistent theme

in the interpretation of quantum mechanics. It is probably best to regard

the terms ‘wave’ and ‘particle’ as remnants of a language based on a false

6 j INTRODUCTION AND ORIENTATION

(classical) model of the universe, and the term ‘duality’ as a late attempt to

bring the language into line with a current (quantum mechanical) model.

The experimental results that confirmed de Broglie’s conjecture are the

observation of the diffraction of electrons by the ranks of atoms in a metal

crystal acting as a diffraction grating. Davisson and Germer, who performed

this experiment in 1925 using a crystal of nickel, found that the diffraction

pattern was consistent with the electrons having a wavelength given by

the de Broglie relation. Shortly afterwards, G.P. Thomson also succeeded

in demonstrating the diffraction of electrons by thin films of celluloid

and gold.2

If electrons—if all particles—have wave-like character, then we should

expect there to be observational consequences. In particular, just as a wave of

definite wavelength cannot be localized at a point, we should not expect

an electron in a state of definite linear momentum (and hence wavelength) to

be localized at a single point. It was pursuit of this idea that led Werner

Heisenberg to his celebrated uncertainty principle, that it is impossible to

specify the location and linear momentum of a particle simultaneously with

arbitrary precision. In other words, information about location is at the

expense of information about momentum, and vice versa. This com-

plementarity of certain pairs of observables, the mutual exclusion of the

specification of one property by the specification of another, is also a major

theme of quantum mechanics, and almost an icon of the difference between it

and classical mechanics, in which the specification of exact trajectories was a

central theme.

The consummation of all this faltering progress came in 1926 when Werner

Heisenberg and Erwin Schrödinger formulated their seemingly different but

equally successful versions of quantum mechanics. These days, we step

between the two formalisms as the fancy takes us, for they are mathematically

equivalent, and each one has particular advantages in different types of cal-

culation. Although Heisenberg’s formulation preceded Schrödinger’s by a few

months, it seemed more abstract and was expressed in the then unfamiliar

vocabulary of matrices. Still today it is more suited for the more formal

manipulations and deductions of the theory, and in the following pages we

shall employ it in that manner. Schrödinger’s formulation, which was in terms

of functions and differential equations, was more familiar in style but still

equally revolutionary in implication. It is more suited to elementary mani-

pulations and to the calculation of numerical results, and we shall employ it in

that manner.

‘Experiments’, said Planck, ‘are the only means of knowledge at our

disposal. The rest is poetry, imagination.’ It is time for that imagination

to unfold.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. It has been pointed out by M. Jammer that J.J. Thomson was awarded the Nobel Prize for

showing that the electron is a particle, and G.P. Thomson, his son, was awarded the Prize for

showing that the electron is a wave. (See The conceptual development of quantum mechanics,

McGraw-Hill, New York (1966), p. 254.)

0.5 THE DUALITY OF MATTER j 7

P R O B L E M S

0.1 Calculate the size of the quanta involved in theexcitation of (a) an electronic motion of period 1.0 fs,(b) a molecular vibration of period 10 fs, and (c) a pendulumof period 1.0 s.

0.2 Find the wavelength corresponding to the maximum inthe Planck distribution for a given temperature, and showthat the expression reduces to the Wien displacement law atshort wavelengths. Determine an expression for the constantin the law in terms of fundamental constants. (This constantis called the second radiation constant, c2.)

0.3 Use the Planck distribution to confirm theStefan–Boltzmann law and to derive an expression forthe Stefan–Boltzmann constant s.

0.4 The peak in the Sun’s emitted energy occurs at about480 nm. Estimate the temperature of its surface on the basisof it being regarded as a black-body emitter.

0.5 Derive the Einstein formula for the heat capacity of acollection of harmonic oscillators. To do so, use thequantum mechanical result that the energy of a harmonicoscillator of force constant k and mass m is one of the values(vþ 12)hv, with v¼ (1/2p)(k/m)

1/2 and v¼ 0, 1, 2, . . . . Hint.Calculate the mean energy, E, of a collection of oscillatorsby substituting these energies into the Boltzmanndistribution, and then evaluate C¼ dE/dT.

0.6 Find the (a) low temperature, (b) high temperatureforms of the Einstein heat capacity function.

0.7 Show that the Debye expression for the heat capacity isproportional to T3 as T! 0.

0.8 Estimate the molar heat capacities of metallic sodium(yD¼ 150 K) and diamond (yD¼ 1860 K) at roomtemperature (300 K).

0.9 Calculate the molar entropy of an Einstein solid atT¼ yE. Hint. The entropy is S ¼

R T0 ðCV=TÞdT. Evaluate the

integral numerically.

0.10 How many photons would be emitted per second by asodium lamp rated at 100 W which radiated all its energywith 100 per cent efficiency as yellow light of wavelength589 nm?

0.11 Calculate the speed of an electron emitted from a cleanpotassium surface (F¼ 2.3 eV) by light of wavelength (a)300 nm, (b) 600 nm.

0.12 When light of wavelength 195 nm strikes a certain metalsurface, electrons are ejected with a speed of 1.23� 106 m s�1.Calculate the speed of electrons ejected from the same metalsurface by light of wavelength 255 nm.

0.13 At what wavelength of incident radiation do therelativistic and non-relativistic expressions for the ejectionof electrons from potassium differ by 10 per cent? That is,find l such that the non-relativistic and relativistic linearmomenta of the photoelectron differ by 10 per cent. UseF¼ 2.3 eV.

0.14 Deduce eqn 0.9 for the Compton effect on the basis ofthe conservation of energy and linear momentum. Hint. Usethe relativistic expressions. Initially the electron is at restwith energy mec

2. When it is travelling with momentum p itsenergy is ðp2c2 þm2e c4Þ1/2. The photon, with initialmomentum h/li and energy hni, strikes the stationaryelectron, is deflected through an angle y, and emerges withmomentum h/lf and energy hnf. The electron is initiallystationary (p¼ 0) but moves off with an angle y 0 to theincident photon. Conserve energy and both components oflinear momentum. Eliminate y 0, then p, and so arrive at anexpression for dl.

0.15 The first few lines of the visible (Balmer) series in thespectrum of atomic hydrogen lie at l/nm¼ 656.46, 486.27,434.17, 410.29, . . . . Find a value of RH, the Rydbergconstant for hydrogen. The ionization energy, I, is theminimum energy required to remove the electron. Find itfrom the data and express its value in electron volts. How isI related to RH? Hint. The ionization limit corresponds ton!1 for the final state of the electron.

0.16 Calculate the de Broglie wavelength of (a) a mass of1.0 g travelling at 1.0 cm s�1, (b) the same at 95 per cent ofthe speed of light, (c) a hydrogen atom at room temperature(300 K); estimate the mean speed from the equipartitionprinciple, which implies that the mean kinetic energy of anatom is equal to 32kT, where k is Boltzmann’s constant, (d)an electron accelerated from rest through a potentialdifference of (i) 1.0 V, (ii) 10 kV. Hint. For the momentumin (b) use p¼mv/(l� v2/c2)1/2 and for the speed in (d) use12mev

2¼ eV, where V is the potential difference.

0.17 Derive eqn 0.12 for the permitted energy levels for theelectron in a hydrogen atom. To do so, use the following(incorrect) postulates of Bohr: (a) the electron moves in acircular orbit of radius r around the nucleus and (b) theangular momentum of the electron is an integral multiple of�h, that is mevr ¼ n�h. Hint. Mechanical stability of theorbital motion requires that the Coulombic force ofattraction between the electron and nucleus equals thecentrifugal force due to the circular motion. The energy ofthe electron is the sum of the kinetic energy and potential(Coulombic) energy. For simplicity, use me rather than thereduced mass m.

8 j INTRODUCTION AND ORIENTATION

The whole of quantum mechanics can be expressed in terms of a small set

of postulates. When their consequences are developed, they embrace the

behaviour of all known forms of matter, including the molecules, atoms, and

electrons that will be at the centre of our attention in this book. This chapter

introduces the postulates and illustrates how they are used. The remaining

chapters build on them, and show how to apply them to problems of chemical

interest, such as atomic and molecular structure and the properties of mole-

cules. We assume that you have already met the concepts of ‘hamiltonian’ and

‘wavefunction’ in an elementary introduction, and have seen the Schrödinger

equation written in the form

Hc ¼ EcThis chapter establishes the full significance of this equation, and provides

a foundation for its application in the following chapters.

Operators in quantum mechanics

An observable is any dynamical variable that can be measured. The principal

mathematical difference between classical mechanics and quantum mechan-

ics is that whereas in the former physical observables are represented by

functions (such as position as a function of time), in quantum mechanics they

are represented by mathematical operators. An operator is a symbol for an

instruction to carry out some action, an operation, on a function. In most of

the examples we shall meet, the action will be nothing more complicated than

multiplication or differentiation. Thus, one typical operation might be

multiplication by x, which is represented by the operator x� . Anotheroperation might be differentiation with respect to x, represented by the

operator d/dx. We shall represent operators by the symbol O (omega) ingeneral, but use A, B, . . . when we want to refer to a series of operators.

We shall not in general distinguish between the observable and the operator

that represents that observable; so the position of a particle along the x-axis

will be denoted x and the corresponding operator will also be denoted x (with

multiplication implied). We shall always make it clear whether we are

referring to the observable or the operator.

We shall need a number of concepts related to operators and functions

on which they operate, and this first section introduces some of the more

important features.

The foundations of quantummechanics

Operators in quantum mechanics

1.1 Linear operators

1.2 Eigenfunctions and eigenvalues

1.3 Representations

1.4 Commutation and

non-commutation

1.5 The construction of operators

1.6 Integrals over operators

1.7 Dirac bracket notation

1.8 Hermitian operators

The postulates of quantum

mechanics

1.9 States and wavefunctions

1.10 The fundamental prescription

1.11 The outcome of measurements

1.12 The interpretation of the

wavefunction

1.13 The equation for the

wavefunction

1.14 The separation of the Schrödinger

equation

The specification and evolution of

states

1.15 Simultaneous observables

1.16 The uncertainty principle

1.17 Consequences of the uncertainty

principle

1.18 The uncertainty in energy and

time

1.19 Time-evolution and conservation

laws

Matrices in quantum mechanics

1.20 Matrix elements

1.21 The diagonalization of the

hamiltonian

The plausibility of the Schrödinger

equation

1.22 The propagation of light

1.23 The propagation of particles

1.24 The transition to quantum

mechanics

1

1.1 Linear operators

The operators we shall meet in quantum mechanics are all linear. A linear

operator is one for which

Oðaf þ bgÞ ¼ aOf þ bOg ð1:1Þwhere a and b are constants and f and g are functions. Multiplication is a

linear operation; so is differentiation and integration. An example of a non-

linear operation is that of taking the logarithm of a function, because it is not

true, for example, that log 2x¼ 2 log x for all x.

1.2 Eigenfunctions and eigenvalues

In general, when an operator operates on a function, the outcome is another

function. Differentiation of sin x, for instance, gives cos x. However, incertain cases, the outcome of an operation is the same function multiplied by

a constant. Functions of this kind are called ‘eigenfunctions’ of the operator.

More formally, a function f (which may be complex) is an eigenfunction of an

operator O if it satisfies an equation of the form

Of ¼ of ð1:2Þwhere o is a constant. Such an equation is called an eigenvalue equation. Thefunction eax is an eigenfunction of the operator d/dx because (d/dx)eax¼ aeax,which is a constant (a) multiplying the original function. In contrast, eax

2is

not an eigenfunction of d/dx, because (d/dx)eax2 ¼ 2axeax2 , which is a con-

stant (2a) times a different function of x (the function xeax2). The constant o

in an eigenvalue equation is called the eigenvalue of the operator O.

Example 1.1 Determining if a function is an eigenfunction

Is the function cos(3xþ 5) an eigenfunction of the operator d2/dx2 and, if so,what is the corresponding eigenvalue?

Method. Perform the indicated operation on the given function and see if

the function satisfies an eigenvalue equation. Use (d/dx)sin ax¼ a cos ax and(d/dx)cos ax¼�a sin ax.

Answer. The operator operating on the function yields

d2

dx2cosð3xþ 5Þ ¼ d

dx�3 sinð3xþ 5Þð Þ ¼ �9 cosð3xþ 5Þ

and we see that the original function reappears multiplied by the eigen-

value �9.

Self-test 1.1. Is the function e3xþ 5 an eigenfunction of the operator d2/dx2

and, if so, what is the corresponding eigenvalue?

[Yes; 9]

An important point is that a general function can be expanded in terms of

all the eigenfunctions of an operator, a so-called complete set of functions.

10 j 1 THE FOUNDATIONS OF QUANTUM MECHANICS

That is, if fn is an eigenfunction of an operator O with eigenvalue on (so Ofn¼on fn), then

1 a general function g can be expressed as the linear combination

g ¼X

n

cnfn ð1:3Þ

where the cn are coefficients and the sum is over a complete set of functions.

For instance, the straight line g¼ ax can be recreated over a certain range bysuperimposing an infinite number of sine functions, each of which is an

eigenfunction of the operator d2/dx2. Alternatively, the same function may be

constructed from an infinite number of exponential functions, which are

eigenfunctions of d/dx. The advantage of expressing a general function as a

linear combination of a set of eigenfunctions is that it allows us to deduce the

effect of an operator on a function that is not one of its own eigenfunctions.

Thus, the effect of O on g in eqn 1.3, using the property of linearity, is simply

Og ¼ OX

n

cnfn ¼X

n

cnOfn ¼X

n

cnonfn

A special case of these linear combinations is when we have a set of

degenerate eigenfunctions, a set of functions with the same eigenvalue. Thus,

suppose that f1, f2, . . . , fk are all eigenfunctions of the operator O, and thatthey all correspond to the same eigenvalue o:

Ofn ¼ ofn with n ¼ 1, 2, . . . , k ð1:4ÞThen it is quite easy to show that any linear combination of the functions fnis also an eigenfunction of O with the same eigenvalue o. The proof is asfollows. For an arbitrary linear combination g of the degenerate set of

functions, we can write

Og ¼ OXkn¼1

cnfn ¼Xkn¼1

cnOfn ¼Xkn¼1

cnofn ¼ oXkn¼1

cnfn ¼ og

This expression has the form of an eigenvalue equation (Og¼og).

Example 1.2 Demonstrating that a linear combination of degenerate

eigenfunctions is also an eigenfunction

Show that any linear combination of the complex functions e2ix and e�2ix is an

eigenfunction of the operator d2/dx2, where i¼ (�1)1/2.Method. Consider an arbitrary linear combination ae2ixþ be�2ix and see if thefunction satisfies an eigenvalue equation.

Answer. First we demonstrate that e2ix and e�2ix are degenerate eigenfunctions.

d2

dx2e�2ix ¼ d

dxð�2ie�2ixÞ ¼ �4e�2ix

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. See P.M. Morse and H. Feschbach, Methods of theoretical physics, McGraw-Hill, New York

(1953).

1.2 EIGENFUNCTIONS AND EIGENVALUES j 11

where we have used i2¼�1. Both functions correspond to the same eigen-value, �4. Then we operate on a linear combination of the functions.

d2

dx2ðae2ix þ be�2ixÞ ¼ �4ðae2ix þ be�2ixÞ

The linear combination satisfies the eigenvalue equation and has the same

eigenvalue (�4) as do the two complex functions.

Self-test 1.2. Show that any linear combination of the functions sin(3x) and

cos(3x) is an eigenfunction of the operator d2/dx2.

[Eigenvalue is �9]

A further technical point is that from n basis functions it is possible to con-struct n linearly independent combinations. A set of functions g1, g2, . . . , gn is

said to be linearly independent if we cannot find a set of constants c1, c2, . . . ,

cn (other than the trivial set c1¼ c2¼ ¼0) for whichXi

cigi ¼ 0

A set of functions that is not linearly independent is said to be linearly

dependent. From a set of n linearly independent functions, it is possible to

construct an infinite number of sets of linearly independent combinations,

but each set can have no more than n members. For example, from three

2p-orbitals of an atom it is possible to form any number of sets of linearlyindependent combinations, but each set has no more than three members.

1.3 Representations

The remaining work of this section is to put forward some explicit forms of

the operators we shall meet. Much of quantum mechanics can be developed in

terms of an abstract set of operators, as we shall see later. However, it is often

fruitful to adopt an explicit form for particular operators and to express them

in terms of the mathematical operations of multiplication, differentiation,

and so on. Different choices of the operators that correspond to a particular

observable give rise to the different representations of quantum mechanics,

because the explicit forms of the operators represent the abstract structure of

the theory in terms of actual manipulations.

One of the most common representations is the position representation,

in which the position operator is represented by multiplication by x (orwhatever coordinate is specified) and the linear momentum parallel to x is

represented by differentiation with respect to x. Explicitly:

Position representation: x! x� px !�h

i

qqx

ð1:5Þ

where �h ¼ h=2p. Why the linear momentum should be represented in pre-cisely this manner will be explained in the following section. For the time

being, it may be taken to be a basic postulate of quantum mechanics.

An alternative choice of operators is the momentum representation, in

which the linear momentum parallel to x is represented by the operation of

12 j 1 THE FOUNDATIONS OF QUANTUM MECHANICS

multiplication by px and the position operator is represented by differentia-tion with respect to px. Explicitly:

Momentum representation: x! � �hi

qqpx

px ! px� ð1:6Þ

There are other representations. We shall normally use the position repres-

entation when the adoption of a representation is appropriate, but we shall

also see that many of the calculations in quantum mechanics can be done

independently of a representation.

1.4 Commutation and non-commutation

An important feature of operators is that in general the outcome of successive

operations (A followed by B, which is denoted BA, or B followed by A,denoted AB) depends on the order in which the operations are carried out.

That is, in general BA 6¼AB. We say that, in general, operators do notcommute. For example, consider the operators x and px and a specific

function x2. In the position representation, (xpx)x2¼x(2�h/i)x¼ (2�h/i)x2,

whereas (pxx)x2¼ pxx3¼ (3�h/i)x2. The operators x and px do not commute.

The quantity AB�BA is called the commutator of A and B and is denoted[A, B]:

½A, B ¼ AB� BA ð1:7Þ

It is instructive to evaluate the commutator of the position and linear

momentum operators in the two representations shown above; the procedure

is illustrated in the following example.

Example 1.3 The evaluation of a commutator

Evaluate the commutator [x,px] in the position representation.

Method. To evaluate the commutator [A,B] we need to remember that the

operators operate on some function, which we shall write f. So, evaluate [A,B]f

for an arbitrary function f, and then cancel f at the end of the calculation.

Answer. Substitution of the explicit expressions for the operators into [x,px]

proceeds as follows:

½x, pxf ¼ ðxpx � pxxÞf ¼ x��h

i

qfqx� �h

i

qðxf Þqx

¼ x� �hi

qfqx� �h

if � x� �h

i

qfqx¼ i�hf

where we have used (1/i)¼�i. This derivation is true for any function f,so in terms of the operators themselves,

½x, px ¼ i�h

The right-hand side should be interpreted as the operator ‘multiply by the

constant i�h’.

Self-test 1.3. Evaluate the same commutator in the momentum representation.

[Same]

1.4 COMMUTATION AND NON-COMMUTATION j 13

1.5 The construction of operators

Operators for other observables of interest can be constructed from the ope-

rators for position and momentum. For example, the kinetic energy operator

T can be constructed by noting that kinetic energy is related to linear

momentum by T¼ p2/2m where m is the mass of the particle. It follows thatin one dimension and in the position representation

T ¼ p2x

2m¼ 1

2m

�h

i

d

dx

� �2¼ � �h

2m

d2

dx2ð1:8Þ

In three dimensions the operator in the position representation is

T ¼ � �h2

2m

q2

qx2þ q

2

qy2þ q

2

qz2

( )¼ � �h

2

2mr2 ð1:9Þ

The operator r2, which is read ‘del squared’ and called the laplacian, is thesum of the three second derivatives.

The operator for potential energy of a particle in one dimension, V(x), is

multiplication by the function V(x) in the position representation. The same is

true of the potential energy operator in three dimensions. For example, in the

position representation the operator for the Coulomb potential energy of an

electron (charge �e) in the field of a nucleus of atomic number Z is themultiplicative operator

V ¼ � Ze2

4pe0r� ð1:10Þ

where r is the distance from the nucleus to the electron. It is usual to omit the

multiplication sign from multiplicative operators, but it should not be for-

gotten that such expressions are multiplications.

The operator for the total energy of a system is called the hamiltonian

operator and is denoted H:

H ¼ T þ V ð1:11Þ

The name commemorates W.R. Hamilton’s contribution to the formulation

of classical mechanics in terms of what became known as a hamiltonian

function. To write the explicit form of this operator we simply substitute the

appropriate expressions for the kinetic and potential energy operators in the

chosen representation. For example, the hamiltonian for a particle of mass m

moving in one dimension is

H ¼ � �h2

2m

d2

dx2þ VðxÞ ð1:12Þ

where V(x) is the operator for the potential energy. Similarly, the hamiltonian

operator for an electron of mass me in a hydrogen atom is

H ¼ � �h2

2mer2 � e

2

4pe0rð1:13Þ

Although eqn 1.9 has explicitly

used Cartesian coordinates, the

relation between the kinetic energy

operator and the laplacian is true

in any coordinate system; for

example, spherical polar

coordinates.

14 j 1 THE FOUNDATIONS OF QUANTUM MECHANICS

The general prescription for constructing operators in the position repres-

entation should be clear from these examples. In short:

1. Write the classical expression for the observable in terms of position

coordinates and the linear momentum.

2. Replace x by multiplication by x, and replace px by (�h/i)q/qx (and likewisefor the other coordinates).

1.6 Integrals over operators

When we want to make contact between a calculation done using operators

and the actual outcome of an experiment, we need to evaluate certain

integrals. These integrals all have the form

I ¼Z

f �mOfn dt ð1:14Þ

where f �m is the complex conjugate of fm. In this integral dt is the volumeelement. In one dimension, dt can be identified as dx; in three dimensions it isdxdydz. The integral is taken over the entire space available to the system,

which is typically from x¼�1 to x¼ þ1 (and similarly for the othercoordinates). A glance at the later pages of this book will show that many

molecular properties are expressed as combinations of integrals of this form

(often in a notation which will be explained later). Certain special cases of this

type of integral have special names, and we shall introduce them here.

When the operator O in eqn 1.14 is simply multiplication by 1, the integralis called an overlap integral and commonly denoted S:

S ¼Z

f �mfn dt ð1:15Þ

It is helpful to regard S as a measure of the similarity of two functions: whenS¼0, the functions are classified as orthogonal, rather like two perpendicularvectors. When S is close to 1, the two functions are almost identical. The

recognition of mutually orthogonal functions often helps to reduce the

amount of calculation considerably, and rules will emerge in later sections

and chapters.

The normalization integral is the special case of eqn 1.15 for m¼n.A function fm is said to be normalized (strictly, normalized to 1) ifZ

f �mfm dt ¼ 1 ð1:16Þ

It is almost always easy to ensure that a function is normalized by multiplying

it by an appropriate numerical factor, which is called a normalization factor,

typically denoted N and taken to be real so that N� ¼N. The procedure isillustrated in the following example.

Example 1.4 How to normalize a function

A certain function f is sin(px/L) between x¼ 0 and x¼L and is zero elsewhere.Find the normalized form of the function.

The complex conjugate of

a complex number z¼ aþ ibis z� ¼ a� ib. Complexconjugation amounts to

everywhere replacing i by � i.The square modulus jzj2 is given byzz� ¼ a2þ b2 since jij2¼ 1.

1.6 INTEGRALS OVER OPERATORS j 15

Method. We need to find the (real) factor N such that N sin(px/L) is norm-alized to 1. To find N we substitute this expression into eqn 1.16, evaluate the

integral, and select N to ensure normalization. Note that ‘all space’ extends

from x¼ 0 to x¼L.

Answer. The necessary integration isZ

f �f dt ¼Z L

0

N2 sin2ðpx=LÞdx ¼ 12LN2

where we have usedR

sin2ax dx¼ (x/2)(sin 2ax)/4aþ constant. For thisintegral to be equal to 1, we require N¼ (2/L)1/2. The normalized function istherefore

f ¼ 2L

� �1=2sinðpx=LÞ

Comment. We shall see later that this function describes the distribution of a

particle in a square well, and we shall need its normalized form there.

Self-test 1.4. Normalize the function f¼ eif, where f ranges from 0 to 2p.[N¼ 1/(2p)1/2]

A set of functions fn that are (a) normalized and (b) mutually orthogonal

are said to satisfy the orthonormality condition:Zf �mfn dt ¼ dmn ð1:17Þ

In this expression, dmn denotes the Kronecker delta, which is 1 when m¼nand 0 otherwise.

1.7 Dirac bracket notation

With eqn 1.14 we are on the edge of getting lost in a complicated notation. The

appearance of many quantum mechanical expressions is greatly simplified by

adopting the Dirac bracket notation in which integrals are written as follows:

hmjOjni ¼Z

f �mOfn dt ð1:18Þ

The symbol jni is called a ket, and denotes the state described by the functionfn. Similarly, the symbol hnj is called a bra, and denotes the complex conjugateof the function, f �n . When a bra and ket are strung together with an operator

between them, as in the bracket hmjOjni, the integral in eqn 1.18 is to beunderstood. When the operator is simply multiplication by 1, the 1 is omitted

and we use the convention

hmjni ¼Z

f �mfn dt ð1:19Þ

This notation is very elegant. For example, the normalization integral

becomes hnjni¼1 and the orthogonality condition becomes hmjni¼0for m 6¼n. The combined orthonormality condition (eqn 1.17) is thenhmjni ¼ dmn ð1:20Þ

16 j 1 THE FOUNDATIONS OF QUANTUM MECHANICS

A final point is that, as can readily be deduced from the definition of a Dirac

bracket,

hmjni ¼ hnjmi�

1.8 Hermitian operators

An operator is hermitian if it satisfies the following relation:Zf �mOfn dt ¼

�Zf �nOfmdt

�ð1:21aÞ

for any two functions fm and fn. An alternative version of this definition isZf �mOfn dt ¼

ZðOfmÞ�fn dt ð1:21bÞ

This expression is obtained by taking the complex conjugate of each term on

the right-hand side of eqn 1.21a. In terms of the Dirac notation, the definition

of hermiticity is

hmjOjni ¼ hnjOjmi� ð1:22Þ

Example 1.5 How to confirm the hermiticity of operators

Show that the position and momentum operators in the position representa-

tion are hermitian.

Method. We need to show that the operators satisfy eqn 1.21a. In some cases

(the position operator, for instance), the hermiticity is obvious as soon as the

integral is written down. When a differential operator is used, it may be

necessary to use integration by parts at some stage in the argument to transfer

the differentiation from one function to another:Zu dv ¼ uv�

Zv du

Answer. That the position operator is hermitian is obvious from inspection:Zf �mxfn dt ¼

Zfnxf

�m dt ¼

Zf �n xfm dt

� �

We have used the facts that (f�)� ¼ f and x is real. The demonstration of thehermiticity of px, a differential operator in the position representation,

involves an integration by parts:

Zf �mpxfn dx ¼

Zf �m

�h

i

d

dxfn dx ¼

�h

i

Zf �m dfn

¼ �hi

�f �mfn �

Zfndf

�m

x¼1

x¼�1

¼ �hi

f �mfnjx¼1x¼�1 �Z 1�1

fnd

dxf �m dx

�

1.8 HERMITIAN OPERATORS j 17

The first term on the right is zero (because when jxj is infinite, a normalizablefunction must be vanishingly small; see Section 1.12). Therefore,

Zf �mpxfn dx ¼ �

�h

i

Zfn

d

dxf �m dx

¼Z

f �n�h

i

d

dxfm dx

� �¼

Zf �n pxfm dx

� �

Hence, the operator is hermitian.

Self-test 1.5. Show that the two operators are hermitian in the momentum

representation.

As we shall now see, the property of hermiticity has far-reaching impli-

cations. First, we shall establish the following property:

Property 1. The eigenvalues of hermitian operators are real.

Proof 1.1 The reality of eigenvalues

Consider the eigenvalue equation

Ojoi ¼ ojoi

The ket joi denotes an eigenstate of the operator O in the sense that thecorresponding function fo is an eigenfunction of the operator O and we arelabelling the eigenstates with the eigenvalue o of the operator O. It is oftenconvenient to use the eigenvalues as labels in this way. Multiplication from the

left by hoj results in the equation

hojOjoi ¼ ohojoi ¼ o

taking joi to be normalized. Now take the complex conjugate of both sides:

hojOjoi� ¼ o�

However, by hermiticity, hojOjoi� ¼ hojOjoi. Therefore, it follows thato¼o�, which implies that the eigenvalue o is real.

The second property we shall prove is as follows:

Property 2. Eigenfunctions corresponding to different eigenvalues of an

hermitian operator are orthogonal.

That is, if we have two eigenfunctions of an hermitian operator O witheigenvalues o and o 0, with o 6¼o 0, then hojo 0i ¼ 0. For example, it follows atonce that all the eigenfunctions of a harmonic oscillator (Section 2.16) are

mutually orthogonal, for as we shall see each one corresponds to a different

energy (the eigenvalue of the hamiltonian, an hermitian operator).

18 j 1 THE FOUNDATIONS OF QUANTUM MECHANICS

Proof 1.2 The orthogonality of eigenstates

Suppose we have two eigenstates joi and jo 0i that satisfy the followingrelations:

Ojoi ¼ ojoi and Ojo0i ¼ o0jo0iThen multiplication of the first relation by ho 0j and the second by hoj givesho0jOjoi ¼ oho0joi and hojOjo0i ¼ o0hojo0i

Now take the complex conjugate of the second relation and subtract it from

the first while using Property 1 (o 0� ¼o 0):ho0jOjoi � hojOjo0i� ¼ oho0joi � o0hojo0i�

Because O is hermitian, the left-hand side of this expression is zero; so (notingthat o 0 is real and using hojo 0i� ¼ ho 0joi as explained earlier) we arrive atðo� o0Þho0joi ¼ 0

However, because the two eigenvalues are different, the only way of satisfying

this relation is for ho 0joi¼ 0, as was to be proved.

The postulates of quantum mechanics

Now we turn to an application of the preceding material, and move into the

foundations of quantum mechanics. The postulates we use as a basis for

quantum mechanics are by no means the most subtle that have been devised,

but they are strong enough for what we have to do.

1.9 States and wavefunctions

The first postulate concerns the information we can know about a state:

Postulate 1. The state of a system is fully described by a function C(r1,r2, . . . , t).

In this statement, r1, r2, . . . are the spatial coordinates of particles 1, 2, . . .

that constitute the system and t is the time. The function C (uppercase psi)plays a central role in quantum mechanics, and is called the wavefunction of

the system (more specifically, the time-dependent wavefunction). When we

are not interested in how the system changes in time we shall denote the

wavefunction by a lowercase psi as c(r1, r2, . . . ) and refer to it as the time-independent wavefunction. The state of the system may also depend on some

internal variable of the particles (their spin states); we ignore that for now

and return to it later. By ‘describe’ we mean that the wavefunction

contains information about all the properties of the system that are open to

experimental determination.

We shall see that the wavefunction of a system will be specified by a set of

labels called quantum numbers, and may then be written ca,b, . . . , wherea, b, . . . are the quantum numbers. The values of these quantum numbers

specify the wavefunction and thus allow the values of various physical

1.9 STATES AND WAVEFUNCTIONS j 19

observables to be calculated. It is often convenient to refer to the state of

the system without referring to the corresponding wavefunction; the state is

specified by listing the values of the quantum numbers that define it.

1.10 The fundamental prescription

The next postulate concerns the selection of operators:

Postulate 2. Observables are represented by hermitian operators chosen to

satisfy the commutation relations

½q, pq0 ¼ i�hdqq0 ½q, q0 ¼ 0 ½pq, pq0 ¼ 0where q and q 0 each denote one of the coordinates x, y, z and pq and pq 0 the

corresponding linear momenta.

The requirement that the operators are hermitian ensures that the observables

have real values (see below). Each commutation relation is a basic, unpro-

vable, and underivable postulate. Postulate 2 is the basis of the selection of

the form of the operators in the position and momentum representations for

all observables that depend on the position and the momentum.2 Thus, if we

define the position representation as the representation in which the position

operator is multiplication by the position coordinate, then as we saw in

Example 1.3, it follows that the momentum operator must involve differ-

entiation with respect to x, as specified earlier. Similarly, if the momentum

representation is defined as the representation in which the linear momentum

is represented by multiplication, then the form of the position operator is

fixed as a derivative with respect to the linear momentum. The coordinates

x, y, and z commute with each other as do the linear momenta px, py, and pz.

1.11 The outcome of measurements

The next postulate brings together the wavefunction and the operators and

establishes the link between formal calculations and experimental observations:

Postulate 3. When a system is described by a wavefunction c, the meanvalue of the observable O in a series of measurements is equal to the expec-tation value of the corresponding operator.

The expectation value of an operator O for an arbitrary state c is denoted hOiand defined as

hOi ¼Rc�Oc dtRc�cdt

¼ hcjOjcihcjci ð1:23Þ

If the wavefunction is chosen to be normalized to 1, then the expectation

value is simply

hOi ¼Z

c�Oc dt ¼ hcjOjci ð1:24Þ

Unless we state otherwise, from now on we shall assume that the wave-

function is normalized to 1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. This prescription excludes intrinsic observables, such as spin (Section 4.8).

20 j 1 THE FOUNDATIONS OF QUANTUM MECHANICS

The meaning of Postulate 3 can be unravelled as follows. First, suppose

that c is an eigenfunction of O with eigenvalue o; then

hOi ¼Z

c�Oc dt ¼Z

c�oc dt ¼ oZ

c�c dt ¼ o ð1:25Þ

That is, a series of experiments on identical systems to determine O will givethe average value o (a real quantity, because O is hermitian). Now supposethat although the system is in an eigenstate of the hamiltonian it is not in an

eigenstate of O. In this case the wavefunction can be expressed as a linearcombination of eigenfunctions of O:

c ¼X

n

cncn where Ocn ¼ oncn

In this case, the expectation value is

hOi ¼Z X

m

cmcm

!�OX

n

cncn

!dt ¼

Xm, n

c�mcn

Zc�mOcn dt

¼Xm,n

c�mcnon

Zc�mcn dt

Because the eigenfunctions form an orthonormal set, the integral in the last

expression is zero if n 6¼m, is 1 if n¼m, and the double sum reduces to asingle sum:

hOi ¼X

n

c�ncnon

Zc�ncn dt ¼

Xn

c�ncnon ¼X

n

cnj j2on ð1:26Þ

That is, the expectation value is a weighted sum of the eigenvalues of O,the contribution of a particular eigenvalue to the sum being determined by the

square modulus of the corresponding coefficient in the expansion of the

wavefunction.

We can now interpret the difference between eqns 1.25 and 1.26 in the

form of a subsidiary postulate:

Postulate 30. When c is an eigenfunction of the operator O, the determina-tion of the property O always yields one result, namely the correspondingeigenvalue o. The expectation value will simply be the eigenvalue o. When cis not an eigenfunction of O, a single measurement of the property yieldsa single outcome which is one of the eigenvalues of O, and the probability thata particular eigenvalue on is measured is equal to jcnj2, where cn is thecoefficient of the eigenfunction cn in the expansion of the wavefunction.

One measurement can give only one result: a pointer can indicate only one

value on a dial at any instant. A series of determinations can lead to a series of

results with some mean value. The subsidiary postulate asserts that a mea-

surement of the observable O always results in the pointer indicating one ofthe eigenvalues of the corresponding operator. If the function that describes

the state of the system is an eigenfunction of O, then every pointer reading isprecisely o and the mean value is also o. If the system has been prepared in astate that is not an eigenfunction of O, then different measurements givedifferent values, but every individual measurement is one of the eigenvalues of

1.11 THE OUTCOME OF MEASUREMENTS j 21

O, and the probability that a particular outcome on is obtained is determinedby the value of jcnj2. In this case, the mean value of all the observations is theweighted average of the eigenvalues. Note that in either case, the hermiticity

of the operator guarantees that the observables will be real.

Example 1.6 How to use Postulate 3 0.

An operator A has eigenfunctions f1, f2, . . . , fn with corresponding eigenvalues

a1, a2, . . . , an. The state of a system is described by a normalized wavefunction

c given by

c ¼ 12 f1 � 38

�1=2

f2 þ 38 i

�1=2

f3

What will be the outcome of measuring the observable A?

Method. First, we need to determine if c is an eigenfunction of the operator A.If it is, then we shall obtain the same eigenvalue of A in every measurement.

If it is not, we shall obtain different values in a series of different measure-

ments. In the latter case, if we have an expression for c in terms of theeigenfunctions of A, then we can determine what different values are possible,

the probabilities of obtaining them, and the average value from a large series

of measurements.

Answer. To test whether c is an eigenfunction of the operator A we proceed asfollows:

Ac ¼ A 12 f1 � 38

�1=2

f2 þ 38 i

�1=2

f3h i

¼ 12a1f1 � 38

�1=2

a2f2 þ 38 i

�1=2

a3f3 6¼ constant� c

Therefore, c is not an eigenfunction of A. However, because c is a linearcombination of f1, f2, and f3 we will obtain, in different measurements,

the values a1, a2, and a3 (the eigenvalues of the eigenfunctions of A that

contribute to c). The probabilities of obtaining a1, a2, and a3 are, respectively,14,

38, and

38. The average value, given by eqn 1.26, is

hAi ¼ 14 a1 þ 38 a2 þ 38 a3Comment. The normalization of c is reflected in the fact that the probabilitiessum to 1. Because the eigenfunctions f4, f5, . . . do not contribute here to c,there is zero probability of finding a4, a5, . . . .

Self-test 1.6. Repeat the problem using c ¼ 13 f2 þ ð79Þ1=2f4� 13 if5:

[hAi ¼ 19 a2 þ 79 a4 þ 19 a5

1.12 The interpretation of the wavefunction

The next postulate concerns the interpretation of the wavefunction itself, and

is commonly called the Born interpretation:

Postulate 4. The probability that a particle will be found in the volume

element dt at the point r is proportional to jc(r)j2dt.

22 j 1 THE FOUNDATIONS OF QUANTUM MECHANICS

As we have already remarked, in one dimension the volume element is dx.In three dimensions the volume element is dxdydz. It follows from this

interpretation that jc(r)j2 is a probability density, in the sense that ityields a probability when multiplied by the volume dt of an infinitesimalregion. The wavefunction itself is a probability amplitude, and has no direct

physical meaning. Note that whereas the probability density is real and non-

negative, the wavefunction may be complex and negative. It is usually con-

venient to use a normalized wavefunction; then the Born interpretation

becomes an equality rather than a proportionality. The implication of the

Born interpretation is that the wavefunction should be square-integrable;

that isZjcj2 dt

In one dimension the equation has the form

HC ¼ � �h2

2m

q2Cqx2þ VðxÞC ¼ i�h qC

qt

Equations of this form can be solved by the technique of separation of

variables, in which a trial solution takes the form

Cðx, tÞ ¼ cðxÞyðtÞ

When this substitution is made, we obtain

� �h2

2my

d2cdx2þ VðxÞcy ¼ i �hc dy

dt

Division of both sides of this equation by cy gives

� �h2

2m

1

cd2cdx2þ VðxÞ ¼ i �h 1

ydydt

Only the left-hand side of this equation is a function of x, so when x changes,

only the left-hand side can change. But as the left-hand side is equal to the

right-hand side, and the latter does not change, the left-hand side must be

equal to a constant. Because the dimensions of the constant are those of an

energy (the same as those of V), we shall write it E. It follows that the time-dependent equation separates into the following two differential equations:

� �h2

2m

d2cdx2þ VðxÞc ¼ Ec ð1:29aÞ

i�hdydt¼ Ey