Molecular QUantum mechanics 4th atkins an freindman

Molecular QUantum mechanics 4th atkins an freindman Molecular QUantum mechanics 4th atkins an freindman Molecular QUantum mechanics 4th atkins an freindman Molecular QUantum mechanics 4th atkins an freindman Molecular QUantum mechanics 4th atkins an freindman

MOLECULAR QUANTUM

MECHANICS, FOURTH EDITION

Peter Atkins

Ronald 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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Table of contents

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 Schro¨dinger 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 Schro¨dinger equation 36

1.22 The propagation of light 36

1.23 The propagation of particles 38

1.24 The transition to quantum mechanics 39

2 Linear motion and the harmonic

The characteristics of acceptable wavefunctions 43

Some general remarks on the Schro¨dinger 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

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

2.12 The solutions 56

2.13 Features of the solutions 57

2.14 The two-dimensional square well 58

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

68

3 Rotational motion and the hydrogen atom 71

Particle on a ring 71

3.1 The hamiltonian and the Schro¨dinger

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 Schro¨dinger equation and

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 Schro¨dinger equation for

hydrogenic atoms 84

3.10 The separation of the relative coordinates 85

3.11 The radial Schro¨dinger equation 85

3.12 Probabilities and the radial

The angular momentum operators 98

4.1 The operators and their commutation

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

4.7 The angular momentum eigenfunctions 108

The angular momenta of composite systems 112

4.9 The specification of coupled states 112

4.10 The permitted values of the total angular

4.11 The vector model of coupled angular

4.12 The relation between schemes 117

4.13 The coupling of several angular momenta 119

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

11.11 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

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

12.8 Polar molecules 420

12.9 Refractive index 422

12.10 Circular birefringence and optical rotation 427

12.11 Magnetically induced polarization 429

12.12 Rotational strength 431

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

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

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

14.18 The Born approximation 504

Appendix 14.1 The derivation of the Breit–Wigner

Further information 513

Classical mechanics 513

2 The canonical momentum 515

3 The virial theorem 516

Solutions of the Schro¨dinger equation 519

5 The motion of wavepackets 519

6 The harmonic oscillator: solution by

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

19 Normal modes: an example 541

The electromagnetic field 543

20 The Maxwell equations 543

21 The dipolar vector potential 546

Many changes have occurred over the editions of this text but we haveretained its essence throughout Quantum mechanics is filled with abstractmaterial that is both conceptually demanding and mathematically challen-ging: we try, wherever possible, to provide interpretations and visualizationsalongside mathematical presentations

One major change since the third edition has been our response to concernsabout the mathematical complexity of the material We have not sacrificedthe mathematical rigour of the previous edition but we have tried innumerous 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 workedexamples to provide the reader with further opportunities to see formulae inaction We have added new problems for each chapter We have expanded thediscussion on numerous occasions within the body of the text to providefurther clarification for or insight into mathematical results We have set asideProofs and Illustrations (brief examples) from the main body of the text sothat 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 tothe 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 www

to signify that on the Website to accompany the text [www.oup.com/uk/booksites/chemistry/] there are opportunities to explore the equations bysubstituting numerical values for variables

We have added new material to a number of chapters, most notably thechapter on electronic structure techniques (Chapter 9) and the chapter onscattering theory (Chapter 14) These two chapters present material that is atthe forefront of modern molecular quantum mechanics; significant advanceshave occurred in these two fields in the past decade and we have tried tocapture their essence Both chapters present topics where comprehensioncould be readily washed away by a deluge of algebra; therefore, we con-centrate on the highlights and provide interpretations and visualizationswherever possible

There are many organizational changes in the text, including the layout ofchapters and the choice of words As was the case for the third edition, thepresent edition is a rewrite of its predecessor In the rewriting, we have aimedfor clarity and precision

We have a deep sense of appreciation for many people who assisted us inthis endeavour We also wish to thank the numerous reviewers of the text-book at various stages of its development In particular, we would like tothank

Charles Trapp, University of Louisville, USA

Ronald Duchovic, Indiana Purdue Fort Wayne, USA

Karl Jalkanen, Technical University of Denmark, DenmarkMark Child, University of Oxford, UK

Ian Mills, University of Reading, UKDavid Clary, University of Oxford, UKStephan Sauer, University of Copenhagen, DenmarkTemer Ahmadi, Villanova University, USA

Lutz Hecht, University of Glasgow, UKScott Kirby, University of Missouri-Rolla, USAAll these colleagues have made valuable suggestions about the content andorganization of the book as well as pointing out errors best spotted in private.Many individuals (too numerous to name here) have offered advice over theyears and we value and appreciate all their insights and advice As always, ourpublishers have been very helpful and understanding

PWA, OxfordRSF, Indiana University Purdue University Fort Wayne

June 2004

There are two approaches to quantum mechanics One is to follow thehistorical development of the theory from the first indications that thewhole fabric of classical mechanics and electrodynamics should be held

in doubt to the resolution of the problem in the work of Planck, Einstein,Heisenberg, Schro¨dinger, and Dirac The other is to stand back at a pointlate 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 ment and intuition jointly determining the form of the theory and, aboveall, 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 islogic and elegance in a scheme that starts from only a few postulates, yetreveals as their implications are unfolded, a rich, experimentally verifiablestructure

experi-This book takes that latter route through the subject However, to set thescene 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 fundamentalconcepts of the nature of matter and its behaviour were overthrown andreplaced by a puzzling but powerful new description

0.1 Black-body radiation

In retrospect—and as will become clear—we can now see that theoreticalphysics hovered on the edge of formulating a quantum mechanical descrip-tion of matter as it was developed during the nineteenth century However, itwas a series of experimental observations that motivated the revolution Ofthese observations, the most important historically was the study of black-body radiation, the radiation in thermal equilibrium with a body that absorbsand emits without favouring particular frequencies A pinhole in an otherwisesealed container is a good approximation (Fig 0.1)

Two characteristics of the radiation had been identified by the end of thecentury and summarized in two laws According to the Stefan–Boltzmannlaw, the excitance, M, the power emitted divided by the area of the emittingregion, is proportional to the fourth power of the temperature:

The Stefan–Boltzmann constant, s, is independent of the material from whichthe body is composed, and its modern value is 56.7 nW m2K4 So, a region

of area 1 cm2of a black body at 1000 K radiates about 6 W if all frequenciesare taken into account Not all frequencies (or wavelengths, with l ¼ 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’sdisplacement law,

with the constant equal to 2.9 mm K

One of the most challenging problems in physics at the end of the teenth century was to explain these two laws Lord Rayleigh, with minor helpfrom James Jeans,1brought his formidable experience of classical physics tobear on the problem, and formulated the theoretical Rayleigh–Jeans law forthe energy density e(l), the energy divided by the volume, in the wavelengthrange l to l þ dl:

where k is Boltzmann’s constant (k ¼ 1.381  10 23J K1) This formulasummarizes the failure of classical physics It suggests that regardless ofthe temperature, there should be an infinite energy density at very shortwavelengths This absurd result was termed by Ehrenfest the ultravioletcatastrophe

At this point, Planck made his historic contribution His suggestion wasequivalent to proposing that an oscillation of the electromagnetic field offrequency n could be excited only in steps of energy of magnitude hn, where

h is a new fundamental constant of nature now known as Planck’s constant.According to this quantization of energy, the supposition that energy can betransferred 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 excitedwith a very small energy: that was the root of the ultraviolet catastrophe.Quantum theory is characterized by discreteness in energies (and, as we shallsee, of certain other properties), and the need for a minimum excitationenergy effectively switches off oscillators of very high frequency, and henceeliminates the ultraviolet catastrophe

When Planck implemented his suggestion, he derived what is now calledthe Planck distribution for the energy density of a black-body radiator:rðlÞ ¼8phc

cata-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).)

Container

at a temperature T

Fig 0.1 A black-body emitter can be

simulated by a heated container with

a pinhole in the wall The

electromagnetic radiation is reflected

many times inside the container and

reaches thermal equilibrium with the

walls.

quantum theory It began the new century as well as a new era, for it waspublished 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 haveexactly the same heat capacity’ of about 25 J K1mol1(in modern units).Dulong and Petit’s rather primitive observations, though, were done at roomtemperature, and it was unfortunate for them and for classical physics whenmeasurements were extended to lower temperatures and to a wider range ofmaterials It was found that all elements had heat capacities lower thanpredicted 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 byassuming that each atom acts as a classical oscillator in three dimensions Thecalculation predicted that the molar isochoric (constant volume) heat capa-city, CV,m, of a monatomic solid should be equal to 3R ¼ 24.94 J K1mol1,where R is the gas constant (R ¼ NAk, with NAAvogadro’s constant) Thatthe heat capacities were smaller than predicted was a serious embarrassment.Einstein recognized the similarity between this problem and black-bodyradiation, for if each atomic oscillator required a certain minimum energybefore 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 electromagneticoscillators to the material, atomic oscillators of the solid, and deduced thefollowing expression:

The importance of Einstein’s contribution is that it complementedPlanck’s Planck had shown that the energy of radiation is quantized;

Fig 0.3 The Einstein and Debye

molar heat capacities The

symbol y denotes the Einstein

and Debye temperatures,

respectively Close to T ¼ 0 the

Debye heat capacity is

proportional to T 3

0.2 HEAT CAPACITIES j 3

Einstein showed that matter is quantized too Quantization appears to beuniversal Neither was able to justify the form that quantization took (withoscillators excitable in steps of hn), but that is a problem we shall solve later

in the text

0.3 The photoelectric and Compton effects

In those enormously productive months of 1905–6, when Einstein lated not only his theory of heat capacities but also the special theory

formu-of relativity, he found time to make another fundamental contribution

to modern physics His achievement was to relate Planck’s quantumhypothesis to the phenomenon of the photoelectric effect, the emission ofelectrons from metals when they are exposed to ultraviolet radiation Thepuzzling features of the effect were that the emission was instantaneous whenthe radiation was applied however low its intensity, but there was no emis-sion, whatever the intensity of the radiation, unless its frequency exceeded athreshold value typical of each element It was also known that the kineticenergy of the ejected electrons varied linearly with the frequency of theincident radiation

Einstein pointed out that all the observations fell into place if the tromagnetic field was quantized, and that it consisted of bundles of energy

elec-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 photoelectriceffect as the outcome of a collision between an incoming projectile, aphoton of energy hn, and an electron buried in the metal This pictureaccounts for the instantaneous character of the effect, because even onephoton can participate in one collision It also accounted for the frequencythreshold, because a minimum energy (which is normally denoted F andcalled the ‘work function’ for the metal, the analogue of the ionizationenergy of an atom) must be supplied in a collision before photoejection canoccur; 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

If photons do have a particle-like character, then they should possess alinear momentum, p The relativistic expression relating a particle’s energy toits mass and momentum is

This linear momentum should be detectable if radiation falls on an electron,for a partial transfer of momentum during the collision should appear as achange in wavelength of the photons In 1923, A.H Compton performed theexperiment with X-rays scattered from the electrons in a graphite target, andfound the results fitted the following formula for the shift in wavelength,

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

where lC¼ h/mec is called the Compton wavelength of the electron(lC¼ 2.426 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, thatelectromagnetic radiation has properties that classically would have beencharacteristic of particles

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

of data that apparently provided unequivocal support for the view thatelectromagnetic radiation is wave-like By following the implications ofexperiments and quantum concepts, we have accounted quantitatively forobservations for which classical physics could not supply even a qualitativeexplanation

0.4 Atomic spectra

There was yet another body of data that classical physics could not elucidatebefore the introduction of quantum theory This puzzle was the observationthat the radiation emitted by atoms was not continuous but consisted ofdiscrete frequencies, or spectral lines The spectrum of atomic hydrogen had avery simple appearance, and by 1885 J Balmer had already noticed that theirwavenumbers, ~nn, where ~nn ¼ n/c, fitted the expression

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

to discrete values, because a transition from one term of energy hcT1 toanother of energy hcT2can be expected to release a photon of energy hc~nn, or

hn, 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 spectroscopictransitions.

But why should the energy of an atom be confined to discrete values? Inclassical physics, all energies are permissible The first attempt to weldtogether Planck’s quantization hypothesis and a mechanical model of an atomwas made by Niels Bohr in 1913 By arbitrarily assuming that the angularmomentum of an electron around a central nucleus (the picture of an atomthat had emerged from Rutherford’s experiments in 1910) was confined tocertain values, he was able to deduce the following expression for the per-mitted energy levels of an electron in a hydrogen atom:

Bohr’s achievement was the union of theories of radiation and models ofmechanics However, it was an arbitrary union, and we now know that it isconceptually untenable (for instance, it is based on the view that an electrontravels in a circular path around the nucleus) Nevertheless, the fact that hewas able to account quantitatively for the appearance of the spectrum ofhydrogen indicated that quantum mechanics was central to any description ofatomic phenomena and properties

0.5 The duality of matter

The grand synthesis of these ideas and the demonstration of the deep linksthat exist between electromagnetic radiation and matter began with Louis deBroglie, who proposed on the basis of relativistic considerations that with anymoving body there is ‘associated a wave’, and that the momentum of the bodyand the wavelength are related by the de Broglie relation:

l¼h

We have seen this formula already (eqn 0.8), in connection with the erties of photons De Broglie proposed that it is universally applicable.The significance of the de Broglie relation is that it summarizes a fusion

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

a property of waves This duality, the possession of properties that in classicalphysics are characteristic of both particles and waves, is a persistent theme

in the interpretation of quantum mechanics It is probably best to regardthe terms ‘wave’ and ‘particle’ as remnants of a language based on a false

(classical) model of the universe, and the term ‘duality’ as a late attempt tobring the language into line with a current (quantum mechanical) model.The experimental results that confirmed de Broglie’s conjecture are theobservation of the diffraction of electrons by the ranks of atoms in a metalcrystal acting as a diffraction grating Davisson and Germer, who performedthis experiment in 1925 using a crystal of nickel, found that the diffractionpattern was consistent with the electrons having a wavelength given bythe de Broglie relation Shortly afterwards, G.P Thomson also succeeded

in demonstrating the diffraction of electrons by thin films of celluloidand gold.2

If electrons—if all particles—have wave-like character, then we shouldexpect there to be observational consequences In particular, just as a wave ofdefinite 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 WernerHeisenberg to his celebrated uncertainty principle, that it is impossible tospecify the location and linear momentum of a particle simultaneously witharbitrary precision In other words, information about location is at theexpense of information about momentum, and vice versa This com-plementarity of certain pairs of observables, the mutual exclusion of thespecification of one property by the specification of another, is also a majortheme of quantum mechanics, and almost an icon of the difference between itand classical mechanics, in which the specification of exact trajectories was acentral theme

The consummation of all this faltering progress came in 1926 when WernerHeisenberg and Erwin Schro¨dinger formulated their seemingly different butequally successful versions of quantum mechanics These days, we stepbetween the two formalisms as the fancy takes us, for they are mathematicallyequivalent, and each one has particular advantages in different types of cal-culation Although Heisenberg’s formulation preceded Schro¨dinger’s by a fewmonths, it seemed more abstract and was expressed in the then unfamiliarvocabulary of matrices Still today it is more suited for the more formalmanipulations and deductions of the theory, and in the following pages weshall employ it in that manner Schro¨dinger’s formulation, which was in terms

of functions and differential equations, was more familiar in style but stillequally revolutionary in implication It is more suited to elementary mani-pulations and to the calculation of numerical results, and we shall employ it inthat manner

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

excitation of (a) an electronic motion of period 1.0 fs,

(b) a molecular vibration of period 10 fs, and (c) a pendulum

of period 1.0 s.

0.2 Find the wavelength corresponding to the maximum in

the Planck distribution for a given temperature, and show

that the expression reduces to the Wien displacement law at

short wavelengths Determine an expression for the constant

in the law in terms of fundamental constants (This constant

is called the second radiation constant, c 2 )

0.3 Use the Planck distribution to confirm the

Stefan–Boltzmann law and to derive an expression for

the Stefan–Boltzmann constant s.

0.4 The peak in the Sun’s emitted energy occurs at about

480 nm Estimate the temperature of its surface on the basis

of it being regarded as a black-body emitter.

0.5 Derive the Einstein formula for the heat capacity of a

collection of harmonic oscillators To do so, use the

quantum mechanical result that the energy of a harmonic

oscillator of force constant k and mass m is one of the values

(v þ 1 )hv, with v ¼ (1/2p)(k/m) 1/2 and v ¼ 0, 1, 2, Hint.

Calculate the mean energy, E, of a collection of oscillators

by substituting these energies into the Boltzmann

distribution, and then evaluate C ¼ dE/dT.

0.6 Find the (a) low temperature, (b) high temperature

forms of the Einstein heat capacity function.

0.7 Show that the Debye expression for the heat capacity is

proportional to T 3 as T ! 0.

0.8 Estimate the molar heat capacities of metallic sodium

(y D ¼ 150 K) and diamond (y D ¼ 1860 K) at room

temperature (300 K).

0.9 Calculate the molar entropy of an Einstein solid at

T ¼ y E Hint The entropy is S ¼ R T

0 ðC V =TÞdT Evaluate the integral numerically.

0.10 How many photons would be emitted per second by a

sodium lamp rated at 100 W which radiated all its energy

with 100 per cent efficiency as yellow light of wavelength

589 nm?

0.11 Calculate the speed of an electron emitted from a clean

potassium surface (F ¼ 2.3 eV) by light of wavelength (a)

300 nm, (b) 600 nm.

0.12 When light of wavelength 195 nm strikes a certain metal

surface, electrons are ejected with a speed of 1.23  106m s 1

Calculate the speed of electrons ejected from the same metal

surface by light of wavelength 255 nm.

0.13 At what wavelength of incident radiation do the relativistic and non-relativistic expressions for the ejection

of electrons from potassium differ by 10 per cent? That is, find l such that the non-relativistic and relativistic linear momenta of the photoelectron differ by 10 per cent Use

F ¼ 2.3 eV.

0.14 Deduce eqn 0.9 for the Compton effect on the basis of the conservation of energy and linear momentum Hint Use the relativistic expressions Initially the electron is at rest with energy m e c2 When it is travelling with momentum p its energy is ðp 2 c 2 þ m 2

e c 4 Þ 1/2 The photon, with initial momentum h/l i and energy hn i , strikes the stationary electron, is deflected through an angle y, and emerges with momentum h/l f and energy hn f The electron is initially stationary (p ¼ 0) but moves off with an angle y 0 to the incident photon Conserve energy and both components of linear momentum Eliminate y 0 , then p, and so arrive at an expression for dl.

0.15 The first few lines of the visible (Balmer) series in the spectrum of atomic hydrogen lie at l/nm ¼ 656.46, 486.27, 434.17, 410.29, Find a value of R H , the Rydberg constant for hydrogen The ionization energy, I, is the minimum energy required to remove the electron Find it from the data and express its value in electron volts How is

I related to R H ? Hint The ionization limit corresponds to

n ! 1 for the final state of the electron.

0.16 Calculate the de Broglie wavelength of (a) a mass of 1.0 g travelling at 1.0 cm s 1 , (b) the same at 95 per cent of the speed of light, (c) a hydrogen atom at room temperature (300 K); estimate the mean speed from the equipartition principle, which implies that the mean kinetic energy of an atom is equal to 3 kT, where k is Boltzmann’s constant, (d)

an electron accelerated from rest through a potential difference of (i) 1.0 V, (ii) 10 kV Hint For the momentum

in (b) use p ¼ mv/(l  v 2 /c 2 ) 1/2 and for the speed in (d) use

1 m e v2¼ eV, where V is the potential difference.

0.17 Derive eqn 0.12 for the permitted energy levels for the electron in a hydrogen atom To do so, use the following (incorrect) postulates of Bohr: (a) the electron moves in a circular orbit of radius r around the nucleus and (b) the angular momentum of the electron is an integral multiple of



h, that is m e vr ¼ nh Hint Mechanical stability of the orbital motion requires that the Coulombic force of attraction between the electron and nucleus equals the centrifugal force due to the circular motion The energy of the electron is the sum of the kinetic energy and potential (Coulombic) energy For simplicity, use m e rather than the reduced mass m.

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

of postulates When their consequences are developed, they embrace thebehaviour of all known forms of matter, including the molecules, atoms, andelectrons that will be at the centre of our attention in this book This chapterintroduces the postulates and illustrates how they are used The remainingchapters build on them, and show how to apply them to problems of chemicalinterest, 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 Schro¨dingerequation 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 principalmathematical difference between classical mechanics and quantum mechan-ics is that whereas in the former physical observables are represented byfunctions (such as position as a function of time), in quantum mechanics theyare represented by mathematical operators An operator is a symbol for aninstruction to carry out some action, an operation, on a function In most ofthe examples we shall meet, the action will be nothing more complicated thanmultiplication or differentiation Thus, one typical operation might bemultiplication by x, which is represented by the operator x  Anotheroperation might be differentiation with respect to x, represented by theoperator 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 operatorthat represents that observable; so the position of a particle along the x-axiswill be denoted x and the corresponding operator will also be denoted x (withmultiplication implied) We shall always make it clear whether we arereferring 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 moreimportant features

The foundations of quantum mechanics

Operators in quantum mechanics

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

1.16 The uncertainty principle

1.17 Consequences of the uncertainty

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 linearoperator is one for which

where a and b are constants and f and g are functions Multiplication is alinear 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 nottrue, for example, that log 2x ¼ 2 log x for all x

1.2 Eigenfunctions and eigenvalues

In general, when an operator operates on a function, the outcome is anotherfunction 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 anoperator O if it satisfies an equation of the form

where o is a constant Such an equation is called an eigenvalue equation Thefunction eaxis an eigenfunction of the operator d/dx because (d/dx)eax¼ aeax,which is a constant (a) multiplying the original function In contrast, eax 2

isnot an eigenfunction of d/dx, because (d/dx)eax 2

¼ 2axeax 2

, which is a stant (2a) times a different function of x (the function xeax 2

con-) 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

what is the corresponding eigenvalue?

Method. Perform the indicated operation on the given function and see ifthe function satisfies an eigenvalue equation Use (d/dx)sin ax ¼ a cos ax and(d/dx)cos ax ¼ a sin ax

Answer. The operator operating on the function yields

That is, if fnis an eigenfunction of an operator O with eigenvalue on(so Ofn¼

onfn), then1a general function g can be expressed as the linear combination

Then it is quite easy to show that any linear combination of the functions fn

is also an eigenfunction of O with the same eigenvalue o The proof is asfollows For an arbitrary linear combination g of the degenerate set offunctions, we can write

Example 1.2 Demonstrating that a linear combination of degenerateeigenfunctions is also an eigenfunction

Method. Consider an arbitrary linear combination ae2ixþ be2ixand see if thefunction satisfies an eigenvalue equation

Answer. First we demonstrate that e2ixand e2ixare degenerate eigenfunctions

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

The linear combination satisfies the eigenvalue equation and has the sameeigenvalue (4) as do the two complex functions

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

[Eigenvalue is 9]

A further technical point is that from n basis functions it is possible to struct n linearly independent combinations A set of functions g1, g2, , gnissaid to be linearly independent if we cannot find a set of constants c1, c2, ,

con-cn(other than the trivial set c1¼ c2¼ ¼ 0) for whichX

i

cigi¼ 0

A set of functions that is not linearly independent is said to be linearlydependent From a set of n linearly independent functions, it is possible toconstruct an infinite number of sets of linearly independent combinations,but each set can have no more than n members For example, from three2p-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 ofthe operators we shall meet Much of quantum mechanics can be developed interms of an abstract set of operators, as we shall see later However, it is oftenfruitful 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 particularobservable give rise to the different representations of quantum mechanics,because the explicit forms of the operators represent the abstract structure ofthe 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 isrepresented by differentiation with respect to x Explicitly:

pre-An alternative choice of operators is the momentum representation, inwhich the linear momentum parallel to x is represented by the operation of

multiplication by pxand the position operator is represented by tion with respect to px Explicitly:

repres-1.4 Commutation and non-commutation

An important feature of operators is that in general the outcome of successiveoperations (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 6¼ AB We say that, in general, operators do notcommute For example, consider the operators x and px and a specificfunction x2 In the position representation, (xpx)x2

¼ x(2h/i)x ¼ (2h/i)x2,whereas (pxx)x2

¼ pxx3

¼ (3h/i)x2 The operators x and pxdo not commute.The quantity AB  BA is called the commutator of A and B and is denoted[A, B]:

ð1:7Þ

It is instructive to evaluate the commutator of the position and linearmomentum operators in the two representations shown above; the procedure

is illustrated in the following example

Example 1.3 The evaluation of a commutator

Method. To evaluate the commutator [A,B] we need to remember that theoperators operate on some function, which we shall write f So, evaluate [A,B]ffor 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:

qðxf Þqx

i

qf

where we have used (1/i) ¼ i This derivation is true for any function f,

so in terms of the operators themselves,

1.5 The construction of operators

Operators for other observables of interest can be constructed from the rators for position and momentum For example, the kinetic energy operator

ope-T can be constructed by noting that kinetic energy is related to linearmomentum by T ¼ p2/2m where m is the mass of the particle It follows that

in one dimension and in the position representation

2

12m

hi

ddx

¼  h2m

The operator for the total energy of a system is called the hamiltonianoperator and is denoted H:

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

of classical mechanics in terms of what became known as a hamiltonianfunction To write the explicit form of this operator we simply substitute theappropriate expressions for the kinetic and potential energy operators in thechosen representation For example, the hamiltonian for a particle of mass mmoving in one dimension is

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.

The general prescription for constructing operators in the position entation should be clear from these examples In short:

repres-1 Write the classical expression for the observable in terms of positioncoordinates and the linear momentum

2 Replace x by multiplication by x, and replace pxby (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 operatorsand the actual outcome of an experiment, we need to evaluate certainintegrals These integrals all have the form

is called an overlap integral and commonly denoted S:

S ¼

Z

f

It is helpful to regard S as a measure of the similarity of two functions: when

S ¼ 0, the functions are classified as orthogonal, rather like two perpendicularvectors When S is close to 1, the two functions are almost identical Therecognition of mutually orthogonal functions often helps to reduce theamount of calculation considerably, and rules will emerge in later sectionsand chapters

The normalization integral is the special case of eqn 1.15 for m ¼ n

A function fmis said to be normalized (strictly, normalized to 1) ifZ

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 N¼ N The procedure isillustrated in the following example

Example 1.4 How to normalize a function

A certain function f is sin(px/L) between x ¼ 0 and x ¼ L and is zero elsewhere.Find the normalized form of the function

The complex conjugate of

Method. We need to find the (real) factor N such that N sin(px/L) is alized to 1 To find N we substitute this expression into eqn 1.16, evaluate theintegral, and select N to ensure normalization Note that ‘all space’ extendsfrom x ¼ 0 to x ¼ L.

norm-Answer. The necessary integration isZ

1.7 Dirac bracket notation

With eqn 1.14 we are on the edge of getting lost in a complicated notation Theappearance of many quantum mechanical expressions is greatly simplified byadopting the Dirac bracket notation in which integrals are written as follows:hmjOjni ¼

Z

The symbol jni is called a ket, and denotes the state described by the function

fn Similarly, the symbol hnj is called a bra, and denotes the complex conjugate

of the function, f When a bra and ket are strung together with an operatorbetween 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 omittedand we use the convention

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

Example 1.5 How to confirm the hermiticity of operators

Show that the position and momentum operators in the position tion are hermitian

representa-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 theintegral is written down When a differential operator is used, it may benecessary to use integration by parts at some stage in the argument to transferthe differentiation from one function to another:

Z

u dv ¼ uv 

Z

v duAnswer. That the position operator is hermitian is obvious from inspection:Z

involves an integration by parts:

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

Hence, the operator is hermitian

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

As we shall now see, the property of hermiticity has far-reaching cations First, we shall establish the following property:

impli-Property 1 The eigenvalues of hermitian operators are real

Proof 1.1 The reality of eigenvaluesConsider the eigenvalue equationOjoi ¼ ojoi

The ket joi denotes an eigenstate of the operator O in the sense that the

labelling the eigenstates with the eigenvalue o of the operator O It is oftenconvenient to use the eigenvalues as labels in this way Multiplication from theleft by hoj results in the equation

hojOjoi ¼ ohojoi ¼ otaking joi to be normalized Now take the complex conjugate of both sides:

The second property we shall prove is as follows:

Property 2 Eigenfunctions corresponding to different eigenvalues of anhermitian operator are orthogonal

That is, if we have two eigenfunctions of an hermitian operator O witheigenvalues o and o0, with o 6¼ o0, then hojo0i ¼ 0 For example, it follows atonce that all the eigenfunctions of a harmonic oscillator (Section 2.16) aremutually orthogonal, for as we shall see each one corresponds to a differentenergy (the eigenvalue of the hamiltonian, an hermitian operator)

Proof 1.2 The orthogonality of eigenstates

relations:

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

Because O is hermitian, the left-hand side of this expression is zero; so (noting

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

The postulates of quantum mechanics

Now we turn to an application of the preceding material, and move into thefoundations of quantum mechanics The postulates we use as a basis forquantum 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 ofthe system (more specifically, the time-dependent wavefunction) When weare not interested in how the system changes in time we shall denote thewavefunction 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 someinternal variable of the particles (their spin states); we ignore that for nowand return to it later By ‘describe’ we mean that the wavefunctioncontains information about all the properties of the system that are open toexperimental determination

We shall see that the wavefunction of a system will be specified by a set oflabels called quantum numbers, and may then be written ca,b, , where

a, b, are the quantum numbers The values of these quantum numbersspecify 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 ofthe system without referring to the corresponding wavefunction; the state isspecified 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 tosatisfy the commutation relations

½q, pq 0 hdqq 0 ½q, q0 ½pq, pq 0where q and q0each denote one of the coordinates x, y, z and pqand pq 0thecorresponding linear momenta

The requirement that the operators are hermitian ensures that the observableshave real values (see below) Each commutation relation is a basic, unpro-vable, and underivable postulate Postulate 2 is the basis of the selection ofthe form of the operators in the position and momentum representations forall observables that depend on the position and the momentum.2Thus, if wedefine the position representation as the representation in which the positionoperator is multiplication by the position coordinate, then as we saw inExample 1.3, it follows that the momentum operator must involve differ-entiation with respect to x, as specified earlier Similarly, if the momentumrepresentation is defined as the representation in which the linear momentum

is represented by multiplication, then the form of the position operator isfixed 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 andestablishes 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 ¼

R

cOcdtR

The meaning of Postulate 3 can be unravelled as follows First, supposethat c is an eigenfunction of O with eigenvalue o; then

We can now interpret the difference between eqns 1.25 and 1.26 in theform of a subsidiary postulate:

Postulate 30 When c is an eigenfunction of the operator O, the tion of the property O always yields one result, namely the correspondingeigenvalue o The expectation value will simply be the eigenvalue o When c

determina-is not an eigenfunction of O, a single measurement of the property yields

a single outcome which is one of the eigenvalues of O, and the probability that

a particular eigenvalue on is measured is equal to jcnj2, where cn is thecoefficient of the eigenfunction cnin the expansion of the wavefunction.One measurement can give only one result: a pointer can indicate only onevalue on a dial at any instant A series of determinations can lead to a series ofresults 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 describesthe 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 onis obtained is determined

by 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 30

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 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

Therefore, c is not an eigenfunction of A However, because c is a linear

Comment. The normalization of c is reflected in the fact that the probabilities

Self-test 1.6. Repeat the problem using c ¼1

3f2þ ð7

9Þ1=2f41

3if5:

[hAi ¼ 1 a 2 þ 7 a 4 þ 1 a 5

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

As we have already remarked, in one dimension the volume element is dx.

In three dimensions the volume element is dxdydz It follows from thisinterpretation 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 directphysical 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 interpretationbecomes an equality rather than a proportionality The implication of theBorn interpretation is that the wavefunction should be square-integrable;that is

Z

jcj2 dt < 1

because there must be a finite probability of finding the particle somewhere inthe whole of space (and that probability is 1 for a normalized wavefunction).This postulate in turn implies that c ! 0 as x ! 1, for otherwise the inte-gral of jcj2would be infinite We shall make frequent use of this implicationthroughout the text

1.13 The equation for the wavefunction

The final postulate concerns the dynamical evolution of the wavefunctionwith time:

Postulate 5 The wavefunction C(r1, r2, , t) evolves in time according

in eqn 1.12, we obtain the time-dependent Schro¨dinger equation in onedimension (x) with a time-independent potential energy for a single particle:

1.14 The separation of the Schro¨dinger equation

The Schro¨dinger equation can often be separated into equations for the timeand space variation of the wavefunction The separation is possible when thepotential energy is independent of time

1.14 THE SEPARATION OF THE SCHRO ¨ DINGER EQUATION j 23

In one dimension the equation has the form

Cðx, tÞ ¼ cðxÞyðtÞWhen this substitution is made, we obtain

h

2

2m

1c

d2c

dx2þ VðxÞ ¼ i h1

y

dydtOnly 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 theright-hand side, and the latter does not change, the left-hand side must beequal to a constant Because the dimensions of the constant are those of anenergy (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:

Hc ¼ EcThis expression is the time-independent Schro¨dinger equation, on whichmuch of the following development will be based

This analysis stimulates several remarks First, eqn 1.29a has the form of astanding-wave equation Therefore, so long as we are interested only in thespatial dependence of the wavefunction, it is legitimate to regard the time-independent Schro¨dinger equation as a wave equation Second, when thepotential energy of the system does not depend on the time, and the system

is in a state of energy E, it is a very simple matter to construct the dependent wavefunction from the time-independent wavefunction simply by

time-multiplying the latter by eiEt/h The time dependence of such a wavefunction

is simply a modulation of its phase, because we can write

eiEt=h ¼ cosðEt=hÞ  i sinðEt=hÞ

It follows that the time-dependent factor oscillates periodically from 1 to i

to 1 to i and back to 1 with a frequency E/h and period h/E This behaviour

is depicted in Fig 1.1 Therefore, to imagine the time-variation of a function of a definite energy, think of it as flickering from positive throughimaginary to negative amplitudes with a frequency proportional to the energy.Although the phase of a wavefunction C with definite energy E oscillates intime, the product CC(or jCj2) remains constant:

wave-CC¼ ðceiEt= hÞðceiEt=hÞ ¼ ccStates of this kind are called stationary states From what we have seen so far,

it follows that systems with a specific, precise energy and in which thepotential energy does not vary with time are in stationary states Althoughtheir wavefunctions flicker from one phase to another in repetitive manner,the value of CCremains constant in time

The specification and evolution of states

Let us suppose for the moment that the state of a system can be specified asja,b, i, where each of the eigenvalues a, b, corresponds to the operatorsrepresenting different observables A, B, of the system If the system is inthe state ja,b, i, then when we measure the property A we shall get exactly

a as an outcome, and likewise for the other properties But can a state bespecified arbitrarily fully? That is, can it be simultaneously an eigenstate of allpossible observables A, B, without restriction? With this question we aremoving into the domain of the uncertainty principle

1.15 Simultaneous observables

As a first step, we establish the conditions under which two observables may

be specified simultaneously with arbitrary precision That is, we establish theconditions for a state jci corresponding to the wavefunction c to be simul-taneously an eigenstate of two operators A and B In fact, we shall prove thefollowing:

Property 3 If two observables are to have simultaneously precisely definedvalues, then their corresponding operators must commute

That is, AB must equal BA, or equivalently, [A,B] ¼ 0

Proof 1.3 Simultaneous eigenstatesAssume that jci is an eigenstate of both operators: Ajci ¼ ajci andBjci ¼ bjci That being so, we can write the following chain of relations:ABjci ¼ Abjci ¼ bAjci ¼ bajci ¼ abjci ¼ aBjci ¼ Bajci ¼ BAjci

We have used Euler’s relation,

corresponding to an energy E rotates

in the complex plane from real to

imaginary and back to real at a

circular frequency E/ h.

1.15 SIMULTANEOUS OBSERVABLES j 25