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MOLECULARQUANTUM
MECHANICS,FOURTH EDITION
Peter AtkinsRonald Friedman
OXFORD UNIVERSITY PRESS
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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
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ACGreat Clarendon Street, Oxford OX2 6DP
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Published in the United Statesby Oxford University Press Inc., New York
# Peter Atkins and Ronald Friedman 2005
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First published 2005
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Oxford University Press, at the address above
You must not circulate this book in any other binding or coverand you must impose this same condition on any acquirer
British Library Cataloguing in Publication Data
Data available
Library of Congress Cataloging in Publication Data
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ISBN 0--19--927498--3
10 9 8 7 6 5 4 3 2 1
Typeset by Newgen Imaging Systems (P) Ltd., Chennai, IndiaPrinted in Great Britain
on acid-free paper by Ashford Colour Press
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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
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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
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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
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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
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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
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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
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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
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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