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- 1. MOLECULAR QUANTUM MECHANICS,FOURTH EDITION Peter Atkins Ronald FriedmanOXFORD UNIVERSITY PRESS
- 2. MOLECULAR QUANTUM MECHANICS
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- 4. MOLECULARQUANTUMMECHANICSFOURTH EDITIONPeter AtkinsUniversity of OxfordRonald FriedmanIndiana Purdue Fort WayneAC
- 5. AC Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao Paulo Shanghai Taipei Tokyo Toronto ˜ Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York # Peter Atkins and Ronald Friedman 2005 The moral rights of the authors have been asserted. Database right Oxford University Press (maker) First published 2005 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press,or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available ISBN 0- 19--927498--3 - 10 9 8 7 6 5 4 3 2 1 Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Ashford Colour Press
- 6. Table of contentsPreface xiiiIntroduction 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 28710 Molecular rotations and vibrations 34211 Molecular electronic transitions 38212 The electric properties of molecules 40713 The magnetic properties of molecules 43614 Scattering theory 473Further information 513Further reading 553Appendix 1 Character tables and direct products 557Appendix 2 Vector coupling coefﬁcients 562Answers to selected problems 563Index 565
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- 8. Detailed ContentsIntroduction and orientation 1 ¨ The plausibility of the Schrodinger equation 36 1.22 The propagation of light 36 0.1 Black-body radiation 1 1.23 The propagation of particles 38 0.2 Heat capacities 3 1.24 The transition to quantum mechanics 39 0.3 The photoelectric and Compton effects 4 PROBLEMS 40 0.4 Atomic spectra 5 0.5 The duality of matter 6 2 Linear motion and the harmonicPROBLEMS 8 oscillator 431 The foundations of quantum mechanics 9 The characteristics of acceptable wavefunctions 43 ¨ Some general remarks on the Schrodinger equation 44Operators in quantum mechanics 9 2.1 The curvature of the wavefunction 45 1.1 Linear operators 10 2.2 Qualitative solutions 45 1.2 Eigenfunctions and eigenvalues 10 2.3 The emergence of quantization 46 1.3 Representations 12 2.4 Penetration into non-classical regions 46 1.4 Commutation and non-commutation 13 Translational motion 47 1.5 The construction of operators 14 2.5 Energy and momentum 48 1.6 Integrals over operators 15 2.6 The signiﬁcance of the coefﬁcients 48 1.7 Dirac bracket notation 16 2.7 The ﬂux density 49 1.8 Hermitian operators 17 2.8 Wavepackets 50The postulates of quantum mechanics 19 Penetration into and through barriers 51 1.9 States and wavefunctions 19 2.9 An inﬁnitely thick potential wall 51 1.10 The fundamental prescription 20 2.10 A barrier of ﬁnite width 52 1.11 The outcome of measurements 20 2.11 The Eckart potential barrier 54 1.12 The interpretation of the wavefunction 22 1.13 The equation for the wavefunction 23 Particle in a box 55 ¨ 1.14 The separation of the Schrodinger equation 23 2.12 The solutions 56 2.13 Features of the solutions 57The speciﬁcation and evolution of states 25 2.14 The two-dimensional square well 58 1.15 Simultaneous observables 25 2.15 Degeneracy 59 1.16 The uncertainty principle 27 1.17 Consequences of the uncertainty principle 29 The harmonic oscillator 60 1.18 The uncertainty in energy and time 30 2.16 The solutions 61 1.19 Time-evolution and conservation laws 30 2.17 Properties of the solutions 63 2.18 The classical limit 65Matrices in quantum mechanics 32 1.20 Matrix elements 32 Translation revisited: The scattering matrix 66 1.21 The diagonalization of the hamiltonian 34 PROBLEMS 68
- 9. viii j CONTENTS3 Rotational motion and the hydrogen atom 71 The angular momenta of composite systems 112 4.9 The speciﬁcation of coupled states 112Particle on a ring 71 4.10 The permitted values of the total angular momentum 113 ¨ 3.1 The hamiltonian and the Schrodinger equation 71 4.11 The vector model of coupled angular momenta 115 3.2 The angular momentum 73 4.12 The relation between schemes 117 3.3 The shapes of the wavefunctions 74 4.13 The coupling of several angular momenta 119 3.4 The classical limit 76 PROBLEMS 120Particle on a sphere 76 ¨ 3.5 The Schrodinger equation and 5 Group theory 122 its solution 76 3.6 The angular momentum of the particle 79 The symmetries of objects 122 3.7 Properties of the solutions 81 5.1 Symmetry operations and elements 123 3.8 The rigid rotor 82 5.2 The classiﬁcation of molecules 124Motion in a Coulombic ﬁeld 84 The calculus of symmetry 129 ¨ 3.9 The Schrodinger equation for hydrogenic atoms 84 5.3 The deﬁnition of a group 129 3.10 The separation of the relative coordinates 85 5.4 Group multiplication tables 130 ¨ 3.11 The radial Schrodinger equation 85 5.5 Matrix representations 131 3.12 Probabilities and the radial 5.6 The properties of matrix representations 135 distribution function 90 5.7 The characters of representations 137 3.13 Atomic orbitals 91 5.8 Characters and classes 138 3.14 The degeneracy of hydrogenic atoms 94 5.9 Irreducible representations 139PROBLEMS 96 5.10 The great and little orthogonality theorems 142 Reduced representations 145 5.11 The reduction of representations 1464 Angular momentum 98 5.12 Symmetry-adapted bases 147The angular momentum operators 98 The symmetry properties of functions 151 4.1 The operators and their commutation 5.13 The transformation of p-orbitals 151 relations 99 5.14 The decomposition of direct-product bases 152 4.2 Angular momentum observables 101 5.15 Direct-product groups 155 4.3 The shift operators 101 5.16 Vanishing integrals 157 5.17 Symmetry and degeneracy 159The deﬁnition of the states 102 4.4 The effect of the shift operators 102 The full rotation group 161 4.5 The eigenvalues of the angular momentum 104 5.18 The generators of rotations 161 4.6 The matrix elements of the angular 5.19 The representation of the full rotation group 162 momentum 106 5.20 Coupled angular momenta 164 4.7 The angular momentum eigenfunctions 108 Applications 165 4.8 Spin 110 PROBLEMS 166
- 10. CONTENTS j ix6 Techniques of approximation 168 7.10 The spectrum of helium 224 7.11 The Pauli principle 225Time-independent perturbation theory 168 Many-electron atoms 229 6.1 Perturbation of a two-level system 169 7.12 Penetration and shielding 229 6.2 Many-level systems 171 7.13 Periodicity 231 6.3 The ﬁrst-order correction to the energy 172 7.14 Slater atomic orbitals 233 6.4 The ﬁrst-order correction to the wavefunction 174 7.15 Self-consistent ﬁelds 234 6.5 The second-order correction to the energy 175 7.16 Term symbols and transitions of 6.6 Comments on the perturbation expressions 176 many-electron atoms 236 6.7 The closure approximation 178 7.17 Hund’s rules and the relative energies of terms 239 6.8 Perturbation theory for degenerate states 180 7.18 Alternative coupling schemes 240Variation theory 183 Atoms in external ﬁelds 242 6.9 The Rayleigh ratio 183 7.19 The normal Zeeman effect 242 6.10 The Rayleigh–Ritz method 185 7.20 The anomalous Zeeman effect 243 7.21 The Stark effect 245The Hellmann–Feynman theorem 187Time-dependent perturbation theory 189 PROBLEMS 246 6.11 The time-dependent behaviour of a two-level system 189 6.12 The Rabi formula 192 8 An introduction to molecular structure 249 6.13 Many-level systems: the variation of constants 193 6.14 The effect of a slowly switched constant The Born–Oppenheimer approximation 249 perturbation 195 8.1 The formulation of the approximation 250 6.15 The effect of an oscillating perturbation 197 8.2 An application: the hydrogen molecule–ion 251 6.16 Transition rates to continuum states 199 6.17 The Einstein transition probabilities 200 Molecular orbital theory 253 6.18 Lifetime and energy uncertainty 203 8.3 Linear combinations of atomic orbitals 253 8.4 The hydrogen molecule 258PROBLEMS 204 8.5 Conﬁguration interaction 2597 Atomic spectra and atomic structure 207 8.6 Diatomic molecules 261 8.7 Heteronuclear diatomic molecules 265The spectrum of atomic hydrogen 207 Molecular orbital theory of polyatomic 7.1 The energies of the transitions 208 molecules 266 7.2 Selection rules 209 8.8 Symmetry-adapted linear combinations 266 7.3 Orbital and spin magnetic moments 212 8.9 Conjugated p-systems 269 7.4 Spin–orbit coupling 214 8.10 Ligand ﬁeld theory 274 7.5 The ﬁne-structure of spectra 216 8.11 Further aspects of ligand ﬁeld theory 276 7.6 Term symbols and spectral details 217 The band theory of solids 278 7.7 The detailed spectrum of hydrogen 218 8.12 The tight-binding approximation 279The structure of helium 219 8.13 The Kronig–Penney model 281 7.8 The helium atom 219 8.14 Brillouin zones 284 7.9 Excited states of helium 222 PROBLEMS 285
- 11. x j CONTENTS9 The calculation of electronic structure 287 10.3 Rotational energy levels 345 10.4 Centrifugal distortion 349The Hartree–Fock self-consistent ﬁeld method 288 10.5 Pure rotational selection rules 349 9.1 The formulation of the approach 288 10.6 Rotational Raman selection rules 351 9.2 The Hartree–Fock approach 289 10.7 Nuclear statistics 353 9.3 Restricted and unrestricted Hartree–Fock The vibrations of diatomic molecules 357 calculations 291 10.8 The vibrational energy levels of diatomic 9.4 The Roothaan equations 293 molecules 357 9.5 The selection of basis sets 296 10.9 Anharmonic oscillation 359 9.6 Calculational accuracy and the basis set 301 10.10 Vibrational selection rules 360Electron correlation 302 10.11 Vibration–rotation spectra of diatomic molecules 362 9.7 Conﬁguration state functions 303 10.12 Vibrational Raman transitions of diatomic molecules 364 9.8 Conﬁguration interaction 303 9.9 CI calculations 305 The vibrations of polyatomic molecules 365 9.10 Multiconﬁguration and multireference methods 308 10.13 Normal modes 365 9.11 Møller–Plesset many-body perturbation theory 310 10.14 Vibrational selection rules for polyatomic 9.12 The coupled-cluster method 313 molecules 368 10.15 Group theory and molecular vibrations 369Density functional theory 316 10.16 The effects of anharmonicity 373 9.13 Kohn–Sham orbitals and equations 317 10.17 Coriolis forces 376 9.14 Exchange–correlation functionals 319 10.18 Inversion doubling 377Gradient methods and molecular properties 321 Appendix 10.1 Centrifugal distortion 379 9.15 Energy derivatives and the Hessian matrix 321 PROBLEMS 380 9.16 Analytical derivatives and the coupled perturbed equations 322 11 Molecular electronic transitions 382Semiempirical methods 325 9.17 Conjugated p-electron systems 326 The states of diatomic molecules 382 9.18 Neglect of differential overlap 329 11.1 The Hund coupling cases 382Molecular mechanics 332 11.2 Decoupling and L-doubling 384 9.19 Force ﬁelds 333 11.3 Selection rules 386 9.20 Quantum mechanics–molecular mechanics 334 Vibronic transitions 386Software packages for 11.4 The Franck–Condon principle 386electronic structure calculations 336 11.5 The rotational structure of vibronic transitions 389PROBLEMS 339 The electronic spectra of polyatomic molecules 39010 Molecular rotations and vibrations 342 11.6 Symmetry considerations 391 11.7 Chromophores 391Spectroscopic transitions 342 11.8 Vibronically allowed transitions 393 10.1 Absorption and emission 342 11.9 Singlet–triplet transitions 395 10.2 Raman processes 344 The fate of excited species 396Molecular rotation 344 11.10 Non-radiative decay 396
- 12. CONTENTS j xi11.11 Radiative decay 397 Magnetic resonance parameters 45211.12 The conservation of orbital symmetry 399 13.11 Shielding constants 45211.13 Electrocyclic reactions 399 13.12 The diamagnetic contribution to shielding 45611.14 Cycloaddition reactions 401 13.13 The paramagnetic contribution to shielding 45811.15 Photochemically induced electrocyclic reactions 403 13.14 The g-value 45911.16 Photochemically induced cycloaddition reactions 404 13.15 Spin–spin coupling 462PROBLEMS 406 13.16 Hyperﬁne interactions 463 13.17 Nuclear spin–spin coupling 46712 The electric properties of molecules 407 PROBLEMS 471The response to electric ﬁelds 407 14 Scattering theory 473 12.1 Molecular response parameters 407 12.2 The static electric polarizability 409 The formulation of scattering events 473 12.3 Polarizability and molecular properties 411 14.1 The scattering cross-section 473 12.4 Polarizabilities and molecular spectroscopy 413 14.2 Stationary scattering states 475 12.5 Polarizabilities and dispersion forces 414 12.6 Retardation effects 418 Partial-wave stationary scattering states 479 14.3 Partial waves 479Bulk electrical properties 418 14.4 The partial-wave equation 480 12.7 The relative permittivity and the electric 14.5 Free-particle radial wavefunctions and the susceptibility 418 scattering phase shift 481 12.8 Polar molecules 420 14.6 The JWKB approximation and phase shifts 484 12.9 Refractive index 422 14.7 Phase shifts and the scattering matrix element 486Optical activity 427 14.8 Phase shifts and scattering cross-sections 48812.10 Circular birefringence and optical rotation 427 14.9 Scattering by a spherical square well 49012.11 Magnetically induced polarization 429 14.10 Background and resonance phase shifts 49212.12 Rotational strength 431 14.11 The Breit–Wigner formula 494 14.12 Resonance contributions to the scatteringPROBLEMS 434 matrix element 49513 The magnetic properties of molecules 436 Multichannel scattering 497 14.13 Channels for scattering 497The descriptions of magnetic ﬁelds 436 14.14 Multichannel stationary scattering states 498 13.1 The magnetic susceptibility 436 14.15 Inelastic collisions 498 13.2 Paramagnetism 437 14.16 The S matrix and multichannel resonances 501 13.3 Vector functions 439 13.4 Derivatives of vector functions 440 The Green’s function 502 13.5 The vector potential 441 14.17 The integral scattering equation and Green’s functions 502Magnetic perturbations 442 14.18 The Born approximation 504 13.6 The perturbation hamiltonian 442 Appendix 14.1 The derivation of the Breit–Wigner 13.7 The magnetic susceptibility 444 formula 508 13.8 The current density 447 Appendix 14.2 The rate constant for reactive 13.9 The diamagnetic current density 450 scattering 50913.10 The paramagnetic current density 451 PROBLEMS 510
- 13. xii j CONTENTSFurther information 513 15 Vector coupling coefﬁcients 535 Spectroscopic properties 537Classical mechanics 513 16 Electric dipole transitions 537 1 Action 513 17 Oscillator strength 538 2 The canonical momentum 515 18 Sum rules 540 3 The virial theorem 516 19 Normal modes: an example 541 4 Reduced mass 518 The electromagnetic ﬁeld 543 ¨Solutions of the Schrodinger equation 519 20 The Maxwell equations 543 5 The motion of wavepackets 519 21 The dipolar vector potential 546 6 The harmonic oscillator: solution by factorization 521 Mathematical relations 547 7 The harmonic oscillator: the standard solution 523 22 Vector properties 547 8 The radial wave equation 525 23 Matrices 549 9 The angular wavefunction 526 10 Molecular integrals 527 11 The Hartree–Fock equations 528 Further reading 553 12 Green’s functions 532 13 The unitarity of the S matrix 533 Appendix 1 557 Appendix 2 562Group theory and angular momentum 534 Answers to selected problems 563 14 The orthogonality of basis functions 534 Index 565
- 14. PREFACEMany 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 bookas 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 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
- 15. xiv j PREFACE 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
- 16. Introduction and orientation0.1 Black-body radiation There are two approaches to quantum mechanics. One is to follow the historical development of the theory from the ﬁrst indications that the0.2 Heat capacities whole fabric of classical mechanics and electrodynamics should be held in doubt to the resolution of the problem in the work of Planck, Einstein,0.3 The photoelectric and Heisenberg, Schrodinger, and Dirac. The other is to stand back at a point ¨ Compton effects late in the development of the theory and to see its underlying theore- tical structure. The ﬁrst is interesting and compelling because the theory0.4 Atomic spectra is seen gradually emerging from confusion and dilemma. We see experi-0.5 The duality of matter 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 sT 4 ð0:1Þ
- 17. 2 j INTRODUCTION AND ORIENTATION The Stefan–Boltzmann constant, s, is independent of the material from which the body is composed, and its modern value is 56.7 nW mÀ2 KÀ4. So, a region Detected of area 1 cm2 of a black body at 1000 K radiates about 6 W if all frequencies radiation are 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’s Pinhole displacement law, lmax T ¼ constant ð0:2Þ Container at a temperature T with the constant equal to 2.9 mm K.Fig. 0.1 A black-body emitter can be One of the most challenging problems in physics at the end of the nine-simulated by a heated container with teenth century was to explain these two laws. Lord Rayleigh, with minor helpa pinhole in the wall. The from James Jeans,1 brought his formidable experience of classical physics toelectromagnetic radiation is reﬂected bear on the problem, and formulated the theoretical Rayleigh–Jeans law formany times inside the container and the energy density e(l), the energy divided by the volume, in the wavelengthreaches thermal equilibrium with thewalls. range l to l þ dl: 8pkT deðlÞ ¼ rðlÞ dl rðlÞ ¼ 4 ð0:3Þ l where k is Boltzmann’s constant (k ¼ 1.381 Â 10 À 23 J KÀ1). This formula summarizes 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, 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 be transferred only in discrete amounts, the oscillator can have the energies 0, 25 hn, 2hn, . . . , and no other energy. Classical physics allowed a continuous variation in energy, so even a very high frequency oscillator could be excited 20 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 /(8π(kT )5/(hc)4) 15 see, of certain other properties), and the need for a minimum excitation energy effectively switches off oscillators of very high frequency, and hence 10 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: 5 8phc eÀhc=lkT rðlÞ ¼ ð0:4Þ 0 l5 1 À eÀhc=lkT 0 0.5 1.0 1.5 2.0 This expression, which is plotted in Fig. 0.2, avoids the ultraviolet cata- kT /hc strophe, and ﬁts the observed energy distribution extraordinarily well if weFig. 0.2 The Planck distribution. take h ¼ 6.626 Â 10À34 J s. Just as the Rayleigh–Jeans law epitomizes the failure 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).)
- 18. 0.2 HEAT CAPACITIES j 3 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 KÀ1 molÀ1 (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 3R ¼ 24.94 J KÀ1 molÀ1, where R is the gas constant (R ¼ NAk, with NA Avogadro’s constant). That the heat capacities were smaller than predicted was a serious embarrassment. Einstein recognized the similarity between this problem and black-body 3 radiation, for if each atomic oscillator required a certain minimum energy Debye 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 Einstein be smaller than expected. He applied Planck’s suggestion for electromagnetic 2 oscillators to the material, atomic oscillators of the solid, and deduced theCV,m /R following expression: & 2 1 yE eyE =2T CV;m ðTÞ ¼ 3RfE ðTÞ fE ðTÞ ¼ Á ð0:5aÞ T 1 À eyE =T where the Einstein temperature, yE, is related to the frequency of atomic 0 oscillators by yE ¼ hn/k. The function CV,m(T)/R is plotted in Fig. 0.3, and 0 0.5 1 1.5 2 T/ closely reproduces the experimental curve. In fact, the ﬁt is not particularly good at very low temperatures, but that can be traced to Einstein’sFig. 0.3 The Einstein and Debye assumption that all the atoms oscillated with the same frequency. When thismolar heat capacities. The restriction was removed by Debye, he obtainedsymbol y denotes the Einsteinand Debye temperatures, 3 Z yD =T T x4 exrespectively. Close to T ¼ 0 the CV;m ðTÞ ¼ 3RfD ðTÞ fD ðTÞ ¼ 3 dx ð0:5bÞ yD ðe x À 1Þ2Debye heat capacity is 0proportional to T3. where the Debye temperature, yD, is related to the maximum frequency of the oscillations 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;
- 19. 4 j INTRODUCTION AND ORIENTATION 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 later in 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 picture accounts 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 and called 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 linear dependence 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 À 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 m2 c4 þ p2 c2 ð0:7Þ where c is the speed of light. In the case of a photon, E ¼ hn and m ¼ 0, so hn h p¼ ¼ ð0:8Þ c l
- 20. 0.4 ATOMIC SPECTRA j 5This 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,dl ¼ lf À li, when the radiation was scattered through an angle y: dl ¼ 2lC sin2 1 y 2 ð0:9Þwhere lC ¼ h/mec is called the Compton wavelength of the electron(lC ¼ 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 accumulationof 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 spectraThere 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, ~, where ~ ¼ n/c, ﬁtted the expression n n 1 1 ~ ¼ RH 2 À 2 n ð0:10Þ 2 nwhere RH has come to be known as the Rydberg constant for hydrogen(RH ¼ 1.097 Â 105 cmÀ1) and n ¼ 3, 4, . . . . Rydberg’s name is commemoratedbecause he generalized this expression to accommodate all the transitions inatomic hydrogen. Even more generally, the Ritz combination principle statesthat the frequency of any spectral line could be expressed as the differencebetween two quantities, or terms: ~ ¼ T1 À T2 n ð0:11ÞThis expression strongly suggests that the energy levels of atoms are conﬁnedto discrete values, because a transition from one term of energy hcT1 toanother of energy hcT2 can be expected to release a photon of energy hc~, or nhn, equal to the difference in energy between the two terms: this argument
- 21. 6 j INTRODUCTION AND ORIENTATION 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: me4 1 En ¼ À Á n ¼ 1, 2, . . . ð0:12Þ 8h2 e2 n2 0 where 1/m ¼ 1/me þ 1/mp and e0 is the vacuum permittivity, a fundamental constant. 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: h l¼ ð0:13Þ p 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
- 22. 0.5 THE DUALITY OF MATTER j 7(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 succeededin 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 expectan electron in a state of deﬁnite linear momentum (and hence wavelength) tobe 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 Schrodinger formulated their seemingly different but ¨equally 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 Schrodinger’s by a few ¨months, 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. Schrodinger’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 imaginationto unfold........................................................................................................ 2. It has been pointed out by M. Jammer that J.J. Thomson was awarded the Nobel Prize forshowing that the electron is a particle, and G.P. Thomson, his son, was awarded the Prize forshowing that the electron is a wave. (See The conceptual development of quantum mechanics,McGraw-Hill, New York (1966), p. 254.)
- 23. 8 j INTRODUCTION AND ORIENTATIONPROBLEMS0.1 Calculate the size of the quanta involved in the 0.13 At what wavelength of incident radiation do theexcitation of (a) an electronic motion of period 1.0 fs, relativistic and non-relativistic expressions for the ejection(b) a molecular vibration of period 10 fs, and (c) a pendulum of electrons from potassium differ by 10 per cent? That is,of period 1.0 s. find l such that the non-relativistic and relativistic linear momenta of the photoelectron differ by 10 per cent. Use0.2 Find the wavelength corresponding to the maximum in F ¼ 2.3 eV.the Planck distribution for a given temperature, and showthat the expression reduces to the Wien displacement law at 0.14 Deduce eqn 0.9 for the Compton effect on the basis ofshort wavelengths. Determine an expression for the constant the conservation of energy and linear momentum. Hint. Usein the law in terms of fundamental constants. (This constant the relativistic expressions. Initially the electron is at restis called the second radiation constant, c2.) with energy mec2. When it is travelling with momentum p its0.3 Use the Planck distribution to confirm the energy is ðp2 c2 þ m2 c4 Þ1/2. The photon, with initial eStefan–Boltzmann law and to derive an expression for momentum h/li and energy hni, strikes the stationarythe Stefan–Boltzmann constant s. electron, is deﬂected through an angle y, and emerges with momentum h/lf and energy hnf. The electron is initially0.4 The peak in the Sun’s emitted energy occurs at about stationary (p ¼ 0) but moves off with an angle y 0 to the480 nm. Estimate the temperature of its surface on the basis incident photon. Conserve energy and both components ofof it being regarded as a black-body emitter. linear momentum. Eliminate y 0 , then p, and so arrive at an0.5 Derive the Einstein formula for the heat capacity of a expression for dl.collection of harmonic oscillators. To do so, use the 0.15 The first few lines of the visible (Balmer) series in thequantum mechanical result that the energy of a harmonic spectrum of atomic hydrogen lie at l/nm ¼ 656.46, 486.27,oscillator of force constant k and mass m is one of the values 434.17, 410.29, . . . . Find a value of RH, the Rydberg(v þ 1)hv, with v ¼ (1/2p)(k/m)1/2 and v ¼ 0, 1, 2, . . . . Hint. 2 constant for hydrogen. The ionization energy, I, is theCalculate the mean energy, E, of a collection of oscillators minimum energy required to remove the electron. Find itby substituting these energies into the Boltzmann from the data and express its value in electron volts. How isdistribution, and then evaluate C ¼ dE/dT. I related to RH? Hint. The ionization limit corresponds to0.6 Find the (a) low temperature, (b) high temperature n ! 1 for the final state of the electron.forms of the Einstein heat capacity function. 0.16 Calculate the de Broglie wavelength of (a) a mass of0.7 Show that the Debye expression for the heat capacity is 1.0 g travelling at 1.0 cm sÀ1, (b) the same at 95 per cent ofproportional to T3 as T ! 0. the speed of light, (c) a hydrogen atom at room temperature (300 K); estimate the mean speed from the equipartition0.8 Estimate the molar heat capacities of metallic sodium principle, which implies that the mean kinetic energy of an(yD ¼ 150 K) and diamond (yD ¼ 1860 K) at room atom is equal to 3kT, where k is Boltzmann’s constant, (d) 2temperature (300 K). an electron accelerated from rest through a potential0.9 Calculate the molar entropy Rof an Einstein solid at difference of (i) 1.0 V, (ii) 10 kV. Hint. For the momentum TT ¼ yE. Hint. The entropy is S ¼ 0 ðCV =TÞdT. Evaluate the in (b) use p ¼ mv/(l À v2/c2)1/2 and for the speed in (d) use 1 2integral numerically. 2mev ¼ eV, where V is the potential difference.0.10 How many photons would be emitted per second by a 0.17 Derive eqn 0.12 for the permitted energy levels for thesodium lamp rated at 100 W which radiated all its energy electron in a hydrogen atom. To do so, use the followingwith 100 per cent efficiency as yellow light of wavelength (incorrect) postulates of Bohr: (a) the electron moves in a589 nm? circular orbit of radius r around the nucleus and (b) the angular momentum of the electron is an integral multiple of0.11 Calculate the speed of an electron emitted from a clean h , that is me vr ¼ n . Hint. Mechanical stability of the hpotassium surface (F ¼ 2.3 eV) by light of wavelength (a) orbital motion requires that the Coulombic force of300 nm, (b) 600 nm. attraction between the electron and nucleus equals the0.12 When light of wavelength 195 nm strikes a certain metal centrifugal force due to the circular motion. The energy ofsurface, electrons are ejected with a speed of 1.23 Â 106 m sÀ1. the electron is the sum of the kinetic energy and potentialCalculate the speed of electrons ejected from the same metal (Coulombic) energy. For simplicity, use me rather than thesurface by light of wavelength 255 nm. reduced mass m.
- 24. 1 The foundations of quantum mechanicsOperators in quantum mechanics The whole of quantum mechanics can be expressed in terms of a small set1.1 Linear operators of postulates. When their consequences are developed, they embrace the1.2 Eigenfunctions and eigenvalues behaviour of all known forms of matter, including the molecules, atoms, and1.3 Representations1.4 Commutation and electrons that will be at the centre of our attention in this book. This chapter non-commutation introduces the postulates and illustrates how they are used. The remaining1.5 The construction of operators chapters build on them, and show how to apply them to problems of chemical1.6 Integrals over operators interest, such as atomic and molecular structure and the properties of mole-1.7 Dirac bracket notation cules. We assume that you have already met the concepts of ‘hamiltonian’ and1.8 Hermitian operatorsThe postulates of quantum ‘wavefunction’ in an elementary introduction, and have seen the Schrodinger ¨mechanics equation written in the form1.9 States and wavefunctions Hc ¼ Ec1.10 The fundamental prescription1.11 The outcome of measurements This chapter establishes the full signiﬁcance of this equation, and provides1.12 The interpretation of the a foundation for its application in the following chapters. wavefunction1.13 The equation for the wavefunction ¨1.14 The separation of the Schrodinger Operators in quantum mechanics equationThe specification and evolution ofstates An observable is any dynamical variable that can be measured. The principal1.15 Simultaneous observables mathematical difference between classical mechanics and quantum mechan-1.16 The uncertainty principle ics is that whereas in the former physical observables are represented by1.17 Consequences of the uncertainty functions (such as position as a function of time), in quantum mechanics they principle are represented by mathematical operators. An operator is a symbol for an1.18 The uncertainty in energy and time instruction to carry out some action, an operation, on a function. In most of1.19 Time-evolution and conservation the examples we shall meet, the action will be nothing more complicated than laws multiplication or differentiation. Thus, one typical operation might beMatrices in quantum mechanics multiplication by x, which is represented by the operator x Â . Another1.20 Matrix elements operation might be differentiation with respect to x, represented by the1.21 The diagonalization of the hamiltonian operator d/dx. We shall represent operators by the symbol O (omega) in ¨The plausibility of the Schrodinger general, but use A, B, . . . when we want to refer to a series of operators.equation We shall not in general distinguish between the observable and the operator1.22 The propagation of light that represents that observable; so the position of a particle along the x-axis1.23 The propagation of particles will be denoted x and the corresponding operator will also be denoted x (with1.24 The transition to quantum mechanics 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.
- 25. 10 j 1 THE FOUNDATIONS OF QUANTUM MECHANICS 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 ð1:1Þ where 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 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 another function. Differentiation of sin x, for instance, gives cos x. However, in certain 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 ð1:2Þ where o is a constant. Such an equation is called an eigenvalue equation. The function eax is an eigenfunction of the operator d/dx because (d/dx)eax ¼ aeax, 2 which is a constant (a) multiplying the original function. In contrast, eax is ax2 ax2 not an eigenfunction of d/dx, because (d/dx)e ¼ 2axe , which is a con- 2 stant (2a) times a different function of x (the function xeax ). 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(3x þ 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 ax ¼ a cos ax and (d/dx)cos ax ¼ Àa sin ax. Answer. The operator operating on the function yields d2 d 2 cosð3x þ 5Þ ¼ ðÀ3 sinð3x þ 5ÞÞ ¼ À9 cosð3x þ 5Þ dx dx and we see that the original function reappears multiplied by the eigen- value À9. Self-test 1.1. Is the function e3x þ 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.
- 26. 1.2 EIGENFUNCTIONS AND EIGENVALUES j 11That is, if fn is an eigenfunction of an operator O with eigenvalue on (so Ofn ¼on fn), then1 a general function g can be expressed as the linear combination X g¼ cn fn ð1:3Þ nwhere the cn are coefﬁcients and the sum is over a complete set of functions.For instance, the straight line g ¼ ax can be recreated over a certain range bysuperimposing an inﬁnite number of sine functions, each of which is aneigenfunction of the operator d2/dx2. Alternatively, the same function may beconstructed from an inﬁnite number of exponential functions, which areeigenfunctions of d/dx. The advantage of expressing a general function as alinear combination of a set of eigenfunctions is that it allows us to deduce theeffect 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 X X X Og ¼ O cn fn ¼ cn Ofn ¼ c n on f n n n n A special case of these linear combinations is when we have a set ofdegenerate 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 offunctions, we can write X k X k X k X k Og ¼ O cn fn ¼ cn Ofn ¼ cn ofn ¼ o cn fn ¼ og n¼1 n¼1 n¼1 n¼1This expression has the form of an eigenvalue equation (Og ¼ og). 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 eÀ2ix is an eigenfunction of the operator d2/dx2, where i ¼ (À1)1/2. Method. Consider an arbitrary linear combination ae2ix þ beÀ2ix and see if the function satisﬁes an eigenvalue equation. Answer. First we demonstrate that e2ix and eÀ2ix are degenerate eigenfunctions. d2 Æ2ix d e ¼ ðÆ2ieÆ2ix Þ ¼ À4eÆ2ix dx2 dx....................................................................................................... 1. See P.M. Morse and H. Feschbach, Methods of theoretical physics, McGraw-Hill, New York(1953).
- 27. 12 j 1 THE FOUNDATIONS OF QUANTUM MECHANICS where we have used i2 ¼ À1. Both functions correspond to the same eigen- value, À4. Then we operate on a linear combination of the functions. d2 ðae2ix þ beÀ2ix Þ ¼ À4ðae2ix þ beÀ2ix Þ dx2 The linear combination satisﬁes the eigenvalue equation and has the same eigenvalue (À4) 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 À9] 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 c1 ¼ c2 ¼ Á Á Á ¼ 0) for which X ci gi ¼ 0 i 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 linearly independent 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 (or whatever coordinate is speciﬁed) and the linear momentum parallel to x is represented by differentiation with respect to x. Explicitly: h q Position representation: x ! x Â px ! ð1:5Þ i qx where ¼ h=2p. Why the linear momentum should be represented in pre- h 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
- 28. 1.4 COMMUTATION AND NON-COMMUTATION j 13multiplication by px and the position operator is represented by differentia-tion with respect to px. Explicitly: h q Momentum representation: x ! À px ! px Â ð1:6Þ i qpxThere are other representations. We shall normally use the position repres-entation when the adoption of a representation is appropriate, but we shallalso see that many of the calculations in quantum mechanics can be doneindependently of a representation.1.4 Commutation and non-commutationAn 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(2 /i)x ¼ (2 /i)x2, h hwhereas (pxx)x2 ¼ pxx3 ¼ (3 /i)x2. The operators x and px do not commute. h The quantity AB À BA is called the commutator of A and B and is denoted[A, B]: ½A, B ¼ AB À BA ð1:7ÞIt is instructive to evaluate the commutator of the position and linearmomentum operators in the two representations shown above; the procedureis 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: h h qf qðxf Þ ½x, px f ¼ ðxpx À px xÞf ¼ x Â À i qx i qx h qf h h qf ¼xÂ À f ÀxÂ ¼ i f h i qx i i qx where we have used (1/i) ¼ Ài. 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]
- 29. 14 j 1 THE FOUNDATIONS OF QUANTUM MECHANICS 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 T ¼ p2/2m where m is the mass of the particle. It follows that in one dimension and in the position representation p2 1 d 2 h d2 h T¼ x ¼ ¼À ð1:8Þ 2m 2m i dx 2m dx2 In three dimensions the operator in the position representation is ( ) 2 q2 h q2 q2 2 2 h T¼À 2 þ 2þ 2 ¼À r ð1:9ÞAlthough eqn 1.9 has explicitly 2m qx qy qz 2mused Cartesian coordinates, therelation between the kinetic energy The operator r2, which is read ‘del squared’ and called the laplacian, is theoperator and the laplacian is true sum of the three second derivatives.in any coordinate system; for The operator for potential energy of a particle in one dimension, V(x), isexample, spherical polar multiplication by the function V(x) in the position representation. The same iscoordinates. 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 Àe) in the ﬁeld of a nucleus of atomic number Z is the multiplicative operator Ze2 V¼À Â ð1:10Þ 4pe0 r 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 ¼TþV 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 2 d2 h H¼À þ VðxÞ ð1:12Þ 2m dx2 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 h2 2 e2 H¼À r À ð1:13Þ 2me 4pe0 r

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