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Molecular Quantum Mechanics
Molecular Quantum Mechanics
Peter Atkins, Ronald S. Friedman
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Quantum mechanics embraces the behavior of all known forms of matter, including the atoms and molecules from which we, and all living organisms, are composed. Molecular Quantum Mechanics leads us through this absorbing yet challenging subject, exploring the fundamental physical principles that explain how all matter behaves.
With the clarity of exposition and extensive learning features that have established the book as a leading text in the field, Molecular Quantum Mechanics takes us from the foundations of quantum mechanics, through quantum models of atomic, molecular, and electronic structure, and on to discussions of spectroscopy, and the electronic and magnetic properties of molecules. Lucid explanations and illuminating artworks help to visualise the many abstract concepts upon which the subject is built.
Fully updated to reflect the latest advances in computational techniques, and enhanced with more mathematical support and worked examples than ever before, Molecular Quantum Mechanics remains the ultimate resource for those wishing to master this important subject.
Online Resource Centre
For students:
Interactive worksheets to help students master mathematical concepts through handson learning
Solutions to selected exercises and problems
For registered adopters of the book:
Figures in electronic format
Solutions to all exercises and problems
With the clarity of exposition and extensive learning features that have established the book as a leading text in the field, Molecular Quantum Mechanics takes us from the foundations of quantum mechanics, through quantum models of atomic, molecular, and electronic structure, and on to discussions of spectroscopy, and the electronic and magnetic properties of molecules. Lucid explanations and illuminating artworks help to visualise the many abstract concepts upon which the subject is built.
Fully updated to reflect the latest advances in computational techniques, and enhanced with more mathematical support and worked examples than ever before, Molecular Quantum Mechanics remains the ultimate resource for those wishing to master this important subject.
Online Resource Centre
For students:
Interactive worksheets to help students master mathematical concepts through handson learning
Solutions to selected exercises and problems
For registered adopters of the book:
Figures in electronic format
Solutions to all exercises and problems
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Year:
2010
Edition:
5
Publisher:
Oxford University Press, USA
Language:
english
Pages:
537 / 552
ISBN 10:
0199541426
ISBN 13:
9780199541423
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PDF, 18.71 MB
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Molecular Quantum Mechanics This page intentionally left blank Molecular Quantum Mechanics Fifth edition Peter Atkins University of Oxford 1 and Ronald Friedman Indiana Purdue Fort Wayne 3 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 Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With ofﬁces in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam 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, 2011 The moral rights of the authors have been asserted Database right Oxford University Press (maker) Second edition 1983 Third edition 1997 Fourth edition 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 the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Graphicraft Limited, Hong Kong Printed in Italy on acidfree paper by L.E.G.O. S.p.A. ISBN 978–0–19–954142–3 10 9 8 7 6 5 4 3 2 1 Brief contents Introduction and orien; tation 1 1 The foundations of quantum mechanics Mathematical background 1 Complex numbers 9 35 2 Linear motion and the harmonic oscillator Mathematical background 2 Differential equations 37 66 3 Rotational motion and the hydrogen atom 69 4 Angular momentum Mathematical background 3 Vectors 99 121 5 Group theory Mathematical background 4 Matrices 125 166 6 Techniques of approximation 170 7 Atomic spectra and atomic structure 210 8 An introduction to molecular structure 258 9 Computational chemistry 295 10 Molecular rotations and vibrations Mathematical background 5 Fourier series and Fourier transforms 338 379 11 Molecular electronic transitions 382 12 The electric properties of molecules 407 13 The magnetic properties of molecules Mathematical background 6 Scalar and vector functions 437 474 14 Scattering theory 476 Resource section 513 Answers to selected exercises and problems 523 Index 529 This page intentionally left blank Detailed contents Introduction and orientation 1 0.1 Blackbody radiation 1 0.2 Heat capacities 2 0.3 The photoelectric and Compton effects 3 0.4 Atomic spectra 4 0.5 1 The duality of matter The foundations of quantum mechanics Operators in quantum mechanics 1.1 Linear operators 5 9 9 10 1.2 Eigenfunctions and eigenvalues 10 1.3 Representations 12 1.4 Commutation and noncommutation 13 1.5 The construction of operators 14 1.6 Integrals over operators 15 1.7 1.8 Dirac bracket and matrix notation (a) Dirac brackets (b) Matrix notation 16 16 17 Hermitian operators (a) The definition of hermiticity (b) The consequences of hermiticity 17 18 19 The postulates of quantum mechanics 2.2 Some general remarks on the Schrödinger equation (a) The curvature of the wavefunction (b) Qualitative solutions (c) The emergence of quantization (d) Penetration into nonclassical regions Translational motion 38 38 39 40 40 41 2.3 Energy and momentum 2.4 The significance of the coefficients 42 2.5 The flux density 43 2.6 Wavepackets 44 Penetration into and through barriers 41 44 2.7 An infinitely thick potential wall 45 2.8 A barrier of finite width (a) The case E < V (b) The case E > V 46 46 48 2.9 The Eckart potential barrier 48 Particle in a box 49 2.10 The solutions 50 2.11 Features of the solutions 51 2.12 The twodimensional square well 52 2.13 Degeneracy 53 The harmonic oscillator 54 20 2.14 The solutions 55 20 2.15 Properties of the solutions 57 1.10 The fundamental prescription 21 2.16 The classical limit 58 1.11 The outcome of measurements 22 1.12 The interpretation of the wavefunction 24 2.1 The motion of wavepackets 60 1.13 The equation for the wavefunction 24 2.2 The harmonic oscillator: solution by factorization 61 1.14 The separation of the Schrödinger equation 25 2.3 The harmonic oscillator: the standard solution 62 26 2.4 The virial theorem 62 1.9 States and wavefunctions The specification and evolution of states Further information 60 1.15 Simultaneous observables 27 Mathematical background 2 Differential equations 66 1.16 The uncertainty principle 28 MB2.1 The structure of differential equations 66 1.17 Consequences of the uncertainty principle 30 MB2.2 The solution of ordinary differential equations 66 1.18 The uncertainty in energy and time 31 MB2.3 The solution of partial differential equations 67 1.19 Timeevolution and conservation laws 31 Mathematical background 1 Complex numbers 35 MB1.1 Definitions 35 MB1.2 Polar representation 35 MB1.3 Operations 36 2 Linear motion and the harmonic oscillator The characteristics of wavefunctions 2.1 Constraints on the wavefunction 37 37 37 3 Rotational motion and the hydrogen atom Particle on a ring 69 69 3.1 The hamiltonian and the Schrödinger equation 69 3.2 The angular momentum 70 3.3 The shapes of the wavefunctions 71 3.4 The classical limit 72 3.5 The circular square well (a) The separation of variables (b) The radial solutions 73 73 73 viii  DETAILED CONTENTS 75 Mathematical background 3 Vectors The Schrödinger equation and its solution (a) The wavefunctions (b) The allowed energies 75 77 78 MB3.1 Definitions 121 MB3.2 Operations 121 3.7 The angular momentum of the particle 78 3.8 Properties of the solutions 80 3.9 The rigid rotor 81 Particle on a sphere 3.6 3.10 Particle in a spherical well Motion in a Coulombic field 83 84 3.11 The Schrödinger equation for hydrogenic atoms 84 3.12 The separation of the relative coordinates 85 3.13 The radial Schrödinger equation (a) The solutions close to the nucleus for l = 0 (b) The solutions close to the nucleus for l ≠ 0 (c) The complete solutions (d) The allowed energies 86 86 86 87 89 3.14 Probabilities and the radial distribution function 121 MB3.3 The graphical representation of vector operations 122 MB3.4 Vector differentiation 123 5 Group theory The symmetries of objects 125 125 5.1 Symmetry operations and elements 126 5.2 The classification of molecules 127 The calculus of symmetry 131 5.3 The definition of a group 5.4 Group multiplication tables 132 89 5.5 Matrix representations 133 3.15 Atomic orbitals (a) sorbitals (b) porbitals (c) d and forbitals (d) The radial extent of orbitals 90 91 91 93 93 5.6 The properties of matrix representations 136 5.7 The characters of representations 138 5.8 Characters and classes 139 5.9 Irreducible representations 140 3.16 The degeneracy of hydrogenic atoms 94 5.10 The great and little orthogonality theorems Further information 95 3.1 The angular wavefunctions 95 3.2 Reduced mass 95 3.3 The radial wave equation 96 4 Angular momentum 99 Reduced representations 4.1 99 The operators and their commutation relations (a) The angular momentum operators (b) The commutation relations 99 100 100 4.2 Angular momentum observables 101 4.3 The shift operators 102 The definition of the states 102 142 146 5.11 The reduction of representations 146 5.12 Symmetryadapted bases (a) Projection operators (b) The generation of symmetryadapted bases 147 148 149 The symmetry properties of functions 5.13 The transformation of porbitals The angular momentum operators 131 151 151 5.14 The decomposition of directproduct bases 152 5.15 Directproduct groups 154 5.16 Vanishing integrals 156 5.17 Symmetry and degeneracy 158 The full rotation group 5.18 The generators of rotations 159 159 5.19 The representation of the full rotation group 161 5.20 Coupled angular momenta 162 4.4 The effect of the shift operators 103 4.5 The eigenvalues of the angular momentum 104 4.6 The matrix elements of the angular momentum 106 Applications 163 4.7 The orbital angular momentum eigenfunctions 108 Mathematical background 4 Matrices 166 4.8 Spin (a) The properties of spin (b) The matrix elements of spin operators 110 110 111 MB4.1 Definitions The angular momenta of composite systems 4.9 111 The specification of coupled states 111 4.10 The permitted values of the total angular momentum 112 4.11 The vector model of coupled angular momenta 114 4.12 The relation between schemes (a) Singlet and triplet coupled states (b) The construction of coupled states (c) States of the configuration d2 115 115 116 117 4.13 The coupling of several angular momenta 118 166 MB4.2 Matrix addition and multiplication 166 MB4.3 Eigenvalue equations 167 6 Techniques of approximation 170 The semiclassical approximation 170 Timeindependent perturbation theory 174 6.1 Perturbation of a twolevel system 174 6.2 Manylevel systems (a) Formulation of the problem 176 177 DETAILED CONTENTS (b) The firstorder correction to the energy (c) The firstorder correction to the wavefunction (d) The secondorder correction to the energy 177 178 180 6.3 Comments on the perturbation expressions (a) The role of symmetry (b) The closure approximation 181 182 183 6.4 Perturbation theory for degenerate states 185 (a) The Hartree–Fock equations (b) Oneelectron energies  ix 235 237 7.17 Restricted and unrestricted Hartree–Fock calculations 238 7.18 Density functional procedures (a) The Thomas–Fermi method (b) The Thomas–Fermi–Dirac method 239 239 242 243 243 244 245 6.5 The Rayleigh ratio 187 7.19 Term symbols and transitions of manyelectron atoms (a) Russell–Saunders coupling (b) Excluded terms (c) Selection rules 6.6 The Rayleigh–Ritz method 189 7.20 Hund’s rules and Racah parameters 245 7.21 Alternative coupling schemes 247 Variation theory 187 The Hellmann–Feynman theorem 191 Timedependent perturbation theory 192 6.7 6.8 6.9 The timedependent behaviour of a twolevel system (a) The solutions (b) The Rabi formula 192 193 195 Manylevel systems: the variation of constants (a) The general formulation (b) The effect of a slowly switched constant perturbation (c) The effect of an oscillating perturbation 196 196 198 199 Transition rates to continuum states 201 Atoms in external fields 7.22 The normal Zeeman effect 249 7.24 The Stark effect 251 Further information The Hartree–Fock equations 253 7.2 Vector coupling schemes 253 Functionals and functional derivatives 254 Solution of the Thomas–Fermi equation 255 202 6.11 Lifetime and energy uncertainty 204 7.4 7 Electric dipole transitions Atomic spectra and atomic structure The spectrum of atomic hydrogen 206 206 210 210 8 The formulation of the approximation 258 An application: the hydrogen moleculeion (a) The molecular potential energy curves (b) The molecular orbitals 260 260 261 210 Selection rules (a) The Laporte selection rule (b) Constraints on Dl (c) Constraints on Dml (d) Higherorder transitions 211 211 212 212 213 7.3 Orbital and spin magnetic moments (a) The orbital magnetic moment (b) The spin magnetic moment 214 214 215 7.4 Spin–orbit coupling 215 7.5 The finestructure of spectra 217 7.6 Term symbols and spectral details 218 7.7 The detailed spectrum of hydrogen 219 8.6 7.9 8.3 262 Linear combinations of atomic orbitals (a) The secular determinant (b) The Coulomb integral (c) The resonance integral (d) The LCAOMO energy levels for the hydrogen moleculeion (e) The LCAOMOs for the hydrogen moleculeion 265 266 8.4 The hydrogen molecule 266 8.5 Configuration interaction 268 Diatomic molecules (a) Criteria for atomic orbital overlap and bond formation (b) Homonuclear diatomic molecules (c) Heteronuclear diatomic molecules 269 269 270 272 Molecular orbital theory of polyatomic molecules 262 263 263 265 274 8.7 Symmetryadapted linear combinations (a) The H2O molecule (b) The NH3 molecule 227 8.8 Conjugated psystems and the Hückel approximation 276 229 8.9 Ligand field theory (a) The SALCs of the octahedral complex (b) The molecular orbitals of the octahedral complex (c) The groundstate configuration: low and highspin complexes (d) Tanabe–Sugano diagrams (e) Jahn–Teller distortion (f) Metal–ligand p bonding 282 282 282 Excited states of helium 224 7.10 The spectrum of helium 225 7.11 The Pauli principle Manyelectron atoms Molecular orbital theory 221 221 221 222 258 8.2 The energies of the transitions The helium atom (a) Atomic units (b) The orbital approximation 258 8.1 7.2 7.8 An introduction to molecular structure The Born–Oppenheimer approximation 7.1 The structure of helium 253 7.1 6.10 The Einstein transition probabilities 6.1 248 7.23 The anomalous Zeeman effect 7.3 Further information 248 7.12 Penetration and shielding 230 7.13 Periodicity 232 7.14 Slater atomic orbitals 233 7.15 Slater determinants and the Condon–Slater rules 234 7.16 Selfconsistent fields 235 274 274 276 283 284 284 285 x  DETAILED CONTENTS The band theory of solids 286 Molecular rotation 340 8.10 The tightbinding approximation 286 10.3 8.11 The Kronig–Penney model 288 8.12 Brillouin zones 290 Further information 292 Rotational energy levels (a) Symmetric rotors (b) Spherical rotors (c) Linear rotors (d) Centrifugal distortion 342 342 344 344 344 10.4 Pure rotational selection rules (a) The gross selection rule (b) The specific selection rules (c) Wavenumbers of allowed transitions 345 345 345 346 8.1 9 Molecular integrals Computational chemistry The Hartree–Fock selfconsistent field method 292 295 296 10.5 Rotational Raman selection rules 347 10.6 Nuclear statistics (a) The case of CO2 (b) The case of H2 (c) A more general case 349 349 350 352 9.1 The formulation of the approach 296 9.2 The Hartree–Fock approach 297 9.3 The Roothaan equations 298 The vibrations of diatomic molecules 353 9.4 The selection of basis sets (a) Gaussiantype orbitals (b) The construction of contracted Gaussians (c) Calculational accuracy and the basis set 302 303 305 306 10.7 The vibrational energy levels of diatomic molecules (a) Harmonic oscillation (b) Anharmonic oscillation 353 353 354 10.8 Vibrational selection rules (a) The gross selection rule (b) The specific selection rule (c) The effect of anharmonicities on allowed transitions 356 356 357 358 10.9 Vibration–rotation spectra of diatomic molecules 358 Electron correlation 307 9.5 Configuration state functions 308 9.6 Configuration interaction 309 9.7 CI calculations 310 9.8 Multiconfiguration methods 312 9.9 Møller–Plesset manybody perturbation theory 313 9.10 The coupledcluster method (a) Formulation of the method (b) The coupledcluster equations Density functional theory 315 315 315 317 9.11 The Hohenberg–Kohn existence theorem 317 9.12 The Hohenberg–Kohn variational theorem 319 9.13 The Kohn–Sham equations 319 9.14 The exchange–correlation challenge (a) Local density approximations (b) More elaborate functionals 321 321 322 Gradient methods and molecular properties 323 9.15 Energy derivatives and the Hessian matrix 324 9.16 Analytical procedures 326 Semiempirical methods 326 9.17 Conjugated πelectron systems (a) The Hückel approximation (b) The Pariser–Parr–Pople method 327 327 328 9.18 General procedures 329 10.10 Vibrational Raman transitions of diatomic molecules 360 The vibrations of polyatomic molecules 361 10.11 Normal modes (a) Potential energy (b) Normal coordinates (c) Vibrational wavefunctions and energies 362 362 363 364 10.12 Vibrational and Raman selection rules for polyatomic molecules (a) Infrared activity (b) Raman activity (c) Group theory and molecular vibrations 365 365 366 366 10.13 Further effects on vibrational and rotational spectra (a) The effects of anharmonicity (b) Coriolis forces (c) Inversion doubling 369 369 372 373 Further information 374 10.1 Centrifugal distortion 374 10.2 Normal modes: an example 375 Mathematical background 5 Fourier series and Fourier transforms 379 MB5.1 Fourier series 379 332 MB5.2 Fourier transforms 380 9.19 Force fields 332 MB5.3 The convolution theorem 381 9.20 Quantum mechanics–molecular mechanics 333 Molecular mechanics 11 Molecular electronic transitions 10 Molecular rotations and vibrations 382 338 The states of diatomic molecules 382 Spectroscopic transitions 338 11.1 The Hund coupling cases 382 10.1 Absorption and emission 338 11.2 Decoupling and Ldoubling 384 10.2 Raman processes 339 11.3 Selection and correlation rules 386 DETAILED CONTENTS Vibronic transitions 387 11.4 The Franck–Condon principle 388 11.5 The rotational structure of vibronic transitions 390 The electronic spectra of polyatomic molecules 391 11.6 Symmetry considerations 391 11.7 Chromophores 392 11.8 Vibronically allowed transitions 393 11.9 Singlet–triplet transitions 395 13 The magnetic properties of molecules  xi 437 The description of magnetic fields 437 13.1 Basic concepts 437 13.2 Paramagnetism 439 13.3 The vector potential (a) The formulation of the vector potential (b) Gauge invariance 440 441 442 396 Magnetic perturbations 443 396 13.4 The perturbation hamiltonian 443 11.11 Radiative decay (a) Fluorescence (b) Phosphorescence 398 398 398 13.5 The magnetic susceptibility (a) Expressions for the susceptibility (b) Contributions to the susceptibility (c) The role of the gauge 444 445 446 448 Excited states and chemical reactions 399 13.6 11.12 The conservation of orbital symmetry 399 The current density (a) Real wavefunctions (b) Orbitally degenerate states, zero field (c) Orbitally nondegenerate states, nonzero field 449 450 450 451 The fates of excited states 11.10 Nonradiative decay 11.13 Electrocyclic reactions 399 11.14 Cycloaddition reactions 401 13.7 The diamagnetic current density 452 11.15 Photochemically induced electrocyclic reactions 402 13.8 The paramagnetic current density 452 11.16 Photochemically induced cycloaddition reactions 404 12 The electric properties of molecules 407 Magnetic resonance parameters 454 13.9 454 454 455 455 457 Shielding constants (a) The nuclear field (b) The hamiltonian (c) The firstorder correction to the energy (d) Contributions to the shielding constant The response to electric fields 407 12.1 Molecular response parameters 407 13.10 The diamagnetic contribution to shielding 12.2 The static electric polarizability (a) The mean polarizability and polarizability volume (b) The polarizability and molecular properties (c) Polarizabilities and molecular spectroscopy (d) Polarizabilities and dispersion interaction (e) Retardation effects 409 409 411 412 413 416 13.11 The paramagnetic contribution to shielding 459 13.12 The gvalue (a) The spin hamiltonian (b) Formulating the gvalue 460 460 461 13.13 Spin–spin coupling 462 13.14 Hyperfine interactions (a) Dipolar coupling (b) The Fermi contact interaction (c) The total interaction 463 464 465 466 13.15 Nuclear spin–spin coupling (a) The formulation of the problem (b) Coupling through a chemical bond 467 468 470 Further information 471 Bulk electrical properties 417 12.3 The relative permittivity and the electric susceptibility (a) Nonpolar molecules (b) Polar molecules 417 418 419 12.4 Refractive index (a) The dynamic polarizability (b) The molar refractivity (c) The refractive index and dispersion 421 422 424 424 13.1 The hamiltonian in the presence of a magnetic field 471 The dipolar vector potential 471 Optical activity 425 13.2 12.5 Circular birefringence and optical rotation 425 12.6 Magnetically induced polarization 427 Mathematical background 6 Scalar and vector functions 12.7 Rotational strength (a) Symmetry properties (b) Optical rotatory dispersion (c) Estimation of rotational strengths 429 429 429 430 Further information 432 12.1 Oscillator strength 432 12.2 Sum rules 432 12.3 The Maxwell equations (a) The general form of the equations (b) The equations for fields in a vacuum (c) The propagation of fields in a polarizable medium (d) Propagation in chiral media 433 433 433 434 434 458 474 MB6.1 Definitions 474 MB6.2 Differentiation 474 14 Scattering theory 476 The fundamental concepts 476 14.1 The scattering matrix 476 14.2 The scattering crosssection 479 xii  DETAILED CONTENTS Elastic scattering 480 14.3 Stationary scattering states (a) The scattering amplitude (b) The differential crosssection 480 481 482 14.4 Scattering by a central potential (a) The partialwave stationary scattering state (b) The partialwave equation (c) The scattering phase shift (d) The scattering matrix element (e) The scattering crosssection 483 483 484 485 487 489 Scattering by a spherical square well (a) The Swave radial wavefunction and phase shift (b) Background and resonance phase shifts (c) The Breit–Wigner formula (d) The resonance contribution to the scattering matrix element 491 491 492 494 Resource section 513 Further reading 513 496 1 Character tables and direct products 516 Methods of approximation (a) The WKB approximation (b) The Born approximation 497 498 499 2 Vector coupling coefficients 520 3 Wigner–Witmer rules 521 Multichannel scattering 503 Answers to selected exercises and problems 523 14.7 504 Index 529 14.5 14.6 The scattering matrix for multichannel processes 14.8 14.9 Inelastic scattering (a) The form of the multichannel stationary scattering state (b) Scattering amplitude and crosssections (c) The closecoupling approximation 504 505 505 506 Reactive scattering 507 14.10 The S matrix and multichannel resonances 508 Further information 509 14.1 509 Green’s functions Preface In this new edition we have sought to reﬂect the changing emphasis in the applications of molecular quantum mechanics and to make the text more accessible without sacriﬁcing rigour. We describe below the key features used to achieve this aim. There are many new organizational and content changes throughout. All the artwork has been redrawn and augmented. We have introduced and placed brief Mathematical background sections following the chapter where a particular mathematical technique is used for the ﬁrst time. The Further information sections of the previous edition have either been incorporated into the Mathematical background sections or moved to the end of the chapter to which they most directly relate. New Further information sections, such as one on the Thomas–Fermi method (Chapter 7), have also been introduced. Subsections have been added to the chapters to help make the material more digestible. A lot of material has been shipped to diCerent locations to make the exposition more systematic, to improve its ﬂow, or to remove diAcult material from early parts of the text. Problem solving is always a diAcult but important area, and we have paid special attention to helping students. In the chapters we have made extensive use of brief illustrations to provide quick and succinct examples of the use of equations, in some cases simply to establish the order of magnitude of a property and not leave it as an abstract entity. As in previous editions, there are numerous Worked examples, which require a more detailed approach; they are accompanied by Selftests, which let readers test their grasp of the approach in a related problem. To provide a more gentle series of tests at the end of each chapter we have divided the questions into straightforward Exercises and more demanding Problems. Answers to numerical questions are given at the end of the book. A Student’s solutions guide provides more detailed solutions to designated Exercises and Problems, and an Instructor’s guide provides detailed solutions to them all. Both guides are available in the book’s Online Resource Centre.1 One almost entirely rewritten chapter on computational chemistry (Chapter 9) deals in detail with density functional theory, one of the most widely used current techniques. We have adopted the novel pedagogical device of developing the theory around the H2 molecule, which though too simple to be of much professional interest has the advantage that the approach can be illustrated in explicit detail, so illustrating exactly what otherwise obscure computer programs are achieving. Computational problems that are best solved by using software are available on the website.1,2 We have encouraged readers to develop their understanding by using interactive spreadsheets on the website, which provide opportunities to explore numerous equations presented in the text by substituting numerical values for variable parameters. 1 www.oxfordtextbooks.co.uk/orc/mqm5e/ Implementation of these and other techniques can be achieved by using the Students’ edition of Spartan software; readers can purchase a copy of this software at special discount by visiting www.wavefunction.com and using the discount code OUPMQM. 2 xiv  PREFACE We have expanded discussion in numerous places in the text to provide more introductory material (for example, classical magnetism in Chapter 13) or a more systematic treatment (for example, Hund coupling cases in Chapter 11). The present edition also contains other signiﬁcant additions, including new or expanded discussions of circular and spherical square wells (Chapter 3); the semiclassical approximation (Chapter 6); Racah parameters, Condon–Slater rules and atomic units (Chapter 7); Wigner–Witmer rules (Chapter 11); gauge invariance (Chapter 13); and reactive scattering (Chapter 14). We are very grateful to all those who have helped in the preparation of this new edition, including the following, who reviewed the textbook at various stages along the way: Temer Ahmadi, Villanova University Arjun Berera, University of Edinburgh Alexander Brown, University of Alberta Fabio Canepa, University of Genoa Jonathan Flynn, University of Southampton Ian Jamie, Macquarie University Karl Jalkanen, Curtin University of Technology Peter Karadakov, University of York Thomas Miller, California Institute of Technology Alejandro Perdomo, Harvard University Charles Trapp, University of Louisville Donald Truhlar, University of Minnesota We also appreciate the insights and advice of the many others, too numerous to name here, who oCered suggestions over the years. Finally, we wish publically to thank our publisher who has been invariably helpful and understanding. PWA RSF Introduction and orientation 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 theoretical structure. The ﬁrst is interesting and compelling because the theory is seen gradually emerging from confusion and dilemma. We see experiment 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 revolutions 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 Blackbody radiation In retrospect—and as will become clear—we can now see that theoretical physics hovered on the edge of formulating a quantum mechanical description 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 blackbody 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 nineteenth 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 = sT 4 (0.1) The Stefan–Boltzmann constant, s, is independent of the material from which the 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 l = 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, 0.1 Blackbody radiation 1 0.2 Heat capacities 2 0.3 The photoelectric and Compton effects 3 0.4 Atomic spectra 4 0.5 The duality of matter 5  2 INTRODUCTION AND ORIENTATION Detected radiation lmaxT = constant (0.2) with the constant equal to 2.9 mm K. One of the most challenging problems in physics at the end of the nineteenth century was to explain these two laws. Each one of them concentrated on ﬁnding an expression for the energy density E(l),the energy in a region divided by the volume of the region, and writing the contribution dE(l) from radiation in the wavelength range l to l + dl as dE(l) = rR(l)dl Container at a temperature T Pinhole Fig. 0.1 A blackbody emitter can be simulated by a heated container with a pinhole in the wall. The electromagnetic radiation is reflected many times inside the container and reaches thermal equilibrium with the walls. Spectral density, r(l)/8π(kT )5/(hc)4 25 20 15 10 5 0 0 0.5 1 1.5 2 Wavelength, l/(hc /kT ) Fig. 0.2 The Planck distribution. Marginal comment Using the Worksheet entitled Equation 0.5 on this text’s website, explore the dependence of the Planck distribution on the temperature. (0.3) where rR(l) is the spectral density of states at the wavelength l. 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 this quantity rR(l) = 8pkT l4 (0.4) where k is Boltzmann’s constant (k = 1.381 × 10−23 J K−1). This formula summarizes the failure of classical physics. Because rR(l) becomes inﬁnite as l approaches zero, eqn 0.4 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, hn, 2hn, . . . , and no other energy. Classical physics allowed a continuous variation 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 since short wavelength radiation could be emitted at even low temperature. 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 eCectively switches oC 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 spectral density of a blackbody radiator: rR(l) = 8phc e−hc/lkT l5 1 − e−hc/lkT (0.5) This expression, which is plotted in Fig. 0.2, avoids the ultraviolet catastrophe, and ﬁts the observed energy distribution extraordinarily well if we take h = 6.626 × 10−34 J s. Just as the Rayleigh–Jeans law epitomizes the failure of classical physics, the Planck distribution epitomizes the inception of 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. The French scientists Dulong and Petit, for example, were able to propose their law that ‘the atoms of all simple 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).) 0.3 THE PHOTOELECTRIC AND COMPTON EFFECTS CV,m(T) = 3RfE(T) fE(T) = ! qE eqE/2T # · @ T 1 − eqE /T $ 2 (0.6a) where the Einstein temperature, qE, is related to the frequency of atomic oscillators by qE = hn/k. The function CV,m(T)/R, which is plotted in Fig. 0.3, provides a reasonable ﬁt to experimental heat capacities except 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,m(T) = 3RfD(T) fD(T) = 3 ATD C qD F 3 qD /T 0 x4ex dx (e − 1)2 x (0.6b) where the Debye temperature, qD, is related to the maximum frequency of the oscillations that can be supported by the solid. This expression gives a very good ﬁt to experimental heat capacities. The importance of Einstein’s contribution is that it complemented Planck’s. Planck had shown that the energy of radiation is quantized; Einstein showed that matter is quantized too. Quantization appeared 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 formulated 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 eIect, the emission of electrons from metals when they are exposed to ultraviolet radiation. The puzzling features of the eCect were that the emission was instantaneous when the radiation was applied however low its intensity, but there was no emission, whatever the intensity of the radiation, unless its frequency exceeded a threshold value typical of each metallic element. It was also known that the kinetic energy of the ejected electrons varied linearly with the frequency of the incident radiation. 3 3 Debye Molar heat capacity, CV,m /R 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 those 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 capacity, CV,m, of a monatomic solid should be equal to 3R = 24.94 J K−1 mol−1, where R is the gas constant (R = 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 blackbody 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:  Einstein 2 1 0 0 0.5 1 1.5 Temperature, T/q 2 Fig. 0.3 The Einstein and Debye molar heat capacities. The symbol q denotes the Einstein and Debye temperatures, respectively. Close to T = 0 the Debye heat capacity is proportional to T 3. Marginal comment Using the Worksheet entitled Equation 0.6a on this text’s website, explore the variation of the Einstein molar heat capacity with temperature for diIerent values of the Einstein temperature. Marginal comment Using the Worksheet entitled Equation 0.6b on this text’s website, explore the variation of the Debye molar heat capacity with temperature for diIerent values of the Debye temperature. 4  INTRODUCTION AND ORIENTATION Einstein pointed out that all the observations fell into place if the electromagnetic 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 eCect 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 eCect, 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 = hn − F (0.7) If photons do have a particlelike character, then they should possess a linear momentum, p. The relativistic expression relating a particle’s energy to its mass and momentum is E2 = m2c4 + p2c 2 (0.8) where c is the speed of light. In the case of a photon, E = hn and m = 0, so p= hn h = c l (0.9) 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 Xrays scattered from the electrons in a graphite target, and found the results ﬁtted the following formula for the shift in wavelength, dl = lf − li, when the radiation was scattered through an angle q: dl = 2lC sin2 12 q (0.10) where lC = h/mec is called the Compton wavelength of the electron (lC = 2.426 pm). This formula is derived on the supposition that a photon does indeed have a linear momentum h/l and that the scattering event is like a collision between two particles (Problem 0.12). 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 wavelike. 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 visible spectrum of atomic hydrogen had a very simple appearance, and by 1885 J. Balmer had already noticed that their wavenumbers, â, where â = n/c, ﬁtted the expression A 1 1D â = RH C 2 − 2F 2 n (0.11) 0.5 THE DUALITY OF MATTER where RH has come to be known as the Rydberg constant for hydrogen (RH = 1.097 × 105 cm−1) and n = 3, 4, . . . . Rydberg’s name is commemorated because he generalized this expression to accommodate all the transitions in atomic hydrogen, not just those in the visible region of the electromagnetic spectrum. Even more generally, the Ritz combination principle states that the frequency of any spectral line could be expressed as the diCerence between two quantities, or terms: â = T1 − T2 (0.12) 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 hcâ, or hn, equal to the diCerence in energy between the two terms: this argument 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 permitted energy levels of an electron in a hydrogen atom (Problem 0.18): En = − me4 1 · 8h2e20 n2 n = 1,2, . . . (0.13) where 1/m = 1/me + 1/mp and e0 is the vacuum permittivity, a fundamental constant. This formula marked the ﬁrst appearance in quantum mechanics of a quantum number, here denoted n, which identiﬁes the state of the system and is used to calculate its energy. Equation 0.13 is consistent with Balmer’s formula (eqn 0.11) 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= h p (0.14) We have seen this formula already (eqn 0.9), in connection with the properties 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  5 6  INTRODUCTION AND ORIENTATION 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 (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 diCraction of electrons by the ranks of atoms in a metal crystal acting as a diCraction grating. Davisson and Germer, who performed this experiment in 1925 using a crystal of nickel, found that the diCraction 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 diCraction of electrons by thin ﬁlms of celluloid and gold.2 If electrons—if all particles—have wavelike 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 complementarity 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 diCerence between it and classical mechanics, in which the speciﬁcation of exact trajectories (positions and momenta) was a central theme. The consummation of all this faltering progress came in 1926 when Werner Heisenberg and Erwin Schrödinger formulated their seemingly diCerent 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 diCerent types of calculation. 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 diCerential equations, was more familiar in style but still equally revolutionary in implication. It is more suited to elementary manipulations and to the calculation of numerical results, and we shall employ it in that manner. You should already be familiar with an application of Schrödinger’s formulation in the case of the ‘particle in a box’, a particle conﬁned by inﬁnite potential energy walls to a ﬁnite region of onedimensional space. Keep the solution of the particle in a box problem (which we treat in detail in Chapter 2) in mind as we unroll the postulates of quantum mechanics in Chapter 1. ‘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, McGrawHill, New York (1966), p. 254.) PROBLEMS  7 Exercises Calculate the size of the quanta involved in the excitation of (a) an electronic motion of period 1.0 fs, (b) a molecular vibration of period 10 fs, and (c) a pendulum of period 1.0 s. *0.1 The peak in the Sun’s emitted energy occurs at about 480 nm. Estimate the temperature of its surface on the basis of it being regarded as a blackbody emitter. *0.2 An unknown metal has a speciﬁc heat capacity of 0.91 J K−1 g−1 at room temperature. Use Dulong and Petit’s law to identify the metal. *0.3 Calculate the energy of 1.00 mol photons of wavelength (a) 510 nm (green), (b) 100 m (radio), (c) 130 pm (Xray). *0.4 *0.5 Calculate the wavelength of the radiation scattered through an angle of 60o when Xrays of wavelength 25.878 pm impinge upon a graphite target. *0.6 Calculate the speed of an electron emitted from a clean potassium surface (F = 2.3 eV) by light of wavelength (a) 300 nm, (b) 600 nm. *0.7 Compute the highest and lowest wavenumbers of the spectral lines in the Balmer series for atomic hydrogen. What are the corresponding wavelengths? *0.8 Compute the energies (in joules and electronvolts) for the two lowest energy levels of an electron in a hydrogen atom. *0.9 Calculate the de Broglie wavelength of a tennis ball of mass 57 g travelling at 80 km h−1. Problems *0.1 Find the wavelength corresponding to the maximum in the Planck distribution for a given temperature, and show that the expression reduces to the Wien displacement law at short wavelengths. Determine an expression for the constant in the law in terms of fundamental constants. (This constant is called the second radiation constant, c2.) Show that the Planck distribution reduces to the Rayleigh–Jeans law at long wavelengths. 0.2 Compute the power emitted by the Sun regarding it as a blackbody radiator at 6 kK; the Sun has a surface area of 6 × 1018 m2. What energy is emitted during a 24hour period? 0.3 Derive the Einstein formula for the heat capacity of a collection of harmonic oscillators. To do so, use the quantum mechanical result that the energy of a harmonic oscillator of force constant kf and mass m is one of the values (n + 1/2)hn with n = (1/2p)(kf /m)1/2 and n = 0, 1, 2, . . . . Hint. Calculate the mean energy, E, of a collection of oscillators by substituting these energies into the Boltzmann distribution, and then evaluate C = dE/dT. *0.4 0.5 Find the (a) low temperature, (b) high temperature forms of the Einstein heat capacity function. Show that the Debye expression for the heat capacity is proportional to T 3 as T → 0. 0.6 *0.7 Estimate the molar heat capacities of metallic sodium (qD = 150 K) and diamond (qD = 1860 K) at room temperature (300 K). Calculate the molar entropy of an Einstein solid at T T = qE. Hint. The entropy is S(T) = 2 0 (CV /T)dT. Evaluate the integral numerically. 0.8 How many photons would be emitted per second by a sodium lamp rated at 100 W which radiated all its energy with 100 per cent eAciency as yellow light of wavelength 589 nm? 0.9 When ultraviolet radiation of wavelength 195 nm strikes a certain metal surface, electrons are ejected at 1.23 Mm s−1. Calculate the speed of electrons ejected from the same metal surface by radiation of wavelength 255 nm. *0.10 At what wavelength of incident radiation do the relativistic and nonrelativistic expressions for the ejection of electrons from potassium diCer by 10 per cent? That is, ﬁnd l such that the nonrelativistic and relativistic linear momenta of the photoelectron diCer by 10 per cent. Use F = 2.3 eV. 0.11 Deduce eqn 0.10 for the Compton eCect on the basis of the conservation of energy and linear momentum. Hint. Use the relativistic expressions. Initially the electron is at rest with energy mec 2. When it is travelling with momentum p its energy is (p2c 2 + me2c4)1/2. The photon, with initial momentum h/li and energy hni, strikes the stationary electron, is deflected through an angle q, and emerges with momentum h/lf and energy hnf. The electron is initially stationary (p = 0) but moves oC with an angle q′ to the incident photon. Conserve energy and both components of linear momentum (parallel and perpendicular to the 0.12 * Indicates that the solution can be found in the Student’s solution manual, which is available in the Online Resource Centre accompanying this book. Go to www.oxfordtextbooks.co.uk/orc/mqm5e/ 8  INTRODUCTION AND ORIENTATION initial momentum). Eliminate q′, then p, and so arrive at an expression for dl. The ﬁrst few lines of the visible (Balmer) series in the spectrum of atomic hydrogen lie at l/nm = 656.46, 486.27, 434.17, 410.29, . . . . Find a value of RH, the Rydberg constant for hydrogen. The ionization energy, I, is the minimum energy required to remove the electron. Find it from the data and express its value in electronvolts (1 eV = 1.602 × 10−19 J). How is I related to RH? Hint. The ionization limit corresponds to n → ∞ for the ﬁnal state of the electron. *0.13 Use eqn 0.13 for the energy levels of an electron in a hydrogen atom to determine an expression for the Rydberg constant (as a wavenumber) in terms of fundamental constants. Evaluate the Rydberg constant (a) using the reduced mass of a hydrogen atom, (b) substituting the mass of the electron for the reduced mass. (c) What is the percentage diCerence between the two expressions? 0.14 Derive an expression that could be used to determine the mass of a deuteron from the shift in spectral lines of 1 H and 2H. 0.15 A measure of the strength of coupling between the electromagnetic ﬁeld and an electric charge is the *0.16 ﬁnestructure constant, a = e2/4pHce0. Express the Rydberg constant (as a wavenumber) in terms of this constant. 0.17 Calculate the de Broglie wavelength of (a) a mass of 1.0 g travelling at 1.0 cm s−1, (b) the same at 95 per cent of the speed of light, (c) a hydrogen atom at room temperature (300 K); estimate the mean speed from the equipartition principle, which implies that the mean kinetic energy of an atom is equal to 32 kT, where k is Boltzmann’s constant, (d) an electron accelerated from rest through a potential diCerence of (i) 1.0 V, (ii) 10 kV. Hint. For the momentum in (b) use p = mv/(l − v 2/c 2)1/2 and for the speed in (d) use 1 /2mev 2 = eV, where V is the potential diCerence. Derive eqn 0.13 for the permitted energy levels for the electron in a hydrogen atom. To do so, use the following (incorrect) postulates of Bohr: (a) the electron moves in a circular orbit of radius r around the nucleus and (b) the angular momentum of the electron is an integral multiple of H, that is mevr = nH. Hint. Mechanical stability of the orbital motion requires that the Coulombic force of attraction between the electron and nucleus equals the centrifugal force due to the circular motion. The energy of the electron is the sum of the kinetic energy and potential (Coulombic) energy. For simplicity, use me rather than the reduced mass m. 0.18 * Indicates that the solution can be found in the Student’s solution manual, which is available in the Online Resource Centre accompanying this book. Go to www.oxfordtextbooks.co.uk/orc/mqm5e/ 1 The foundations of quantum mechanics 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 molecules. 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 Hy = Ey This chapter establishes the full signiﬁcance of this equation and provides a foundation for its application in the following chapters. It will also be helpful to bear in mind the solutions of the Schrödinger equation for a particle in a box, which we also presume to be generally familiar. In brief, for a particle of mass m in a onedimensional box of length L: • The energies are quantized, with En = n h /8mL , n = 1,2, . . . 2 2 Operators in quantum mechanics 9 1.1 Linear operators 1.2 Eigenfunctions and eigenvalues 10 10 1.3 Representations 12 1.4 Commutation and noncommutation 13 1.5 The construction of operators 14 1.6 Integrals over operators 15 1.7 Dirac bracket and matrix notation 16 1.8 Hermitian operators 17 2 • The normalized wavefunctions are yn(x) = (2/L)1/2 sin(n px/L) We use these solutions to illustrate some of the points made in this chapter (they are developed formally in Chapter 2). A ﬁnal preparatory point is that quantum mechanics makes extensive use of complex numbers: they are reviewed in Mathematical background 1 following this chapter. Operators in quantum mechanics An observable is any dynamical variable that can be measured. The principal mathematical diCerence between classical mechanics and quantum mechanics 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 diCerentiation. Thus, one typical operation might be multiplication by x, which is represented by the operator x ×. Another operation might be diCerentiation with respect to x, represented by the operator d /dx. We shall represent operators by the symbol W (uppercase omega) in general, 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 xaxis 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. The postulates of quantum mechanics 20 1.9 States and wavefunctions 20 1.10 The fundamental prescription 21 1.11 The outcome of measurements 22 1.12 The interpretation of the wavefunction 24 1.13 The equation for the wavefunction 24 1.14 The separation of the Schrödinger equation 25 The speciﬁcation and evolution of states 26 1.15 Simultaneous observables 27 1.16 The uncertainty principle 28 1.17 Consequences of the uncertainty principle 30 1.18 The uncertainty in energy and time 31 1.19 Timeevolution and conservation laws 31 10  1 THE FOUNDATIONS OF QUANTUM MECHANICS 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. 1.1 Linear operators The operators we shall meet in quantum mechanics are all linear. A linear operator is one for which W(af ) = aWf (1.1) where a is a constant and f is a function. Multiplication is a linear operation; so are diCerentiation and integration. An example of a nonlinear operation is that of taking the logarithm of a function, because it is not true, for example, that log 2x = 2 log x for all x. The operation of taking a square is also nonlinear, because it is not true, for example, that (2x)2 = 2x2 for all x. 1.2 Eigenfunctions and eigenvalues In general, when an operator operates on a function, the outcome is another function. DiCerentiation 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 W if it satisﬁes an equation of the form Wf = wf (1.2) where w 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 = aeax, which is a constant (a) multiplying the original function. In contrast, eax is not an eigenfunction of d /dx, because (d /dx)eax = 2axeax , which is a constant (2a) times a diFerent function of x (the function xeax ). The constant w in an eigenvalue equation is called the eigenvalue of the operator W. 2 2 2 2 Example 1.1 Determining if a function is an eigenfunction Is the wavefunction y1(x) = (2/L)1/2 sin(px/L) of a particle in a box 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 = a cos ax and (d /dx)cos ax = −a sin ax. Answer The operation on the function yields d2 y1(x) A 2 D 1/2 d2sin(px/L) A 2 D 1/2A p D dcos(px/L) = = C LF C LF C LF dx2 dx dx2 =− A pD2 A 2 D 1/2A p D 2 sin(px/L) = − y (x) C LF 1 C LF C LF and we see that the original function reappears multiplied by a constant, so y1(x) is an eigenfunction of d2/dx2, and its eigenvalue is −(p/L)2. Selftest 1.1 Is the function e3x+5 an eigenfunction of the operator d2/dx2 and, if so, what is the corresponding eigenvalue? [Yes; 9] 1.2 EIGENFUNCTIONS AND EIGENVALUES An important point is that a general function can be expanded in terms of all the eigenfunctions of an operator, a socalled complete set of functions. The functions used to construct a general function are called basis functions. That is, if fn is an eigenfunction of an operator W with eigenvalue wn (so Wfn = wn fn), then a general function g can be expressed as the linear combination g = ∑ cn fn (1.3) n where the cn are coeAcients and the sum is over a complete set of basis functions fn. For instance, the straight line g = ax can be recreated over a certain range (−L/2 ≤ x ≤ L/2) by superimposing an inﬁnite number of sine functions, each of which is an eigenfunction of the operator d2/dx2: ∞ g(x) = A aL D {(−1)n+1/n}sin(2npx/L) C p F∑ n=1 (The formulation and illustration of expressions like this are described in Mathematical background 5 following Chapter 10.) The same function may also 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 eCect of an operator on a function that is not one of its own eigenfunctions. Thus, the eCect of W on g in eqn 1.3, using the property of linearity, is simply Wg = W ∑ cn fn = ∑ cn Wfn = ∑ cn wn fn n n (1.4) n 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 W, and that they all correspond to the same eigenvalue w: Wfn = wfn with n = 1,2, . . . , k (1.5) Then it is quite easy to show that any linear combination of the functions fn is also an eigenfunction of W with the same eigenvalue w. The proof is as follows. For an arbitrary linear combination g of the degenerate set of functions, we can write k Wg = W ∑ cn fn = n=1 k k ∑ c Wf = ∑ c wf n n=1 n n n n=1 k = w ∑ cn fn = wg (1.6) n=1 This expression has the form of an eigenvalue equation (Wg = wg). 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 i = (−1)1/2. Method Consider an arbitrary linear combination ae2ix + be−2ix and see if the func tion satisﬁes an eigenvalue equation. Answer First we demonstrate that e2ix and e−2ix are degenerate eigenfunctions: d2 ±2ix d e = (±2ie±2ix) = −4e±2ix dx2 dx ››  11 12  1 THE FOUNDATIONS OF QUANTUM MECHANICS where we have used i2 = −1. Both functions correspond to the same eigenvalue, − 4. Then we operate on a linear combination of the functions: d2 (ae2ix + be−2ix) = −4(ae2ix + be−2ix) dx2 The linear combination satisﬁes the eigenvalue equation and has the same eigenvalue (− 4) as do the two exponential functions. Selftest 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 construct 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 = c2 = · · · = 0) for which ∑c g = 0 (1.7) i i i A set of functions that are not linearly independent are 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. A brief illustration Consider an H1s orbital on each hydrogen atom in NH3, and denote them sA, sB, and sC. The three linear combinations 2sA − sB − sC 2sB − sC − sA 2sC − sA − sB are not linearly independent (their sum is zero). Put another way: the third can be expressed as the sum of the ﬁrst two. On the other hand, the linear combinations 2sA − sB − sC sA + sB + sC sB − sC are linearly independent, and any one cannot be expressed as a sum or diCerence of the other two. The three p orbitals (px, py, pz) of a shell of an atom are linearly independent. It is possible to form any number of sets of linearly independent combinations of them, but each set has no more than three members. One such set (which will be discussed further in Section 3.15) is p+1 = − 1.3 1 (px + ipy) 2 1/2 p−1 = 1 (px − ipy) 2 1/2 p0 = p z 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, diCerentiation, and so on. DiCerent choices of the operators that correspond to a particular observable give rise to the diCerent representations of quantum mechanics, because the explicit 1.4 COMMUTATION AND NONCOMMUTATION 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 diCerentiation with respect to x. Explicitly: Position representation: x → x × px → H [ i [x (1.8) where H = h/2π. We replace the partial derivative, [/[x, by an ordinary derivative, d /dx, when considering onedimensional systems in which x is the only variable. Why the linear momentum should be represented in precisely this manner is 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 multiplication by px and the position operator is represented by diCerentiation with respect to px. Explicitly: Momentum representation: x → − H [ i [px px → px × (1.9) There are other representations. We shall normally use the position representation 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 noncommutation 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 ≠ AB. We say that, in general, operators do not commute. A brief illustration Consider the operators x and px and a speciﬁc function x2. In the position representation, (xpx)x2 = x × H d 2 x = −2iHx 2 i dx whereas (px x)x2 = H d x × x 2 = −3iHx 2 i dx We see that because the outcomes are diCerent, 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]: [A,B] = AB − BA (1.10) 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.  13 14  1 THE FOUNDATIONS OF QUANTUM MECHANICS Example 1.3 Evaluating 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] pro ceeds as follows: [x,px]f = (xpx − px x)f = x × =x× H [f H [(xf ) − i [x i [x H [f H H [f − f−x× = iHf i [x i i [x where we have used (1/i) = −i. This derivation is true for any function f, so in terms of the operators themselves, [x,px] = iH. The righthand side of this expression should be interpreted as the operator ‘multiply by the constant iH’. Selftest 1.3 Evaluate the same commutator in the momentum representation. [Same] The noncommutation of operators is highly reminiscent of the noncommutation of matrix multiplication. Indeed, Heisenberg formulated his version of quantum mechanics, which is called matrix mechanics, by representing position and linear momentum by the matrices x and px , and requiring that xpx − px x = iH1 where 1 is the unit matrix, a square matrix with all diagonal elements equal to 1 and all others 0. (Matrices are discussed in Mathematical background 4 following Chapter 5.) 1.5 The construction of operators Operators for other observables of interest can be constructed from the operators 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 = p2/ 2m, where m is the mass of the particle and p2 (in general W 2) means that the operator is applied twice in succession. It follows that in one dimension and in the position representation 2 T= A brief comment Although eqn 1.11b 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. These alternative versions of the laplacian are given in Mathematical background 3 following Chapter 4. p2x 1 AH d D H2 d2 = =− C F 2m 2m i dx 2m dx2 (1.11a) In three dimensions the operator in the position representation is T=− H2 ! [2 [2 [2 # H2 2 + + = − ∇ 2m @ [x2 [y 2 [z 2 $ 2m (1.11b) The operator ∇2, which is read ‘del squared’ and called the laplacian, is the sum of the three second derivatives. Because the potential energy depends only on position coordinates, 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 and charge Ze is the multiplicative operator 1.6 INTEGRALS OVER OPERATORS V=− Ze2 4pe0r (1.12) where r is the distance from the nucleus to the electron. As here, it is usual to omit the multiplication sign from multiplicative operators, but it should not be forgotten that such expressions imply multiplications of whatever stands on their right. The operator for the total energy of a system is called the hamiltonian operator and is denoted H: H=T+V (1.13) 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 operator for a particle of mass m moving in one dimension is H=− H2 d2 + V(x) 2m dx2 (1.14) where V(x) is the operator for the potential energy. Similarly, the hamiltonian operator (from now on, just ‘the hamiltonian’) for an electron of mass me in a hydrogen atom is H=− H2 2 e2 ∇ − 2me 4pe0r (1.15) The general prescription for constructing operators in the position representation 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)[/[x (and likewise for 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, it will turn out that we shall need to evaluate certain integrals. These integrals all have the form I= f*Wf dt m n (1.16) where f* m is the complex conjugate (Mathematical background 1) of fm. In this integral dt is the volume element. In one dimension, dt can be identiﬁed as dx; in three dimensions it is dx dy dz. The integral is taken over the entire space available to the system, which is typically from x = −∞ to x = +∞ (and similarly for the other coordinates). 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 W in eqn 1.16 is simply multiplication by 1, the integral is called an overlap integral and commonly denoted S: S= f *f dt m n (1.17)  15 16  1 THE FOUNDATIONS OF QUANTUM MECHANICS It is helpful to regard S as a measure of the similarity of two functions: when S = 0, the functions are classiﬁed as orthogonal, rather like two perpendicular vectors. 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 1.17 for m = n. A function fm is said to be normalized (strictly, normalized to 1) if f*f dt = 1 (1.18) m m The integration here, as (by convention) it always is when dt is used to denote the volume element, is over all space. 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* = N. We could take N to have any complex phase, but because all observables are proportional to N*N, the phase cancels and it is simply convenient to make N real. The procedure is illustrated in the following example. Example 1.4 Normalizing a function The ground state wavefunction of a particle in a box is y1(x) = N sin(px/L) between x = 0 and x = L and is zero elsewhere. Conﬁrm that N = (2/L)1/2. Method To ﬁnd N we substitute this expression into eqn 1.18, evaluate the inte gral, and select N to ensure normalization. Note that ‘all space’ in eCect extends from x = 0 to x = L because the function is identically zero outside this region. Answer The necessary integration is L f *f dt = N sin (px/L)dx = 2 2 1 2 LN 2 0 where we have used ∫ sin2 ax dx = (x/2) − (sin 2ax)/4a + constant. For this integral to be equal to 1, we require N = (2/L)1/2. Selftest 1.4 Normalize the function f = eij, where j ranges from 0 to 2p. [N = 1/(2p)1/2] A set of functions fn that are (a) normalized and (b) mutually orthogonal are said to satisfy the orthonormality condition: f*f dt = d m n (1.19) mn In this expression, dmn denotes the Kronecker delta, which is 1 when m = n and 0 otherwise. 1.7 Dirac bracket and matrix notation The appearance of many quantum mechanical expressions is greatly simpliﬁed by adopting a simpliﬁed notation. (a) Dirac brackets In the Dirac bracket notation integrals are written as follows: 〈m  W  n〉 = f* Wf dt m n (1.20) 1.8 HERMITIAN OPERATORS The symbol  n〉 is called a ket, and denotes the state described by the function fn. Similarly, the symbol 〈n  is called a bra, and denotes the complex conjugate of the function, f*n. When a bra and ket are strung together with an operator between them, as in the bracket 〈m  W  n〉, the integral in eqn 1.20 is to be understood. When the operator is simply multiplication by 1, the 1 is omitted and we use the convention 〈m  n〉 = f*f dt (1.21) m n This notation is very elegant. For example, the normalization integral becomes 〈n  n〉 = 1 and the orthogonality condition becomes 〈m  n〉 = 0 for m ≠ n. The combined orthonormality condition (eqn 1.19) is then 〈m  n〉 = dmn (1.22) A further point is that, as can readily be deduced from the deﬁnition of a Dirac bracket, 〈m  n〉 = 〈n  m〉* (1.23) (b) Matrix notation A matrix, M, is an array of numbers (which may be complex), called matrix elements. Each element is speciﬁed by quoting the row (r) and column (c) that it occupies, and denoting the matrix element as Mrc. The rules of matrix algebra are set out in Mathematical background 4 following Chapter 5, where they are centre stage. Dirac brackets are commonly abbreviated to Wmn, which immediately suggests that they are elements of a matrix. For this reason, the Dirac bracket 〈m  W  n〉 is often called a matrix element of the operator W. A diagonal matrix element Wnn is then a bracket of the form 〈n  W  n〉 with the bra and the ket referring to the same state. We shall often encounter sums over products of Dirac brackets that have the form ∑ 〈r  A  s〉〈s  B  c〉 s If the brackets that appear in this expression are interpreted as matrix elements, then we see that it has the form of a matrix multiplication, and we may write ∑ 〈r  A  s〉〈s  B  c〉 = ∑ A rs s Bsc = (AB)rc = 〈r  AB  c〉 (1.24) s That is, the sum is equal to the single matrix element (bracket) of the product of operators AB. Comparison of the ﬁrst and last terms in this line of equations also allows us to write the symbolic relation ∑  s〉〈s  = 1 (1.25) s This completeness relation (or closure relation) is exceptionally useful for developing quantum mechanical equations. It is often used in reverse: the matrix element 〈r  AB  c〉 can always be split into a sum of two factors by regarding it as 〈r  A1B  c〉 and then replacing the 1 by a sum over a complete set of states of the form in eqn 1.25. 1.8 Hermitian operators ‘Hermitian operators’ are central to the development of quantum theory. Here we deﬁne what it means to be Hermitian and then unfold the consequences of that property.  17 18  1 THE FOUNDATIONS OF QUANTUM MECHANICS (a) The deﬁnition of hermiticity An operator is Hermitian if it satisﬁes the following relation: 1 5* f* m Wfn dt = 2 f* n Wfm dt6 3 7 (1.26a) for any two functions fm and fn. An alternative version of this deﬁnition is f*Wf dt = (Wf )*f dt m n m (1.26b) n This expression is obtained by taking the complex conjugate of each term on the righthand side of eqn 1.26a. In terms of the Dirac notation, the deﬁnition of hermiticity is 〈m  W  n〉 = 〈n  W  m〉* (1.26c) Example 1.5 Conﬁrming the hermiticity of operators Show that the position and momentum operators in the position representation are Hermitian. Method We need to show that the operators satisfy eqn 1.26a. In some cases (the position operator, for instance), the hermiticity is obvious as soon as the integral is written down. When a diCerential operator is used, it may be necessary to use integration by parts at some stage in the argument to transfer the diCerentiation from one function to another: u dv = uv − v du Answer That the position operator is Hermitian is obvious from inspection: 1 5* f* m xfn dt = fn xf* m dt = 2 f* n xfm dt6 3 7 We have used the facts that (f *)* = f and x is real. The demonstration of the hermiticity of px, a diCerential operator in the position representation, involves an integration by parts and we consider the deﬁnite integral over all space to show the disappearance of one of the terms: ∞ f* m px fn dx = −∞ = ∞ f* m −∞ 1 H dfn H dx = i dx i H2 f* m fn − i3 5 ∞ f* m dfn −∞ ∞ fn df* m6 7 = −∞ 1 H2 ∞ f* m fn  −∞ − i3 ∞ fn −∞ 5 df* m dx6 dx 7 The ﬁrst term on the right is zero (because when  x  is inﬁnite, a normalizable function must be vanishingly small; see Section 1.12). Therefore, reverting for notational simplicity to indeﬁnite integrals: 1 5* 1 5* f*p f dx = − Hi f dxd f * dx = 23 f*Hi dxd f dx67 = 23 f*p f dx67 m x n n m n m n x m Hence, the operator is Hermitian. Selftest 1.5 Show that the two operators are Hermitian in the momentum representation. 1.8 HERMITIAN OPERATORS (b) The consequences of hermiticity As we shall now see, the property of hermiticity has farreaching implications. 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 W  w〉 = w  w〉 The ket  w〉 denotes an eigenstate of the operator W in the sense that the corresponding function fw is an eigenfunction of the operator W and we are labelling the eigenstates with the eigenvalue w of the operator W. It is often convenient to use the eigenvalues as labels in this way. Multiplication from the left by 〈w  results in the equation 〈w  W  w〉 = w〈w  w〉 = w taking w〉 to be normalized. Now take the complex conjugate of both sides: 〈w  W  w〉* = w* However, by hermiticity, 〈w  W  w〉* = 〈w  W  w〉. Therefore, it follows that w = w*, which implies that the eigenvalue w is real. The second property we shall prove is as follows: Property 2. Eigenfunctions corresponding to diCerent eigenvalues of an Hermitian operator are orthogonal. That is, if we have two eigenfunctions of an Hermitian operator W with eigenvalues w and w′, with w ≠ w′, then 〈w  w′〉 = 0. For example, it follows at once that all the eigenfunctions of a particle in a box are mutually orthogonal, for as we shall see each one corresponds to a diCerent energy (the eigenvalue of the hamiltonian, an Hermitian operator). A brief illustration The (real) wavefunctions y1(x) = (2/L)1/2 sin(px/L) and y2(x) = (2/L)1/2 sin(2px/L) of a particle in a box correspond to diCerent energies (h2/8mL2 and h2/2mL2, respectively). That they are mutually orthogonal is veriﬁed by writing L y1(x)y2(x)dx = 0 2 L L sin(px/L)sin(2px/L)dx = 0 0 We have made use of a standard result to evaluate the integral; alternatively, note that sin(px/L) is an even function with respect to reﬂection in x = 1/2L whereas sin(2px/L) is odd. Proof 1.2 The orthogonality of eigenstates Suppose we have two eigenstates  w〉 and  w′〉 that satisfy the following relations: W  w〉 = w  w〉 and W  w′〉 = w′  w′〉 ››  19 20  1 THE FOUNDATIONS OF QUANTUM MECHANICS Then multiplication of the ﬁrst relation by 〈w′  and the second by 〈w  gives 〈w′  W  w〉 = w〈w′  w〉 and 〈w  W  w′〉 = w′〈w  w′〉 Now take the complex conjugate of the second relation and subtract it from the ﬁrst while using Property 1 (w′* = w′): 〈w′  W  w〉 − 〈w  W  w′〉* = w〈w′  w〉 − w′〈w  w′〉* Because W is Hermitian, the lefthand side of this expression is zero; so (noting that w′ is real and using 〈w  w′〉* = 〈w′  w〉 as explained earlier) we arrive at (w − w′)〈w′  w〉 = 0 However, because the two eigenvalues are diCerent, the only way of satisfying this relation is for 〈w′  w〉 = 0, as was to be proved. Example 1.6 Making use of the completeness relation Use the completeness relation to prove that the eigenvalues of the square of an Hermitian operator are nonnegative. Method We have to prove, for W 2  w〉 = a(w)  w〉, that a(w) ≥ 0 if W is Hermitian. If both sides of the eigenvalue equation are multiplied by 〈w , converting it to 〈w  W 2  w〉 = a(w), we see that the proof requires us to show that the expectation value on the left is nonnegative. As it has the form 〈w  WW  w〉, it suggests that the completeness relation might provide a way forward. The hermiticity of W implies that it will be appropriate to use the property 〈m  W  n〉 = 〈n  W  m〉* at some stage in the argument. Answer The diagonal matrix element 〈w  W 2  w〉 can be developed as follows: 〈w  W 2  w〉 = 〈w  WW  w〉 = ∑ 〈w  W  s〉〈s  W  w〉 s = ∑ 〈w  W  s〉〈w  W  s〉* = ∑  〈w  W  s〉  2 s ≥0 s The ﬁnal inequality follows from the fact that all the terms in the sum are nonnegative. Selftest 1.6 Show that if (Wf )* = −Wf *, then 〈W〉 = 0 for any real function f. 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 Y(r1, r2, . . . , t). 1.10 THE FUNDAMENTAL PRESCRIPTION In this statement, r1, r2, . . . are the spatial coordinates of particles 1, 2, . . . that constitute the system and t is the time, a variable parameter common to the entire system. The function Y (uppercase psi) plays a central role in quantum mechanics, and is called the wavefunction of the system (more speciﬁcally, the timedependent wavefunction). When we are not interested in how the system changes in time we shall denote the wavefunction by a lowercase psi as y(r1, r2, . . .) and refer to it as the timeindependent 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. The wavefunction must also behave in a certain way (speciﬁcally, change sign or not change sign) when the labels of identical particles are interchanged. This is the realm of the ‘Pauli principle’. The principle is properly considered to be an additional postulate of quantum mechanics, but as it requires concepts that are beyond the scope of this chapter (speciﬁcally, the classiﬁcation of particles according to their spin, their intrinsic angular momentum), we delay its introduction until Section 7.11, where it ﬁrst plays a role. 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 ya,b, . . . , where a, b, . . . are the quantum numbers. For a particle in a onedimensional box, the single quantum number is n = 1, 2, . . . . The values of these quantum numbers specify the wavefunction and thus allow the values of various physical observables to be calculated (for instance, from En = n2h2/8mL2). 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 [q,pq′] = iHdqq′ [q,q′] = 0 [pq,pq′] = 0 (1.27) where q and q′ each denote one of the coordinates x, y, z and pq and pq′ the corresponding linear momenta. The requirement that the operators are Hermitian ensures that the observables have real values (see below). Each commutation relation can be regarded as a basic, unprovable, and underivable postulate. This postulate 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.1 A brief illustration If we choose the operator for position along the coordinate q as q ×, then (as we saw in Example 1.3), the appropriate operator for pq is (H/i)[/[q, for these two operators satisfy the ﬁrst of the three commutation relations in eqn 1.27. The second of the commutation relations implies, trivially, xy = yx and the third implies px py = py px, which also follows from the properties of partial diCerentials, [2/[x[y = [2/[y[x. Similarly, if 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. 1 This prescription excludes intrinsic observables, such as spin (Section 4.8).  21 22  1 THE FOUNDATIONS OF QUANTUM MECHANICS 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 y, the mean value of the observable W in a series of measurements is equal to the expectation value of the corresponding operator. The expectation value of an operator W for an arbitrary state y is denoted 〈W〉 and deﬁned as 2y*Wy dt 〈W〉 = 2y*y dt = 〈y  W  y〉 〈y  y〉 (1.28a) If the wavefunction is chosen to be normalized to 1, then the expectation value is simply 〈W〉 = y*Wy dt = 〈y  W  y〉 (1.28b) Unless we state otherwise, from now on we shall assume that the wavefunction is normalized to 1. A brief illustration The average value of the position of a particle in the ground state of a onedimensional box is 〈x〉 = L 0 1/2 1/2 ! A 2 D sin(px/L)# x ! A 2 D sin(px/L)# dx = 2 @ C LF $ @ C LF $ L L x sin (px/L)dx 2 0 The integral evaluates to L2/4 (use mathematical software), so 〈x〉 = 1/2L: the average value of x is half the length of the box. Similarly, the average value of the linear momentum along the xaxis is 2 〈px〉 = L L 2pH H d sin(px/L) sin(px/L)dx = 2 i dx Li 0 L sin(px/L)cos(px/L)dx 0 The integral on the right is zero (use software, or note that the sine function is symmetric and the cosine function is antisymmetric around the centre of the range), so we conclude that the average linear momentum is zero: in the classical picture, the particle travels to the right as often as it travels to the left. The meaning of Postulate 3 can be unravelled as follows. First, suppose that y is an eigenfunction of W with eigenvalue w; then 〈W〉 = y*Wy dt = y*wy dt = w y*y dt = w (1.29) That is, for an ensemble of identically prepared systems all in the state y (an eigenstate of the operator W), each measurement of the property W will give the same outcome w (a real quantity, because W is Hermitian), and that outcome will therefore also be the average value of the observations. Now suppose that although the system is in an eigenstate of the hamiltonian it is not in an eigenstate of W. In this case the wavefunction can be expressed as a linear combination of eigenfunctions of W: y = ∑ cn yn n where Wyn = wn yn (1.30) 1.11 THE OUTCOME OF MEASUREMENTS In this case, the expectation value is 〈W〉 = AC∑ c y DF *W AC∑ c y DF dt = ∑ c*c y*Wy dt = ∑ c*c w y* y dt m m m n n n m n m m,n n m n n m n m,n Because the eigenfunctions form an orthonormal set, the integral in the last expression is zero if n ≠ m, is 1 if n = m, and the double sum reduces to a single sum: 〈W〉 = ∑ c*n cn wn y*n yn dt = ∑ c*n cn wn = ∑  cn 2wn n n (1.31) n That is, the expectation value is a weighted sum of the eigenvalues of W, the contribution of a particular eigenvalue to the sum being determined by the square modulus of the corresponding coeAcient in the expansion of the wavefunction. We can now interpret the diCerence between eqns 1.29 and 1.31 in the form of a subsidiary postulate: Postulate 3′. When y is an eigenfunction of the operator W, the determination of the property W always yields one result, namely the corresponding eigenvalue w. The expectation value will simply be the eigenvalue w. When y is not an eigenfunction of W, a single measurement of the property yields a single outcome which is one of the eigenvalues of W, and the probability that a particular eigenvalue wn is measured is equal to  cn 2, where cn is the coeAcient of the eigenfunction yn in the expansion of the wavefunction. Moreover, immediately after that measurement, the state of the system will be yn. That the measurement of the property W forces a general wavefunction to become an eigenfunction of the operator W, and speciﬁcally that the observation of the eigenvalue wn forces the wavefunction to become yn, is called the collapse of the wavefunction. One measurement can give only one result: a pointer can indicate only one value on a dial at any instant. In an ensemble of systems all identically prepared in some particular state y, a series of determinations will result in a series of values. The subsidiary postulate asserts that a measurement of the observable W in each case results in the pointer indicating one of the eigenvalues of the corresponding operator. If the function that describes the state of the system is an eigenfunction of W, then every pointer reading is precisely w and the mean value is also w. If the systems have been prepared in a state that is not an eigenfunction of W, then diCerent measurements give diCerent values, but each individual measurement is one of the eigenvalues of W, and the probability that a particular outcome wn is obtained is determined by the value of  cn 2. In this case, the mean value of all the observations is the weighted average of the eigenvalues. Note that in either case, the hermiticity of the operator guarantees that the observables are real. A brief illustration The wavefunction for the ground state of a particle in a box is not an eigenfunction of the linear momentum operator px = (H/i)d /dx. However, by using Euler’s relation, eix = cos x + i sin x we note that sin(px/L) = (eipx/L − e−ipx/L)/2i; therefore, we recognize that the wavefunction is the linear combination, with equal weights, of two functions that are eigenfunctions of px with eigenvalues +Hp/L and −Hp/L, respectively. Therefore in a series of measurements of the linear momentum along x, we obtain one of these two values in each measurement with equal probability. (The average, as we saw in the preceding brief illustration, is zero.)  23 24  1 THE FOUNDATIONS OF QUANTUM MECHANICS 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  y(r) 2 dt. As we have already remarked, in one dimension the volume element is dx. In three dimensions the volume element is dx dy dz. It follows from this postulate that  y(r) 2 is a probability density, in the sense that it yields a probability when multiplied by the volume dt of an inﬁnitesimal region (just as a mass density gives a mass when multiplied by a volume element). The wavefunction itself is a probability amplitude, and has no direct physical meaning. Note that whereas the probability density is real and nonnegative, the wavefunction may be complex and negative. It is usually convenient to use a normalized wavefunction; then the Born interpretation becomes an equality rather than a proportionality. A brief illustration We continue to use the groundstate wavefunction of the particle in a box, y1(x) = (2/L)1/2 sin(px/L). We can infer that the probability of ﬁnding the particle in the range x to x + dx is (2/L) sin2(px/L)dx. At the centre of the box, x = 1/2L, and at that point the probability density is (2/L) sin2(p/2) = 2/L and the probability itself is 2 dx/L. At x = 1/4L, the probability density has fallen to (2/L) sin2(p/4) = 1/L and the probability itself is dx/L. If we approximate the inﬁnitesimal quantity dx by 10−3L (so, in a box of length 1 m, we are interested in a region of length 1 mm), then the two probabilities are 0.002 (that is, in an ensemble of 500 identically prepared systems all in the state y1(x), in only one of these systems the particle will be found in the region inspected) and 0.001 (1 in 1000 inspections), respectively. The implication of the Born interpretation is that the wavefunction should be squareintegrable; that is  y  dt < ∞ 2 (1.32) because there must be a ﬁnite probability of ﬁnding the particle somewhere in the whole of space (and that probability is 1 for a normalized wavefunction). This postulate in turn implies that y → 0 as x → ± ∞, for otherwise the integral of  y 2 would be inﬁnite. We shall make frequent use of this implication throughout the text. 1.13 The equation for the wavefunction The ﬁnal postulate concerns the dynamical evolution of the wavefunction with time: Postulate 5. The wavefunction Y(r1, r2, . . . , t) evolves in time according to the equation iH [Y = HY [t (1.33) This partial diCerential equation is the celebrated Schrödinger equation introduced by Erwin Schrödinger in 1926. The operator H in the Schrödinger 1.14 THE SEPARATION OF THE SCHRÖDINGER EQUATION equation is the hamiltonian operator for the system, the operator corresponding to the total energy. A brief illustration The hamiltonian for the motion of a particle of mass m free to move in one dimension in a region where its potential energy varies with position but not time is speciﬁed in eqn 1.14. The corresponding timedependent Schrödinger equation is therefore iH [Y H2 [2Y =− + V(x)Y [t 2m [x2 (1.34) If the potential energy is that of a particle in a box, then V is inﬁnite outside the box and zero within the box (between 0 and L). Needless to say, we shall have a great deal to say about the Schrödinger equation and its solutions in the rest of the text. The separation of the Schrödinger equation 1.14 The Schrödinger equation can often be separated into equations for the time and space variation of the wavefunction. The separation is possible when the potential energy is independent of time. In one dimension the equation has the form HY = − H2 [2Y [Y + V(x)Y = iH 2m [x2 [t Equations of this form can be solved by the technique of separation of variables, in which a trial solution takes the form Y(x,t) = y(x)q(t) (1.35) When this substitution is made, we obtain − H2 d2y dq q + V(x)yq = iHy 2m dx2 dt Division of both sides of this equation by yq gives − H2 1 d2y 1 dq + V(x) = iH q dt 2m y dx2 Only the lefthand side of this equation is a function of x, so when x changes, only the lefthand side can change. But as the lefthand side is equal to the righthand side, and the latter does not change, the lefthand 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 timedependent equation separates into the following two diCerential equations: − H2 d2y + V(x)y = Ey 2m dx2 (1.36a) dq = Eq dt (1.36b) iH The second of these equations has the solution q ∝ e−iEt/H (1.37)  25 26  1 THE FOUNDATIONS OF QUANTUM MECHANICS Therefore, the complete wavefunction (Y = yq) has the form Re Y(x,t) = y(x)e−iEt/H Re (1.38) The constant of proportionality in eqn 1.37 has been absorbed into the normalization constant for y. The timeindependent wavefunction satisﬁes eqn 1.36a, which may be written in the form Hy = Ey Im Im Fig. 1.1 A wavefunction corresponding to an energy E rotates in the complex plane from real to imaginary and back to real at a frequency E/h (and circular frequency E/H). (1.39) This expression is the timeindependent Schrödinger equation, on which much of the following development will be based. This analysis stimulates several remarks. First, eqn 1.36a has the form of a standingwave equation. Therefore, as long as we are interested only in the spatial dependence of the wavefunction, it is legitimate to regard the timeindependent Schrödinger equation as a wave equation. Second, when the potential energy of the system does not depend on the time, and the system is in a state of energy E, it is a very simple matter to construct the timedependent wavefunction from the timeindependent wavefunction simply by multiplying the latter by e−iEt/H. The time dependence of such a wavefunction is simply a modulation of its phase, because we can use Euler’s relation, eix = cos x + i sin x to write e−iEt/H = cos(Et/H) − i sin(Et/H) (1.40) It follows that the timedependent factor oscillates periodically from 1 to −i to −1 to i and back to 1 with a frequency E/h and period h/E. This behaviour is depicted in Fig. 1.1. Therefore, to imagine the time variation of a wavefunction of a deﬁnite energy, think of it as rotating from positive through imaginary to negative amplitudes with a frequency proportional to the energy. A brief illustration The timeindependent groundstate wavefunction of a particle in a box is y(x) = (2/L)1/2 sin(px/L