MOLECULAR QUANTUM MECHANICS, FOURTH EDITION Peter Atkins Ronald Friedman OXFORD UNIVERSITY PRESS
Transcript
1. MOLECULAR QUANTUM MECHANICS,FOURTH EDITION Peter Atkins
Ronald FriedmanOXFORD UNIVERSITY PRESS
2. MOLECULAR QUANTUM MECHANICS
3. This page intentionally left blank
4. MOLECULARQUANTUMMECHANICSFOURTH EDITIONPeter
AtkinsUniversity of OxfordRonald FriedmanIndiana Purdue Fort
WayneAC
5. AC Great Clarendon Street, Oxford OX2 6DP Oxford University
Press is a department of the University of Oxford. It furthers the
Universitys objective of excellence in research, scholarship, and
education by publishing worldwide in Oxford New York Auckland
Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong
Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico
City Mumbai Nairobi Sao Paulo Shanghai Taipei Tokyo Toronto Oxford
is a registered trade mark of Oxford University Press in the UK and
in certain other countries Published in the United States by Oxford
University Press Inc., New York # Peter Atkins and Ronald Friedman
2005 The moral rights of the authors have been asserted. Database
right Oxford University Press (maker) First published 2005 All
rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any
means, without the prior permission in writing of Oxford University
Press,or as expressly permitted by law, or under terms agreed with
the appropriate reprographics rights organization. Enquiries
concerning reproduction outside the scope of the above should be
sent to the Rights Department, Oxford University Press, at the
address above You must not circulate this book in any other binding
or cover and you must impose this same condition on any acquirer
British Library Cataloguing in Publication Data Data available
Library of Congress Cataloging in Publication Data Data available
ISBN 0- 19--927498--3 - 10 9 8 7 6 5 4 3 2 1 Typeset by Newgen
Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain
on acid-free paper by Ashford Colour Press
6. Table of contentsPreface xiiiIntroduction and orientation 1
1 The foundations of quantum mechanics 9 2 Linear motion and the
harmonic oscillator 43 3 Rotational motion and the hydrogen atom 71
4 Angular momentum 98 5 Group theory 122 6 Techniques of
approximation 168 7 Atomic spectra and atomic structure 207 8 An
introduction to molecular structure 249 9 The calculation of
electronic structure 28710 Molecular rotations and vibrations 34211
Molecular electronic transitions 38212 The electric properties of
molecules 40713 The magnetic properties of molecules 43614
Scattering theory 473Further information 513Further reading
553Appendix 1 Character tables and direct products 557Appendix 2
Vector coupling coefcients 562Answers to selected problems 563Index
565
7. This page intentionally left blank
8. Detailed ContentsIntroduction and orientation 1 The
plausibility of the Schrodinger equation 36 1.22 The propagation of
light 36 0.1 Black-body radiation 1 1.23 The propagation of
particles 38 0.2 Heat capacities 3 1.24 The transition to quantum
mechanics 39 0.3 The photoelectric and Compton effects 4 PROBLEMS
40 0.4 Atomic spectra 5 0.5 The duality of matter 6 2 Linear motion
and the harmonicPROBLEMS 8 oscillator 431 The foundations of
quantum mechanics 9 The characteristics of acceptable wavefunctions
43 Some general remarks on the Schrodinger equation 44Operators in
quantum mechanics 9 2.1 The curvature of the wavefunction 45 1.1
Linear operators 10 2.2 Qualitative solutions 45 1.2 Eigenfunctions
and eigenvalues 10 2.3 The emergence of quantization 46 1.3
Representations 12 2.4 Penetration into non-classical regions 46
1.4 Commutation and non-commutation 13 Translational motion 47 1.5
The construction of operators 14 2.5 Energy and momentum 48 1.6
Integrals over operators 15 2.6 The signicance of the coefcients 48
1.7 Dirac bracket notation 16 2.7 The ux density 49 1.8 Hermitian
operators 17 2.8 Wavepackets 50The postulates of quantum mechanics
19 Penetration into and through barriers 51 1.9 States and
wavefunctions 19 2.9 An innitely thick potential wall 51 1.10 The
fundamental prescription 20 2.10 A barrier of nite width 52 1.11
The outcome of measurements 20 2.11 The Eckart potential barrier 54
1.12 The interpretation of the wavefunction 22 1.13 The equation
for the wavefunction 23 Particle in a box 55 1.14 The separation of
the Schrodinger equation 23 2.12 The solutions 56 2.13 Features of
the solutions 57The specication and evolution of states 25 2.14 The
two-dimensional square well 58 1.15 Simultaneous observables 25
2.15 Degeneracy 59 1.16 The uncertainty principle 27 1.17
Consequences of the uncertainty principle 29 The harmonic
oscillator 60 1.18 The uncertainty in energy and time 30 2.16 The
solutions 61 1.19 Time-evolution and conservation laws 30 2.17
Properties of the solutions 63 2.18 The classical limit 65Matrices
in quantum mechanics 32 1.20 Matrix elements 32 Translation
revisited: The scattering matrix 66 1.21 The diagonalization of the
hamiltonian 34 PROBLEMS 68
9. viii j CONTENTS3 Rotational motion and the hydrogen atom 71
The angular momenta of composite systems 112 4.9 The specication of
coupled states 112Particle on a ring 71 4.10 The permitted values
of the total angular momentum 113 3.1 The hamiltonian and the
Schrodinger equation 71 4.11 The vector model of coupled angular
momenta 115 3.2 The angular momentum 73 4.12 The relation between
schemes 117 3.3 The shapes of the wavefunctions 74 4.13 The
coupling of several angular momenta 119 3.4 The classical limit 76
PROBLEMS 120Particle on a sphere 76 3.5 The Schrodinger equation
and 5 Group theory 122 its solution 76 3.6 The angular momentum of
the particle 79 The symmetries of objects 122 3.7 Properties of the
solutions 81 5.1 Symmetry operations and elements 123 3.8 The rigid
rotor 82 5.2 The classication of molecules 124Motion in a Coulombic
eld 84 The calculus of symmetry 129 3.9 The Schrodinger equation
for hydrogenic atoms 84 5.3 The denition of a group 129 3.10 The
separation of the relative coordinates 85 5.4 Group multiplication
tables 130 3.11 The radial Schrodinger equation 85 5.5 Matrix
representations 131 3.12 Probabilities and the radial 5.6 The
properties of matrix representations 135 distribution function 90
5.7 The characters of representations 137 3.13 Atomic orbitals 91
5.8 Characters and classes 138 3.14 The degeneracy of hydrogenic
atoms 94 5.9 Irreducible representations 139PROBLEMS 96 5.10 The
great and little orthogonality theorems 142 Reduced representations
145 5.11 The reduction of representations 1464 Angular momentum 98
5.12 Symmetry-adapted bases 147The angular momentum operators 98
The symmetry properties of functions 151 4.1 The operators and
their commutation 5.13 The transformation of p-orbitals 151
relations 99 5.14 The decomposition of direct-product bases 152 4.2
Angular momentum observables 101 5.15 Direct-product groups 155 4.3
The shift operators 101 5.16 Vanishing integrals 157 5.17 Symmetry
and degeneracy 159The denition of the states 102 4.4 The effect of
the shift operators 102 The full rotation group 161 4.5 The
eigenvalues of the angular momentum 104 5.18 The generators of
rotations 161 4.6 The matrix elements of the angular 5.19 The
representation of the full rotation group 162 momentum 106 5.20
Coupled angular momenta 164 4.7 The angular momentum eigenfunctions
108 Applications 165 4.8 Spin 110 PROBLEMS 166
10. CONTENTS j ix6 Techniques of approximation 168 7.10 The
spectrum of helium 224 7.11 The Pauli principle 225Time-independent
perturbation theory 168 Many-electron atoms 229 6.1 Perturbation of
a two-level system 169 7.12 Penetration and shielding 229 6.2
Many-level systems 171 7.13 Periodicity 231 6.3 The rst-order
correction to the energy 172 7.14 Slater atomic orbitals 233 6.4
The rst-order correction to the wavefunction 174 7.15
Self-consistent elds 234 6.5 The second-order correction to the
energy 175 7.16 Term symbols and transitions of 6.6 Comments on the
perturbation expressions 176 many-electron atoms 236 6.7 The
closure approximation 178 7.17 Hunds rules and the relative
energies of terms 239 6.8 Perturbation theory for degenerate states
180 7.18 Alternative coupling schemes 240Variation theory 183 Atoms
in external elds 242 6.9 The Rayleigh ratio 183 7.19 The normal
Zeeman effect 242 6.10 The RayleighRitz method 185 7.20 The
anomalous Zeeman effect 243 7.21 The Stark effect 245The
HellmannFeynman theorem 187Time-dependent perturbation theory 189
PROBLEMS 246 6.11 The time-dependent behaviour of a two-level
system 189 6.12 The Rabi formula 192 8 An introduction to molecular
structure 249 6.13 Many-level systems: the variation of constants
193 6.14 The effect of a slowly switched constant The
BornOppenheimer approximation 249 perturbation 195 8.1 The
formulation of the approximation 250 6.15 The effect of an
oscillating perturbation 197 8.2 An application: the hydrogen
moleculeion 251 6.16 Transition rates to continuum states 199 6.17
The Einstein transition probabilities 200 Molecular orbital theory
253 6.18 Lifetime and energy uncertainty 203 8.3 Linear
combinations of atomic orbitals 253 8.4 The hydrogen molecule
258PROBLEMS 204 8.5 Conguration interaction 2597 Atomic spectra and
atomic structure 207 8.6 Diatomic molecules 261 8.7 Heteronuclear
diatomic molecules 265The spectrum of atomic hydrogen 207 Molecular
orbital theory of polyatomic 7.1 The energies of the transitions
208 molecules 266 7.2 Selection rules 209 8.8 Symmetry-adapted
linear combinations 266 7.3 Orbital and spin magnetic moments 212
8.9 Conjugated p-systems 269 7.4 Spinorbit coupling 214 8.10 Ligand
eld theory 274 7.5 The ne-structure of spectra 216 8.11 Further
aspects of ligand eld theory 276 7.6 Term symbols and spectral
details 217 The band theory of solids 278 7.7 The detailed spectrum
of hydrogen 218 8.12 The tight-binding approximation 279The
structure of helium 219 8.13 The KronigPenney model 281 7.8 The
helium atom 219 8.14 Brillouin zones 284 7.9 Excited states of
helium 222 PROBLEMS 285
11. x j CONTENTS9 The calculation of electronic structure 287
10.3 Rotational energy levels 345 10.4 Centrifugal distortion
349The HartreeFock self-consistent eld method 288 10.5 Pure
rotational selection rules 349 9.1 The formulation of the approach
288 10.6 Rotational Raman selection rules 351 9.2 The HartreeFock
approach 289 10.7 Nuclear statistics 353 9.3 Restricted and
unrestricted HartreeFock The vibrations of diatomic molecules 357
calculations 291 10.8 The vibrational energy levels of diatomic 9.4
The Roothaan equations 293 molecules 357 9.5 The selection of basis
sets 296 10.9 Anharmonic oscillation 359 9.6 Calculational accuracy
and the basis set 301 10.10 Vibrational selection rules 360Electron
correlation 302 10.11 Vibrationrotation spectra of diatomic
molecules 362 9.7 Conguration state functions 303 10.12 Vibrational
Raman transitions of diatomic molecules 364 9.8 Conguration
interaction 303 9.9 CI calculations 305 The vibrations of
polyatomic molecules 365 9.10 Multiconguration and multireference
methods 308 10.13 Normal modes 365 9.11 MllerPlesset many-body
perturbation theory 310 10.14 Vibrational selection rules for
polyatomic 9.12 The coupled-cluster method 313 molecules 368 10.15
Group theory and molecular vibrations 369Density functional theory
316 10.16 The effects of anharmonicity 373 9.13 KohnSham orbitals
and equations 317 10.17 Coriolis forces 376 9.14
Exchangecorrelation functionals 319 10.18 Inversion doubling
377Gradient methods and molecular properties 321 Appendix 10.1
Centrifugal distortion 379 9.15 Energy derivatives and the Hessian
matrix 321 PROBLEMS 380 9.16 Analytical derivatives and the coupled
perturbed equations 322 11 Molecular electronic transitions
382Semiempirical methods 325 9.17 Conjugated p-electron systems 326
The states of diatomic molecules 382 9.18 Neglect of differential
overlap 329 11.1 The Hund coupling cases 382Molecular mechanics 332
11.2 Decoupling and L-doubling 384 9.19 Force elds 333 11.3
Selection rules 386 9.20 Quantum mechanicsmolecular mechanics 334
Vibronic transitions 386Software packages for 11.4 The FranckCondon
principle 386electronic structure calculations 336 11.5 The
rotational structure of vibronic transitions 389PROBLEMS 339 The
electronic spectra of polyatomic molecules 39010 Molecular
rotations and vibrations 342 11.6 Symmetry considerations 391 11.7
Chromophores 391Spectroscopic transitions 342 11.8 Vibronically
allowed transitions 393 10.1 Absorption and emission 342 11.9
Singlettriplet transitions 395 10.2 Raman processes 344 The fate of
excited species 396Molecular rotation 344 11.10 Non-radiative decay
396
12. CONTENTS j xi11.11 Radiative decay 397 Magnetic resonance
parameters 45211.12 The conservation of orbital symmetry 399 13.11
Shielding constants 45211.13 Electrocyclic reactions 399 13.12 The
diamagnetic contribution to shielding 45611.14 Cycloaddition
reactions 401 13.13 The paramagnetic contribution to shielding
45811.15 Photochemically induced electrocyclic reactions 403 13.14
The g-value 45911.16 Photochemically induced cycloaddition
reactions 404 13.15 Spinspin coupling 462PROBLEMS 406 13.16 Hyperne
interactions 463 13.17 Nuclear spinspin coupling 46712 The electric
properties of molecules 407 PROBLEMS 471The response to electric
elds 407 14 Scattering theory 473 12.1 Molecular response
parameters 407 12.2 The static electric polarizability 409 The
formulation of scattering events 473 12.3 Polarizability and
molecular properties 411 14.1 The scattering cross-section 473 12.4
Polarizabilities and molecular spectroscopy 413 14.2 Stationary
scattering states 475 12.5 Polarizabilities and dispersion forces
414 12.6 Retardation effects 418 Partial-wave stationary scattering
states 479 14.3 Partial waves 479Bulk electrical properties 418
14.4 The partial-wave equation 480 12.7 The relative permittivity
and the electric 14.5 Free-particle radial wavefunctions and the
susceptibility 418 scattering phase shift 481 12.8 Polar molecules
420 14.6 The JWKB approximation and phase shifts 484 12.9
Refractive index 422 14.7 Phase shifts and the scattering matrix
element 486Optical activity 427 14.8 Phase shifts and scattering
cross-sections 48812.10 Circular birefringence and optical rotation
427 14.9 Scattering by a spherical square well 49012.11
Magnetically induced polarization 429 14.10 Background and
resonance phase shifts 49212.12 Rotational strength 431 14.11 The
BreitWigner formula 494 14.12 Resonance contributions to the
scatteringPROBLEMS 434 matrix element 49513 The magnetic properties
of molecules 436 Multichannel scattering 497 14.13 Channels for
scattering 497The descriptions of magnetic elds 436 14.14
Multichannel stationary scattering states 498 13.1 The magnetic
susceptibility 436 14.15 Inelastic collisions 498 13.2
Paramagnetism 437 14.16 The S matrix and multichannel resonances
501 13.3 Vector functions 439 13.4 Derivatives of vector functions
440 The Greens function 502 13.5 The vector potential 441 14.17 The
integral scattering equation and Greens functions 502Magnetic
perturbations 442 14.18 The Born approximation 504 13.6 The
perturbation hamiltonian 442 Appendix 14.1 The derivation of the
BreitWigner 13.7 The magnetic susceptibility 444 formula 508 13.8
The current density 447 Appendix 14.2 The rate constant for
reactive 13.9 The diamagnetic current density 450 scattering
50913.10 The paramagnetic current density 451 PROBLEMS 510
13. xii j CONTENTSFurther information 513 15 Vector coupling
coefcients 535 Spectroscopic properties 537Classical mechanics 513
16 Electric dipole transitions 537 1 Action 513 17 Oscillator
strength 538 2 The canonical momentum 515 18 Sum rules 540 3 The
virial theorem 516 19 Normal modes: an example 541 4 Reduced mass
518 The electromagnetic eld 543 Solutions of the Schrodinger
equation 519 20 The Maxwell equations 543 5 The motion of
wavepackets 519 21 The dipolar vector potential 546 6 The harmonic
oscillator: solution by factorization 521 Mathematical relations
547 7 The harmonic oscillator: the standard solution 523 22 Vector
properties 547 8 The radial wave equation 525 23 Matrices 549 9 The
angular wavefunction 526 10 Molecular integrals 527 11 The
HartreeFock equations 528 Further reading 553 12 Greens functions
532 13 The unitarity of the S matrix 533 Appendix 1 557 Appendix 2
562Group theory and angular momentum 534 Answers to selected
problems 563 14 The orthogonality of basis functions 534 Index
565
14. PREFACEMany changes have occurred over the editions of this
text but we haveretained its essence throughout. Quantum mechanics
is lled with abstractmaterial that is both conceptually demanding
and mathematically challen-ging: we try, wherever possible, to
provide interpretations and visualizationsalongside mathematical
presentations. One major change since the third edition has been
our response to concernsabout the mathematical complexity of the
material. We have not sacricedthe mathematical rigour of the
previous edition but we have tried innumerous ways to make the
mathematics more accessible. We have intro-duced short commentaries
into the text to remind the reader of the mathe-matical
fundamentals useful in derivations. We have included more
workedexamples to provide the reader with further opportunities to
see formulae inaction. We have added new problems for each chapter.
We have expanded thediscussion on numerous occasions within the
body of the text to providefurther clarication for or insight into
mathematical results. We have set asideProofs and Illustrations
(brief examples) from the main body of the text sothat readers may
nd key results more readily. Where the depth of pre-sentation
started to seem too great in our judgement, we have sent material
tothe back of the chapter in the form of an Appendix or to the back
of the bookas a Further information section. Numerous equations are
tabbed with wwwto signify that on the Website to accompany the text
[www.oup.com/uk/booksites/chemistry/] there are opportunities to
explore the equations bysubstituting numerical values for
variables. We have added new material to a number of chapters, most
notably thechapter on electronic structure techniques (Chapter 9)
and the chapter onscattering theory (Chapter 14). These two
chapters present material that is atthe forefront of modern
molecular quantum mechanics; signicant advanceshave occurred in
these two elds in the past decade and we have tried tocapture their
essence. Both chapters present topics where comprehensioncould be
readily washed away by a deluge of algebra; therefore, we
con-centrate on the highlights and provide interpretations and
visualizationswherever possible. There are many organizational
changes in the text, including the layout ofchapters and the choice
of words. As was the case for the third edition, thepresent edition
is a rewrite of its predecessor. In the rewriting, we have aimedfor
clarity and precision. We have a deep sense of appreciation for
many people who assisted us inthis endeavour. We also wish to thank
the numerous reviewers of the text-book at various stages of its
development. In particular, we would like tothank Charles Trapp,
University of Louisville, USA Ronald Duchovic, Indiana Purdue Fort
Wayne, USA
15. xiv j PREFACE Karl Jalkanen, Technical University of
Denmark, Denmark Mark Child, University of Oxford, UK Ian Mills,
University of Reading, UK David Clary, University of Oxford, UK
Stephan Sauer, University of Copenhagen, Denmark Temer Ahmadi,
Villanova University, USA Lutz Hecht, University of Glasgow, UK
Scott Kirby, University of Missouri-Rolla, USA All these colleagues
have made valuable suggestions about the content and organization
of the book as well as pointing out errors best spotted in private.
Many individuals (too numerous to name here) have offered advice
over the years and we value and appreciate all their insights and
advice. As always, our publishers have been very helpful and
understanding. PWA, Oxford RSF, Indiana University Purdue
University Fort Wayne June 2004
16. Introduction and orientation0.1 Black-body radiation There
are two approaches to quantum mechanics. One is to follow the
historical development of the theory from the rst indications that
the0.2 Heat capacities whole fabric of classical mechanics and
electrodynamics should be held in doubt to the resolution of the
problem in the work of Planck, Einstein,0.3 The photoelectric and
Heisenberg, Schrodinger, and Dirac. The other is to stand back at a
point Compton effects late in the development of the theory and to
see its underlying theore- tical structure. The rst is interesting
and compelling because the theory0.4 Atomic spectra is seen
gradually emerging from confusion and dilemma. We see experi-0.5
The duality of matter ment and intuition jointly determining the
form of the theory and, above all, we come to appreciate the need
for a new theory of matter. The second, more formal approach is
exciting and compelling in a different sense: there is logic and
elegance in a scheme that starts from only a few postulates, yet
reveals as their implications are unfolded, a rich, experimentally
veriable structure. This book takes that latter route through the
subject. However, to set the scene we shall take a few moments to
review the steps that led to the revo- lutions of the early
twentieth century, when some of the most fundamental concepts of
the nature of matter and its behaviour were overthrown and replaced
by a puzzling but powerful new description. 0.1 Black-body
radiation In retrospectand as will become clearwe can now see that
theoretical physics hovered on the edge of formulating a quantum
mechanical descrip- tion of matter as it was developed during the
nineteenth century. However, it was a series of experimental
observations that motivated the revolution. Of these observations,
the most important historically was the study of black- body
radiation, the radiation in thermal equilibrium with a body that
absorbs and emits without favouring particular frequencies. A
pinhole in an otherwise sealed container is a good approximation
(Fig. 0.1). Two characteristics of the radiation had been identied
by the end of the century and summarized in two laws. According to
the StefanBoltzmann 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
17. 2 j INTRODUCTION AND ORIENTATION The StefanBoltzmann
constant, s, is independent of the material from which the body is
composed, and its modern value is 56.7 nW m2 K4. So, a region
Detected of area 1 cm2 of a black body at 1000 K radiates about 6 W
if all frequencies radiation are taken into account. Not all
frequencies (or wavelengths, with l c/n), though, are equally
represented in the radiation, and the observed peak moves to
shorter wavelengths as the temperature is raised. According to
Wiens Pinhole displacement law, lmax T constant 0:2 Container at a
temperature T with the constant equal to 2.9 mm K.Fig. 0.1 A
black-body emitter can be One of the most challenging problems in
physics at the end of the nine-simulated by a heated container with
teenth century was to explain these two laws. Lord Rayleigh, with
minor helpa pinhole in the wall. The from James Jeans,1 brought his
formidable experience of classical physics toelectromagnetic
radiation is reected bear on the problem, and formulated the
theoretical RayleighJeans law formany times inside the container
and the energy density e(l), the energy divided by the volume, in
the wavelengthreaches thermal equilibrium with thewalls. range l to
l dl: 8pkT del rl dl rl 4 0:3 l where k is Boltzmanns constant (k
1.381 10 23 J K1). This formula summarizes the failure of classical
physics. It suggests that regardless of the temperature, there
should be an innite 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 eld 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 Plancks constant. According to this
quantization of energy, the supposition that energy can be
transferred only in discrete amounts, the oscillator can have the
energies 0, 25 hn, 2hn, . . . , and no other energy. Classical
physics allowed a continuous variation in energy, so even a very
high frequency oscillator could be excited 20 with a very small
energy: that was the root of the ultraviolet catastrophe. Quantum
theory is characterized by discreteness in energies (and, as we
shall /(8(kT )5/(hc)4) 15 see, of certain other properties), and
the need for a minimum excitation energy effectively switches off
oscillators of very high frequency, and hence 10 eliminates the
ultraviolet catastrophe. When Planck implemented his suggestion, he
derived what is now called the Planck distribution for the energy
density of a black-body radiator: 5 8phc ehc=lkT rl 0:4 0 l5 1
ehc=lkT 0 0.5 1.0 1.5 2.0 This expression, which is plotted in Fig.
0.2, avoids the ultraviolet cata- kT /hc strophe, and ts the
observed energy distribution extraordinarily well if weFig. 0.2 The
Planck distribution. take h 6.626 1034 J s. Just as the
RayleighJeans law epitomizes the failure of classical physics, the
Planck distribution epitomizes the inception of
.......................................................................................................
1. It seems to me, said Jeans, that Lord Rayleigh has introduced an
unnecessary factor 8 by counting negative as well as positive
values of his integers. (Phil. Mag., 91, 10 (1905).)
18. 0.2 HEAT CAPACITIES j 3 quantum theory. It began the new
century as well as a new era, for it was published in 1900. 0.2
Heat capacities In 1819, science had a deceptive simplicity. Dulong
and Petit, for example, were able to propose their law that the
atoms of all simple bodies have exactly the same heat capacity of
about 25 J K1 mol1 (in modern units). Dulong and Petits rather
primitive observations, though, were done at room temperature, and
it was unfortunate for them and for classical physics when
measurements were extended to lower temperatures and to a wider
range of materials. It was found that all elements had heat
capacities lower than predicted by Dulong and Petits law and that
the values tended towards zero as T ! 0. Dulong and Petits law was
easy to explain in terms of classical physics by assuming that each
atom acts as a classical oscillator in three dimensions. The
calculation predicted that the molar isochoric (constant volume)
heat capa- city, CV,m, of a monatomic solid should be equal to 3R
24.94 J K1 mol1, where R is the gas constant (R NAk, with NA
Avogadros constant). That the heat capacities were smaller than
predicted was a serious embarrassment. Einstein recognized the
similarity between this problem and black-body 3 radiation, for if
each atomic oscillator required a certain minimum energy Debye
before it would actively oscillate and hence contribute to the heat
capacity, then at low temperatures some would be inactive and the
heat capacity would Einstein be smaller than expected. He applied
Plancks suggestion for electromagnetic 2 oscillators to the
material, atomic oscillators of the solid, and deduced theCV,m /R
following expression: & 2 1 yE eyE =2T CV;m T 3RfE T fE T 0:5a
T 1 eyE =T where the Einstein temperature, yE, is related to the
frequency of atomic 0 oscillators by yE hn/k. The function
CV,m(T)/R is plotted in Fig. 0.3, and 0 0.5 1 1.5 2 T/ closely
reproduces the experimental curve. In fact, the t is not
particularly good at very low temperatures, but that can be traced
to EinsteinsFig. 0.3 The Einstein and Debye assumption that all the
atoms oscillated with the same frequency. When thismolar heat
capacities. The restriction was removed by Debye, he obtainedsymbol
y denotes the Einsteinand Debye temperatures, 3 Z yD =T T x4
exrespectively. Close to T 0 the CV;m T 3RfD T fD T 3 dx 0:5b yD e
x 12Debye heat capacity is 0proportional to T3. where the Debye
temperature, yD, is related to the maximum frequency of the
oscillations that can be supported by the solid. This expression
gives a very good t with observation. The importance of Einsteins
contribution is that it complemented Plancks. Planck had shown that
the energy of radiation is quantized;
19. 4 j INTRODUCTION AND ORIENTATION Einstein showed that
matter is quantized too. Quantization appears to be universal.
Neither was able to justify the form that quantization took (with
oscillators excitable in steps of hn), but that is a problem we
shall solve later in the text. 0.3 The photoelectric and Compton
effects In those enormously productive months of 19056, when
Einstein formu- lated not only his theory of heat capacities but
also the special theory of relativity, he found time to make
another fundamental contribution to modern physics. His achievement
was to relate Plancks quantum hypothesis to the phenomenon of the
photoelectric effect, the emission of electrons from metals when
they are exposed to ultraviolet radiation. The puzzling features of
the effect were that the emission was instantaneous when the
radiation was applied however low its intensity, but there was no
emis- sion, whatever the intensity of the radiation, unless its
frequency exceeded a threshold value typical of each element. It
was also known that the kinetic energy of the ejected electrons
varied linearly with the frequency of the incident radiation.
Einstein pointed out that all the observations fell into place if
the elec- tromagnetic eld was quantized, and that it consisted of
bundles of energy of magnitude hn. These bundles were later named
photons by G.N. Lewis, and we shall use that term from now on.
Einstein viewed the photoelectric effect as the outcome of a
collision between an incoming projectile, a photon of energy hn,
and an electron buried in the metal. This picture accounts for the
instantaneous character of the effect, because even one photon can
participate in one collision. It also accounted for the frequency
threshold, because a minimum energy (which is normally denoted F
and called the work function for the metal, the analogue of the
ionization energy of an atom) must be supplied in a collision
before photoejection can occur; hence, only radiation for which hn
> F can be successful. The linear dependence of the kinetic
energy, EK, of the photoelectron on the frequency of the radiation
is a simple consequence of the conservation of energy, which
implies that EK hn F 0:6 If photons do have a particle-like
character, then they should possess a linear momentum, p. The
relativistic expression relating a particles energy to its mass and
momentum is E2 m2 c4 p2 c2 0:7 where c is the speed of light. In
the case of a photon, E hn and m 0, so hn h p 0:8 c l
20. 0.4 ATOMIC SPECTRA j 5This linear momentum should be
detectable if radiation falls on an electron,for a partial transfer
of momentum during the collision should appear as achange in
wavelength of the photons. In 1923, A.H. Compton performed
theexperiment with X-rays scattered from the electrons in a
graphite target, andfound the results tted the following formula
for the shift in wavelength,dl lf li, when the radiation was
scattered through an angle y: dl 2lC sin2 1 y 2 0:9where lC h/mec
is called the Compton wavelength of the electron(lC 2.426 pm). This
formula is derived on the supposition that a photondoes indeed have
a linear momentum h/l and that the scattering event is like
acollision between two particles. There seems little doubt,
therefore, thatelectromagnetic radiation has properties that
classically would have beencharacteristic of particles. The photon
hypothesis seems to be a denial of the extensive accumulationof
data that apparently provided unequivocal support for the view
thatelectromagnetic radiation is wave-like. By following the
implications ofexperiments and quantum concepts, we have accounted
quantitatively forobservations for which classical physics could
not supply even a qualitativeexplanation.0.4 Atomic spectraThere
was yet another body of data that classical physics could not
elucidatebefore the introduction of quantum theory. This puzzle was
the observationthat the radiation emitted by atoms was not
continuous but consisted ofdiscrete frequencies, or spectral lines.
The spectrum of atomic hydrogen had avery simple appearance, and by
1885 J. Balmer had already noticed that theirwavenumbers, ~, where
~ n/c, tted the expression n n 1 1 ~ RH 2 2 n 0:10 2 nwhere RH has
come to be known as the Rydberg constant for hydrogen(RH 1.097 105
cm1) and n 3, 4, . . . . Rydbergs name is commemoratedbecause he
generalized this expression to accommodate all the transitions
inatomic hydrogen. Even more generally, the Ritz combination
principle statesthat the frequency of any spectral line could be
expressed as the differencebetween two quantities, or terms: ~ T1
T2 n 0:11This expression strongly suggests that the energy levels
of atoms are connedto discrete values, because a transition from
one term of energy hcT1 toanother of energy hcT2 can be expected to
release a photon of energy hc~, or nhn, equal to the difference in
energy between the two terms: this argument
21. 6 j INTRODUCTION AND ORIENTATION leads directly to the
expression for the wavenumber of the spectroscopic transitions. But
why should the energy of an atom be conned to discrete values? In
classical physics, all energies are permissible. The rst attempt to
weld together Plancks 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 Rutherfords
experiments in 1910) was conned to certain values, he was able to
deduce the following expression for the per- mitted energy levels
of an electron in a hydrogen atom: me4 1 En n 1, 2, . . . 0:12 8h2
e2 n2 0 where 1/m 1/me 1/mp and e0 is the vacuum permittivity, a
fundamental constant. This formula marked the rst appearance in
quantum mechanics of a quantum number, n, which identies the state
of the system and is used to calculate its energy. Equation 0.12 is
consistent with Balmers formula and accounted with high precision
for all the transitions of hydrogen that were then known. Bohrs
achievement was the union of theories of radiation and models of
mechanics. However, it was an arbitrary union, and we now know that
it is conceptually untenable (for instance, it is based on the view
that an electron travels in a circular path around the nucleus).
Nevertheless, the fact that he was able to account quantitatively
for the appearance of the spectrum of hydrogen indicated that
quantum mechanics was central to any description of atomic
phenomena and properties. 0.5 The duality of matter The grand
synthesis of these ideas and the demonstration of the deep links
that exist between electromagnetic radiation and matter began with
Louis de Broglie, who proposed on the basis of relativistic
considerations that with any moving body there is associated a
wave, and that the momentum of the body and the wavelength are
related by the de Broglie relation: h l 0:13 p We have seen this
formula already (eqn 0.8), in connection with the prop- erties of
photons. De Broglie proposed that it is universally applicable. The
signicance of the de Broglie relation is that it summarizes a
fusion of opposites: the momentum is a property of particles; the
wavelength is a property of waves. This duality, the possession of
properties that in classical physics are characteristic of both
particles and waves, is a persistent theme in the interpretation of
quantum mechanics. It is probably best to regard the terms wave and
particle as remnants of a language based on a false
22. 0.5 THE DUALITY OF MATTER j 7(classical) model of the
universe, and the term duality as a late attempt tobring the
language into line with a current (quantum mechanical) model. The
experimental results that conrmed de Broglies conjecture are
theobservation of the diffraction of electrons by the ranks of
atoms in a metalcrystal acting as a diffraction grating. Davisson
and Germer, who performedthis experiment in 1925 using a crystal of
nickel, found that the diffractionpattern was consistent with the
electrons having a wavelength given bythe de Broglie relation.
Shortly afterwards, G.P. Thomson also succeededin demonstrating the
diffraction of electrons by thin lms of celluloidand gold.2 If
electronsif all particleshave wave-like character, then we
shouldexpect there to be observational consequences. In particular,
just as a wave ofdenite wavelength cannot be localized at a point,
we should not expectan electron in a state of denite linear
momentum (and hence wavelength) tobe localized at a single point.
It was pursuit of this idea that led WernerHeisenberg to his
celebrated uncertainty principle, that it is impossible tospecify
the location and linear momentum of a particle simultaneously
witharbitrary precision. In other words, information about location
is at theexpense of information about momentum, and vice versa.
This com-plementarity of certain pairs of observables, the mutual
exclusion of thespecication of one property by the specication of
another, is also a majortheme of quantum mechanics, and almost an
icon of the difference between itand classical mechanics, in which
the specication of exact trajectories was acentral theme. The
consummation of all this faltering progress came in 1926 when
WernerHeisenberg and Erwin Schrodinger formulated their seemingly
different but equally successful versions of quantum mechanics.
These days, we stepbetween the two formalisms as the fancy takes
us, for they are mathematicallyequivalent, and each one has
particular advantages in different types of cal-culation. Although
Heisenbergs formulation preceded Schrodingers by a few months, it
seemed more abstract and was expressed in the then
unfamiliarvocabulary of matrices. Still today it is more suited for
the more formalmanipulations and deductions of the theory, and in
the following pages weshall employ it in that manner. Schrodingers
formulation, which was in terms of functions and differential
equations, was more familiar in style but stillequally
revolutionary in implication. It is more suited to elementary
mani-pulations and to the calculation of numerical results, and we
shall employ it inthat manner. Experiments, said Planck, are the
only means of knowledge at ourdisposal. The rest is poetry,
imagination. It is time for that imaginationto
unfold........................................................................................................
2. It has been pointed out by M. Jammer that J.J. Thomson was
awarded the Nobel Prize forshowing that the electron is a particle,
and G.P. Thomson, his son, was awarded the Prize forshowing that
the electron is a wave. (See The conceptual development of quantum
mechanics,McGraw-Hill, New York (1966), p. 254.)
23. 8 j INTRODUCTION AND ORIENTATIONPROBLEMS0.1 Calculate the
size of the quanta involved in the 0.13 At what wavelength of
incident radiation do theexcitation of (a) an electronic motion of
period 1.0 fs, relativistic and non-relativistic expressions for
the ejection(b) a molecular vibration of period 10 fs, and (c) a
pendulum of electrons from potassium differ by 10 per cent? That
is,of period 1.0 s. find l such that the non-relativistic and
relativistic linear momenta of the photoelectron differ by 10 per
cent. Use0.2 Find the wavelength corresponding to the maximum in F
2.3 eV.the Planck distribution for a given temperature, and
showthat the expression reduces to the Wien displacement law at
0.14 Deduce eqn 0.9 for the Compton effect on the basis ofshort
wavelengths. Determine an expression for the constant the
conservation of energy and linear momentum. Hint. Usein the law in
terms of fundamental constants. (This constant the relativistic
expressions. Initially the electron is at restis called the second
radiation constant, c2.) with energy mec2. When it is travelling
with momentum p its0.3 Use the Planck distribution to confirm the
energy is p2 c2 m2 c4 1/2. The photon, with initial
eStefanBoltzmann law and to derive an expression for momentum h/li
and energy hni, strikes the stationarythe StefanBoltzmann constant
s. electron, is deected through an angle y, and emerges with
momentum h/lf and energy hnf. The electron is initially0.4 The peak
in the Suns emitted energy occurs at about stationary (p 0) but
moves off with an angle y 0 to the480 nm. Estimate the temperature
of its surface on the basis incident photon. Conserve energy and
both components ofof it being regarded as a black-body emitter.
linear momentum. Eliminate y 0 , then p, and so arrive at an0.5
Derive the Einstein formula for the heat capacity of a expression
for dl.collection of harmonic oscillators. To do so, use the 0.15
The first few lines of the visible (Balmer) series in thequantum
mechanical result that the energy of a harmonic spectrum of atomic
hydrogen lie at l/nm 656.46, 486.27,oscillator of force constant k
and mass m is one of the values 434.17, 410.29, . . . . Find a
value of RH, the Rydberg(v 1)hv, with v (1/2p)(k/m)1/2 and v 0, 1,
2, . . . . Hint. 2 constant for hydrogen. The ionization energy, I,
is theCalculate the mean energy, E, of a collection of oscillators
minimum energy required to remove the electron. Find itby
substituting these energies into the Boltzmann from the data and
express its value in electron volts. How isdistribution, and then
evaluate C dE/dT. I related to RH? Hint. The ionization limit
corresponds to0.6 Find the (a) low temperature, (b) high
temperature n ! 1 for the final state of the electron.forms of the
Einstein heat capacity function. 0.16 Calculate the de Broglie
wavelength of (a) a mass of0.7 Show that the Debye expression for
the heat capacity is 1.0 g travelling at 1.0 cm s1, (b) the same at
95 per cent ofproportional to T3 as T ! 0. the speed of light, (c)
a hydrogen atom at room temperature (300 K); estimate the mean
speed from the equipartition0.8 Estimate the molar heat capacities
of metallic sodium principle, which implies that the mean kinetic
energy of an(yD 150 K) and diamond (yD 1860 K) at room atom is
equal to 3kT, where k is Boltzmanns constant, (d) 2temperature (300
K). an electron accelerated from rest through a potential0.9
Calculate the molar entropy Rof an Einstein solid at difference of
(i) 1.0 V, (ii) 10 kV. Hint. For the momentum TT yE. Hint. The
entropy is S 0 CV =TdT. Evaluate the in (b) use p mv/(l v2/c2)1/2
and for the speed in (d) use 1 2integral numerically. 2mev eV,
where V is the potential difference.0.10 How many photons would be
emitted per second by a 0.17 Derive eqn 0.12 for the permitted
energy levels for thesodium lamp rated at 100 W which radiated all
its energy electron in a hydrogen atom. To do so, use the
followingwith 100 per cent efficiency as yellow light of wavelength
(incorrect) postulates of Bohr: (a) the electron moves in a589 nm?
circular orbit of radius r around the nucleus and (b) the angular
momentum of the electron is an integral multiple of0.11 Calculate
the speed of an electron emitted from a clean h " , that is me vr
n" . Hint. Mechanical stability of the hpotassium surface (F 2.3
eV) by light of wavelength (a) orbital motion requires that the
Coulombic force of300 nm, (b) 600 nm. attraction between the
electron and nucleus equals the0.12 When light of wavelength 195 nm
strikes a certain metal centrifugal force due to the circular
motion. The energy ofsurface, electrons are ejected with a speed of
1.23 106 m s1. the electron is the sum of the kinetic energy and
potentialCalculate the speed of electrons ejected from the same
metal (Coulombic) energy. For simplicity, use me rather than
thesurface by light of wavelength 255 nm. reduced mass m.
24. 1 The foundations of quantum mechanicsOperators in quantum
mechanics The whole of quantum mechanics can be expressed in terms
of a small set1.1 Linear operators of postulates. When their
consequences are developed, they embrace the1.2 Eigenfunctions and
eigenvalues behaviour of all known forms of matter, including the
molecules, atoms, and1.3 Representations1.4 Commutation and
electrons that will be at the centre of our attention in this book.
This chapter non-commutation introduces the postulates and
illustrates how they are used. The remaining1.5 The construction of
operators chapters build on them, and show how to apply them to
problems of chemical1.6 Integrals over operators interest, such as
atomic and molecular structure and the properties of mole-1.7 Dirac
bracket notation cules. We assume that you have already met the
concepts of hamiltonian and1.8 Hermitian operatorsThe postulates of
quantum wavefunction in an elementary introduction, and have seen
the Schrodinger mechanics equation written in the form1.9 States
and wavefunctions Hc Ec1.10 The fundamental prescription1.11 The
outcome of measurements This chapter establishes the full
signicance of this equation, and provides1.12 The interpretation of
the a foundation for its application in the following chapters.
wavefunction1.13 The equation for the wavefunction 1.14 The
separation of the Schrodinger Operators in quantum mechanics
equationThe specification and evolution ofstates An observable is
any dynamical variable that can be measured. The principal1.15
Simultaneous observables mathematical difference between classical
mechanics and quantum mechan-1.16 The uncertainty principle ics is
that whereas in the former physical observables are represented
by1.17 Consequences of the uncertainty functions (such as position
as a function of time), in quantum mechanics they principle are
represented by mathematical operators. An operator is a symbol for
an1.18 The uncertainty in energy and time instruction to carry out
some action, an operation, on a function. In most of1.19
Time-evolution and conservation the examples we shall meet, the
action will be nothing more complicated than laws multiplication or
differentiation. Thus, one typical operation might beMatrices in
quantum mechanics multiplication by x, which is represented by the
operator x . Another1.20 Matrix elements operation might be
differentiation with respect to x, represented by the1.21 The
diagonalization of the hamiltonian operator d/dx. We shall
represent operators by the symbol O (omega) in The plausibility of
the Schrodinger general, but use A, B, . . . when we want to refer
to a series of operators.equation We shall not in general
distinguish between the observable and the operator1.22 The
propagation of light that represents that observable; so the
position of a particle along the x-axis1.23 The propagation of
particles will be denoted x and the corresponding operator will
also be denoted x (with1.24 The transition to quantum mechanics
multiplication implied). We shall always make it clear whether we
are referring to the observable or the operator. We shall need a
number of concepts related to operators and functions on which they
operate, and this rst section introduces some of the more important
features.
25. 10 j 1 THE FOUNDATIONS OF QUANTUM MECHANICS 1.1 Linear
operators The operators we shall meet in quantum mechanics are all
linear. A linear operator is one for which Oaf bg aOf bOg 1:1 where
a and b are constants and f and g are functions. Multiplication is
a linear operation; so is differentiation and integration. An
example of a non- linear operation is that of taking the logarithm
of a function, because it is not true, for example, that log 2x 2
log x for all x. 1.2 Eigenfunctions and eigenvalues In general,
when an operator operates on a function, the outcome is another
function. Differentiation of sin x, for instance, gives cos x.
However, in certain cases, the outcome of an operation is the same
function multiplied by a constant. Functions of this kind are
called eigenfunctions of the operator. More formally, a function f
(which may be complex) is an eigenfunction of an operator O if it
satises an equation of the form Of of 1:2 where o is a constant.
Such an equation is called an eigenvalue equation. The function eax
is an eigenfunction of the operator d/dx because (d/dx)eax aeax, 2
which is a constant (a) multiplying the original function. In
contrast, eax is ax2 ax2 not an eigenfunction of d/dx, because
(d/dx)e 2axe , which is a con- 2 stant (2a) times a different
function of x (the function xeax ). The constant o in an eigenvalue
equation is called the eigenvalue of the operator O. Example 1.1
Determining if a function is an eigenfunction Is the function
cos(3x 5) an eigenfunction of the operator d2/dx2 and, if so, what
is the corresponding eigenvalue? Method. Perform the indicated
operation on the given function and see if the function satises an
eigenvalue equation. Use (d/dx)sin ax a cos ax and (d/dx)cos ax a
sin ax. Answer. The operator operating on the function yields d2 d
2 cos3x 5 3 sin3x 5 9 cos3x 5 dx dx and we see that the original
function reappears multiplied by the eigen- value 9. Self-test 1.1.
Is the function e3x 5 an eigenfunction of the operator d2/dx2 and,
if so, what is the corresponding eigenvalue? [Yes; 9] An important
point is that a general function can be expanded in terms of all
the eigenfunctions of an operator, a so-called complete set of
functions.
26. 1.2 EIGENFUNCTIONS AND EIGENVALUES j 11That is, if fn is an
eigenfunction of an operator O with eigenvalue on (so Ofn on fn),
then1 a general function g can be expressed as the linear
combination X g cn fn 1:3 nwhere the cn are coefcients and the sum
is over a complete set of functions.For instance, the straight line
g ax can be recreated over a certain range bysuperimposing an
innite number of sine functions, each of which is aneigenfunction
of the operator d2/dx2. Alternatively, the same function may
beconstructed from an innite number of exponential functions, which
areeigenfunctions of d/dx. The advantage of expressing a general
function as alinear combination of a set of eigenfunctions is that
it allows us to deduce theeffect of an operator on a function that
is not one of its own eigenfunctions.Thus, the effect of O on g in
eqn 1.3, using the property of linearity, is simply X X X Og O cn
fn cn Ofn c n on f n n n n A special case of these linear
combinations is when we have a set ofdegenerate eigenfunctions, a
set of functions with the same eigenvalue. Thus,suppose that f1,
f2, . . . , fk are all eigenfunctions of the operator O, and
thatthey all correspond to the same eigenvalue o: Ofn ofn with n 1,
2, . . . , k 1:4Then it is quite easy to show that any linear
combination of the functions fnis also an eigenfunction of O with
the same eigenvalue o. The proof is asfollows. For an arbitrary
linear combination g of the degenerate set offunctions, we can
write X k X k X k X k Og O cn fn cn Ofn cn ofn o cn fn og n1 n1 n1
n1This expression has the form of an eigenvalue equation (Og og).
Example 1.2 Demonstrating that a linear combination of degenerate
eigenfunctions is also an eigenfunction Show that any linear
combination of the complex functions e2ix and e2ix is an
eigenfunction of the operator d2/dx2, where i (1)1/2. Method.
Consider an arbitrary linear combination ae2ix be2ix and see if the
function satises an eigenvalue equation. Answer. First we
demonstrate that e2ix and e2ix are degenerate eigenfunctions. d2
2ix d e 2ie2ix 4e2ix dx2
dx.......................................................................................................
1. See P.M. Morse and H. Feschbach, Methods of theoretical physics,
McGraw-Hill, New York(1953).
27. 12 j 1 THE FOUNDATIONS OF QUANTUM MECHANICS where we have
used i2 1. Both functions correspond to the same eigen- value, 4.
Then we operate on a linear combination of the functions. d2 ae2ix
be2ix 4ae2ix be2ix dx2 The linear combination satises the
eigenvalue equation and has the same eigenvalue (4) as do the two
complex functions. Self-test 1.2. Show that any linear combination
of the functions sin(3x) and cos(3x) is an eigenfunction of the
operator d2/dx2. [Eigenvalue is 9] A further technical point is
that from n basis functions it is possible to con- struct n
linearly independent combinations. A set of functions g1, g2, . . .
, gn is said to be linearly independent if we cannot nd a set of
constants c1, c2, . . . , cn (other than the trivial set c1 c2 0)
for which X ci gi 0 i A set of functions that is not linearly
independent is said to be linearly dependent. From a set of n
linearly independent functions, it is possible to construct an
innite number of sets of linearly independent combinations, but
each set can have no more than n members. For example, from three
2p-orbitals of an atom it is possible to form any number of sets of
linearly independent combinations, but each set has no more than
three members. 1.3 Representations The remaining work of this
section is to put forward some explicit forms of the operators we
shall meet. Much of quantum mechanics can be developed in terms of
an abstract set of operators, as we shall see later. However, it is
often fruitful to adopt an explicit form for particular operators
and to express them in terms of the mathematical operations of
multiplication, differentiation, and so on. Different choices of
the operators that correspond to a particular observable give rise
to the different representations of quantum mechanics, because the
explicit forms of the operators represent the abstract structure of
the theory in terms of actual manipulations. One of the most common
representations is the position representation, in which the
position operator is represented by multiplication by x (or
whatever coordinate is specied) and the linear momentum parallel to
x is represented by differentiation with respect to x. Explicitly:
h " q Position representation: x ! x px ! 1:5 i qx where " h=2p.
Why the linear momentum should be represented in pre- h cisely this
manner will be explained in the following section. For the time
being, it may be taken to be a basic postulate of quantum
mechan