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Quantum Mechanics
A New Introduction
K. Konishi
University of Pisa and INFN, Pisa
G. Paffuti
University of Pisa and INFN, Pisa
CLARENDON PRESS
2009
.
OXFORD
Preface
A student’s first encounter with quantum mechanics could be a traumatic one. Instead of the solid differential equation with respect to time
(t) which is Newton’s classical equation of motion, with its inevitable
consequences, she or he learns that the new mechanics predicts as a rule
only certain probabilities (!), and that electrons behave like a sort of
wave, a bizarre notion—but an empirical fact.1
When the student makes some progress in her or his study, however,
she (he) will realize that, after all, things are not that bad: the fundamental equation of the new mechanics—the Schrödinger equation—is a
well-defined, perfectly respectable linear differential equation in t, and
when left alone, the microscopic system evolves in a rigorously deterministic fashion. Not only that, but due to the quantization of finite
motions, and intimately related to this, to the existence of a new fundamental constant of Nature, the Planck constant, h, quantum mechanics
provides a much sharper (and sometimes, far simpler) explanation of
the properties of atoms than does classical mechanics. For instance, all
atoms of the same kind, in their normal state, have rigorously identical
properties. This fact is fundamental, for instance, to the regular structures, and in the working, of the macroscopic world (solids, crystals,
biological phenomena, etc). The advantage of the new mechanics over
the classical one is, of course, not limited to atoms. There are many
phenomena in our daily life, such as electrical conduction, the laser,
electronics, quantum optics, and all other related contemporary technologies, which require quantum mechanics for a proper understanding.
Later on in her or his study, the student might find out that physicists today are debating the validity of the standard model predictions
sometimes to the eleventh digit, for instance concerning the anomalous
magnetic moment of the muon (a kind of heavy electron). Of course, here
we are comparing a particular model of Nature with experiments; however, the standard model of fundamental interactions—quantum chromodynamics for strong interactions and the Glashow–Weinberg–Salam
theory of electroweak interactions—are all based on relativistic quantum
mechanics. In atomic physics, the agreement between theory and experiment can be equally good and sometimes even more impressive. All
this, finally, will convince her (him) that we are indeed dealing with one
of the most precise and perhaps most elegant theories ever known in
physics.
One day she or he might become a researcher or a teacher, and may
start giving a course on quantum mechanics. Perhaps, after many years,
1
The background picture on the frontcover page represents electron wave ripples, formed by 50 kV electron beams
going through a collodion thin film
with tiny holes. The magnification is
such that the full page width corresponds to about 0.6 microns. (Courtesy
of Dr. Akira Tonomura, Hitachi Advanced Research Laboratory, Saitama,
Japan.)
vi Preface
she (he) will continue to marvel at the simplicity and beauty of quantum mechanics, and at the same time its subtle and far-reaching consequences.
One of the main aims of this book is to try to convey this sense of
wonder to young students who are starting to appreciate the beauty of
physics.
This book is, in fact, meant to be an introductory textbook on quantum mechanics: it should be adequate for those who are learning it for
the first time, as well as for slightly more advanced students. Standard
courses on classical physics, including classical mechanics, electromagnetism, statistical mechanics and thermodynamics, plus basic mathematics, should provide a sufficient background.
At the same time, however, we hope that this book, with its many
examples of solved problems, and the diverse subjects discussed, will be
a useful reference tool for more advanced students, active researchers
and teachers alike.
Let us illustrate some of the innovative features of this book. We took
great pains to try to present quantum mechanics pedagogically, and at
the same time with as much logical clarity and organization as possible.
Concepts and methods are introduced gradually, and each of them is
elaborated better and more precisely as the pages go on. We start, in
fact, from the very basic concepts illustrated by elementary applications,
and move on to more structural issues such as symmetry, statistics, and
formal aspects of quantum mechanics, and then explore several standard
approximation methods. Various applications of physical interest are
then discussed, taking full advantage of the artillery we have armed
ourselves with.
As far as the content goes, for the most part, it is fairly standard,
even though some of the discussions in the main text, e.g. in Chapter 12 (systems with general time-dependent Hamiltonians), in Chapter
13 (metastable states), and in Chapter 15 (atoms), and several topics
treated in Supplements (Chapter 20), may often not be found in a standard textbook.
At the end of each chapter, there are a number of problems to be
solved analytically, as well as some others to be solved by numerical
methods. The solutions to both types of problems are provided in an
accompanying CD, in the form of PDF files (analytical problems) or
in the form of Mathematica notebooks (there are 88 of these). The
latter contain self-explanatory expositions of the solutions proposed, as
well as an elementary guide to the Mathematica commands used, so that
they should be easily usable even by those who are seeing a Mathematica
program for the first time. The reader is encouraged to run the program,
enjoy observing how the wave functions evolve, for example, modify and
extend the problems as she (he) pleases, try to improve the precision
of the calculation, etc. (Here are practical tips for the beginner: first,
carefully read the ReadmeFirst file before starting; second, make a copy
of each nb file before proceeding, and keep the original intact. Use a
copy, when actually running the program, and making modifications
vii
and extensions.)
In some cases the analysis is pushed a little deeper into the heart of
the problem than is ordinarily done in a quantum mechanics textbook
(such as the problem of the divergences of perturbation series and resummation; the study of metastable systems; concrete determination
of atomic spectra for general elements, etc.), but always in a concrete,
physical fashion, never going too much into mathematics.
All in all, this is meant to be a contemporary, but at the same time
relatively self-contained and comprehensive, textbook on quantum mechanics.
The book is organized as follows. Part I is an elementary introduction
to the basics of quantum mechanics. Together with some initial sections
on perturbation theory and variational methods in Part II, Part I could
correspond to standard material for an introductory semester course
on quantum mechanics in most universities. Part II is dedicated to the
three standard methods of approximation, perturbation theory, the variational method, and the semiclassical approximation, through which the
concepts in the theory are further developed and the range of applicability vastly increased. In Part III the formalism and methods of analyses
developed are applied to various physical situations, from general timedependent Hamiltonians, general discussions of metastable systems, the
motion of electrically charged particles in electromagnetic fields, atoms,
the scattering problem, atomic nuclei, and elementary particles.
Part IV is dedicated to two fundamental issues of a conceptual nature:
quantum entanglement and the measurement problems.
Part V—the Supplements—is a collection of discussions of various
natures, ranging from a review of useful formulas and tables, to some
advanced topics, technical issues, and mathematical appendices. They
are independent of each other, there is no ordering among them, and
many are even independent of the main text, so that each of them can
be read at leisure in a convenient moment for each reader.
The accompanying CD, as already anticipated, contains the Mathematica notebooks and PDF files in which the problems proposed at the
end of each chapter are solved and discussed. The subfiles for each chapter contain all the notebooks of that chapter, accompanied by a file called
Guide-to-NB.nb. In this file a list of all the Mathematica notebooks of
that chapter is given, as well as a brief description of each notebook. All
analyses have been done by using Mathematica 6, Wolfram Research,
and tested with Mathematica 7, which has just come out.
For updates and corrections, consult our webpages:
http://www.df.unipi.it/e konishi
http://www.df.unipi.it/e paffuti
We are grateful to Mark Seymour of OUP for his brave attempt at
polishing our English and for his invaluable help in improving the look of
the whole book. Of course, the responsibility for any errors in the text or
formulas, or for any misleading expressions, which may undoubtedly still
remain or might have been introduced during the course of corrections,
viii Preface
is ours and ours only. Thanks are also due to Charlotte Green of OUP
for her crisp approach to editorial help, and to Sonke Adlung, the senior
physics editor, for his admirable patience during this book’s long period
of gestation.
Our hearty gratitude goes to our friends and colleagues who helped us
at various moments and in various—small and big—ways. A short list
includes: A. Bonaccorso, D. M. Brink, P. Calabrese, P. Cecchi, R. Collina, E. D’Emilio, A. Di Giacomo, G. Dunne, T. Elze, M. Fukugita, C.
Giannessi, V. Gracco, R. Guida, R. Jackiw, F. Maccarrone, G. Marchesini, M. Matone, P. Menotti, M. Mintchev, F. M. Miranda, G. Morchio,
E. Onofri, L. Picasso, M. Rocca, S. Shore, A. Toncelli, M. Tonelli, A.
Tonomura, P. Truini, A. Vainshtein, and G. Veneziano.
A final message to all of you (especially to the young):
Read and Enjoy!
Pisa, February 2009
K. Konishi and G. Paffuti
Contents
I
Basic quantum mechanics
1 Introduction
1.1 The quantum behavior of the electron
1.1.1 Diffraction and interference—visualizing the quantum world
1.1.2 The stability and identity of atoms
1.1.3 Tunnel effects
1.2 The birth of quantum mechanics
1.2.1 From the theory of specific heat to Planck’s formula
1.2.2 The photoelectric effect
1.2.3 Bohr’s atomic model
1.2.4 The Bohr–Sommerfeld quantization condition;
de Broglie’s wave
Further reading
Guide to the Supplements
Problems
Numerical analyses
2 Quantum mechanical laws
2.1 Quantum states
2.1.1 Composite systems
2.1.2 Photon polarization and the statistical nature of
quantum mechanics
2.2 The uncertainty principle
2.3 The fundamental postulate
2.3.1 The projection operator and state vector reduction
2.3.2 Hermitian operators
2.3.3 Products of operators, commutators, and compatible observables
2.3.4 The position operator, the momentum operator,
fundamental commutators, and Heisenberg’s relation
2.3.5 Heisenberg’s relations
2.4 The Schrödinger equation
2.4.1 More about the Schrödinger equations
2.4.2 The Heisenberg picture
2.5 The continuous spectrum
2.5.1 The delta function
2.5.2 Orthogonality
1
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x Contents
2.5.3
The position and momentum eigenstates; momentum as a translation operator
2.6 Completeness
Problems
Numerical analyses
3 The Schrödinger equation
3.1 General properties
3.1.1 Boundary conditions
3.1.2 Ehrenfest’s theorem
3.1.3 Current density and conservation of probability
3.1.4 The virial and Feynman–Hellman theorems
3.2 One-dimensional systems
3.2.1 The free particle
3.2.2 Topologically nontrivial space
3.2.3 Special properties of one-dimensional Schrödinger
equations
3.3 Potential wells
3.3.1 Infinitely deep wells (walls)
3.3.2 The finite square well
3.3.3 An application
3.4 The harmonic oscillator
3.4.1 The wave function and Hermite polynomials
3.4.2 Creation and annihilation operators
3.5 Scattering problems and the tunnel effect
3.5.1 The potential barrier and the tunnel effect
3.5.2 The delta function potential
3.5.3 General aspects of the scattering problem
3.6 Periodic potentials
3.6.1 The band structure of the energy spectrum
3.6.2 Analysis
Guide to the Supplements
Problems
Numerical analyses
4 Angular momentum
4.1 Commutation relations
4.2 Space rotations
4.3 Quantization
4.4 The Stern–Gerlach experiment
4.5 Spherical harmonics
4.6 Matrix elements of J
4.6.1 Spin- 12 and Pauli matrices
4.7 The composition rule
4.7.1 The Clebsch–Gordan coefficients
4.8 Spin
4.8.1 Rotation matrices for spin 21
Guide to the Supplements
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Contents xi
Problems
109
5 Symmetry and statistics
111
5.1 Symmetries in Nature
111
5.2 Symmetries in quantum mechanics
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5.2.1 The ground state and symmetry
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5.2.2 Parity (P)
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5.2.3 Time reversal
121
5.2.4 The Galilean transformation
123
5.2.5 The Wigner–Eckart theorem
125
5.3 Identical particles: Bose–Einstein and Fermi–Dirac statistics
127
5.3.1 Identical bosons
130
5.3.2 Identical fermions and Pauli’s exclusion principle 132
Guide to the Supplements
133
Problems
134
6 Three-dimensional problems
6.1 Simple three-dimensional systems
6.1.1 Reduced mass
6.1.2 Motion in a spherically symmetric potential
6.1.3 Spherical waves
6.2 Bound states in potential wells
6.3 The three-dimensional oscillator
6.4 The hydrogen atom
Guide to the Supplements
Problems
Numerical analyses
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7 Some finer points of quantum mechanics
7.1 Representations
7.1.1 Coordinate and momentum representations
7.2 States and operators
7.2.1 Bra and ket; abstract Hilbert space
7.3 Unbounded operators
7.3.1 Self-adjoint operators
7.4 Unitary transformations
7.5 The Heisenberg picture
7.5.1 The harmonic oscillator in the Heisenberg picture
7.6 The uncertainty principle
7.7 Mixed states and the density matrix
7.7.1 Photon polarization
7.8 Quantization in general coordinates
Further reading
Guide to the Supplements
Problems
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8 Path integrals
8.1 Green functions
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183
xii Contents
8.2
Path integrals
8.2.1 Derivation
8.2.2 Mode expansion
8.2.3 Feynman graphs
8.2.4 Back to ordinary (Minkowski) time
8.2.5 Tunnel effects and instantons
Further reading
Numerical analyses
II
Approximation methods
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9 Perturbation theory
207
9.1 Time-independent perturbations
207
9.1.1 Degenerate levels
212
9.1.2 The Stark effect on the n = 2 level of the hydrogen
atom
214
9.1.3 Dipole interactions and polarizability
217
9.2 Quantum transitions
219
9.2.1 Perturbation lasting for a finite interval
221
9.2.2 Periodic perturbation
223
9.2.3 Transitions in a discrete spectrum
223
9.2.4 Resonant oscillation between two levels
225
9.3 Transitions in the continuum
226
9.3.1 State density
228
9.4 Decays
228
9.5 Electromagnetic transitions
233
9.5.1 The dipole approximation
234
9.5.2 Absorption of radiation
237
9.5.3 Induced (or stimulated) emission
238
9.5.4 Spontaneous emission
239
9.6 The Einstein coefficients
240
Guide to the Supplements
242
Problems
242
Numerical analyses
244
10 Variational methods
10.1 The variational principle
10.1.1 Lower limits
10.1.2 Truncated Hilbert space
10.2 Simple applications
10.2.1 The harmonic oscillator
10.2.2 Helium: an elementary variational calculation
10.2.3 The virial theorem
10.3 The ground state of the helium
Guide to the Supplements
Problems
Numerical analyses
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Contents xiii
11 The semi-classical approximation
11.1 The WKB approximation
11.1.1 Connection formulas
11.2 The Bohr–Sommerfeld quantization condition
11.2.1 Counting the quantum states
11.2.2 Potentials defined for x > 0 only
11.2.3 On the meaning of the limit ~ → 0
11.2.4 Angular variables
11.2.5 Radial equations
11.2.6 Examples
11.3 The tunnel effect
11.3.1 The double well
11.3.2 The semi-classical treatment of decay processes
11.3.3 The Gamow–Siegert theory
11.4 Phase shift
Further reading
Guide to the Supplements
Problems
Numerical analyses
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III
303
Applications
12 Time evolution
307
12.1 General features of time evolution
307
12.2 Time-dependent unitary transformations
309
12.3 Adiabatic processes
311
12.3.1 The Landau–Zener transition
313
12.3.2 The impulse approximation
315
12.3.3 The Berry phase
316
12.3.4 Examples
318
12.4 Some nontrivial systems
320
12.4.1 A particle within moving walls
320
12.4.2 Resonant oscillations
324
12.4.3 A particle encircling a solenoid
327
12.4.4 A ring with a defect
328
12.5 The cyclic harmonic oscillator: a theorem
331
12.5.1 Inverse linear variation of the frequency
335
12.5.2 The Planck distribution inside an oscillating cavity 336
12.5.3 General power-dependent frequencies
338
12.5.4 Exponential dependence
339
12.5.5 Creation and annihilation operators; coupled oscillators
340
Guide to the Supplements
341
Problems
341
Numerical analyses
342
13 Metastable states
343
xiv Contents
13.1 Green functions
343
13.1.1 Analytic properties of the resolvent
345
13.1.2 Free particles
349
13.1.3 The free Green function in general dimensions
351
13.1.4 Expansion in powers of HI
352
13.2 Metastable states
356
13.2.1 Formulation of the problem
356
13.2.2 The width of a metastable state; the mean halflifetime
358
13.2.3 Formal treatment
361
13.3 Examples
368
13.3.1 Discrete–continuum coupling
368
13.4 Complex scale transformations
370
13.4.1 Analytic continuation
372
13.5 Applications and examples
374
13.5.1 Resonances in helium
375
13.5.2 The potential V0 r2 e−r
375
13.5.3 The unbounded potential; the Lo Surdo–Stark effect376
Further reading
379
Problems
379
Numerical analyses
379
14 Electromagnetic interactions
14.1 The charged particle in an electromagnetic field
14.1.1 Classical particles
14.1.2 Quantum particles in electromagnetic fields
14.1.3 Dipole and quadrupole interactions
14.1.4 Magnetic interactions
14.1.5 Relativistic corrections: LS coupling
14.1.6 Hyperfine interactions
14.2 The Aharonov–Bohm effect
14.2.1 Superconductors
14.3 The Landau levels
14.3.1 The quantum Hall effect
14.4 Magnetic monopoles
Guide to the Supplements
Problems
Numerical analyses
381
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381
383
385
388
388
390
392
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15 Atoms
405
15.1 Electronic configurations
405
15.1.1 The ionization potential
408
15.1.2 The spectrum of alkali metals
410
15.1.3 X rays
410
15.2 The Hartree approximation
412
15.2.1 Self-consistent fields and the variational principle 415
15.2.2 Some results
417
15.3 Multiplets
418
Contents xv
15.3.1 Structure of the multiplets
15.4 Slater determinants
15.5 The Hartree–Fock approximation
15.5.1 Examples
15.6 Spin–orbit interactions
15.6.1 The hydrogen atom
15.7 Atoms in external electric fields
15.7.1 Dipole interaction and polarizability
15.7.2 Quadrupole interactions
15.8 The Zeeman effect
15.8.1 The Zeeman effect in quantum mechanics
Further reading
Guide to the Supplements
Problems
Numerical analyses
16 Elastic scattering theory
16.1 The cross section
16.2 Partial wave expansion
16.2.1 The semi-classical limit
16.3 The Lippman–Schwinger equation
16.4 The Born approximation
16.5 The eikonal approximation
16.6 Low-energy scattering
16.7 Coulomb scattering: Rutherford’s formula
16.7.1 Scattering of identical particles
Further reading
Guide to the Supplements
Problems
Numerical analyses
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17 Atomic nuclei and elementary particles
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17.1 Atomic nuclei
477
17.1.1 General features
477
17.1.2 Isospin
478
17.1.3 Nuclear forces, pion exchange, and the Yukawa
potential
480
17.1.4 Radioactivity
482
17.1.5 The deuteron and two-nucleon forces
483
17.2 Elementary particles: the need for relativistic quantum
field theories
485
17.2.1 The Klein–Gordon and Dirac equations
487
17.2.2 Quantization of the free Klein–Gordon fields
490
17.2.3 Quantization of the free Dirac fields and the spin–
statistics connection
491
17.2.4 Causality and locality
492
17.2.5 Self-interacting scalar fields
494
17.2.6 Non-Abelian gauge theories: the Standard Model 495
xvi Contents
Further reading
IV
Entanglement and Measurement
496
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18 Quantum entanglement
503
18.1 The EPRB Gedankenexperiment and quantum entanglement
503
18.2 Aspect’s experiment
508
18.3 Entanglement with more than two particles
511
18.4 Factorization versus entanglement
512
18.5 A measure of entanglement: entropy
514
Further reading
516
19 Probability and measurement
517
19.1 The probabilistic nature of quantum mechanics
517
19.2 Measurement and state preparation: from PVM to POVM 519
19.3 Measurement “problems”
521
19.3.1 The EPR “paradox”
522
19.3.2 Measurement as a physical process: decoherence
and the classical limit
525
19.3.3 Schrödinger’s cat
527
19.3.4 The fundamental postulate versus Schrödinger’s
equation
529
19.3.5 Is quantum mechanics exact?
530
19.3.6 Cosmology and quantum mechanics
531
19.4 Hidden-variable theories
532
19.4.1 Bell’s inequalities
532
19.4.2 The Kochen–Specker theorem
535
19.4.3 “Quantum non-locality” versus “locally causal theories” or “local realism”
538
Further reading
539
Guide to the Supplements
539
V
Supplements
541
20 Supplements for Part I
545
20.1 Classical mechanics
545
20.1.1 The Lagrangian formalism
545
20.1.2 The Hamiltonian (canonical) formalism
547
20.1.3 Poisson brackets
549
20.1.4 Canonical transformations
550
20.1.5 The Hamilton–Jacobi equation
552
20.1.6 Adiabatic invariants
552
20.1.7 The virial theorem
554
20.2 The Hamiltonian of electromagnetic radiation field in the
vacuum
554
Contents xvii
20.3 Orthogonality and completeness in a system with a onedimensional delta function potential
20.3.1 Orthogonality
20.3.2 Completeness
20.4 The S matrix; the wave packet description of scattering
20.4.1 The wave packet description
20.5 Legendre polynomials
20.6 Groups and representations
20.6.1 Group axioms; some examples
20.6.2 Group representations
20.6.3 Lie groups and Lie algebras
20.6.4 The U (N ) group and the quarks
20.7 Formulas for angular momentum
20.8 Young tableaux
20.9 N -particle matrix elements
20.10 The Fock representation
20.10.1 Bosons
20.10.2 Fermions
20.11 Second quantization
20.12 Supersymmetry in quantum mechanics
20.13 Two- and three-dimensional delta function potentials
20.13.1 Bound states
20.13.2 Self-adjoint extensions
20.13.3 The two-dimensional delta-function potential: a
quantum anomaly
20.14 Superselection rules
20.15 Quantum representations
20.15.1 Weyl’s commutation relations
20.15.2 Von Neumann’s theorem
20.15.3 Angular variables
20.15.4 Canonical transformations
20.15.5 Self-adjoint extensions
20.16 Gaussian integrals and Feynman graphs
21 Supplements for Part II
21.1 Supplements on perturbation theory
21.1.1 Change of boundary conditions
21.1.2 Two-level systems
21.1.3 Van der Waals interactions
21.1.4 The Dalgarno–Lewis method
21.2 The fine structure of the hydrogen atom
21.2.1 A semi-classical model for the Lamb shift
21.3 Hydrogen hyperfine interactions
21.4 Divergences of perturbative series
21.4.1 Perturbative series at large orders: the anharmonic
oscillator
21.4.2 The origin of the divergence
21.4.3 The analyticity domain
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xviii Contents
21.4.4 Asymptotic series
21.4.5 The dispersion relation
21.4.6 The perturbative–variational approach
21.5 The semi-classical approximation in general systems
21.5.1 Introduction
21.5.2 Keller quantization
21.5.3 Integrable systems
21.5.4 Examples
21.5.5 Caustics
21.5.6 The KAM theorem and quantization
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22 Supplements for Part III
22.1 The K 0 –K 0 system and CP violation
22.2 Level density
22.2.1 The free particle
22.2.2 g(E) and the partition function
22.2.3 g(E) and short-distance behavior
22.2.4 Level density and scattering
22.2.5 The stabilization method
22.3 Thomas precession
22.4 Relativistic corrections in an external field
22.5 The Hamiltonian for interacting charged particles
22.5.1 The interaction potentials
22.5.2 Spin-dependent interactions
22.5.3 The quantum Hamiltonian
22.5.4 Electron–electron interactions
22.5.5 Electron–nucleus interactions
22.5.6 The 1/M corrections
22.6 Quantization of electromagnetic fields
22.6.1 Matrix elements
22.7 Atoms
22.7.1 The Thomas–Fermi approximation
22.7.2 The Hartree approximation
22.7.3 Slater determinants and matrix elements
22.7.4 Hamiltonians for closed shells
22.7.5 Mean energy
22.7.6 Hamiltonians for incomplete shells
22.7.7 Eigenvalues of H
22.7.8 The elementary theory of multiplets
22.7.9 The Hartree–Fock equations
22.7.10 The role of Lagrange multipliers
22.7.11 Koopman’s theorem
22.8 H2+
22.9 The Gross–Pitaevski equation
22.10 The semi-classical scattering amplitude
22.10.1 Caustics and rainbows
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23 Supplements for Part IV
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Contents xix
23.1 Speakable and unspeakable in quantum mechanics
23.1.1 Bell’s toy model for hidden variables
23.1.2 Bohm’s pilot waves
23.1.3 The many-worlds interpretation
23.1.4 Spontaneous wave function collapse
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24 Mathematical appendices and tables
24.1 Mathematical appendices
24.1.1 Laplace’s method
24.1.2 The saddle-point method
24.1.3 Airy functions
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References
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Index
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