com This Page Intentionally Left Blank www.com Contents Preface xi Introduction: The Phenomena of Quantum Mechanics 1 Chapter 1 Mathematical Prerequisites 7 1.1 Linear Vector Space 7 1.3 Self-Adjoint Operators 15 1.4 Hilbert Space and Rigged Hilbert Space 26 1.5 Probability Theory 29 Problems 38 Chapter 2 The Formulation of Quantum Mechanics 42 2.1 Basic Theoretical Concepts 42 2.2 Conditions on Operators 48 2.3 General States and Pure States 50 2.4 Probability Distributions 55 Problems 60 Chapter 3 Kinematics and Dynamics 63 3.1 Transformations of States and Observables 63 3.2 The Symmetries of Space–Time 66 3.3 Generators of the Galilei Group 68 3.4 Identification of Operators with Dynamical Variables 76 3.7 Equations of Motion 89 3.8 Symmetries and Conservation Laws 92 Problems 94 Chapter 4 Coordinate Representation and Applications 97 4.2 The Wave Equation and Its Interpretation 98 4.3 Galilei Transformation of Schrödinger’s Equation 102 www.com v vi Contents 4.5 Conditions on Wave Functions 106 4.6 Energy Eigenfunctions for Free Particles 109 4.8 Path Integrals 116 Problems 123 Chapter 5 Momentum Representation and Applications 126 5.2 Momentum Distribution in an Atom 128 5.4 Diffraction Scattering: Theory 133 5.5 Diffraction Scattering: Experiment 139 5.6 Motion in a Uniform Force Field 145 Problems 149 Chapter 6 The Harmonic Oscillator 151 6.2 Solution in Coordinate Representation 154 6.3 Solution in H Representation 157 Problems 158 Chapter 7 Angular Momentum 160 7.1 Eigenvalues and Matrix Elements 160 7.2 Explicit Form of the Angular Momentum Operators 164 7.3 Orbital Angular Momentum 166 7.7 Addition of Angular Momenta 185 7.8 Irreducible Tensor Operators 193 7.9 Rotational Motion of a Rigid Body 200 Problems 203 Chapter 8 State Preparation and Determination 206 8.3 States of Composite Systems 216 8.4 Indeterminacy Relations 223 Problems 227 www.com Contents vii Chapter 9 Measurement and the Interpretation of States 230 9.1 An Example of Spin Measurement 230 9.2 A General Theorem of Measurement Theory 232 9.3 The Interpretation of a State Vector 234 9.4 Which Wave Function? 238 9.5 Spin Recombination Experiment 241 9.6 Joint and Conditional Probabilities 244 Problems 254 Chapter 10 Formation of Bound States 258 10.1 Spherical Potential Well 258 10.2 The Hydrogen Atom 263 10.3 Estimates from Indeterminacy Relations 271 10.4 Some Unusual Bound States 273 10.5 Stationary State Perturbation Theory 276 10.6 Variational Method 290 Problems 304 Chapter 11 Charged Particle in a Magnetic Field 307 11.3 Motion in a Uniform Static Magnetic Field 314 11.4 The Aharonov–Bohm Effect 321 11.5 The Zeeman Effect 325 Problems 330 Chapter 12 Time-Dependent Phenomena 332 12.2 Exponential and Nonexponential Decay 338 12.3 Energy–Time Indeterminacy Relations 343 12.5 Time-Dependent Perturbation Theory 349 12.7 Adiabatic Approximation 363 Problems 367 Chapter 13 Discrete Symmetries 370 13.3 Time Reversal 377 Problems 386 www.com viii Contents Chapter 14 The Classical Limit 388 14.1 Ehrenfest’s Theorem and Beyond 389 14.2 The Hamilton–Jacobi Equation and the Quantum Potential 394 14.4 The Large Quantum Number Limit 400 Problems 404 Chapter 15 Quantum Mechanics in Phase Space 406 15.1 Why Phase Space Distributions? 406 15.2 The Wigner Representation 407 15.3 The Husimi Distribution 414 Problems 420 Chapter 16 Scattering 421 16.2 Scattering by a Spherical Potential 427 16.3 General Scattering Theory 433 16.4 Born Approximation and DWBA 441 16.7 Diverse Topics 462 Problems 468 Chapter 17 Identical Particles 470 17.2 Indistinguishability of Particles 472 17.3 The Symmetrization Postulate 474 17.4 Creation and Annihilation Operators 478 Problems 492 Chapter 18 Many-Fermion Systems 493 18.2 The Hartree–Fock Method 499 18.4 Fundamental Consequences for Theory 513 18.5 BCS Pairing Theory 514 Problems 525 Chapter 19 Quantum Mechanics of the Electromagnetic Field 526 19.1 Normal Modes of the Field 526 19.2 Electric and Magnetic Field Operators 529 www.com Contents ix 19.3 Zero-Point Energy and the Casimir Force 533 19.4 States of the EM Field 539 19.9 Optical Homodyne Tomography — Determining the Quantum State of the Field 578 Problems 581 Chapter 20 Bell’s Theorem and Its Consequences 583 20.1 The Argument of Einstein, Podolsky, and Rosen 583 20.4 A Stronger Proof of Bell’s Theorem 591 20.6 Bell’s Theorem Without Probabilities 602 20.7 Implications of Bell’s Theorem 607 Problems 610 Appendix A Schur’s Lemma 613 Appendix B Irreducibility of Q and P 615 Appendix C Proof of Wick’s Theorem 616 Appendix D Solutions to Selected Problems 618 Bibliography 639 Index 651 www.com This Page Intentionally Left Blank www.com Preface Although there are many textbooks that deal with the formal apparatus of quantum mechanics and its application to standard problems, before the first edition of this book (Prentice–Hall, 1990) none took into account the devel- opments in the foundations of the subject which have taken place in the last few decades. There are specialized treatises on various aspects of the founda- tions of quantum mechanics, but they do not integrate those topics into the standard pedagogical material. I hope to remove that unfortunate dichotomy, which has divorced the practical aspects of the subject from the interpreta- tion and broader implications of the theory. This book is intended primarily as a graduate level textbook, but it will also be of interest to physicists and philosophers who study the foundations of quantum mechanics.
Parts of the book could be used by senior undergraduates. The first edition introduced several major topics that had previously been found in few, if any, textbooks. They included: – A review of probability theory and its relation to the quantum theory. – Discussions about state preparation and state determination.
– The Aharonov–Bohm effect. – Some firmly established results in the theory of measurement, which are useful in clarifying the interpretation of quantum mechanics. – A more complete account of the classical limit. – Introduction of rigged Hilbert space as a generalization of the more familiar Hilbert space.
It allows vectors of infinite norm to be accommodated within the formalism, and eliminates the vagueness that often surrounds the question whether the operators that represent observables possess a complete set of eigenvectors. – The space–time symmetries of displacement, rotation, and Galilei transfor- mations are exploited to derive the fundamental operators for momentum, angular momentum, and the Hamiltonian. – A charged particle in a magnetic field (Landau levels).com xi xii Preface – Basic concepts of quantum optics. – Discussion of modern experiments that test or illustrate the fundamental aspects of quantum mechanics, such as: the direct measurement of the momentum distribution in the hydrogen atom; experiments using the sin- gle crystal neutron interferometer; quantum beats; photon bunching and antibunching.
– Bell’s theorem and its implications. This edition contains a considerable amount of new material. Some of the newly added topics are: – An introduction describing the range of phenomena that quantum theory seeks to explain. – Feynman’s path integrals.
– The adiabatic approximation and Berry’s phase. – Expanded treatment of state preparation and determination, including the no-cloning theorem and entangled states. – A new treatment of the energy–time uncertainty relations. – A discussion about the influence of a measurement apparatus on the envi- ronment, and vice versa.
– A section on the quantum mechanics of rigid bodies. – A revised and expanded chapter on the classical limit. – The phase space formulation of quantum mechanics. – Expanded treatment of the many new interference experiments that are being performed.
– Optical homodyne tomography as a method of measuring the quantum state of a field mode. – Bell’s theorem without inequalities and probability. The material in this book is suitable for a two-semester course. Chapter 1 consists of mathematical topics (vector spaces, operators, and probability), which may be skimmed by mathematically sophisticated readers.
These topics have been placed at the beginning, rather than in an appendix, because one needs not only the results but also a coherent overview of their theory, since they form the mathematical language in which quantum theory is expressed. The amount of time that a student or a class spends on this chapter may vary widely, depending upon the degree of mathematical preparation. A mathe- matically sophisticated reader could proceed directly from the Introduction to Chapter 2, although such a strategy is not recommended.com Preface xiii The space–time symmetries of displacement, rotation, and Galilei trans- formations are exploited in Chapter 3 in order to derive the fundamental operators for momentum, angular momentum, and the Hamiltonian. This approach replaces the heuristic but inconclusive arguments based upon analogy and wave–particle duality, which so frustrate the serious student.
It also introduces symmetry concepts and techniques at an early stage, so that they are immediately available for practical applications. This is done without requiring any prior knowledge of group theory. Indeed, a hypothetical reader who does not know the technical meaning of the word “group”, and who interprets the references to “groups” of transformations and operators as meaning sets of related transformations and operators, will lose none of the essential meaning. A purely pedagogical change in this edition is the dissolution of the old chapter on approximation methods.
Instead, stationary state perturbation theory and the variational method are included in Chapter 10 (“Formation of Bound States”), while time-dependent perturbation theory and its applications are part of Chapter 12 (“Time-Dependent Phenomena”). I have found this to be a more natural order in my teaching. Finally, this new edition contains some additional problems, and an updated bibliography. Solutions to some problems are given in Appendix D.
The solved problems are those that are particularly novel, and those for which the answer or the method of solution is important for its own sake (rather than merely being an exercise). At various places throughout the book I have segregated in double brackets, [[ · · · ]], comments of a historical comparative, or critical nature. Those remarks would not be needed by a hypothetical reader with no previous exposure to quantum mechanics. They are used to relate my approach, by way of comparison or contrast, to that of earlier writers, and sometimes to show, by means of criticism, the reason for my departure from the older approaches.
Acknowledgements The writing of this book has drawn on a great many published sources, which are acknowledged at various places throughout the text. However, I would like to give special mention to the work of Thomas F. Jordan, which forms the basis of Chapter 3. Many of the chapters and problems have been “field-tested” on classes of graduate students at Simon Fraser University.
A special mention also goes to my former student Bob Goldstein, who discovered www.com xiv Preface a simple proof for the theorem in Sec.3, and whose creative imagination was responsible for the paradox that forms the basis of Problem 9. The data for Fig.4 was taken by Jeff Rudd of the SFU teaching laboratory staff. In preparing Sec.5 on probability theory, I benefitted from discussions with Prof. I would also like to thank Hans von Baeyer for the key idea in the derivation of the orbital angular momentum eigenvalues in Sec.
Unruh for point out interesting features of the third example in Sec. Ballentine Simon Fraser University www.com Introduction The Phenomena of Quantum Mechanics Quantum mechanics is a general theory. It is presumed to apply to every- thing, from subatomic particles to galaxies. But interest is naturally focussed on those phenomena that are most distinctive of quantum mechanics, some of which led to its discovery.
Rather than retelling the historical develop- ment of quantum theory, which can be found in many books,∗ I shall illustrate quantum phenomena under three headings: discreteness, diffraction, and coherence. It is interesting to contrast the original experiments, which led to the new discoveries, with the accomplishments of modern technology. It was the phenomenon of discreteness that gave rise to the name “quan- tum mechanics”. Certain dynamical variables were found to take on only a Fig.1 Current through a tube of Hg vapor versus applied voltage, from the data of Franck and Hertz (1914).
[Figure reprinted from Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, R.] ∗ See,for example, Eisberg and Resnick (1985) for an elementary treatment, or Jammer (1966) for an advanced study.