A Modern Approach to Quantum Mechanics 2nd Edition PDF

A Modem Approach to QUANTUM MECHANICS Second Edition John S. Townsend University Science Books www.uscibooks.com Production Manager: Paul C. Anagnostopoulos, Windfall Software Copyeditor: Lee A. Young Proofreader: MaryEllen N. Oliver Text Design: Yvonne Tsang Cover Design: Genette Itoko Mc...

A Modem Approach to QUANTUM MECHANICS Second Edition John S. Townsend University Science Books www.uscibooks.com Production Manager: Paul C. Anagnostopoulos, Windfall Software Copyeditor: Lee A. Young Proofreader: MaryEllen N. Oliver Text Design: Yvonne Tsang Cover Design: Genette Itoko McGrew Illustrator: Lineworks Compositor: Windfall Software Printer & Binder: Edwards Brothers, Inc. This book was set in Times Roman and Gotham and composed with ZzTgX, a macro package for Donald Knuth’s TgX typesetting system. This book is printed on acid-free paper. Copyright © 2012 by University Science Books Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission o f the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, University Science Books. ISBN: 978-1-891389-78-8 Library of Congress Cataloging-in-Publication Data Townsend, John S. A modem approach to quantum mechanics / John S. Townsend. — 2nd ed. p. cm. Includes index. ISBN 978-1-891389-78-8 (alk. paper) 1. Quantum theory—Textbooks. I. Title. QC174.12.T69 2012 530.12— dc23 2011049655 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 Contents Preface xi CHAPTER 1 Stern-Gerlach Experiments 1 1.1 The Original Stern-Gerlach Experiment 1 1.2 Four Experiments 5 1.3 The Quantum State Vector 10 1.4 Analysis of Experiment 3 14 1.5 Experiment5 18 1.6 Summary 21 Problems 25 CHAPTER 2 Rotation of Basis States and Matrix Mechanics 29 2.1 The Beginnings of Matrix Mechanics 29 2.2 Rotation Operators 33 2.3 The Identity and Projection Operators 41 2.4 Matrix Representations of Operators 46 2.5 Changing Representations 52 2.6 Expectation Values 58 2.7 Photon Polarization and the Spin of the Photon 59 2.8 Summary 65 -* Problems 70 CHAPTER 3 Angular Momentum 75 3.1 Rotations Do Not Commute and Neither Do the Generators 75 3.2 Commuting Operators 80 3.3 The Eigenvalues and Eigenstates of Angular Momentum 82 3.4 The Matrix Elements of the Raising and Lowering Operators 90 3.5 Uncertainty Relations and Angular Momentum 91 3.6 The Spin-^ Eigenvalue Problem 94 3.7 A Stern-Gerlach Experiment with Spin-1 Particles 100 3.8 Summary 104 Problems 106 v vi | Contents CHAPTER 4 Time Evolution 111 4.1 The Hamiltonian and the Schrodinger Equation 111 4.2 Time Dependence of Expectation Values 114 4.3 Precession of a Spin-| Particle in a Magnetic Field 115 4.4 Magnetic Resonance 124 4.5 The Ammonia Molecule and the Ammonia Maser 128 4.6 The Energy-Time Uncertainty Relation 134 4.7 Summary 137 Problems 138 CHAPTER 5 A System of Two Spin-1/2 Particles 141 5.1 The Basis States for a System of Two Spin-| Particles 141 5.2 The Hyperfine Splitting of the Ground State of Hydrogen 143 5.3 The Addition of Angular Momenta for Two Spin- ^ Particles 147 5.4 The Einstein-Podolsky-Rosen Paradox 152 5.5 A Nonquantum Model and the Bell Inequalities 156 5.6 Entanglement and Quantum Teleportation 165 5.7 The Density Operator 171 5.8 Summary 181 Problems 183 CHAPTER 6 Wave Mechanics in One Dimension 191 6.1 Position Eigenstates and the Wave Function 191 6.2 The Translation Operator 195 6.3 The Generator of Translations 197 6.4 The Momentum Operator in the Position Basis 201 6.5 Momentum Space 202 6.6 A Gaussian Wave Packet 204 6.7 The Double-Slit Experiment 210 6.8 General Properties of Solutions to the Schrodinger Equation in Position Space 213 6.9 The Particle in a Box 219 6.10 Scattering in One Dimension 224 6.11 Summary 234 Problems 237 CHAPTER 7 The One-Dimensional Harmonic Oscillator 245 7.1 The Importance of the Harmonic Oscillator 245 7.2 Operator Methods 247  Contents I vii 7.3 Matrix Elements of the Raising and Lowering Operators 252 7.4 Position-Space Wave Functions 254 7.5 The Zero-Point Energy 257 7.6 The Large-n Limit 259 7.7 Time Dependence 261 7.8 Coherent States 262 7.9 Solving the Schrodinger Equation in Position Space 269 7.10 Inversion Symmetry and the Parity Operator 273 7.11 Summary 274 Problems 276 c h a p te r 8 Path Integrals 281 8.1 The Multislit, Multiscreen Experiment 281 8.2 The Transition Amplitude 282 8.3 Evaluating the Transition Amplitude for Short Time Intervals 284 8.4 The Path Integral 286 8.5 Evaluation of the Path Integral for a Free Particle 289 8.6 Why Some Particles Follow the Path of Least Action 291 8.7 Quantum Interference Due to Gravity 297 8.8 Summary 299 Problems 301 c h a p te r 9 Translational and Rotational Symmetry in the Two-Body Problem 303 9.1 The Elements of Wave Mechanics in Three Dimensions 303 9.2 Translational Invariance and Conservation of Linear Momentum 307 t. 9.3 Relative and Center-of-Mass Coordinates 311 9.4 Estimating Ground-State Energies Using the Uncertainty Principle 313 9.5 Rotational Invariance and Conservation of Angular Momentum 314 9.6 A Complete Set of Commuting Observables 317 9.7 Vibrations and Rotations of a Diatomic Molecule 321 9.8 Position-Space Representations of L in Spherical Coordinates 328 9.9 Orbital Angular Momentum Eigenfunctions 331 9.10 Summary 337 Problems 339 viii | Contents CHAPTER 10 Bound States of Central Potentials 345 10.1 The Behavior of the Radial Wave Function Near the Origin 345 10.2 The Coulomb Potential and the Hydrogen Atom 348 10.3 The Finite Spherical Well and the Deuteron 360 10.4 The Infinite Spherical Well 365 10.5 The Three-Dimensional Isotropic Harmonic Oscillator 369 10.6 Conclusion 375 Problems 376 CHAPTER 11 Time-Independent Perturbations 381 11.1 Nondegenerate Perturbation Theory 381 11.2 Degenerate Perturbation Theory 389 11.3 The Stark Effect in Hydrogen 391 11.4 The Ammonia Molecule in an External Electric Field Revisited 395 11.5 Relativistic Perturbations to the Hydrogen Atom 398 11.6 The Energy Levels of Hydrogen 408 11.7 The Zeeman Effect in Hydrogen 410 11.8 Summary 412 Problems 413 CHAPTER 12 Identical Particles 419 12.1 Indistinguishable Particles in Quantum Mechanics 419 12.2 The Helium Atom 424 12.3 Multielectron Atoms and the Periodic Table 437 12.4 Covalent Bonding 441 12.5 Conclusion 448 Problems 448 CHAPTER 13 Scattering 451 13.1 The Asymptotic Wave Function and the Differential Cross Section 451 13.2 The Bom Approximation 458 13.3 An Example of the Bom Approximation: The Yukawa Potential 463 13.4 The Partial Wave Expansion 465 13.5 Examples of Phase-Shift Analysis 469 13.6 Summary 477 Problems 478  Contents I ix c h a p te r 14 Photoas and Atoms 483 14.1 The Aharonov-Bohm Effect 483 14.2 The Hamiltonian for the Electromagnetic Field 488 14.3 Quantizing the Radiation Field 493 14.4 The Hamiltonian of the Atom and the Electromagnetic Field 501 14.5 Time-Dependent Perturbation Theory 504 14.6 Fermi’s Golden Rule 513 14.7 Spontaneous Emission 518 14.8 Cavity Quantum Electrodynamics 526 14.9 Higher Order Processes and Feynman Diagrams 530 Problems 533 Appendix A Electromagnetic Units 539 Appendix B The Addition of Angular Momenta 545 Appendix C Dirac Delta Functions 549 Appendix D Gaussian Integrals 553 Appendix E The Lagrangian for a Charge q in a Magnetic Field 557 Appendix F Values of Physical Constants 561 Appendix G Answers to Selected Problems 563 Index 565 Preface There have been two revolutions in the way we view the physical world in the twentieth century: relativity and quantum mechanics. In quantum mechanics the revolution has been both profound—requiring a dramatic revision in the structure of the laws of mechanics that govern the behavior of all particles, be they electrons or photons—and far-reaching in its impact—determining the stability of matter itself, shaping the interactions of particles on the atomic, nuclear, and particle physics level, and leading to macroscopic quantum effects ranging from lasers and superconductivity to neutron stars and radiation from black holes. Moreover, in a triumph for twentieth-century physics, special relativity and quantum mechanics have been joined together in the form of quantum field theory. Field theories such as quantum electrodynamics have been tested with an extremely high precision, with agreement between theory and experiment verified to better than nine significant figures. It should be emphasized that while our understanding of the laws of physics is continually evolving, always being subjected to experimental scrutiny, so far no confirmed discrepancy between theory and experiment for quantum mechanics has been detected. This book is intended for an upper-division course in quantum mechanics. The most likely audience for the book consistfrof students who have completed a course in modem physics that includes an introduction to quantum mechanics that emphasizes wave mechanics. Rather than continue with a similar approach in a second course, I have chosen to introduce the fundamentals of quantum mechanics through a detailed discussion of the physics of intrinsic spin. Such an approach has a number of significant advantages. First, students find starting a course with something “new” such as intrinsic spin both interesting and exciting, and they enjoy making the connections with what they have seen before. Second, spin systems provide us with many beautiful but straightforward illustrations of the essential structure of quantum mechanics, a structure that is not obscured by the mathematics of wave mechanics. Quantum mechanics can be presented through concrete examples. I believe that most physicists learn through specific examples and then find it easy to generalize. By xii | Preface starting with spin, students are given plenty of time to assimilate this novel and striking material. I have found that they seem to learn this key introductory material easily and well—material that was often perceived to be difficult when I came to it midway through a course that began with wave mechanics. Third, when we do come to wave mechanics, students see that wave mechanics is only one aspect of quantum mechanics, not the fundamental core of the subject. They see at an early stage that wave mechanics and matrix mechanics are just different ways of calculating based on the same underlying quantum mechanics and that the approach they use depends on the particular problem they are addressing. I have been inspired by two sources, an “introductory” treatment in Volume III of The Feynman Lectures on Physics and an advanced exposition in J. J. Sakurai’s Modern Quantum Mechanics. Overall, I believe that wave mechanics is probably the best way to introduce students to quantum mechanics. Wave mechanics makes the largest overlap with what students know from classical mechanics and shows them the strange behavior of quantum mechanics in a familiar environment. This is probably why students find their first introduction to quantum mechanics so stimulating. However, starting a second course with wave mechanics runs the risk of diminishing much of the excitement and enthusiasm for the entirely new way of viewing nature that is demanded by quantum mechanics. It becomes sort of old hat, material the students has seen before, repeated in more depth. It is, I believe, with the second exposure to quantum mechanics that something like Feynman’s approach has its best chance to be effective. But to be effective, a quantum mechanics text needs to make lots of contact with the way most physicists think and calculate in quantum mechanics using the language of kets and operators. This is Sakurai’s approach in his graduate-level textbook. In a sense, the approach that I am presenting here can be viewed as a superposition of these two approaches, but at the junior-senior level. Chapter 1 introduces the concepts of the quantum state vector, complex proba­ bility amplitudes, and the probabilistic interpretation of quantum mechanics in the context of analyzing a number of Stem-Gerlach experiments carried out with spin- | particles. By introducing ket vectors at the beginning, we have the framework for thinking about states as having an existence quite apart from the way we happen to choose to represent them, whether it be with matrix mechanics, which is discussed at length in Chapter 2, or, where appropriate, with wave mechanics, which is in­ troduced in Chapter 6. Moreover, there is a natural role for operators; in Chapter 2 they rotate spin states so that the spin “points” in a different direction. I do not fol­ low a postulatory approach, but rather I allow the basic physics of this spin system to drive the introduction of concepts such as Hermitian operators, eigenvalues, and eigenstates. In Chapter 3 the commutation relations of the generators of rotations are deter­ mined from the behavior of ordinary vectors under rotations. Most of the material in this chapter is fairly conventional; what is not so conventional is the introduc-  Preface | xiii tion of operator techniques for determining the angular momentum eigenstates and eigenvalue spectrum and the derivation of the uncertainty relations from the com­ mutation relations at such an early stage. Since so much of our initial discussion of quantum mechanics revolves around intrinsic spin, it is important for students to see how quantum mechanics can be used to determine from first principles the spin states that have been introduced in Chapters 1 and 2, without having to appeal only to experimental results. Chapter 4 is devoted to time evolution of states. The natural operation in time development is to translate states forward in time. The Hamiltonian enters as the generator of time translations, and the states are shown to obey the Schrodinger equation. Most of the chapter is devoted to physical examples. In Chapter 5 another physical system, the spin-spin interaction of an electron and proton in the ground state of hydrogen, is used to introduce the spin states of two spin-^ particles. The total-spin-0 state serves as the basis for a discussion of the Einstein-Podolsky-Rosen (EPR) paradox and the Bell inequalities. The main theme of Chapter 6 is making contact with the usual formalism of wave mechanics. The special problems in dealing with states such as position and momen­ tum states that have a continuous eigenvalue spectrum are analyzed. The momentum operator enters naturally as the generator of translations. Sections 6.8 through 6.10 include a general discussion with examples of solutions to the Schrodinger equation that can serve as a review for students with a good background in one-dimensional wave mechanics. Chapter 7 is devoted to the one-dimensional simple harmonic oscillator, which merits a chapter all its own. Although the material in Chapter 8 on path integrals can be skipped without affecting subsequent chapters (with the exception of Sec­ tion 14.1, on the Aharonov-Bohm effect), I believe that path integrals should be discussed, if possible, since this formalism provides real insight into quantum dy­ namics. However, I have found it difficult to fit this material into our one-semester course, which is taken by all physics majors as well as some students majoring in other disciplines. Rather, I have choserfto postpone path integrals to a second course and then to insert the material in Chapter 8 before Chapter 14. Incidentally, the ma­ terial on path integrals is the only part of the book that may require students to have had an upper-division classical mechanics course, one in which the principle of least action is discussed. Chapters 9 through 13 cover fully three-dimensional problems, including the two-body problem, orbital angular momentum, central potentials, time-independent perturbations, identical particles, and scattering. An effort has been made to include as many physical examples as possible. Although this is a textbook on nonrelativistic quantum mechanics, I have chosen to include a discussion of the quantized radiation field in the final chapter, Chapter 14. The use of ket and bra vectors from the beginning and the discussion of solutions xiv | Preface to problems such as angular momentum and the harmonic oscillator in terms of abstract raising and lowering operators should have helped to prepare the student for the exciting jump to a quantized electromagnetic field. By quantizing this field, we can really understand the properties of photons, we can calculate the lifetimes for spontaneous emission from first principles, and we can understand why a laser works. By looking at higher order processes such as photon-atom scattering, we can also see the essentials of Feynman diagrams. Although the atom is treated nonrelativistically, it is still possible to gain a sense of what quantum field theory is all about at this level without having to face the complications of the relativistic Dirac equation. For the instructor who wishes to cover time-dependent perturbation theory but does not have time for all of the chapter, Section 14.5 stands on its own. Although SI units are the standard for undergraduate education in electricity and magnetism, I have chosen in the text to use Gaussian units, which are more commonly used to describe microscopic phenomena. However, with the possible exception of the last chapter, with its quantum treatment of the electromagnetic field, the choice of units has little impact. My own experience suggests that students who are generally at home with SI units are comfortable (as indicated in a number of footnotes through the text) replacing e2 with £2/47T€0 or ignoring the factor of c in the Bohr magneton whenever they need to carry out numerical calculations. In addition, electromagnetic units are discussed in Appendix A. In writing the second edition, I have added two sections to Chapter 5, one on entanglement and quantum teleportation and the other on the density operator. Given the importance of entanglement in quantum mechanics, it may seem strange, as it does to me now, to have written a quantum mechanics textbook without explicit use of the word entanglement. The concept of entanglement is, of course, at the heart of the discussion of the EPR paradox, which focused on the entangled state of two spin-j particles in a spin-singlet state. Nonetheless, it wasn’t until the early 1990s, when topics such as quantum teleportation came to the fore, that the importance of entanglement as a fundamental resource that can be utilized in novel ways was fully appreciated and the term entanglement began to be widely used. I am also somewhat embarrassed not to have included a discussion of the density operator in the first edition. Unlike a textbook author, the experimentalist does not necessarily have the luxury of being able to focus on pure states. Thus there is good reason to introduce the density operator (and the density matrix) as a systematic way to deal with mixed states as well as pure states in quantum mechanics. I have added a section on coherent states of the harmonic oscillator to Chapter 7. Coherent states were first derived by Schrodinger in his efforts to find states that satisfy the correspondence principle. The real utility of these states is most apparent in Chapter 14, where it is seen that coherent states come closest to representing classical electromagnetic waves with a well-defined phase. I have also added a section to Chapter 14 on cavity quantum electrodynamics, showing how the interaction of the quantized electromagnetic  Preface | xv field with atoms is modified by confinement in a reflective cavity. Like quantum teleportation, cavity quantum electrodynamics is a topic that really came to the fore in the 1990s. In addition to these new sections, I have added numerous worked example problems to the text, with the hope that these examples will help students in mastering quantum mechanics. I have also increased the end-of-chapter problems by 25 percent. There is almost certainly enough material here for a full-year course. For a one- semester course, I have covered the material through Chapter 12, omitting Sections 6.7 through 6.10 and, as noted earlier, Chapter 8. The material in the latter half of Chapter 6 is covered thoroughly in our introductory course on quantum physics. See John S. Townsend, Quantum Physics: A Fundamental Approach to Modem Physics, University Science Books, 2010. In addition to Chapter 8, other sections that might be omitted in a one-semester course include parts of Chapter 5, Section 9.7, and Sections 11.5 through 11.9. Or one might choose to go as far as Chapter 10 and reserve the remaining material for a later course. A comprehensive solutions manual for the instructor is available from the pub­ lisher, upon request of the instructor. Finally, some grateful acknowledgments are certainly in order. Students in my quantum mechanics classes have given me useful feedback as I have taught from the book over the years. Colleagues at Harvey Mudd College who have offered valuable comments as well as encouragement include Bob Cave, Chih-Yung Chen, Tom Don­ nelly, Tom Helliwell, Theresa Lynn, and Peter Saeta. Art Weldon of West Virginia University suggested a number of ways to improve the accuracy and effectiveness of the first edition. This text was initially published in the McGraw-Hill Interna­ tional Series in Pure and Applied Physics. I have benefited from comments from the following reviewers: William Dalton, St. Cloud State University; Michael Grady, SUNY-Fredonia; Richard Hazeltine, University of Texas at Austin; Jack Mochel, University of Illinois at Urbana-Champaign; and Jae Y. Park, North Carolina State University. For the first edition, the Pew Science Program provided support for Doug Dunston and Doug Ridgway, two Harvey-Mudd College students, who helped in the preparation of the text and figures, respectively, and Helen White helped in checking the galley proofs. A number of people have kindly given me feedback on the material for the second edition, including Rich Holman, Carnegie Mellon University; Randy Hulet, Rice University; Jim Napolitano, RPI; Tom Moore and David Tanenbaum, Pomona College; and John Taylor, University of Colorado. I have been fortunate to have the production of the book carried out by a very capable group of individuals headed by Paul Anagnostopoulos, the project manager. In addition to Paul, I want to thank Lee Young for copyediting, Joe Snowden for entering the copyedits and laying out the pages, Tom Webster for the artwork, MaryEllen Oliver for her amazingly thorough job of proofreading, Yvonne Tsang for text design, and Genette Itoko McGrew for her creative cover design. I also wish xvi | Preface to thank Jane Ellis and Bruce Armbruster of University Science Books not only for their assistance but also for the care and attention to detail they have taken in preparing this new edition of the book. And I especially want to thank ray wife, Ellen, for cheerfully letting me devote so much time to this project. Please do not hesitate to contact me if you find errors or have suggestions that might improve the book. John S. Townsend Department of Physics Harvey Mudd College Claremont, CA 91711 [email protected] A Modern Approach to Quantum Mechanics CHAPTER 1 Stern-Gerlach Experiments We begin our discussion of quantum mechanics with a conceptually simple experi­ ment in which we measure a component of the intrinsic spin angular momentum of an atom. This experiment was first carried out by O. Stem and W. Gerlach in 1922 using a beam of silver atoms. We will refer to the measuring apparatus as a Stern- Gerlach device. The results of experiments with a number of such devices are easy to describe but, as we shall see, nonetheless startling in their consequences. 1.1 The Original Stern-Gerlach Experiment Before analyzing the experiment, we need to know something about the relationship between the intrinsic spin angular momentum of a particle and its corresponding magnetic moment. To the classical physicist, angular momentum is always orbital angular momentum, namely, L = r x p. Although the Earth is said to have spin angular momentum Ia> due to its rotation about its axis as well as orbital angular momentum due to its revolution about the Sun, both types of angular momentum are just different forms of L. The intrinsic spin angular momentum S of a microscopic particle is not at all of the same sort as orbital angular momentum, but it is real angular momentum nonetheless. To get a feeling for the relationship that exists between the angular momentum of a charged particle and its corresponding magnetic moment, we first use a classical example and then point out some of its limitations. Consider a point particle with charge q and mass m moving in a circular orbit of radius r with speed v. The magnetic moment ft is given by (1.1) 2 | 1. Stern-Gerlach Experiments where A is the area of the circle formed by the orbit, the current I is the charge q divided by the period T = (2n r/v), and L = mvr is the orbital angular momentum of the particle.1 Since the magnetic moment and the orbital angular momentum are parallel or antiparallel depending on the sign of the charge q , we may express this relationship in the vector form M= ~ L (1.2) 2me This relationship between L and ft turns out to be generally true whenever the mass and charge coincide in space. One can obtain different constants of proportionality by adjusting the charge and mass distributions independently. For example, a solid spherical ball of mass m rotating about an axis through its center with the charge q distributed uniformly only on the surface of the ball has a constant of proportionality of 5q/6mc. When we come to intrinsic spin angular momentum of a particle, we write = ^ S (1.3) 2me where the value of the constant g is experimentally determined to be g = 2.00 for an electron, g = 5.58 for a proton, or even g = —3.82 for a neutron.2 One might be tempted to presume that g is telling us about how the charge and mass are distributed for the different particles and that intrinsic spin angular momentum is just orbital angular momentum of the particle itself as it spins about its axis. We will see as we go along that such a simple classical picture of intrinsic spin is entirely untenable and that the intrinsic spin angular momentum we are discussing is a very different beast indeed. In fact, it appears that even a point particle in quantum mechanics may have intrinsic spin angular momentum.3 Although there are no classical arguments that we can give to justify (1.3), we can note that such a relationship between the 1 If you haven’t seen them before, the Gaussian units we are using for electromagnetism may take a little getting used to. A comparison of SI and Gaussian units is given in Appendix A. In SI units the magnetic moment is just I A, so you can ignore the factor o f c, the speed of light, in expressions such as (1.1) if you wish to convert to SI units. 2 Each o f these g factors has its own experimental uncertainty. Recent measurements by B. Odom, D. Hanneke, B. D ’Urso, and G, Gabrielse, Phys. Rev. Lett. 97, 030801 (2006), have shown that g /2 for an electron is 1.00115965218085(76), where the factor o f 76 reflects the uncertainty in the last two places. Relativistic quantum mechanics predicts that g = 2 for an electron. The deviations from this value can be accounted for by quantum field theory. The much larger deviations from g = 2 for the proton and the (neutral) neutron are due to the fact that these particles are not fundamental but are composed o f charged constituents called quarks. 3 It is amusing to note that in 1925 S. Goudsmit and G. Uhlenbeck as graduate students “discovered” the electron’s spin from an analysis of atomic spectra. They were trying to understand why the optical spectra o f alkali atoms such as sodium are composed o f a pair o f closely spaced lines, such as the sodium doublet. Goudsmit and Uhlenbeck realized that an additional degree o f freedom (an independent coordinate) was required, a degree o f freedom that they could understand only if they assumed the electron was a small ball o f charge that could rotate about an axis.  1.1 The Original Stern-Gerlach Experiment | 3 (a) Figure 1.1 (a) A schematic diagram o f the Stern-Gerlach experiment, (b) A cross-sectional view o f the pole pieces o f the magnet depicting the inhom ogeneous magnetic field they produce. magnetic moment and the intrinsic spin angular momentum is at least consistent with dimensional analysis. At this stage, you can think of g as a dimensionless factor that has been inserted to make the magnitudes as well as the units come out right. Let’s turn to the Stern-Gerlach experiment itself. Figure 1.1a shows a schematic diagram of the apparatus. A collimated beam of silver atoms is produced by evap­ orating silver in a hot oven and selecting those atoms that pass through a series of narrow slits. The beam is then directed between the poles of a magnet. One of the pole pieces is flat; the other has a sharp tip. Such a magnet produces an inhomoge­ neous magnetic field, as shown in Fig. 1.1b. When a neutral atom with a magnetic moment fi enters the magnetic field B, it experiences a force F = V (/t B), since —fi B is the energy of interaction of a magnetic dipole with an external magnetic field. If we call the direction in which the inhomogeneous magnetic field gradient is large the z direction, we see that „ 9B dBz Fz = V- — - V z — (1.4) oz oz In this way they could account for the electron’s spin angular momentum and magnetic dipole moment. The splitting o f the energy levels that was needed to account for the doublet could then be understood as due to the potential energy o f interaction o f the electron’s magnetic moment in the internal magnetic field o f the atom (see Section 11.5). Goudsmit and Uhlenbeck wrote up their results for their advisor P. Ehrenfest, who then advised them to discuss the matter with H. Lorentz. When Lorentz showed them that a classical model o f the electron required that the electron must be spinning at a speed on the surface approximately ten times the speed o f light, they went to Ehrenfest to tell him of their foolishness. He informed them that he had already submitted their paper for publication and that they shouldn’t worry since they were “both young enough to be able to afford a stupidity.” Physics Today, June 1976, pp. 40-48. 4 | 1. Stern-Gerlach Experiments Notice that we have taken the magnetic field gradient dBz/ ‘dz in the figure to be neg­ ative, so that if i±z is negative as well, then Fz is positive and the atoms are deflected in the positive z direction. Classically, \xz — \fi\ cos 6 , where 6 is the angle that the magnetic moment fi makes with the z axis. Thus n z should take on a continuum of values ranging from + n to —fi. Since the atoms coming from the oven are not polar­ ized with their magnetic moments pointing in a preferred direction, we should find a corresponding continuum of deflections. In the original Stern-Gerlach experiment, the silver atoms were detected by allowing them to build up to a visible deposit on a glass plate. Figure 1.2 shows the results of this original experiment. The surprising result is that fiz takes on only two values, corresponding to the values ± h f 2 for Sz. Numerically, h = h /liz = 1.055 x 10“ 27 erg s = 6.582 x 10“ 16 eV s, where h is Planck’s constant. 1 f /fV jt = (+z|*>* and J O-23) where we have written the complex numbers multiplying the kets |+z) and |- z ) in such a way as to ensure that there is a 50 percent probability of obtaining Sz = h/2 and a 50 percent probability of obtaining Sz = —h/2. Note that in the last step we pulled out in front an overall phase factor ely+ for future computational convenience. Moreover, since in Experiment 5 there is a 50 percent probability of finding a particle No/2 /V 2 (a) N()I2 N()/2 (b) Figure 1.9 Block diagrams showing the last two SG devices in (a) Experiment 3 and in (b) Experiment 5.  1.5 Experiment 5 I 19 with Sy = h/2 when |t exits the SGx device in the state |+x), we must have |(+ y |+ x ) | 2 = l (1.24) Now the bra corresponding to the ket (1.23) is e - ‘ Y+ e ~ ‘ Y- e ~ iY + r -i I z) (2.5) S'-basis \ (—Z|—Z) )-C) although the label under the arrow is really superfluous in (2.4) and (2.5) given the form of the column vectors on the right. Using (1.29), we can also write, for example, |+ x >------- \. /l\ ( 2. 6) 5, basis \ (-Z|+X) / y/2 \ 1/ How do we represent bra vectors? We know that the bra vector corresponding to the ket vector (. ) is 2 1 (f\ = (Vr|+z)(+z| + (\ff\ z) ( z| = C+(+z| + C*_{ z| (2.7) We can express ( f \ f ) = (V'l+zX+zIVO + (V ^ l-z )(-z|^ ) = 1 (. ) 2 8 conveniently as / (+z|V^) \ m * ) = «1M+Z>, (V^l-z)) ( , # 1= 1 (2.9) ------------*-----------' V { - i W ) / bravector v v—— ket vector where we are using the usual rules of matrix multiplication for row and column vectors. This suggests that we represent the bra (V/| by the row vector m ------- > ( ( f \ + z ) , ( f \ - z ) ) (. 2 1 0 ) Sz basis Since (VM+z) = (+z|V^)* and (\j/ —z) = (—z 1 | (2.10) can also be expressed as W ------- ►((+Z|^r)*, { - Z\ \ j / ) * ) = (C* c*_) (2.11) Sz basis Comparing (2.11) with (2.3), we see that the row vector that re p re se n ts the bra is the complex conjugate and transpose of the column vector that re p re se n ts the corresponding ket. In this representation, an inner product such as (2.9) is carried out using the usual rules of matrix multiplication.  2.1 The Beginnings of Matrix Mechanics I 31 As an example, Ve may determine the representation for the ket |—x) in the Sz basis. We know from the Stem-Gerlach experiments that there is zero amplitude to obtain Sx = —h/2 for a state with Sx = h/2, that is, (—x|+ x) = 0. Making the amplitude (—x|+ x) vanish requires that eiS ( 1 \ l - x ) -- ► - = ( J (. 2 1 2 ) Sz basis y j2 \ — 1 / since then e~iS 1 / 1\ < - « , + « > ( 2 , 3 ) Note that the \/y/2 in front of the column vector in (2.12) has been chosen so that the ket |—x) is properly normalized: -is is / i \ (_ ,) = , ( 2 ,4 ) The common convention, and the one that we will generally follow, is to choose the overall phase 8 = so that 0 | - x ) --- (215) S- basis y j2 V — 1 / However, in Section 2.5 we will see that an interesting case can be made forehoosing 8 = 71. As another example, (1.30) indicates that the state with Sv = h/2 is (2.16) '+y) = T i ' +z) + i i ' ~ z) f which may be represented in the S "basis by 2 l+y) (2.17a) V 2 The bra corresponding to this ket is represented in the same basis by (+yl - p (l, - o (2 ,7 b ) Ji Note the appearance of the —i in this representation for the bra vector. Using these representations, we can check that (2 ,8 ) SOLUTION K+ylVOI = -— a - o -x y/2 2 ,>3) 1 \ /A „ /- 1 V3 (1 + V3) = - (4 + 2V3 = - + — 2 V 2 8 2 4 Compare this relatively compact derivation with the use of kets and bras in Example 1.3. FREEDOM OF REPRESENTATION It is often convenient to use a number of different basis sets to express a particular state | VO- Just as we can write the electric field in a particular coordinate system as  2.2 Rotation Operators I 33 (. ), we could use | different coordinate system with unit vectors i', j', and k' to 2 2 write the same electric field as E = ExX + Eyij' + Ez>k' (2.21a) or E (Exi, Ey>, Ez>) (2.2lb) Of course, the electric field E hasn’t changed. It still has the same magnitude and direction, but we have chosen a different set of unit vectors, or basis vectors, to express it. Similarly, we can take the quantum state IVO in (2.1) and write it in terms of the basis states |+x) and |—x) as \ f ) = +xX+xlVO + I—x>(—x|V 1 0 (. 2 2 2 ) which expresses the state as a superposition of the states with Sx = ±ti/2 multiplied by the amplitudes for the particle to be found in these states. We can then construct a column vector representing \ijf) in this basis using these amplitudes: (+XIV \ 0 W) — ► (2.23) Sx basis < - x| *>/ Thus the column vector representing the ket |+x) is which is to be compared with the column vector (2.6). The ket |+x) is the same state in the two cases; we have just written it out using the Sz basis in the first case and the Sx basis in the second case. Which basis we use is determined by what is convenient, such as what measurements we are going to perform on the state |+x). t. 2.2 Rotation Operators There is a nice physical way to transform the kets themselves from one basis set to another Recall that within classical physics a magnetic moment placed in a. 1 uniform magnetic field precesses about the direction of the field. When we discuss time evolution in Chapter 4, we will see that the interaction of the magnetic moment of a spin-^ particle with the magnetic field also causes the quantum spin state of the particle to rotate about the direction of the field as time progresses. In particular, if 1 You may object to calling anything dealing directly with kets physical since ket vectors are abstract vectors specifying the quantum state o f the system and involve, as we have seen, complex numbers. 34 | 2. Rotation of Basis States and Matrix Mechanics the magnetic field points in the y direction and the particle is initially in the state |+ z), the spin will rotate in the x-z plane. At some later time the particle will be in the state |+x). With this example in mind, it is useful at this stage to introduce a rotation operator R ( yj) that acts on the ket |+z), a state that is spin up along the z axis, and transforms it into the ket |+ x), a state that is spin up along the x axis: |+x) = fi(fj)|+ z> (2.25) Changing or transforming a ket in our vector space into a different ket requires an operator. To distinguish operators from ordinary numbers, we denote all operators with a hat. What is the nature of the transformation effected by the operator fl(y j)? This operator just rotates the ket |+z) by n /2 radians, or 90°, about the y axis (indicated by the unit vector j) in a counterclockwise direction as viewed from the positive y axis, turning, or rotating, it into the ket |+ x), as indicated in Fig. 2.1a. The same rotation operator should rotate |—z) into |—x). In fact, since the most general state of a spin particle may be expressed in the form of (. ), the operator rotates this - 1 2 1 ket as well: K ( f j)IV^) = K ( f j) (c+ l+Z) + c_|-z> ) = C+R{\j)|+z> + c _ f i( f j ) |—z> = c+ +x) + c_ | - x ) 1 (2.26) Note that the operator acts on kets, not on the complex numbers.2 THE ADJOINT OPERATOR What is the bra equation corresponding to the ket equation (2.25)? You may be tempted to guess that (+x| = (+z| ( yj), but we can quickly see that this cannot be correct, for if it were, we could calculate 3 (+x|+x> = [] = (+ z |fl(fj)fl(fj)|+ z > We know that (+ x|+ x) = 1, but since /?(yj) rotates by 90° around the y axis, ^ ( f j ) ^ ( f j) = performs a rotation of 180° about the y axis. But as indicated 2 An operator A satisfying A(a\\!r) +b\, we can carry out a rotation by any finite angle by compounding an infinite number of 0 infinitesimal rotations with d(f> = lim — N - *o o /V The rotation operator R(k) is then given by R(d>k) = lim (2.32) N^-oo The last identity in (2.32) can be established by expanding both sides in a Taylor series and showing that they agree term by term (see Problem 2.1). In fact, a series expansion is really the only way to make sense of an expression such as an exponential of an operator. EIGENSTATES AND EIGENVALUES What happens to a ket |+z) if we rotate it about the z axis—that is, what is R(k)|+z)? If you were to rotate a classical spinning top about its axis of rota­ tion, it would still be in the same state with its angular momentum pointing in the same direction. Similarly, rotating a state of a spin-| particle that is spin up along z about the z axis should still yield a state that is spin up along z , as illustrated in 5 Now you can see one reason for introducing the i in the defining relation (2.29) for an infinitesimal rotation operator. Without it, the generator J. would not have turned out to be Hermitian. 38 | 2. Rotation of Basis States and Matrix Mechanics z z (a) (b) Figure 2.3 (a) Rotating |+ z ) by angle 0 about the z axis with the operator /?(k) does not change the state, in contrast to the action o f the operator fl(0j), which rotates |+ z ) by angle 0 about the y axis, producing a different state, as indicated in (b). Fig. 2.3. In Chapter 1 we saw that the overall phase of a state does not enter into the calculation of probabilities, such as in (1.24). This turns out to be quite a general feature: two states that differ only by an overall phase are really the same state. We will now show that in order for R(k)|+z) to differ from |+z) only by an overall phase, it is necessary that Jz\+z) = (constant) | +z) (2.33) In general, when an operator acting on a state yields a constant times the state, we call the state an eigenstate of the operator and the constant the corresponding eigenvalue. First we will establish the eigenstate condition (2.33). If we expand the exponen­ tial in the rotation operator (2.32) in a Taylor series, we have fl(0k)|+z> = 1 _. 1 I+Z) (2.34) n 2 ! If (2.33) is not satisfied and Jz |+z) is something other than a constant times |+z), such as |+x), the first two terms in the series will yield |+z) plus a term involving |+ x), which would mean that R (0k) |+z) differs from |+z) by other than a mul­ tiplicative constant. Note that other terms in the series cannot cancel this unwanted |+x) term, since each term involving a different power of is linearly independent 0 from the rest. Thus we deduce that the ket |+z) must be an eigenstate, or eigenket, of the operator Jz. Let’s now turn our attention to the value of the constant, the eigenvalue, in (2.33). We will give a self-consistency argument to show that we will have agreement with  2.2 Rotation Operators I 39 the analysis of the St^m-Gerlach experiments in Chapter 1 provided h |± ) = ± 4 ± 2 ) 2 (2.35) This equation asserts that the eigenvalues for the spin-up and spin-down states are the values of Sz that these states are observed to have in the Stem-Gerlach experiments.6 First consider the spin-up state. If JI \+z) = ^\+ z) (2.36a) then 'y jj\+ z) = j\^ \+ z ) = ^ l j + z ) = ^ J 1+2) (2.36b) and so on. From (2.34), we obtain R(k)|+z> = \+z) = e~i - z ) = e,f2\-z) (2.40) 6 You can start to see why we introduced a factor of \ / h in the defining relation (2.29) between the infinitesimal rotation operator and the generator of rotations. 40 | 2. Rotation of Basis States and Matrix Mechanics Using (2.37) and (2.40), we see that e-W 2 e^ = l+z> + -Z) ~7T ~7T 1 el \ = e~i/2 (2.41) T 2 ' +I) + T 2 ' - Z)) which is clearly a different state from (2.38) for ^ 0. In particular, with the choice (j>= n / , we obtain 2 r/2 J?(*k)|+x> = * l+z> + l- z ) ^ >J = e~i7t/4\+y) (2.42) where we have replaced the term in the brackets by the state |+y) that we determined in (1.30). Since two states that differ only by an overall phase are the same state, we see that rotating the state |+x) by 90° counterclockwise about the z axis does generate the state |+y) when (2.35) holds. Thus we are led to a striking conclusion: When the operator that generates rotations about the z axis acts on the spin-up-along- z and spin-down-along-z states, it throws out a constant (the eigenvalue) times the state (the eigenstate); the eigenvalues for the two states are just the values of the z component of the intrinsic spin angular momentum that characterize these states. Finally, let us note something really perplexing about the effects of rotations on spin-^ particles: namely. /?(27rk)|+z) = e ,7r|+z) = - |+ z ) (2.43a) and /?(27rk)|+z) = e in\-z ) = - |- z > (2.43b) Thus, if we rotate a spin-^ state by 360° and end up right where we started, we find that the state picks up an overall minus sign. Earlier we remarked that we could actually perform these rotations on our spin systems by inserting them in a magnetic field. When we come to time evolution in Chapter 4, we will see how this strange prediction (2.43) for spin-^ particles may be verified experimentally. EXAMPLE 2.2 Show that rotating the spin-up-along-jc state |+ x ) by 180' about the z axis yields the spin-down-along-jc state.  2.3 The Identity and Projection Operators | 41 SOLUTION /?0rk)|+ x) = /?(7rk) ( -J=|+z> + ~j= l-z> - in / I 7i n / l ) t t m + v r ' “ z> = e-i*/2 / 7 1 I I+Z>+v! 1-z) e1" ) = ^ / 2 ( J _ | +z) _ J _ l - z ) = e - ‘*n-\- x> Ji V 2 ) where in the last line we have used the phase convention for the state | - x ) given in (2.15). 2.3 The Identity and Projection Operators In general, the operator R(0 n) changes a ket into a different ket by rotating it by an angle 0 around the axis specified by the unit vector n. Most operators tend to do something when they act on ket vectors, but it is convenient to introduce an operator that acts on a ket vector and does nothing: the identity operator. Surprisingly, we will see that this operator is a powerful operator that will be very useful to us. We have expressed the spin state | \fr) of a spin-| particle in the Sz basis as \\fr) = \+z)(+z\\fr) + |-z )(-z |V 0. Wecan think of the rather strange-looking object |+ z)(+ z| + |- z ) ( - z | (2.44).1*.*.’ as the identity operator. It is an operator because when it is applied to a ket, it yields another ket. Moreover, if we apply it to the ket \ijf), we obtain (|+ z)(+ z| + |-z>+/>_ = (|+ z > (+ z |)(|-z > (-z |) = |+ z )(+ z |—z)(—z| = 0 (2.52a) P_P+ = (|-Z )(-Z |)(|+ Z )(+ Z |) = —z)(—z|+ z)(+ z | 1 = 0 (2.52b) These results are illustrated in Fig. 2.5. Our discussion of the identity operator and the projection operators has arbitrarily been phrased in terms of the Sz basis. We could as easily have expressed the same state | \j/) in terms of the Sx basis as | \j/) = |+x)(+x|V^) + |-x )(-x |V ^ ). Thus we can also express the identity operator as |+ x )(+ x | + I—x> 'M > I = 1 (2.57) i where the sum is from / = to i = 2. The straightforward generalization of this 1 relationship to larger dimensional bases will be very useful to us later. 2.4 Matrix Representations of Operators In order to change, or transform, kets, operators are required. Although one can discuss concepts such as the adjoint operator abstractly in terms of its action on the bra vectors, it is helpful to construct matrix representations for operators, making concepts such as adjoint and Hermitian operators more concrete, as well as providing the framework for matrix mechanics. Equation (2.25) is a typical equation of the form m ) = \cp) (2.58)  2.4 Matrix Representations of Operators | 47 where A is an operator and | \fr) and \(p) are, in general, different kets. We can also think of the eigenvalue equation (2.35) as being of this form with | + (+z|A |-z)(-z|V '-> = (+z| = \ / ( + z | ^ ) \ _ /(+ z|< p> \ \ (—z|/4|+z) (—z|A|—z) / V (-z\ijf) ) \ (—z|^p) / f' In the same way that we can represent a ket \ijf) in the Sz basis by the column vector / (+Z|V \0 IV^) — (2.64) Sr basis we can also represent the operator A in the Sz basis by the 2 x 2 matrix in (2.63). Just as for states, we indicate a representation of an operator with an arrow: ^ (265) Sxbasis \ (—z|/4 |+z) (—Z|/4 | —z) / \ A2\ A22 ) If we label our basis vectors by 11) and |2) for the states |+z) and |- z ) , respectively, we can express the matrix elements Ajj in the convenient form Aij = (i\A\j) (2.66) 48 I 2. Rotation of Basis States and Matrix Mechanics where i labels the rows and j labels the columns of the matrix. Note that knowing the four matrix elements in (2.63) allows us to determine the action of the operator A on any state \\fr). MATRIX REPRESENTATIONS OF THE PROJECTION OPERATORS As an example, the matrix representation of the projection operator P+ is given by P+ ------- / < + ^ + l+ ‘ > < + ^ +l - * > W i o\ (267a) Sz basis \( - z |P + |+ Z ) < — Z|P+ | —Z > / \0 0 / where we have taken advantage of (2.49) in evaluating the matrix elements. Similarly, the matrix representation of the projection operator P_ is given by Thus, the completeness relation P+ + P_ = 1 in matrix form becomes ( 2. 68) where I is the identity matrix. The action of the projection operator P+ on the basis states is given by c :)C)=o c :)o =o in agreement with equations (2.49a) and (2.49b), respectively. MATRIX REPRESENTATION OF J , As another example, consider the operator Jz, the generator of rotations about the z axis. With the aid of (2.35), we can evaluate the matrix elements: j ____^ / (+Zl'/zl+z) (+zlAl~z) \ 5: ba.sis \ ( - Z|7z|+ Z) {-Z\Jz\ - Z ) j _ / (fi/2) { - h i 2 ) \ \ (/i/2 )(-z |+ z ) (-h /2 ){-z\-z)) / h/2 0 \ (2.70) i o -up)  2.4 Matrix Representations of Operators | 49 The matrix is diagonal with the eigenvalues as the diagonal matrix elements be­ cause we are using the eigenstates of the operator as a basis and these eigenstates are orthogonal to each other. The eigenvalue equations j z\+z) = {h/ 2)|+z) and Jz |—z) = (—h/2)\-z) may be expressed in matrix mechanics as (T -;„)CHC) and (T JUKD-IC) respectively. Incidentally, we can write the matrix representation (2.70) in the form - < m 0 NA / l 0 X W 0 0\ Szbasis V 0 - h i 2 ) 2 VO 0 / 2 VO 1/ which indicates that t h * h * h.... h..... Jz = - p + - - P - = - |+ z ) + z - —|—z) (—z| (2.73b) 2 2 2 2 We could have also obtained this result directly in terms of bra and ket vectors by applying Jz to the identity operator (2.48). EXAMPLE 2.3 Obtain the matrix representation of the rotation operator R (< f> k) in the Sz basis. SOLUTION Since R ((f> k) = e ~ ' ^ z/h and = e T ,< t> /2 \ ± z ) ( e " ' * / 2 0 \ R((f>k) — *-» I n I st biffs V 0 e'M2 ) This matrix is diagonal because we are using the eigenstates of Jz as a basis. MATRIX ELEMENTS OF THE ADJOINT OPERATOR We next form the matrix representing the adjoint operator If an operator A acting on a ket | V^) satisfies A |V0 = I (2.89)  2.5 Changing Representations | 53 that is by a rotation of the state |- z ) by 90° around the y axis. Following this procedure, as Problem 3.5 shows, we find that l-x) = — 5=|+z> + -— I—*) (2.90) V2 V2 which differs from (2.15) by an overall minus sign. We will use the states |+x) and | - x ) shown in (2.87) and (2.90) for the remainder of this section since it is convenient to focus our discussion on basis states that are related to the states by |+z) and | —z) by application of a rotation operator, specifically |±x) = /?(fj)|± z> (2.91a) and therefore (+x|y2|-x> \ / (+x\f) \ (Sz) = ((iM+x>, (iM-x>) (-x |7 2|+x) ( - x |i 2|-x ) / \ (-xlVX / You can verify that we can also go from (2.105) in the Sz basis to (2.107) in the Sx basis by inserting the identity operator §§+ before and after the 2 x 2 matrix in (2.105), provided we use the S-matrix (2.99) that transforms between these two basis sets. As an example, let’s return to (1.20), where we evaluated the expectation value of Sz for the state |+x). Substituting the column vector representation (2.6) for this  2.7 Photon Polarization and the Spin of the Photon | 59 ket in the S2 basis jnto (2.105), we see that the expectation value may be written in matrix form as (2.108) EXAMPLE 2.6 Use matrix mechanics to evaluate the expectation value (Sz) for the state |+x) in the Sx basis states l+x> = 7 I I+Z> + T i ' ~ z) ' ~ x) = 7 ! l+z) “ SOLUTION In Example 2.5 we saw that in this basis n /o i 2 \ 1 0 / then for the state |+x) h xh / 0 1 This result agrees of course with (2.108). In (2.108) the matrix form for the operator is especially straightforward, while here it is the representation for the state that is especially simple. EXAMPLE 2.7 Use matrix mechanics to determine (Sz) for the state m = ^\+*) + ‘- Y \ - z ) Compare your result with thatfef Example 1.2. SOLUTION 1 r h ( 1 0 \ 1 h (5; ) = W 5z W = - ( l , - , V 3 ) I ( o _ J - 4 in agreement with Example 1.2. 2.7 Photon Polarization and the Spin of the Photon The previous discussion about representations of states and operators may seem somewhat mathematical in nature. The usefulness of this type of mathematics is just 60 | 2. Rotation of Basis States and Matrix Mechanics Figure 2.10 Two sets o f transmission axes o f a polarizer that may be used to create polarization states o f photons traveling in the z direction. a reflection of the fundamental underlying linear-vector-space structure of quantum mechanics. We conclude this chapter by looking at how we can apply this formalism to another physical two-state system, the polarization of the electromagnetic field. Many polarization effects can be described by classical physics, unlike the physics of spin-^ particles, which is a purely quantum phenomenon. Nonetheless, analyzing polarization effects using quantum mechanics can help to illuminate the differences between classical and quantum physics and at the same time tell us something fundamental about the quantum nature of the electromagnetic field. Instead of a beam of spin-^ atoms passing through a Stem-Gerlach device, we consider a beam of photons, traveling in the z direction, passing through a linear polarizer. Those photons that pass through a polarizer with its transmission axis horizontal, that is, along the x axis, are said to be in the state |x), and those photons that pass through a polarizer with its transmission axis vertical are said to be in the state |y ). 12These two polarization states form a basis and the basis states satisfy {x\y) = , since a beam of photons that passes through a polarizer 0 whose transmission axis is vertical will be completely absorbed by a polarizer whose transmission axis is horizontal. Thus none of the photons will be found to be in the state |x) if they are put into the state |y) by virtue of having passed through the initial polarizer (assuming that our polarizers function with percent efficiency). 1 0 0 We can also create polarized photons by sending the beam through a polarizer whose transmission axis is aligned at some angle to our original x-y axes. If the transmission axis is along the x' axis or y' axis shown in Fig. 2.10, the corresponding polarization states may be written as a superposition of the |x) and |y) polarization states as |x') = |x)(x|x') + |y)(y|x') |y') = |x)(x|y') + |y )(y |y ') (2.109) What are the amplitudes such as (x|x'), the amplitude for a photon linearly polarized along the jc' axis to be found with its polarization along the x axis? 12 These states are often referred to as |.v) and |y). A different typeface is used to help distinguish these polarization states from position states, which will be introduced in Chapter 6.  2.7 Photon Polarization and the Spin of the Photon | 61 Figure 2.11 An x' polarizer follow ed by an x polarizer. A classical physicist asked to determine the intensity of light passing through a polarizer with its transmission axis along either the x or the y axis after it has passed through a polarizer with its transmission axis along x \ as pictured in Fig. 2.11, would calculate the component of the electric field along the x or the y axis and would square the amplitude of the field to determine the intensity passing through the second polarizer. If we denote the electric field after passage through the initial polarizer by Ex>, then the components of the field along the x and y axes are given by Ex = Ex>cos (j> E v = Ex>sin Thus the intensity of the light after passing through the second polarizer with its transmission axis along the x or y axis is proportional to cos or sin , 2 0 2 respectively. We can duplicate the classical results if we choose {x\x') = cos and 0 {y\x') = sin. Similarly, if the first polarizer has its transmission axis along the 0 / axis and we denote the electric field after passage through this polarizer by £ y then the components of the field along the x and y axes are given by Ex = —E Y>sin (j> E v = E v>cos (j> Again, we can duplicate the classical results if we choose {x\y') = — sin 0 and (y \y') = cos (j>. Of course, the experiments outlined here alone do not give us any information about the phases of the amplitudes. However, since classical electromag­ netic theory can account for interference phenomena such as the Young double-slit experiment, it is perhaps not too surprising that our conjectures about the amplitudes based on classical physics yield a valid quantum mechanical set, including phases: \x ') = cos \x) + sin \y) \y') = —sin (f>\x) + cos \y) (. 2 1 1 0 ) Where do the quantum effects show up? Classical physics cannot account for the granular nature of the measurements, that a photomultiplier can detect photons coming in single lumps. Nor can it account for the inherently probabilistic nature of the measurements; we cannot do more than give a probability that a single photon in the state \x ') will pass through a polarizer with its transmission axis along x. For 62 | 2. Rotation of Basis States and Matrix Mechanics example, if the angle (j>= 60°, then a single photon after having passed through an x' polarizer has a probability of | (x\x')\2 = cos 60° = 0.25 of passing through a 2 second x polarizer. Knowing the polarization state of the photon does not, in general, determine whether it will pass through a subsequent polarizer. All we can determine is the probability, much to the discomfiture of the classical physicist who would like to believe that such results should be completely determined if enough information is known about the state of the system. The classical and quantum predictions are, however, in complete accord when the intensity of the beams is high so that the number of photons is large. We can use (2.110) to calculate the matrix § + that transforms from the |x)-|y) basis to the \x')-\y') basis: / (x'|x> (x'|y) \ _ / cos\ Sf = l " 1 = 1 I (2.111) V( y » (y'ly) / \ - sin cos 0 / The matrix § that transforms from the \x')-\y') basis to the |x)-|y) basis is given by , , , , , j')\ / c o s (/> - s in < £ \ s=l ; )= (. (2.112) ') / \ sin (j> cos (j> / You can check that these matrices satisfy § +§ = I. All the elements of the matrix § are real. In fact, it is an example of an orthogonal matrix familiar from classical physics for rotating a vector in the x-y plane counterclockwise about the z axis by an angle.We can express § in terms of the rotation operator R(k) that rotates the ket vectors themselves in this direction (|x') = /?( k)|x) and |y') = J?( k)|y)): 0 0 / {x\R((f>k)\x) (x|/?(0k)|y) \ _ / c o s 0 - s in < £ \ \ (y|J?( k)|x) 0 (y|J?( k)|y) / 0 \s in 0 cos 0 / There is another set of basis vectors that have a great deal of physical significance but cannot be obtained from the |x)-|y) basis by a simple rotation. We introduce \R) = ~^=(\x) + /|y » (2.114a) V 2 |L) = - j= ( |x ) - i|y > ) (2.114b) V 2 These states are referred to as right-circularly polarized and left-circularly polarized, respectively. First, let’s ask what the classical physicist would make of a right-circularly polarized electromagnetic plane wave of amplitude E0 traveling in the z direction. E = E0iei(kz- OJt) + iE 0y (kz~/c)z. Thus a beam of linearly polarized light incident on such a crystal with its polarization axis inclined at 45° to the x axis will have equal magnitudes for the x and y components of the electric field, as indicated in Fig. 2.12, and there will be a phase difference [(nx —n v)co/c]z between these two components that grows as the light passes through a distance z in the crystal. The crystal can be cut to a particular thickness, called a quarter-wave plate, so that the phase difference is 90° when the light of a particular wavelength exi&the crystal, thus producing circularly polarized light. What does the quantum physicist make of these circular polarization states (2.114)? Following the formalism of Section 2.2, it is instructive to ask how these states change under a rotation about the z axis. If we consider a right-circularly Figure 2.12 Plane-polarized light incident on a quarter-wave plate with its direction o f polarization oriented at 45° to the optic axis will produce circularly polarized light. 64 | 2. Rotation of Basis States and Matrix Mechanics polarized state that has been rotated by an angle 0 counterclockwise about the z axis, we see that it can be expressed as \R , ) = j = ( \ x,) + i\y ')) = —^[cos 0|x) + sin 0| y) + / ( - sin 0|x) + cos0|y))] V 2 (cos 0 —i sin 0 ) (|^r) + /'!>')) = n = e~i4>\R) (2.116) Thus this state picks up only an overall phase factor when the state is rotated about the z axis. Based on our experience with the behavior of spin-1 states under rotations, (2.116) indicates that the state is one with definite angular momentum in the z direction. Since (2.32) shows that |/?') = fl(0k)|/?> = e~iJ=*n\R) (2.117) consistency with the preceding equation requires that Jz\R) = h\R) (2.118) Similarly, if we rotate the left-circularly polarized state by angle 0 counterclockwise about the z axis, we obtain \L')=ei*\L) (2.119) telling us that 13 Jz\L) = -h\L) (2.120) Thus the right-circularly and left-circularly polarized states are eigenstates of Jz, the operator that generates rotations about the z axis, but with eigenvalues ±h, not the ± h / 2 characteristic of a spin-^ particle. In Chapter 3 we will see that the eigenvalues of Jz for a spin-1 particle are +h, 0, and —h. Photons have intrinsic spin of 1 instead of 4 The absence of the 0 eigenvalue for Jz for a photon turns out to be a special characteristic of a massless particle, which moves at speed c. 13 A particle with a positive (negative) projection of the intrinsic angular momentum along the direction of motion is said to have positive (negative) helicity. Photons thus come in two types, with both positive and negative helicity, corresponding to right- and left-circularly polarized light, respectively.  2.8 Summary | 65 EXAMPLE 2.8 Determine the matrix representation of the angular mo­ mentum operator Jz using both the circular polarization vectors | R) and \L) and the linear polarization vectors |x) and |y) as a basis. SOLUTION Let’s start with the easy one first. Since the states |R) and |Z_) are eigenstates of Jz with eigenvalues h and —h, respectively J _______ ( { R \ J Z\R) {R\Jz\L)\ = / h 0 \ w-wbasis' V (L\JZ\R) (L\JZ\L) ) 2 VO - h ) The matrix is diagonal in this basis with the eigenvalues of the basis states on the diagonal. Switching to the linear polarization states |x) and |y) : j _______ ^ ( (x\R) (x| L) \ / (R\JZ\R) (R\JZ\L) \ / {R\x) (R\y) \ 2 \x ).\y) b a s is ' V (y\R) (y\L) ) V (L\JZ\R) (L\JZ\L) ) v (L\X) (L\y)) -U', -X -JiC I X In this basis, the matrix has only off-diagonal elements. Since a Hermitian matrix is equal to its transpose, complex conjugate, both of these represen­ tations for Jz satisfy this condition, as they must. 2.8 Summary In this chapter we have introduced operators in order to change a state into a different state. Since we sire dealing here primarily with states of angular momentum, the natural operation is to rotate these states so that a state in which a component of the angular momentum has a definite value in a particular direction is rotated into a state in which the angular momentum t$s the same value in a different direction The. 14 operator that rotates states counterclockwise by angle 4>about the z axis is R{(t>k) = e - iJ^ /n (. 2 1 2 1 ) where the operator Jz is called the generator of rotations about the z axis. In general, for an arbitrary operator A , the bra corresponding to the ket m ) = \ = ± ||± x ) (2.127) where Jx is the generator of rotations about the x axis. In Chapter 3 we will argue on more general grounds that we should identify the generator of rotations with the component of the angular momentum along the axis about which the rotation is taking place. In subsequent chapters we will see that the operator that generates displacements in space is the linear momentum operator and the operator that generates time translations (moves the state forward in time) is the energy operator. Thus we will see repeated a pattern in which a Hermitian operator A is associated with a physical observable and the result an of a measurement for a particular state |an) satisfies A \an) = a n\a„) (2.128)  2.8 Summary I 67 Note that for a Hermitian, or self-adjoint, operator (A = A*), the bra equation corresponding to (2.128) is (an\A = (an\a* (2.129) An equation in which an operator acting on a state yields a constant times the state is called an eigenvalue equation. In this case, the constant an in (2.128) is called the eigenvalue and the state \an) [or (an\ in (2.129)] is called the eigenstate. We will now show that the eigenvalues of a Hermitian operator are real. Taking the inner product of the eigenvalue equation (2.128) with the bra {ak |, we obtain (ak\A\an) = a n(ak\an) (2.130) Taking advantage of (2.129), this equation becomes a*k(ak\an) = a„(ak\an) (2.131a) or (a*k - an)(ak\an) = 0 (2.131b) Note that if we take k = n, we find ( a * - a n)(an\an) = 0 (2.132) and therefore the eigenvalues of a Hermitian operator are real (a* = an), ac ce ssa ry condition if these are to be the values that we obtain for a measurement. Moreover, (2.131b) shows that (ak\an) = 0 ak ^ a n (2.133) as we argued in Chapter 1 must be tfpe based on the fact that (ak\a„) is the amplitude. >- to obtain ak for a particle in the state \an). This shows that the eigenstates of a Hermitian operator corresponding to distinct eigenvalues are orthogonal. Thus our association of Hermitian operators with observables such as angular momentum forms a nice, self-consistent physical picture. We also see that we can express the expectation value (A) of the observable A in terms of the operator A as (A) = ( f \ A \ f ) (2.134) For simplicity, let’s consider the case where there are two eigenstates 1^) and \a2) with a i ^ a 2, as is the case for spin Since a general state can be written as IVO = C\\ax) + c 2\a2) (2.135) 68 | 2. Rotation of Basis States and Matrix Mechanics then W \ A \ t ) = (c\(ax\ + cl(a2\)A{cx\ax) + c |a2» 2 = (c\(a{\ +cl{a2\)(c{a {\a{) + c2a2\a2)) = \c\\2a\ + |c |~a 2 2 = (A) (2.136) where the last step follows since the penultimate line of (2.136) is just the sum of the eigenvalues weighted by the probability of obtaining each of those values, which is just what we mean by the expectation value. Also note that, as in (1.40), (2.135) can be expressed in the form IVO = ItfiX^ilVO + \a2)(a2\ f ) (2.137) This suggests that we can write the identity operator in the form Ifli>(flil + |fl >(fl l = l 2 2 (2.138) which is also known as a completeness relation, because it is equivalent to saying that we can express an arbitrary state \xf/) as a superposition of the states ^ ) and 1 |a2), as shown in (2.137). The identity operator can be decomposed into projection operators ^ = l^iXail and P2 = \a2){a2\ (2.139) that project out of the state \xf/) the component of the vector in the direction of the eigenvector. For example, P i m = \al)(alm (2.140) If we insert the identity operator (2.138) between theket and the bra in the amplitude {(pity), we obtain ((p\f) = ( 0, as we would expect physically since X specifies the magnitude squared of the angular momentum in the state | A., m). Consider (X, m\J2\X, m) = Xh2(X, m\X, m) (3.26) Like all physical states, the eigenstates satisfy (A,m|A,m) = l. A typical term in the left-hand side of (3.26) is of the form (X, m) = W \ f ) (3.27) where we have defined 7JA, m) = |V0, and (VM = (A., m\Jx since Jx is Hermitian. Although the ket | VO is not normalized, we can always write it as | VO = c\(p), where c is a complex constant (that must have the dimensions of h) and \q>) is a physical state satisfying ( 0. AN EXAMPLE: SPIN 1 t To illustrate what we have discovered so far and suggest the next step, let’s take the specific example involving the following three 3 x 3 matrices: (0 1 °\ /0 -i 0 \ (i 0 0 \ _h_ n_ 1 0 i i 0 —i Jz —►h 0 0 0 V 2 V2 \o i 0 / ^0 / 0 ) \0 0 -1 / (3.28) 4 Because Jx is the generator of rotations about the x axis, the ket (1 — iJxd/h)\X, m) is just the ket that is produced by rotating the ket \X, m) by angle d about the x axis. Thus the ket |V0 can be viewed as a linear combination o f the rotated ket and the ket |X, m), that is, a superposition o f two physical states. 84 | 3. Angular Momentum For now, don’t worry about how we have obtained these matrices. Later in this chapter we will see how we can deduce the form of these matrices (see Example 3.3 and Problem 3.14). In the meantime, let’s see what we can leam from the matrices themselves. To begin, how can we be sure that these three matrices really represent angular momentum operators? Following our earlier discussion, it is sufficient to check (see Problem 3.13) that these matrices do indeed satisfy the commutation relations (3.14). We next calculate /I 0 °\ j 2 = j - j = 3 ] + py + ; 2 0 1 0 (3.29) \ 0 0 l) We see explicitly that J is just a constant times the identity matrix and thus com- 2 mutes with each of the components of J. The operator Jz is diagonal as well, sug­ gesting that the matrix representations (3.28) sire formed using the eigenstates of Jz as well as J as a basis. The column vectors representing these eigenstates arc 2 given by 5 / \ (0\ 1 0 1 and (°\ I 0 \o J \o j \ 1 ) which have eigenvalues ft, , and —ft, respectively, as can be verified by operating 0 on them with the matrix representing Jz. For example, 1 0 0 \ /1\ ( 1\ 0 0 0 0 =h 0 (3.31) 0 0 -1) \oj w Similarly, we see that each of these states is an eigenstate of J with eigenvalue 2 ti1. 2 Since the matrix representations of Jx and Jy are not diagonal, the states (3.30) are not eigenstates of these operators. It is straightforward to evaluate the action of the operators Jx and Jy on the basis states. There is, however, a linear combination of these two operators, namely, (0 1 °\ Jx + I Jy y/lti 0 0 1 (3.32) \0 0 0/ whose action on the basis states exhibits an interesting pattern. Applying this oper­ ator to the basis states (3.30), we obtain 5 Compare these results with (2.70), (2.71), and (2.72) for a spin-^ particle.  3.3 The Eigenvalues and Eigenstates of Angular Momentum | 85 i (0 1 o\ (0\ (0\ y/2ti 0 0 1 0 = y/lh 1 (3.33) U 0 o) w (0 1 °\ (0\ / 1\ y/2ti 0 0 1 1 = y/2h 0 (3.34) \0 0 oj \oj \ 0 / (0 1 °\ ( l) ( 0\ Vin 0 0 1 0 = 0 (3.35) u 0 oj \ 0 y U / according to (3.33), the operator Jx + i Jy acting on the —h for Jz turns it into a state with eigenvalue 0, multiplied by y/lft. Similarly, as (3.34) shows, when the operator acts on the state with eigenvalue 0 for Jz, it turns it into a state with eigenvalue h, multiplied by y/lft. This raising action terminates when the operator Jx + i JY acts on the state with eigenvalue fi, the maximum eigenvalue for Jz. See (3.35). It can be similarly verified that the operator (0 0 °\ Vin i 0 0 (3.36) \0 1 0/ has a lowering action when it acts on the states with eigenvalues h and , turning 0 them into states with eigenvalues 0 and —h , respectively. In this case, the lowering action terminates when the operator (3.36) acts on the state with eigenvalue —h, the lowest eigenvalue for Jz. RAISING AND LOWERING OPERATORS Let’s return to our general analysis £>f angular momentum. The example suggests that it is convenient to introduce the two operators J± = Jx ± i J y (3.37) in the general case. Notice that these sire not Hermitian operators since j'l = j ] + ( - i ) / J = j x - iJy = j_ (3.38) The utility of these operators derives from their commutation relations with Jz: [Jz, 4 ] = [Jz, Jx ± iJy] = ifijy ± i { - i h J x) = ± h J ± (3.39) To see the effect of J+ on the eigenstates, we evaluate JZJ+\X, m). We can use the commutation relation (3.39) to invert the order of the operators so that Jz can act 86 | 3. Angular Momentum directly on its eigenstate \X, m). However, since the commutator of Jz and J+ is not zero but rather is proportional to the operator J+ itself, we pick up an additional contribution: Jzj +|A., m) = ( j +Jz + fiJ+)|A., m) = ( J+mfi + fiJ+)\X, m) = (m + \)hJ+\X, m) (3.40a) Inserting some parentheses to help guide the eye: JZ(J+\X, m » = (m + \)h(J+\X, m » (3.40b) we see that 7+ |A., m) is an eigenstate of Jz with eigenvalue (m + 1)fi. Hence J+ is referred to as a raising operator. The action of J+ on the state \X,m) is to produce a new state with eigenvalue (m + )h. 1 Also JZJ-\X, m) = (J_JZ — hJ_)\X , m) = ( J_mh — HJ_)\X, m) = (m — \)h.J_\X, m) (3.41a) Again, inserting some parentheses, JZ(J_\X, m)) = (m — \)fi(J_\X, m)) (3.41b) showing that 7_|A., m) is an eigenstate of Jz with eigenvalue (m — 1)h\ hence J_ is a lowering operator. Notice that since J+ and J_ commute with J2, the states J±\X, m) are still eigenstates of the operator J2 with eigenvalue Xfi2: J2(7± |A., m)) = J±J2|A, m) = Xfi2(J±\X, m)) (3.42) THE EIGENVALUE SPECTRUM We now have enough information to determine the eigenvalues X and m, because there sire bounds on how far we can raise or lower m. Physically (see Fig. 3.4), we expect that the square of the projection of the angular momentum on any axis should not exceed the magnitude of J2 and hence m2 < X (3.43) Formally, since {X, m\(J2 + y )|A., m) > 0 2 (3.44)  3.3 The Eigenvalues and Eigenstates of Angular Momentum | 87 z Figure 3.4 The projection o f the angular momentum on the axis never exceeds the magnitude o f the angular momentum. Caution: This is a classical picture; the angular momentum cannot point in any definite direction. we have (A., m |(J — J 2) |A, m) = (A —m2)h2(A, m|A, m) > 0 2 (3.45) establishing (3.43). Let’s call the maximum m value j. Then we must have J+| A , » = 0 (3.46) since otherwise J+ would create a state |A, j + 1), violating our assumption that j is the maximum eigenvalue for 7 Using ,. 6 y_ J+ = (Jx —i Jv)(jX + iJy) = J 2 + J 2 + i[JXi Jy] = j 2- J 2- h J z (3.47) we see that J-J+ |X, j) = (J - 2 / 2 - hJz)\\, j) = & - j 2 - m 2\ l. j ) = 0 (3-48) orA = j ( j + ). 1 Similarly, if we call the minimum m value j \ then 7_|A, j') = 0 (3.49) and we find that /) = y 2 - y + hJz)\\, j') 2 = a - j 12 + j ') n 2ia, / ) = o (3.50) 6 Equation (3.35) demonstrates how this works for the special case o f spin I. 88 | 3. Angular Momentum j j ~ I 7 -2 --------------------- - j + 2 j +^ Figure 3.5 The possible m values for a fixed magnitude ----------------------- j y / j ( j + 1)^ o f the angular momentum. In deriving this result, we have used J+J- = (Jx + iJy)(Jx - iJ y) = P + i ) - i[ i„ yv] = j 2 - J 2 + hJz (3.51) Thus X = j a — j'. The solutions to the equation j + j = j ' 2 — j \ which results 2 from setting these two values of X equal to each other, sire j ' = —j and j ' = j + 1. The second solution violates our assumption that the maximum m value is j. Thus we find the minimum m value is —j. If we start at the m = j state, the state with the maximum m, and apply the lowering operator a sufficient number of times, we must reach the state with m = —j, the state with the minimum m. If this were not the case, we would either reach a state with an m value not equal to —j for which (3.49) is satisfied or we would violate the bound on the m values. But (3.49) determines uniquely the value of j ' to be —j. Since we lower an integral number of times, j — j' = j — (—j ) = 2j = an integer, and we deduce that the allowed values of j are given by j= 0 ' ' i 2’ ” - (3-52) As indicated in Fig. 3.5, the m values for each j run from j to —j in integral steps: m = j , j - 1, j - 2......... - j + 1, - j (3.53) “““““““ 2j + 1states Given these results, we now change our notation slightly. It is conventional to denote a simultaneous eigenstate of the operators J and Jz by \j, m) instead of 2 |A., m) = | j (j + 1), m). It is important to remember in this shorthand notation that J 2| j, m) = j ( j + \)fi2\j, m) (3.54a) as well as Jz\j, m) = m h\ jt m) (3.54b)  3.3 T