Preface
There were two revolutions in the way we viewed the physical world in the twentieth century: relativity and quantum mechanics. In quantum mechanics the revolution was 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 consists of students who have completed a course in modem physics that includes an introduction to quantum mechanics in the form of one-dimensional wave mechanics. Such a modern physics course is likely to emphasize the historical development of the subject. 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 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 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 physics and shows them the strange behavior of quantum mechanics in a familiar environment. This is probably why students find their first introduction to wave 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 student 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 a quantum state vector, complex probability amplitudes, and the probabilistic interpretation of quantum mechanics in the context of analyzing a number of Stern-Gerlach experiments carried out with spin-1/2 particles. By introducing ket vectors at the beginning, we have the framework for matrix mechanics and for thinking about states as having an existence quite apart from the way we happen to choose to represent them. 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 follow 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 determined 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 introduction of the operator techniques for determining the angular momentum eigenstates and eigenvalue spectrum and the derivation of the uncertainty relations from the communtation 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-1/2particles. The total-spin-0 state serves as the basis for a discussion of the Einstein-Podolsky-Rosen 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 states that have a spectrum in the continuum 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, which can serve as a review for students with a good background in one-dimensional wave mechanics.
Chapter 7 is devoted to the simple harmonic oscillator, which merits a chapter all its own. Although the material in Chapter 8 on path integrals may be skipped without affecting subsequent chapters (with the exception of Section 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 dynamics. 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 chosen to postpone path integrals to a second course and then to insert the material in Chapter 8 before Chapter 14. Incidentally, the material on path integrals is the only part of the book that may require students to have had an upper-division 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 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 this chapter. Section 14.6 stands on its own.Although SI units are the standard for undergraduate education in electricity and magnetism, I have chosen in this 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 electromagnectic field, the choice of units has little impact.
A comprehensive solutions manual for the instructor is available from the publisher, 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 tried out versions of this book in manuscript form. One of my colleagues at Harvey Mudd, Tom Helliwell, read the first draft of the manuscript and offered some valuable comments as well as encouragement. Art Weldon of West Virginia University suggested a number of ways to improve the accuracy and effectiveness of the manuscript. This text was initially published in the McGraw-Hill International Series in Pure and Applied Physics. I have benefited from the comments of 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. I wish to thank Jane Ellis for her help in the reprinting of this book by University Science Books. The Pew Science Program provided support for two Harvey Mudd students, Doug Dunston and Doug Ridgway, who helped in the preparation of the text and figures, respectively. Helen White helped in checking the galley proofs. And I especially want to thank my family for cheerfully letting me devote so much time to this project.John S. Townnsend