Modern Quantum Mechanics 3rd Edition

Book Preface

This book covers the material on quantum mechanics typically found in a first year graduate physics curriculum. The approach emphasizes states, operators, eigenvalues, and representations from the start. Building on these foundations, the reader sees, for example, how the Schr¨odinger representation is just one of several ways to realize quantum dynamics, and how classical physics emerges as an approximation. This approach also helps the reader gain an appreciation of purely quantum-mechanical phenomena, for example the magnetic moment and spin of an electron, that have no classical analogue.

The intended audience is the same as for earlier editions, that is, students having taken upper level undergraduate coursework in quantum physics, classical mechanics and electromagnetism, multivariable calculus, and ordinary and partial differential equations.
Professor Jun John Sakurai originally conceived the idea for this textbook, I think inspired by Dirac’s monograph. Sakurai’s life was cut short suddenly, as he was preparing the first manuscript. His colleague San Fu Tuan took over as Editor, completing a seven chapter manuscript for Addison-Wesley, who published the First Edition in 1985 and a Revised Edition in 1993. Some time later, I started work on the Second Edition for Pearson (who had since acquired Addison-Wesley). This volume contained a lot of new material, including an eighth chapter, and was published in 2010. The text was reissued by Cambridge University Press in 2017, which was also when I started work on the Third Edition.

Quantum mechanics has always fascinated me, but it was the First Edition of Modern Quantum Mechanics that finally explained to me the logical progression from fundamental assumptions to practical applications, with classical physics emerging as an approximation. When I first taught this material at Rensselaer Polytechnic Institute, I used the Revised Edi-tion, but found myself supplementing with my own notes on solutions of the Schr¨odinger equation and other topics. I also tried to use my course to prepare students for quantum field theory, introducing second quantization and relativistic quantum mechanics, neither of which were included in Sakurai’s book.
I was therefore pleased to be asked to take on the Second Edition. Sections were added to Chapters Two and Three on solutions to the Schr¨odinger equation. I reversed the order of Chapters Six and Seven, so that Scattering Theory came first, and I reworked the treatment so that it was based on the formal theory of time-dependent perturbations. The following chapter on Identical Particles was augmented to include second quantization and the quantization of the free electromagnetic field, and I added a new chapter on Relativistic Quantum Mechanics. I also included several connections throughout the book to experimental measurements, and worked to fix a number of idiosyncrasies that I found when I taught out of the book.

The result was a text that, I thought, achieved my goal of a high level treatment respect-ing Sakurai’s vision, adding reference to additional modern concepts and experiment, and preparing the reader for quantum field theory and beyond. The first two chapters lay the mathematical and physical foundations for the rest of the book, and connect the reader to undergraduate topics in wave mechanics. Chapter Three covers angular momentum from the perspective of the rotation operator, with strong connections to important concepts such as the density operator, central potentials, and Bell’s inequality. Groups are also introduced here, with further exposition in Chapter Four. Applications to “real world” problems are the focus of Chapters Five and Six, all the while keeping to the focus of building on the fundamentals. Chapters Seven and Eight move the discussion towards the “next” course in quantum mechanics, covering many-body formalism and the inclusion of special relativity.

The Third Edition keeps the same ordering of the eight chapters. Significant new material has been added, but I also worked to clarify some of the discussions and to fix various issues that I discovered after teaching out of the Second Edition. In fact, I compiled a long list of “Typographical Errors, Mistakes, and Comments” based on covering nearly the entire book in class, and working through all of the end-of-chapter problems. The Third Edition addresses all of the errors. It also addresses most of the comments, having to give up on some only for lack of time.

There are three new sections of new material. Despite its increasing use in condensed matter physics, I found no treatments of density functional theory in any quantum mechanics textbook. So, I added Section 7.6 to introduce the subject and take it through to its application in the helium atom. A reviewer’s suggestion inspired me to add Section 8.1.5 to show how the Klein–Gordon field, built using second quantization, fixes the problems of negative energies and nonpositive definite probability currents in the Klein–Gordon wave equation. The Second Edition treated spontaneous emission only as an end-of-chapter problem, but Section 5.8.4 now goes through the derivation, with some details and numerical calculations left as problems.

I added new appendices on the Hamiltonian for a Charge in an Electromagnetic Field, Notes on Complex Analysis, and Calculating Clebsch–Gordan Coefficients. The appendix on Electromagnetic Units has been significantly revised, and I updated the appendix on Elementary Solutions to Schr¨odinger’s Wave Equation to better connect to the discussions in the text.
Instructors may elect to pick and choose from topics in the book, and not necessarily in the order of presentation. Chapter One should be covered first, since it lays down the notation and fundamental assumptions. One could then, for example, take parts of Chapters Three and Four to expand on operators, observables, and symmetries, prior to discussing dynamics in Chapter Two. Many other combinations are possible. Indeed, throughout the book, I have tried to refer to other places in the text where relevant related material is covered or discussed.
As befits a graduate level textbook, the strategy here is to lay down the principles, following up with implications by deduction. Some example calculations are carried through in the text, but the end-of-chapter problems are generally meant to extend the discussion, and not simply practice what was covered. As such, I recommend that instructors choose problems, from the text or otherwise, that follow this idea, including connection to experimental measurements, where practical.

In several places in the book, either explicitly or implicitly, computer calculations are necessary to completely follow the arguments or to work the problems. I worked through these using MATHEMATICA, and am happy to share the code with anyone who would like to see it, but any other programming language or application can also be used, of course.

Producing the Second Edition was a long process that would not have been possible without help from many, many people. Colleagues in physics include John Cummings, Jack Fishburn, Joel Giedt, David Hertzog, Barry Holstein, Bob Jaffe, Matthew Kirby, Joe Levinger, Alan Litke, Kam-Biu Luk, Bob McKeown, Harry Nelson, Joe Paki, Murray Peshkin, Olivier Pfister, Mike Snow, John Townsend, San Fu Tuan, David Van Baak, Dirk Walecka, and Tony Zee. The people at Addison-Wesley/Pearson who guided me included Adam Black, Ashley Eklund, Deb Greco, Dyan Menezes, John Rogosich, and Jim Smith.

So many others were very helpful to me as I developed the Third Edition. This includes colleagues Kieron Burke, Mark Caprio, Carl Carlson, Benjamin Chandran, Chris Cocuzza, Martha Constantinou, Patrick Fasano, Jeremias Gonzalez, Aaron Kaplan (with special thanks for helping me learn DFT), Toh-Ming Lu, Carl Maes, Andreas Metz, Jerry Miller, Djordje Minic, Adilson Motter, Nick Murphy, Steve Naculich, Celso Nishi, John Perdew, Jon Rosner, and Roland Winkler. I am forever grateful to Simon Capelin at Cambridge University Press, for first bringing to me the possibility of republishing the Second Edition, and encouraging me to consider a Third Edition. Other key people at CUP include Jane Adams, Nick Gibbons, Lisa Pinto, and Ilaria Tassistro.

I can only offer my sincere apologies to people I should have listed, but whose name doesn’t appear because I’ve been careless with note keeping. There are also the very many people who, over the past several years, offered comments, some of which I’ve not been able to incorporate.

Finally, I give a special acknowledgement for Stuart Freedman, my mentor, colleague, and friend. Stuart’s Ph.D. thesis experiment was the first verification of the violation of Bell’s inequality, and he used this to stoke my interest in quantum mechanics. His guidance during my years as a graduate student and young scientist shaped my career, and he remained my friend and counselor until his untimely passing.