Quantum mechanics fascinates me. It describes a wide variety of phenomena based on very few assumptions. It starts with a framework so unlike the differential equations of classical physics, yet it contains classical physics within it. It provides quantitative predictions for many physical situations, and these predictions agree with experiments. In short, quantum mechanics is the ultimate basis, today, by which we understand the physical world.

Thus, I was very pleased to be asked to write the next revised edition of Modern Quantum Mechanics, by J. J. Sakurai. I had taught this material out of this book for a few years and found myself very in tune with its presentation. Like many other instructors, however, I found some aspects of the book lacking and therefore introduced material from other books and from my own background and research. My hybrid class notes form the basis for the changes in this new edition.

Of course, my original proposal was more ambitious than could be realized, and it still took much longer than I would have liked. So many excellent suggestions found their way to me through a number of reviewers, and I wish I had been able to incorporate all of them. I am pleased with the result, however, and I have tried hard to maintain the spirit of Sakurai's original manuscript.

Chapter 1 is essentially unchanged. Some of the figures were updated, and reference is made to Chapter 8, where the relativistic origin of the Dirac magnetic moment is laid out.

Material was added to Chapter 2. This includes a new section on elementary solutions including the free particle in three dimensions; the simple harmonic oscillator in the Schrödinger equation using generating functions; and the linear potential as a way of introducing Airy functions. The linear potential solution is used to feed into the discussion of the WKB approximation, and the eigenvalues are compared to an experiment measuring “bouncing neutrons.” Also included is a brief discussion of neutrino oscillations as a demonstration of quantummechanical interference.

Since 1989 the editor has enthusiastically pursued a revised edition of Modern Quantum Mechanics by his late great friend J. J. Sakurai, in order to extend this text's usefulness into the twenty-first century. Much consultation took place with the panel of Sakurai friends who helped with the original edition, but in particular with Professor Yasuo Hara of Tsukuba University and Professor Akio Sakurai of Kyoto Sangyo University in Japan.

This book is intended for the first-year graduate student who has studied quantum mechanics at the junior or senior level. It does not provide an introduction to quantum mechanics for the beginner. The reader should have had some experience in solving time-dependent and time-independent wave equations. A familiarity with the time evolution of the Gaussian wave packet in a force-free region is assumed, as is the ability to solve one-dimensional transmission-reflection problems. Some of the general properties of the energy eigenfunctions and the energy eigenvalues should also be known to the student who uses this text.

The major motivation for this project is to revise the main text. There are three important additions and/or changes to the revised edition, which otherwise preserves the original version unchanged. These include a reworking of certain portions of Section 5.2 on time-independent perturbation theory for the degenerate case, by Professor Kenneth Johnson of M.I.T., taking into account a subtle point that has not been properly treated by a number of texts on quantum mechanics in this country. Professor Roger Newton of Indiana University contributed refinements on lifetime broadening in Stark effect and additional explanations of phase shifts at resonances, the optical theorem, and the non-normalizable state. These appear as “remarks by the editor” or “editor's note” in the revised edition. Professor Thomas Fulton of the Johns Hopkins University reworked his Coulomb scattering contribution (Section 7.13); it now appears as a shorter text portion emphasizing the physics, with the mathematical details relegated to Appendix C.

Though not a major part of the text, some additions were deemed necessary to take into account developments in quantum mechanics that have become prominent since November 1, 1982.

J. J. Sakurai was always a very welcome guest here at CERN, for he was one of those rare theorists to whom the experimental facts are even more interesting than the theoretical game itself. Nevertheless, he delighted in theoretical physics and in its teaching, a subject on which he held strong opinions. He thought that much theoretical physics teaching was both too narrow and too remote from application: “…we see a number of sophisticated, yet uneducated, theoreticians who are conversant in the LSZ formalism of the Heisenberg field operators, but do not know why an excited atom radiates, or are ignorant of the quantum theoretic derivation of Rayleigh's law that accounts for the blueness of the sky.” And he insisted that the student must be able to use what has been taught: “The reader who has read the book but cannot do the exercises has learned nothing.”

He put these principles to work in his fine book Advanced Quantum Mechanics (1967) and in Invariance Principles and Elementary Particles (1964), both of which have been very much used in the CERN library. This new book, Modern Quantum Mechanics, should be used even more, by a larger and less specialized group. The book combines breadth of interest with a thorough practicality. Its readers will find here what they need to know, with a sustained and successful effort to make it intelligible.

J. J. Sakurai's sudden death on November 1, 1982 left this book unfinished. Reinhold Bertlmann and I helped Mrs. Sakurai sort out her husband's papers at CERN. Among them we found a rough, handwritten version of most of the book and a large collection of exercises. Though only three chapters had been completely finished, it was clear that the bulk of the creative work had been done. It was also clear that much work remained to fill in gaps, polish the writing, and put the manuscript in order.

That the book is now finished is due to the determination of Noriko Sakurai and the dedication of San Fu Tuan. Upon her husband's death, Mrs. Sakurai resolved immediately that his last effort should not go to waste. With great courage and dignity she became the driving force behind the project, overcoming all obstacles and setting the high standards to be maintained.

So far we have not discussed how physical systems change with time. This chapter is devoted exclusively to the dynamic development of state kets and/or observables. In other words, we are concerned here with the quantum mechanical analogue of Newton's (or Lagrange's or Hamilton's) equations of motion.

TIME-EVOLUTION AND THE SCHRÖDINGER EQUATION

The first important point we should keep in mind is that time is just a parameter in quantum mechanics, not an operator. In particular, time is not an observable in the language of the previous chapter. It is nonsensical to talk about the time operator in the same sense as we talk about the position operator. Ironically, in the historical development of wave mechanics both L. de Broglie and E. Schrödinger were guided by a kind of covariant analogy between energy and time on the one hand and momentum and position (spatial coordinate) on the other. Yet when we now look at quantum mechanics in its finished form, there is no trace of a symmetrical treatment between time and space. The relativistic quantum theory of fields does treat the time and space coordinates on the same footing, but it does so only at the expense of demoting position from the status of being an observable to that of being just a parameter.

Time-Evolution Operator

Our basic concern in this section is, How does a state ket change with time? Suppose we have a physical system whose state ket at is represented by. At later times, we do not, in general, expect the system to remain in the same state. Let us denote the ket corresponding to the state at some later time by

where we have written α,

to remind ourselves that the system used to be in state

at some earlier reference time. Because time is assumed to be a continuous parameter, we expect

and we may as well use a shorthand notation,

for this. Our basic task is to study the time evolution of a state ket:

To put it another way, we are interested in asking how the state ket changes under a time displacement.