1. This book is a sequel to its authors' recently published Angular momentum in quantum physics: Theory and application ; it treats various advanced topics that could not be covered in the earlier volume without making it inconveniently long. My purpose is to explain the subject matter from a mathematician's point of view, but it would be awkward and difficult to do this without taking into account the contents of both books. Thus, in spite of its tardy appearance, this essay will, in effect, be an introduction to the two-volume work as a whole. When a physicist speaks of “angular momentum theory,” he is alluding to a theory that a mathematician would be more likely to describe as “the theory of rotational invariance.” This theory, whatever we call it, is concerned ( a ) with a technique for exploiting the fact that many physical laws are independent of orientation in space, and ( b ) with the many important consequences of this fact. The physicists' choice of the words “angular momentum theory” illustrates a tendency that is one of the many factors inhibiting communication between mathematicians and physicists. This is the tendency physicists have to avoid thinking in the abstract and instead to keep a concrete physical problem constantly in mind and use physical terminology whenever possible. From the mathematician's point of view, the physicist is behaving like a beginner who will not take the step from “two oranges and two oranges is four oranges” to “two plus two equals four.”
This book is a sequel to its authors' recently published Angular momentum in quantum physics: Theory and application ; it treats various advanced topics that could not be covered in the earlier volume without making it inconveniently long. My purpose is to explain the subject matter from a mathematician's point of view, but it would be awkward and difficult to do this without taking into account the contents of both books. Thus, in spite of its tardy appearance, this essay will, in effect, be an introduction to the two-volume work as a whole. When a physicist speaks of “angular momentum theory,” he is alluding to a theory that a mathematician would be more likely to describe as “the theory of rotational invariance.” This theory, whatever we call it, is concerned ( a ) with a technique for exploiting the fact that many physical laws are independent of orientation in space, and ( b ) with the many important consequences of this fact. The physicists' choice of the words “angular momentum theory” illustrates a tendency that is one of the many factors inhibiting communication between mathematicians and physicists. This is the tendency physicists have to avoid thinking in the abstract and instead to keep a concrete physical problem constantly in mind and use physical terminology whenever possible. From the mathematician's point of view, the physicist is behaving like a beginner who will not take the step from “two oranges and two oranges is four oranges” to “two plus two equals four.”
Abstract This book introduces the quantum theory of angular momentum to students who are unfamiliar with it and develops it to a stage useful for research. The first part contains the basic theory of rotations and angular momentum. As the book aims to emphasize applications, mathematical details are avoided and difficult theorems stated without proof. The second part contains examples of applications to a wide range of physical phenomena and presents a collection of results helpful in solving problems.
Many students find quantum mechanics conceptually difficult when they first encounter the subject. In this book, the postulates and key applications of quantum mechanics are well illustrated by means of a carefully chosen set of problems, complete with detailed, step-by-step solutions. Beginning with a chapter on orders of magnitude, a variety of topics are then covered, including the mathematical foundations of quantum mechanics, Schrödinger's equation, angular momentum, the hydrogen atom, the harmonic oscillator, spin, time-independent and time-dependent perturbation theory, the variational method, multielectron atoms, transitions and scattering. Throughout, the physical interpretation or application of certain results is highlighted, thereby providing useful insights into a wide range of systems and phenomena. This approach will make the book invaluable to anyone taking an undergraduate course in quantum mechanics.
For many years the physical interpretation of quantum theory has been dominated by the “wave-particle duality” attitude of the “Copenhagen school.” This point of view is eloquently described in Bohr's collection of essays on the subject. Despite persistent concerns with apparent paradoxes and limitations to this interpretation (as exemplified by Schrodinger's cat, the paradox of Einstein, Podolsky and Rosen, among others), the Copenhagen view persists de facto in the daily life of the modern physicist. Traditional quantum theory has so successfully explained such a vast amount of data in atomic and molecular physics, solid state physics (and to a lesser extent, elementary particle physics) that little doubt can exist concerning its essential validity. As a consequence of the overwhelming practical success of the “Copenhagen interpretation, ” the latter has acquired the status of dogma. For many years, therefore, most physicists have found it expedient to relegate the puzzling aspects of the theory to philosophers, and mathematicians. Nevertheless, a persistent interest in this subject has produced a significant and fascinating literature, reviewed by Jammer. In recent years, concerns over the proper meaning to be ascribed to quantum theory have produced an increasingly deep and incisive series of investigations. These investigations fall generally into two categories: either (1) discussion of the philosophical content of the theory, or (2) analyses of the mathematical variants of the theory and their connection with differing interpretational schemes. Physicists tend to be detached from a commitment to philosophical issues, because of their realization of the transient character of the meanings attached to theories of the day. As a simple illustration of this we can mention the profound differences between nonrelativistic and relativistic quantum mechanics.
This book is concerned with the practical aspects of solving angular momentum problems. The novel but fully tested-out method (the Invariant Graph Method) allows one to write down from a single graph the complete final result of the problem. The drawing of the graph involves very few simple, essentially self-evident rules. Still it is a powerful tool to easily solve the most involved physical problems. The method is introduced step-by-step in a sequence of examples, beginning with the simplest matrix elements, and ending with the most general case of a reaction including angular distributions and correlations. The many-body and particle anti-particle systems are fully developed. All aspects: wave functions, vectors, operators, Fock space state vectors and operators, etc., are treated on the same footing. All concepts of angular momentum theory acquire a transparent meaning. Hence the book is valuable not only as a handbook in problem solving, but extremely so as an adjunct in any course on advanced qunatum physics, atomic, molecular, nuclear and particle physics
Written in a lucid and engaging style, the author takes readers from an overview of classical mechanics and the historical development of quantum theory through to advanced topics. The mathematical aspects of quantum theory necessary for a firm grasp of the subject are developed in the early chapters, but an effort is made to motivate that formalism on physical grounds. Including animated figures and their respective Mathematica® codes, this book provides a complete and comprehensive text for students in physics, maths, chemistry and engineering needing an accessible introduction to quantum mechanics. Supplementary Mathematica codes available within Book Information
Index notation is the modern and easy way to work through problems such as we now face. It has always seemed to the author that many good, even great, physicists have never learned this topic and therefore find sections of books and papers that do use it very hard or even impossible to follow. It is truly easy to learn and takes only a little practice to become competent in its use. Parts of this section will appear again later, sometimes as problems, so the equations here will be numbered I1, I2, and so forth to emphasize the point.