An elementary approach to the quantum theory of angular momentum is sketched. The quantization of angular momentum components, Heisenberg-type inequalities for their spectral dispersions, and the quantization of the angular momentum modulus are introduced without using operators or commutation relations. The purpose is to focus the attention on angular momemtum as a very important practical concept of quantum physics, as well as on its specifically quantum-theoretical aspects, right from the beginning of an introductory course. Since the heuristic arguments employed here closely parallel the rigorous derivations of a formal approach, the intuitive presentation does not contradict, but rather supports, the theoretical concepts to be introduced at a later stage.
An elementary approach to the quantum theory of angular momentum is sketched. The quantization of angular momentum components, Heisenberg-type inequalities for their spectral dispersions, and the quantization of the angular momentum modulus are introduced without using operators or commutation relations. The purpose is to focus the attention on angular momemtum as a very important practical concept of quantum physics, as well as on its specifically quantum-theoretical aspects, right from the beginning of an introductory course. Since the heuristic arguments employed here closely parallel the rigorous derivations of a formal approach, the intuitive presentation does not contradict, but rather supports, the theoretical concepts to be introduced at a later stage.
THE QUANTUM THEORY OF ANGULAR MOMENTUM began with Bohr’s enunciation of the rules for the quantization of orbital angular momentum in 1913, and the refinement of these rules by Sommerfeld [ 1 ] and Landé [ 2 ] within the framework of the old quantum theory. A basic result for angular momentum in modern quantum theory, which we call The First Fundamental Result, was given in the third foundational paper in 1926 by Born, Heisenberg, and Jordan [ 3 ], who not only gave the basic commutation relations for the components of the total angular momentum J, but all finite-dimensional Hermitian matrix representations of these components. Dirac [ 4 ] simultaneously deduced these basic commutation relations using his algebraic approach to quantum mechanics, and later gave a marvelously brief derivation of their matrix representations in his book [ 5 ]. Further developments, including the famous Pauli matrices and results relating to angular momentum selection rules for transitions in atomic systems, are given in the papers by Heisenberg and Jordan [ 6 ], Pauli [ 7 ], and Güttinger and Pauli [ 8 ]. These works, and the books by Born and Jordan [ 9 ] and Condon and Shortley [ 10 ], fixed the framework for much of the developments in angular momentum theory, which was hardly distinguishable from the general quantum theory in the context of which it was developed for the purpose of giving a consistent description of the spectra of atoms. Little emphasis was given to the universal and distinctive role of symmetry.
Elementary particles are found in two different situations: (i) bound to metastable states of matter, for which angular momentum is quantized, and (ii) free, for which, due to their high energy-momentum and leaving aside inner a.m. or spin, the h-quantization step is completely harmless. Perhaps Quantum Mechanics can be seen just as the simplest mathematical formalism where angular momentum (the magnitude of each of its three orthogonal projections) is by construction quantized: all possible values are taken from a discrete set. Indeed: (i) This idea finds support in very reasonable, completely classical physical arguments, if we place ourselves in the framework of Stochastic Electrodynamics (SED): there, all sustained periodic movement of a charge must satisfy a power balance that restricts the value of the average angular momentum, on each of its projections. (ii) It gives a natural explanation of the concept of "photon", as a constraint on the observable spectrum of energy-momentum exchanges between metastable physical states, in particular also for its discreteness. QM would be, in this picture, a semi-static theory, transparent to all the (micro)-dynamics taking place between apparently "discrete" events (transitions in the state of the system). For instance, (the magnitude of the projections of) quantum angular momentum would only reflect average values over a (classical) cyclic trajectory, a fact that we regard as almost obvious given the particularity of the corresponding addition rules in QM.
Designed for a two-semester advanced undergraduate or graduate level course, this distinctive and modern textbook provides students with the physical intuition and mathematical skills to tackle even complex problems in quantum mechanics with ease and fluency. Beginning with a detailed introduction to quantum states and Dirac notation, the book then develops the overarching theoretical framework of quantum mechanics, before explaining physical quantum mechanical properties such as angular momentum and spin. Symmetries and groups in quantum mechanics, important components of current research, are covered at length. The second part of the text focuses on applications, and includes a detailed chapter on quantum entanglement, one of the most exciting modern applications of quantum mechanics, and of key importance in quantum information and computation. Numerous exercises are interspersed throughout the text, expanding upon key concepts and further developing students' understanding. A fully worked solutions manual and lecture slides are available for instructors.
It has been the purpose of the monograph Angular Momentum in Quantum Physics (AMQP) to develop comprehensively those aspects of angular momentum theory that are required in carrying out research in modern physics and chemistry. The emphasis there was principally on physical concepts and the symmetry principles that underlie the general applicability of angular momentum theory to a broad area of physical phenomena. Mathematical techniques were introduced within the context of the physical concepts themselves. Physical applications of angular momentum theory are to a large extent applications of specific properties of Wigner or Racah functions (more generally of 3 n-j coefficients) or of the representation functions themselves. Such applications tend to emphasize the “numerical” aspects of these functions and thus obscure the relationships to other areas of mathematics. The purpose of the present monograph is to show by specific examples the many interrelations that exist between concepts originating in angular momentum theory and various areas of mathematics. It turns out that this is a substantial task—the diversity of the interrelations is far greater than might be thought. Even a brief acquaintance with the contents of AMQP will show that the concept that recurs again and again in physical applications is that of an irreducible tensor operator with respect to a group [in our case, SU (2), the quantal rotation group]. Indeed, it is this concept that carries the mathematical apparatus of the physical theory beyond the related but much simpler results of the Lie algebraic theory of the angular momentum itself and the implied representation theory of the group SU (2).
This book offers a concise introduction to the angular momentum, one of the most fundamental quantities in all of quantum mechanics. Beginning with the quantization of angular momentum, spin angular momentum, and the orbital angular momentum, the author goes on to discuss the Clebsch-Gordan coefficients for a two-component system. After developing the necessary mathematics, specifically spherical tensors and tensor operators, the author then investigates the 3- j , 6- j , and 9- j symbols. Throughout, the author provides practical applications to atomic, molecular, and nuclear physics. These include partial-wave expansions, the emission and absorption of particles, the proton and electron quadrupole moment, matrix element calculation in practice, and the properties of the symmetrical top molecule.
Containing basic definitions and theorems as well as relations, tables of formulas and numerical tables which are essential for applications to many physical problems, the book is useful for specialists in nuclear and particle physics, atomic and molecular spectroscopy, plasma physics, collision and reaction theory, quantum chemistry, etc. The authors write many formulas in different coordinate systems. Each chapter opens with a list of its contents. New results relating to different aspects of the angular momentum theory are included. This book gathers together many useful formulas besides those related to angular momentum, and compares different notations used by previous authors.