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Physics Bulletin

Scattering Theory in Quantum Mechanics: Physical Principles and Mathematical Methods

J.A.D. Matthew

Published Dec 1, 1978• 25 citations
Publisher

Abstract

Werner O Amrein, Josef M Jauch and Kalyan Sinha Reading, Massachusetts: W A Benjamin 1911 pp xxiii + 691 price $17.50 (paperback) The late Professor J M Jauch and his coworkers in Geneva have published an authoritative account of the mathematical foundations of quantum mechanical scattering theory in terms of functional analysis in Hilbert space. Following an extensive mathematical introduction, scattering theory is first approached from a timedependent viewpoint before moving to the time-independent methods that are most useful in practical application.

DOI
10.1088/0031-9112/29/12/038

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Werner O Amrein, Josef M Jauch and Kalyan Sinha Reading, Massachusetts: W A Benjamin 1911 pp xxiii + 691 price $17.50 (paperback) The late Professor J M Jauch and his coworkers in Geneva have published an authoritative account of the mathematical foundations of quantum mechanical scattering theory in terms of functional analysis in Hilbert space. Following an extensive mathematical introduction, scattering theory is first approached from a timedependent viewpoint before moving to the time-independent methods that are most useful in practical application.
After a fairly rigorous mathematical formulation of quantum mechanics based on transformation theory by Dirac in which he cast Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics on a common operator theoretic setting and successfully explained their equivalence, it was John Von Neumann in monumental treatise “Mathematical foundations of quantum mechanics” who proved the spectral theorem for all self adjoint operators in an infinite dimensional Hilbert space and thereby cast the masterly work of Dirac “The principles of quantum mechanics“ in a way that even pure mathematicians were able to understand. Dirac in his book had also indicated the work of Jordan and Heisenberg on quantum scattering theory but without the modern method of wave operators. Lippmann and Schwinger gave a more rigorous discussion of scattering theory via the introduction of free particle states and scattered states. Actually, free particle states are not square integrable and hence cannot strictly be termed as states. However by the use of wave operators the equations of Lippmann and Schwinger could be made rigorous using the setting of the continuous spectrum of a self-adjoint opearator introduced by Von Neumann, Crucial to this development is the notion of the domain of definition of an unbounded operator. This leads to the fact that the wave operators that appear in scattering theory are not defined on all vectors but usually on those vectors that belong to the continuous spectrum of the free particle Hamiltonian. Today scattering theory is a well developed subject and the interested reader can get a good grasp of this subject from the books 1 Tosio Kato, “Perturbation theory for linear operators”, Springer. 2 Werner O. Amrein, “Hilbert space methods in quantum mechanics”, CRC press, 2012.
Scattering is one of the most powerful methods used to study the structure of matter, and many of the most important breakthroughs in physics have been made by means of scattering. Nearly a century has passed since the first investigations in this field, and the work undertaken since then has resulted in a rich literature encompassing both experimental and theoretical results. In scattering, one customarily studies collisions among nuclear, sub-nuclear, atomic or molecular particles, and as these are intrinsically quantum systems, it is logical that quantum mechanics is used as the basis for modern scattering theory. In Principles of Quantum Scattering Theory, the author judiciously combines physical intuition and mathematical rigour to present various selected principles of quantum scattering theory. As always in physics, experiment should be used to ultimately validate physical and mathematical modelling, and the author presents a number of exemplary illustrations, comparing theoretical and experimental cross sections in a selection of major inelastic ion-atom collisions at high non-relativistic energies. Quantum scattering theory, one of the most beautiful theories in physics, is also very rich in mathematics. Principles of Quantum Scattering Theory is intended primarily for graduate physics students, but also for non-specialist physicists for whom the clarity of exposition should aid comprehension of these mathematical complexities.
Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators. Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory. This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics. Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature. This new edition has additions and improvements throughout the book to make the presentation more student friendly.
H J Lipkin Amsterdam: North-Holland 1973 pp xv + 465 price Dfl 95 The author himself explains that this volume can be considered as a collection of self contained monographs, dealing with such topics as the Mossbauer effect (introducing ‘form factor physics‘), many body theory (second quantization techniques), kaon physics (applications of time-dependent perturbation theory), scattering theory, Feynman diagrams, and relativistic quantum mechanics. Concepts and techniques are presented with clarity, and their use well illustrated through application to problems of real physical interest.