A Complete Set Of Functions In The Quantum Mechanical Three Body Problem

The Quantum Mechanical Three-Body Problem deals with the three-body problem in quantum mechanics. Topics include the two- and three-particle problem, the Faddeev equations and their solution, separable potentials, and variational methods. This book has eight chapters; the first of which introduces the reader to the quantum mechanical three-body problem, its difficulties, and its importance in nuclear physics. Scattering experiments with three-particle breakup are presented. Attention then turns to some concepts of quantum mechanics, with emphasis on two-particle scattering and the Hamiltonian for three particles. The chapters that follow are devoted to the Faddeev equations, including those for scattering states and transition operators, and how such equations can be solved in practice. The solution of the Faddeev equations for separable potentials and local potentials is presented, along with the use of Padé approximation to solve the Faddeev equations. This book concludes with an appraisal of variational methods for bound states, elastic and rearrangement scattering, and the breakup reaction. A promising variational method for solving the Faddeev equations is described. This book will be of value to students interested in three-particle physics and to experimentalists who want to understand better how the theoretical data are derived.

Few-body systems are both technically relatively simple and physically non trivial enough to test theories quantitatively. For instance the He-atom played historically an important role in verifying predictions of QED. A similar role is contributed nowadays to the three-nucleon system as a testing ground far nuclear dynamics and maybe in the near future to few-quark systems. They are also often the basic building blocks for many-body systems like to some extent nuclei, where the real many-body aspect is not the dominant feature. The presentation of the subject given here is based on lectures held at var ious places in the last ten years. The selection of the topics is certainly subjec tive and influenced by my own research interests. The content of the book is simply organized according to the increasing nu mb er of particles treated. Be cause of its conceptual simplicity single particle motion is very suitable for in troducing the basic elements of scattering theory. Using these elements the two-body system is treated for the specific case of two nucleons, which is of great importance in the study of the nuclear interaction. Great space is devoted to the less trivial few-body system consisting of three particles. Again physical examples are taken solely from nuclear physics. Finally the four particle system is discussed so as to familiarize the reader with the techniques required for the formulations of n-bodies in general.

The quantum-mechanical ground-state problem for three identical particles bound by attractive inter-particle potentials is discussed. For this problem it has previously been shown that it is advantageous to write the wave function in a special functional form, form which an integral equation which is equivalent to the Schrodinger equation was derived. In this paper a new method for solving this equation is presented. The method involves an expansion of a two-body problem with a potential of the same shape as the inter-particle potential in the three-body problem, but of enhanced strength.

This invaluable collection of memoirs and reviews on scientific activities of the most prominent theoretical physicists belonging to the Landau School — Landau, Migdal, Zeldovich, Smorodinsky, Ter-Martirosyan, Kirzhnits, Gribov, Larkin and Anselm — are being published in English for the first time. The main goal is to acquaint readers with the life and work of outstanding Soviet physicists who, to a large extent, shaped theoretical physics in the 1950s–70s. Many intriguing details have remained unknown beyond the “Iron Curtain” which was dismantled only with the fall of the USSR. Contents:Lev LandauArkady MigdalYakov ZeldovichYakov SmorodinskyKaren Ter-MartirosyanDavid KirzhnitsVladimir GribovAnatoly LarkinAlexei AnselmTime and DestiniesSome Details for Experts Readership: Students and teachers of physics in high schools and colleges. Keywords:Theoretical Physics;Landau and His Circle;Soviet UnionReviews: “This volume is filled with fascinating reminiscences of the Landau school of theoretical physics that flourished in the then Soviet Union. Life was often difficult but the science was top-notch. We can all be grateful that these reminiscences have been written down and collected and are now being published in English.” Professor Edward Witten Institute for Advanced Study, Princeton “In this book we get to know about not only Landau but many from his circle, persons who are now gone. People who are still active give very personal and detailed narrations of the masters of the Landau circle. They come back to life for us and we get a very good understanding of how these people worked and lived. We can follow the history of theoretical physics through the achievements of these people. In this sense the book is a must for anyone interested in the history of science.”(See Full Review) Professor Lars Brink Chalmers Univ. of Tech., Sweden “The presented material is highly interesting for all those, who had the chance to take part in seminars and conferences in the Soviet Union or even to cooperate with Soviet colleagues.” Zentralblatt MATH

A technique is presented for treating a general type of three-body bound-state problem for situations where the interaction may be written as the sum of three pair potentials. The method is based on the work of Eyges and consists of writing the total wavefunction for the three-body problem in a special form. Some simple one-dimensional applications are examined. The first involves delta-function pair-potentials including an example where one of the pair interactions is repulsive. The second involves the one-dimensional problem of three identical spinless particles interacting through attractive, square-well potentials. The helium atom is discussed from a three-body point of view. Each orbital is expanded into a complete set of two-body Sturmian functions for the Coulomb potential. For states in atomic helium of the form L=0 the equations assume a simplified form. For these states, the infinite set of integral equations generated by the expansion is truncated at several orders and solved numerically. Rapid convergence is demonstrated for low-lying states of the helium-like atoms using this method. In particular, results are reported for the 1s1s(1)S, 1s2s(1)S, and 2s2s(1)S states of He, Li(+), and H( - ).