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A numerical technique is presented which yields an exact solution to the one dimensional scattering problem. The algorithm is used to compute phase shift error curves associated with the WKB approximation. A variety of scattering potentials are considered and cover cases for which the WKB solution varies from extremely good to poor. It is shown that some commonly assumed forms for ionospheric electron density profiles yield phase responses which are discontinuous functions of wavenumber. Implications of these results to the inverse scattering problem are discussed. Keywords: Wave Equation; and Numerical Methods.
Electron Scattering from Complex Nuclei, Part A covers the historical phases of experimental development in elastic and inelastic electron scattering. This five-chapter text presents the logical development of the underlying theory of electron scattering. After briefly discussing the history of electron scattering from nuclei, this book goes on describing the theory of elastic scattering from a point nucleus, both with Born approximation and the accurate solution of the Dirac equation, as well as the corresponding experiments. The following chapter considers the analysis of nuclear charge distributions experiments using Born cross section and phase-shift methods. A chapter is devoted to the complete elastic and inelastic Born theory. This chapter also deals with the derivation of a theorem on the general form of the electron-nucleus scattering cross section, with an emphasis on the influence of the neglected transverse interaction on the cross section. The last chapter presents the status of elastic scattering along with some topics in muonic atoms that also determine nuclear charge densities. This book will be of great benefit to physicists, researchers, and graduate students who are interested in nuclear structure problems.
This book discusses some ways of doing mathematical work and the subject matter that is being worked upon and created. It argues that the conventions we adopt, the subject areas we delimit, what we can prove and calculate about the physical world, and the analogies that work for mathematicians — all depend on mathematics, what will work out and what won't. And the mathematics, as it is done, is shaped and supported, or not, by convention, subject matter, calculation, and analogy. The cases studied include the central limit theorem of statistics, the sound of the shape of a drum, the connection between algebra and topology, the stability of matter, the Ising model, and the Langlands Program in number theory and representation theory.
The second edition of this book consists of three parts. The first one is dedicated to the WKB methods and the semi-classical limit before the formation of caustics. The second part treats the semi-classical limit in the presence of caustics, in the special geometric case where the caustic is reduced to a point (or to several isolated points). The third part is new in this edition, and addresses the nonlinear propagation of coherent states. The three parts are essentially independent.Compared with the first edition, the first part is enriched by a new section on multiphase expansions in the case of weakly nonlinear geometric optics, and an application related to this study, concerning instability results for nonlinear Schrödinger equations in negative order Sobolev spaces.The third part is an overview of results concerning nonlinear effects in the propagation of coherent states, in the case of a power nonlinearity, and in the richer case of Hartree-like nonlinearities. It includes explicit formulas of an independent interest, such as generalized Mehler's formula, generalized lens transform.
Advances in One-Dimensional Wave Mechanics provides a comprehensive description of the motion of microscopic particles in one-dimensional, arbitrary-shaped potentials based on the analogy between Quantum Mechanics and Electromagnetism. Utilizing a deeper understanding of the wave nature of matter, this book introduces the concept of the scattered sub-waves and a series of new analytical results using the Analytical Transfer Matrix (ATM) method. This work will be useful for graduate students majoring in physics, mainly in basic quantum theory, as well as for academic researchers exploring electromagnetism, particle physics, and wave mechanics and for experts in the field of optical waveguide and integrated optics. Prof. Zhuangqi Cao is a Professor of Physics at Shanghai Jiao Tong University, China. Dr. Cheng Yin is a teacher at Jiangsu Key Laboratory of Power Transmission and Distribution Equipment Technology, Hohai University, China.