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Nuovo cimento della Società italiana di fisica. A, Nuclei, particles and fields

On the application of a gauss transformation in nonlinear quantum mechanics. II: Problems with spin

W. Ulmer

Published May 1, 1979• 6 citations
Publisher
DOI
10.1007/bf02775428

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Journal of Liaoning University•Jan 1, 2009•0 citations
This 2004 textbook provides a pedagogical introduction to the formalism, foundations and applications of quantum mechanics. Part I covers the basic material which is necessary to understand the transition from classical to wave mechanics. Topics include classical dynamics, with emphasis on canonical transformations and the Hamilton-Jacobi equation, the Cauchy problem for the wave equation, Helmholtz equation and eikonal approximation, introduction to spin, perturbation theory and scattering theory. The Weyl quantization is presented in Part II, along with the postulates of quantum mechanics. Part III is devoted to topics such as statistical mechanics and black-body radiation, Lagrangian and phase-space formulations of quantum mechanics, and the Dirac equation. This book is intended for use as a textbook for beginning graduate and advanced undergraduate courses. It is self-contained and includes problems to aid the reader's understanding.
We present an efficient quantum algorithm for estimating Gauss sums over finite fields and finite rings. This is a natural problem as the description of a Gauss sum can be done without reference to a black box function. With a reduction from the discrete logarithm problem to Gauss sum estimation we also give evidence that this problem is hard for classical algorithms. The workings of the quantum algorithm rely on the interaction between the additive characters of the Fourier transform and the multiplicative characters of the Gauss sum.
In this paper we propose a new method of nonlinear transformation(Eigen-Mapping) to treat the eigenvalue problems in quantum mechanics as examples.The basic idea is to make the matrix of some operator we concerned diagonalization through introduction of small undetermined parameters and establishment equations for matrix elements.These problems can be devided into two types based on that there are or not 1st order approximate solution for them(in sense of the perturbation theory).In either type we consider also two cases of degenerate and nondegenerate.To obtain solution or approximate solution for eigenvalue problems,it is given the procedures of the method and usable formulas exhaustively.Seven examples in this paper are a common sight in general course of quantum mechanics.The results from them are compared with that from other approximate methods(for examples: perturbation method and variation method).After checking the results obtained,we can see that this new method has many advantages.