INTRODUCTION TO QUANTUM MECHANICSSCHRODINGER EQUATION AND PATH INTEGRAL

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Author : H. J. W. Mller-Kirsten
ISBN : 9789812566911
Genre : Science
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After a consideration of basic quantum mechanics, this introduction aims at a side by side treatment of fundamental applications of the Schrdinger equation on the one hand and the applications of the path integral on the other. Different from traditional texts and using a systematic perturbation method, the solution of Schrdinger equations includes also those with anharmonic oscillator potentials, periodic potentials, screened Coulomb potentials and a typical singular potential, as well as the investigation of the large order behavior of the perturbation series. On the path integral side, after introduction of the basic ideas, the expansion around classical configurations in Euclidean time, such as instantons, is considered, and the method is applied in particular to anharmonic oscillator and periodic potentials. Numerous other aspects are treated on the way, thus providing the reader an instructive overview over diverse quantum mechanical phenomena, e.g. many other potentials, Green's functions, comparison with WKB, calculation of lifetimes and sojourn times, derivation of generating functions, the Coulomb problem in various coordinates, etc. All calculations are given in detail, so that the reader can follow every step.

Author : Harald J W MÃ¼ller-Kirsten
ISBN : 9789814397766
Genre : Science
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This text on quantum mechanics begins by covering all the main topics of an introduction to the subject. It then concentrates on newer developments. In particular it continues with the perturbative solution of the Schrödinger equation for various potentials and thereafter with the introduction and evaluation of their path integral counterparts. Considerations of the large order behavior of the perturbation expansions show that in most applications these are asymptotic expansions. The parallel consideration of path integrals requires the evaluation of these around periodic classical configurations, the fluctuation equations about which lead back to specific wave equations. The period of the classical configurations is related to temperature, and permits transitions to the thermal domain to be classified as phase transitions. In this second edition of the text important applications and numerous examples have been added. In particular, the chapter on the Coulomb potential has been extended to include an introduction to chemical bonds, the chapter on periodic potentials has been supplemented by a section on the band theory of metals and semiconductors, and in the chapter on large order behavior a section has been added illustrating the success of converging factors in the evaluation of asymptotic expansions. Detailed calculations permit the reader to follow every step.

Author : World Scientific Press
ISBN :
Genre : Mathematics
File Size : 71.93 MB
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With the discovery of quantization by Planck in 1900, quantum mechanics is now more than a hundred years old. However, a proper understanding of the phenomenon was gained only later in 1925 with the fundamental Heisenberg commutation relation or phase space algebra and the associated uncertainty principle. The resulting Schrodinger equation has ever since been the theoretical basis of atomic physics. The alternative formulation by Feynman in terms of path integrals appeared two to three decades later. Although the two approaches are basically equivalent, the Schrodinger equation has found much wider usefulness, particularly in applications, presumably, in view of its simpler mathematics. However, the realization that solutions of classical equations, notably in field theory, play an important role in our understanding of a large number of physical phenomena, intensified the interest in Feynman's formulation of quantum mechanics, so that today this method must be considered of equal basic significance. Thus there are two basic approaches to the solution of a quantum mechanical problem, and an understanding of both and their usefulness in respective domains calls for their application to exemplary problems and their comparison. This is our aim here on an introductory level. Throughout the development of theoretical physics two types of forces played an exceptional role: That of the restoring force of simple harmonic motion proportional to the displacement, and that in the Kepler problem proportional to the inverse square of the distance, i.e. Newton's gravitational force like that of the Coulomb potential. In the early development of quantum mechanics again oscillators appeared (though not really those of harmonic type) in Planck's quantization and the Coulomb potential in the Bohr model of the hydrogen atom. Again after the full and proper formulation of quantum mechanics with Heisenberg's phase space algebra and Born's wave function interpretation the oscillator and the Coulomb potentials provided the dominant and fully solvable models with a large number of at least approximate applications. To this day these two cases of interaction with nonresonant spectra feature as the standard and most important xv XVI illustrative examples in any treatise on quantum mechanics and — excepting various kinds of square well and rectangular barrier potentials — leave the student sometimes puzzled about other potentials that he encounters soon thereafter, like periodic potentials, screened Coulomb potentials and maybe singular potentials, but also about complex energies that he encounters in a parallel course on nuclear physics. Excluding spin, any problem more complicated is frequently dispensed with by referring to cumbersome perturbation methods. Diverse and more detailed quantum mechanical investigations in the second half of the last century revealed that perturbation theory frequently does permit systematic procedures (as is evident e.g. in Feynman diagrams in quantum electrodynamics), even though the expansions are mostly asymptotic. With various techniques and deeper studies, numerous problems could, in fact, be treated to a considerable degree of satisfaction perturbatively. With the growing importance of models in statistical mechanics and in field theory, the path integral method of Feynman was soon recognized to offer frequently a more general procedure of enforcing first quantization instead of the Schrodinger equation. To what extent the two methods are actually

This is the fifth, expanded edition of the comprehensive textbook published in 1990 on the theory and applications of path integrals. It is the first book to explicitly solve path integrals of a wide variety of nontrivial quantum-mechanical systems, in particular the hydrogen atom. The solutions have been made possible by two major advances. The first is a new euclidean path integral formula which increases the restricted range of applicability of Feynman''s time-sliced formula to include singular attractive 1/r- and 1/r2-potentials. The second is a new nonholonomic mapping principle carrying physical laws in flat spacetime to spacetimes with curvature and torsion, which leads to time-sliced path integrals that are manifestly invariant under coordinate transformations. In addition to the time-sliced definition, the author gives a perturbative, coordinate-independent definition of path integrals, which makes them invariant under coordinate transformations. A consistent implementation of this property leads to an extension of the theory of generalized functions by defining uniquely products of distributions. The powerful Feynman-Kleinert variational approach is explained and developed systematically into a variational perturbation theory which, in contrast to ordinary perturbation theory, produces convergent results. The convergence is uniform from weak to strong couplings, opening a way to precise evaluations of analytically unsolvable path integrals in the strong-coupling regime where they describe critical phenomena. Tunneling processes are treated in detail, with applications to the lifetimes of supercurrents, the stability of metastable thermodynamic phases, and the large-order behavior of perturbation expansions. A variational treatment extends the range of validity to small barriers. A corresponding extension of the large-order perturbation theory now also applies to small orders. Special attention is devoted to path integrals with topological restrictions needed to understand the statistical properties of elementary particles and the entanglement phenomena in polymer physics and biophysics. The ChernoOe1/4OC Simons theory of particles with fractional statistics (anyons) is introduced and applied to explain the fractional quantum Hall effect. The relevance of path integrals to financial markets is discussed, and improvements of the famous BlackoOe1/4OC Scholes formula for option prices are developed which account for the fact, recently experienced in the world markets, that large fluctuations occur much more frequently than in Gaussian distributions."

This course contains a brief and mathematically precise introduction to the conceptual body, and calculation tools, of Quantum Mechanics and Quantum Physics. It is addressed to undergraduate students. The introduction is performed using the semiclassical framework, where Newton´s Classical Mechanics and Relativity as the reference points. Feynman's propagation is used as an axiomatic basis for Quantum Mechanics, completed with the generally admitted ideas about the measurement problem.

Author : R. Shankar
ISBN : 9781475705768
Genre : Science
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R. Shankar has introduced major additions and updated key presentations in this second edition of Principles of Quantum Mechanics. New features of this innovative text include an entirely rewritten mathematical introduction, a discussion of Time-reversal invariance, and extensive coverage of a variety of path integrals and their applications. Additional highlights include: - Clear, accessible treatment of underlying mathematics - A review of Newtonian, Lagrangian, and Hamiltonian mechanics - Student understanding of quantum theory is enhanced by separate treatment of mathematical theorems and physical postulates - Unsurpassed coverage of path integrals and their relevance in contemporary physics The requisite text for advanced undergraduate- and graduate-level students, Principles of Quantum Mechanics, Second Edition is fully referenced and is supported by many exercises and solutions. The book’s self-contained chapters also make it suitable for independent study as well as for courses in applied disciplines.

This book aims to provide a quick pedagogical introduction to path integrals. It contains original material that never before has appeared in a book, for example the path integrals for the Wigner functions and for Classical Mechanics. This application to Classical Mechanics connects different fields like Hamiltonian mechanics and differential geometry, so the book is suitable for students and researchers from various disciplines.

Author : Jean Zinn-Justin
ISBN : 9780198566755
Genre : Mathematics
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Quantum field theory is hardly comprehensible without path integrals: the goal of this book is to introduce students to this topic within the context of ordinary quantum mechanics and non-relativistic many-body theory, before facing the problems associated with the more involved quantum field theory formalism.