Direct and Inverse Problems : Potentials in Quantum Scattering

Boris N. Zakhariev

ISBN 10: 3540524843 / ISBN 13: 9783540524847

Editore: Springer Berlin Heidelberg, 1990

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Druck auf Anfrage Neuware - Printed after ordering - This textbook can almost be viewed as a 'how-to' manual for solving quantum inverse problems, that is, for deriving the potential from spectra or scattering data and also, as somewhat of a quantum 'picture book' which should enhance the reader's quantum intuition. The formal exposition of inverse methods is paralleled by a discussion of the direct problem. Differential and finite-difference equations are presented side by side. The common features and (dis)advantages of a variety of solution methods are analyzed. To foster a better understanding, the physical meaning of the mathematical quantities are discussed explicitly. Wave confinement in continuum bound states, resonance and collective tunneling, energy shifts and the spectral and phase equivalence of various interactions are some of the physical problems covered. Dieses Buch ist fast ein 'Rezeptbuch' für die Lösung des quantenmechanischen inversen Problems, d.h. für die Ermittlung des Potentials aus Spektren oder Streudaten, sowie eine Art 'Bilderbuch' der Quantenmechanik, welches das Gefühl des Lesers für die Quantenmechanik verbessern soll. Die formale Darstellung der inversen Methoden erfolgt parallel zu der des direkten Problems. Differential- und Differenzengleichungen werden Seite an Seite entwickelt. Eine Vielfalt von Lösungsmethoden wird präsentiert. Zur Vertiefung des Verständnisses werden die physikalischen Bedeutungen der mathematischen Größen im Detail diskutiert.

Codice articolo 9783540524847

Dati bibliografici

• Titolo: Direct and Inverse Problems : Potentials in Quantum Scattering
• Editore: Springer Berlin Heidelberg
• Data di pubblicazione: 1990
• Lingua: Inglese
• ISBN 10: 3540524843
• ISBN 13: 9783540524847
• Rilegatura: Taschenbuch
• Condizione: Neu
• Dimensioni: 235 L x 155 A x 14 P (millimetri) 235 millimetri di larghezza per 155 millimetri di altezza per 14 millimetri di profondità
• Peso: 371 grammi

Descrizione articolo

Riassunto

Rapid progress in quantum theory brings us new important results which are often not immediately clear to all who need them. But fortunately, this is also followed by simplifications and unifications of our previous concepts. The inverse problem method ("The most beautiful idea of the XX-th century" - Zakharov et aI., 1980) has just both these aspects. It is rather astonishing that it took 50 years after the foundation of quantum mechanics for the creation of the "pictures" showing the direct connection of obser­ vables with interactions. Recently, illustrations of this type began to appear in the literature (e. g., how potentials are deformed with thc shift of one energy level or change of some resonance reduced width). Although they are transparent to those studying the quantum world and can be included within the necessary elements of quantum literacy, they are still largely unknown even to many specialists. For the first time, the most interesting of these pictures enriching our quantum intuition are col­ lected here and placed at your disposal. The readers of this monograph have the advantage of getting the latest information which became available after the publication of the Russian edition. It has been incor­ porated here in the simplest presentation possible. For example, new sections con­ cerning exactly solvable models, including the multi-channel, multi-dimensional ones and with time dependent potentials have been added. The first attempts in solving the three-body inverse problem are also mentioned.

Contenuti

I: One-dimensional, One-channel Systems.- 1 The principal Equations of Scattering Theory.- 1.1 General Remarks.- 1.2 Elements of the Direct and Inverse Problems.- 1.2.1 The Simplest Difference Schrödinger Equation.- 1.2.2 Potential Wells of Infinite Depth.- 1.2.3 The Direct Problem.- 1.2.4 The Inverse Problem.- 1.2.5 Scattering by a Potential of Finite Range.- 1.2.6 The Finite-Difference Analogue of the R-matrix Scattering Theory.- 1.2.7 Conditions of Orthonormality and Completeness of the Eigenfunctions of the Finite-Difference Schrödinger Operator on [O, ?].- 1.2.8 Relations Between the Scattering Parameters {E? and ??}.- 1.2.9 Reconstruction of the Potential on the Semi-axis 0 ?x Upper k Walf-plane.- 2.2.3 Potentials with S (k) with Two Poles in the Upper k Half-Plane.- 2.3 More General Models.- 2.3 Multi-term Degenerate Kernels of the Inverseproblem Equations.- 2.3.2 Models of One-Dimensional Motion on the Whole Axis.- 2.3.3 The Finite-Difference-Approach.- 2.3.4 The Rational Reflection Coefficient (no Bound States).- 2.3.5 The Finite-difference Approach.- 2.4 Potentials of the Finite-Range and Infinitely Deep Wells. R-matrix Models.- 2.5 Potentials Allowing Exact Solutions for Variable Angular Momenta.- 2.5.1 Potentials from Spectral Data at Fixed Energy and Variable l.- 2.5.2 Newton-Sabatier Potentials.- 2.5.3 The Generalized Crum-Krein Transformations.- 2.5.4 Lipperheide-Fidelday Potentials.- 2.6 Notes on the Literature.- 2.7 Exercises.- 3 Approximate Solutions.- 3.1 Convergence of the Approximations, Stability and Regularization of Solutions.- 3.2 Solutions Using Bargmann Potentials.- 3.4 Approximation of Datentials by Steps, at Discrete Points, and by Splines. The Role of the Upper Part of the Spectrum.- 3.5 Method of Multiple Solutions of the Direct Problem.- 3.6 Notes on the Literature.- 4 The Levinson Theorem.- 4.1 General Remarks.- 4.2 Simple Examples.- 4.3 The Coulomb Potential and Other Singular Interactions.- 4.4 Other Types of Interactions.- 4.4.1 Potentials Depending on Velocity.- 4.4.2 Potentials Depending on Energy.- 4.4.3 The Finite-Difference Schrödinger Equation.- 4.4.4 Motion Along the Axis.- 4.4.5 Nonlocal Potentials.- 4.5 Notes on the Literature.- II. Multi-channel, Multi-dimensional, Multi-particle Problems.- 5 Multi-channel Equations.- 5.1 General Remarks.- 5.2 The Formalism of Multi-channel Coupling.- 5.3 Finite-Difference Equations of Motion.- 5.4 Exactly Solvable Models.- 5.5 Notes on the Literature.- 5.6 Exercises.- 6 Multi-dimensional Problems.- 6.1 General Remarks.- 6.2 The Finite-Difference Formalism.- 6.3 Reduction of Multi-dimensional problems to Multichannel problems.- 6.4 The Multi-dimensional Inverse Problem.- 6.5 Separation of Variables in Noncentral Field.- 6.6 Exactly Solvable Models.- 6.7 Notes on the Literature.- 7 Multi-particle Systems.- 7.1 General Remarks.- 7.2 Asymptotic Hamiltonians and Boundary Conditions.- 7.3 Tunnelling Through Potential Barriers by Several Particles.- 7.4 Excitation of Collective Degrees of Freedom of Multi-particle Systems..- 7.4.1 Transformation of Amplitudes in Transition to the Reference Frame of the Target.- 7.5 The Method of Hyperspherical Functions (K-harmonics).- 7.6 The Levinson Theorem.- 7.7 Three-Particle Potentials.- 7.8 Notes on the Literature.- References.