Factorization Method in Quantum Mechanics

by Dong, Shi-hai

• ISBN13: 9781402057953
• ISBN10: 1402057954
• eBook ISBN(s): 9781402057960
• Format: Hardcover
• Copyright: 2007-03-22
• Publisher: Springer Verlag

Summary

This work introduces the factorization method in quantum mechanics at an advanced level with an aim to put mathematical and physical concepts and techniques like the factorization method, Lie algebras, matrix elements and quantum control at the reader's disposal. For this purpose a comprehensive description is provided of the factorization method and its wide applications in quantum mechanics which complements the traditional coverage found in the existing quantum mechanics textbooks. Related to this classic method are the supersymmetric quantum mechanics, shape invariant potentials and group theoretical approaches. It is no exaggeration to say that this method has become the milestone of these approaches. In fact the author's driving force has been his desire to provide a comprehensive review volume that includes some new and significant results about the factorization method in quantum mechanics since the literature is inundated with scattered articles in this field, and to pave the reader's way into this territory as rapidly as possible. The result: clear and understandable derivations with the necessary mathematical steps included so that the intelligent reader should be able to follow the text with relative ease, in particular when mathematically difficult material is presented.

Table of Contents

Introduction
Introduction	p. 3
Method
Theory	p. 15
Lie algebras SU(2) and SU(1,1)	p. 17
Applications in non-relativistic quantum mechanics
Harmonic oscillator	p. 35
Infinitely deep square-well potential	p. 57
Morse potential	p. 73
Poschl-Teller potential	p. 95
Pseudoharmonic oscillator	p. 111
Algebraic approach to an electron in a uniform magnetic field	p. 137
Ring-shaped non-spherical oscillator	p. 143
Generalized Laguerre functions	p. 151
New noncentral ring-shaped potential	p. 157
Poschl-Teller like potential	p. 169
Postition-dependent mass Schrodinger equation for a singular oscillator	p. 177
Applications in relativistic quantum mechanics
SUSYQM and SWKB approach to the Dirac equation with a coulomb potential in 2+1 dimensions	p. 187
Realization of dynamic group for the Dirac hydrogen-like atom in 2+1 dimensions	p. 201
Algebraic approach to Klein-Gordon equation with the hydrogen-like atom in 2+1 dimensions	p. 207
SUSYQM and SWKB approaches to relativistic Dirac and Klein-Gordon equations with hyperbolic potential	p. 213
Quantum control
Controllability of quantum systems for the Morse and PT potentials with dynamic group SU(2)	p. 225
Controllability of quantum system for the PT-like potential with dynamic group SU(1,1)	p. 229
Conclusions and outlooks
Conclusions and outlooks	p. 237
Integral formulas of the confluent hypergeometric functions	p. 239
Mean values [actual symbol not reproducible] for hydrogen-like atom	p. 243
Commutator identities	p. 247
Angular momentum operators in spherical coordinates	p. 249
Confluent hypergeometric function	p. 251
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