FACTORIZATION METHOD IN QUANTUM MECHANICS
CONTENTS PART I INTRODUCTION 1 INTRODUCTION BASIC REVIEW MOTIVATIONS AND AIMS PART II METHOD 2 THEORY INTRODUCTION FORMALISM 3 LIE ALGEBRAS SU(2) AND SU(1, 1) ABSTRACT GROUPS MATRIX REPRESENTATION PROPERTIES OF GROUPS SU(2) AND SO(3) PROPERTIES OF NON-COMPACT GROUPS SO(2, 1) AND SU(1, 1) GENERATORS OF LIE GROUPS SU(2) AND SU(1, 1) IRREDUCIBLE REPRESENTATIONS IRREDUCIBLE UNITARY REPRESENTATIONS CONCLUDING REMARKS PART III APPLICATIONS IN NON-RELATIVISTIC QUANTUM MECHANICS 4 HARMONIC OSCILLATOR EXACT SOLUTIONS LADDER OPERATORS BARGMANN-SEGAL TRANSFORM SINGLE MODE REALIZATION OF DYNAMIC GROUP SU(1, 1) MATRIX ELEMENTS COHERENT STATES FRANCK-CONDON FACTORS 5 INFINITELY DEEP SQUARE-WELL POTENTIAL LADDER OPERATORS FOR IN?NITELY DEEP SQUARE-WELL POTENTIAL REALIZATION OF DYNAMIC GROUP SU(1, 1) AND MATRIX ELEMENTS LADDER OPERATORS FOR IN?NITELY DEEP SYMMETRIC WELL POTENTIAL SUSYQM APPROACH TO IN?NITELY DEEP SQUARE WELL POTENTIAL PERELOMOV COHERENT STATES BARUT-GIRARDELLO COHERENT STATES 6 MORSE POTENTIAL LADDER OPERATORS FOR THE MORSE POTENTIAL REALIZATION OF DYNAMIC GROUP SU(2) HARMONIC LIMIT TRANSITION PROBABILITY REALIZATION OF DYNAMIC GROUP SU(1, 1) 7 P¨OSCHL-TELLER POTENTIAL ALTERNATIVE APPROACH TO DERIVE LADDER OPERATORS EXPANSIONS OF THE COORDINATE X AND MOMENTUM P FROM THE SU(2) GENERATORS 8 PSEUDOHARMONIC OSCILLATOR EXACT SOLUTIONS IN ONE DIMENSION THERMODYNAMIC PROPERTIES PSEUDOHARMONIC OSCILLATOR IN ARBITRARY DIMENSIONS RECURRENCE RELATIONS AMONG MATRIX ELEMENTS 9 ALGEBRAICAPPROACHTOANELECTRONINAUNIFORM MAGNETIC FIELD 10 RING-SHAPED NON-SPHERICAL OSCILLATOR REALIZATION OF DYNAMIC GROUP 11 GENERALIZED LAGUERRE FUNCTIONS GENERALIZED LAGUERRE FUNCTIONS LADDER OPERATORS AND REALIZATION OF DYNAMIC GROUP SU(1, 1) 12 NEW NONCENTRAL RING-SHAPED POTENTIAL BOUND STATES MEAN VALUES CONTINUUM STATES 13 P¨OSCHL-TELLER LIKE POTENTIAL REALIZATION OF DYNAMIC GROUP AND MATRIX ELEMENTS IN?NITELY SQUARE WELL AND HARMONIC LIMITS 14 POSITION-DEPENDENTMASSSCHR¨ODINGEREQUATION FOR A SINGULAR OSCILLATOR POSITION-DEPENDENT EFFECTIVE MASS SCHR¨ODINGER EQUATION FOR HARMONIC OSCILLATOR SINGULAR OSCILLATOR WITH A POSITION-DEPENDENT EFFECTIVE MASS COMPLETE SOLUTIONS ANOTHER POSITION-DEPENDENT EFFECTIVE MASS PART IV APPLICATIONS IN RELATIVISTIC QUANTUM MECHANICS 15 SUSYQMANDSWKBAPPROACHTOTHEDIRACEQUATION WITH A COULOMB POTENTIAL IN 2+1 DIMENSIONS DIRAC EQUATION IN 2 +1 DIMENSIONS SUSYQM AND SWKB APPROACHES TO COULOMB PROBLEM ALTERNATIVE METHOD TO DERIVE EXACT EIGENFUNCTIONS 16 REALIZATION OF DYNAMIC GROUP FOR THE DIRAC HYDROGEN-LIKE ATOM IN 2+1 DIMENSIONS 17 ALGEBRAIC APPROACH TO KLEIN-GORDON EQUATION WITH THE HYDROGEN-LIKE ATOM IN 2+1 DIMENSIONS 18 SUSYQM AND SWKB APPROACHES TO RELATIVISTIC DIRACANDKLEIN-GORDONEQUATIONSWITHHYPERBOLIC POTENTIAL RELATIVISTIC KLEIN-GORDON AND DIRAC EQUATIONS WITH HYPERBOLIC POTENTIAL V0 TANH2(R/D) SUSYQM AND SWKB APPROACHES TO OBTAIN EIGENVALUES COMPLETE SOLUTIONS BY TRADITIONAL METHOD PART V QUANTUM CONTROL 19 CONTROLLABILITY OF QUANTUM SYSTEMS FOR THE MORSE AND PT POTENTIALS WITH DYNAMIC GROUP SU(2) PRELIMINARIES ON CONTROL THEORY ANALYSIS OF THE CONTROLLABILITY 20 CONTROLLABILITY OF QUANTUM SYSTEM FOR THE PT-LIKE POTENTIAL WITH DYNAMIC GROUP SU(1, 1) PRELIMINARIES ON THE CONTROL THEORY ANALYSIS OF CONTROLLABILITY PART VI CONCLUSIONS AND OUTLOOKS 21 CONCLUSIONS AND OUTLOOKS CONCLUSIONS OUTLOOKS