Factorization Method in Quantum Mechanics

PART I - Introduction. 1: Introduction. 1.1 Basic review. 1.2. Motivations and aims. PART II -- Method. 2: Theory. 2.1. Introduction. 2.2. Formalism. 3: Lie Algebras SU(2) and SU(1,1). 3.1. Introduction. 3.2. Abstract groups. 3.3. Matrix representation. 3.4. properties of groups SU(2) and SO(3). 3.5. Properties of non-compact groups SO(2,1) and SU(1,1). 3.6. Generators of Lie groups SU(2) and SU(1,1). 3.7. Irreducible representations. 3.8. Irreducible unitary representations. 3.9. Concluding remarks. PART III -- Applications in Non-Relativistic Quantum mechanics. 4: Harmonic Oscillator. 4.1. Introduction. 4.2. Exact solutions. 4.3. Ladder operators. 4.4. Bargmann-Segal transformations. 4.5. Single mode realization of dynamic group SU(1,1). 4.6. Matrix elements. 4.7. Coherent states. 4.8. Franck-Condon factors. 4.9. Concluding remarks. 5: Infinitely Deep Square-Well Potential. 5.1. Introduction. 5.2. Ladder operators for infinitely deep square-well potential. 5.3. Realization of dynamic group SU(1,1) and matrix elements. 5.4. Ladder operators for infinitely deep symmetric well potential. 5.5. SUSYQM approach to infinitely deep square-well potential. 5.6. Perelomov coherent states. 5.7. Barut-Girardello coherent states. 5.8. Concluding remarks. 6: Morse Potential. 6.1. Introduction. 6.2. Exact solutions. 6.3. Ladder operators for the Morse potential. 6.4. Realization of dynamic group SU(2). 6.5. Matrix elements. 6.6. Harmonic limit. 6.7. Franck-Condon factors. 6.8. Transition probability. 6.9. Realization of dynamic group SU(1,1). 6.10. Concluding remarks. 7: Poschl-Teller Potential. 7.1. Introduction. 7.2. Exact solutions. 7.3. Ladder operators. 7.4. Realization of dynamic group SU(2). 7.5. Alternative approach to derive ladder operators. 7.6. Harmonic limit. 7.7. Expansions of the coordinate x and momentum p from the SU(2) generators. 7.8. Concluding remarks. 8: Pseudoharmonic Oscillator. 8.1. Introduction. 8.2. Exact solutions in one dimension. 8.3. Ladder operators. 8.4. Barut-Girardello coherent states. 8.5. Thermodynamic properties. 8.6. Pseudoharmonic oscillator in arbitrary dimensions. 8.7. Recurrence relations among matrix elements. 8.8. Concluding remarks. 9: Algebraic Approach to an Electron in a Uniform Magnetic Field. 9.1. Introduction. 9.2. Exact solutions. 9.3. Ladder operators. 9.4. Concluding remarks. 10: Ring-Shaped Non-Spherical Oscillator. 10.1. Introduction. 10.2. Exact solutions. 10.3. Ladder operators. 10.4. Realization of dynamic group. 10.5. Concluding remarks. 11: Generalized Laguerre Functions. 11.1. Introduction. 11.2. generalized Laguerre functions. 11.3. Ladder operators and realization of dynamic group SU(1,1). 11.4. Concluding remarks. 12: New Non-Central Ring-Shaped Potential. 12.1. Introduction. 12.2. Bound states. 12.3. Ladder operators. 12.4. Mean values. 12.5. Continuum states. 12.6. Concluding remarks. 13: Poschl-Teller Like Potential. 13.1. Introduction. 13.2. Exact solutions. 13.3. Ladder operators. 13.4. Realization of dynamic group and matrix elements. 13.5. Infinitely square-well and harmonic limits. 13.6. Concluding remarks. 14: Position-Dependent Mass Schrodinger Equation for a Singular Oscillator. 14.1. Introduction. 14.2. Position-dependent effective mass Schrodinger equation for harmonic oscillator. 14.3. Singular oscillator with a position-dependent effective mass. 14.4. Complete solutions. 14.5. Another position-dependent effective mass. 14.6. Concluding remarks. PART IV -- Applications in Relativistic Quantum Mechanics. 15: SUSYQM and SKWB Approach to the Dirac Equation with a Coulomb Potential in 2+1 Dimensions. 15.1. Introduction. 15.2. Dirac equation in 2+1 dimensions. 15.3. Exact solutions. 15.4. SUSYQM and SKWB approaches to Coulomb problem. 15.5. Alternative method to derive exact eigenfunctions. 15.6. Concluding remarks. 16: Realization of Dynamic Group for the Dirac Hydrogen-like Atom in 2+1 Dimensions. 16.1. Introduction. 16.2. realization of Dynamic group SU(1,1). 16.3. Concluding remarks. 17: Algebraic Approach to Klein-Gordon Equation with the Hydrogen-like Atom in 1+2 Dimensions. 17.1. Introduction. 17.2. Exact solutions. 17.3. Realization of dynamic group SU(1,1). 17.4. Concluding remarks. 18: SUSYQM and SKWB Approaches to Relativistic Dirac and Klein-Gordon Equations with Hyperbolic Potential. 18.1. Introduction. 18.2. Relativistic Klein-Gordon and Dirac equations with hyperbolic potential V0tanh2(r/d). 18.3. SUSYQM and SKWB approaches to obtain eigenvalues. 18.4. Complete solutions by traditional method. 18.5. Harmonic limit. 18.6. Concluding remarks. PART V -- Quantum Control. 19: Controllability of Quantum Systems for the Morse and PT Potentials with Dynamic Group SU(2). 19.1. Introduction. 19.2. Preliminaries on control theory. 19.3. Analysis of the controllability. 19.4. Concluding remarks. 20: Controllability of Quantum System for the PT-like Potential with Dynamic Group SU(1,1). 20.1. Introduction. 20.2. Preliminaries on control theory. 20.3. Analysis of controllability. 20.4. Concluding remarks. PART VI - Conclusions and Outlooks. 21: Conclusions and outlooks. 21.1. Conclusions. 21.2. Outlooks. APPENDICES A Integral formulas of the confluent hypergeometric functions B Mean values rk for hydrogen-like atom C Commutator identities D Angular momentum operators in spherical coordinates E Confluent hypergeometric function References Index