A fresh outlook and new ways of handling the important quantum systems in all branches of physics and chemistry
This book introduces the factorization method in quantum mechanics at an advanced level, with the aim of putting 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, the text provides a comprehensive deillegalscription of the factorization method and its wide applications in quantum mechanics which complements the traditional coverage found in quantum mechanics textbooks.
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: Pöschl-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: Pöschl-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 Schrödinger Equation for a Singular Oscillator. 14.1. Introduction. 14.2. Position-dependent effective mass Schrödinger 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
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 deillegalscription 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.
Audience:
Researchers and students of physics, mathematics, chemistry and electrical engineering.
From the reviews:
"An up-to-date organized account of material that can be addressed to an interdisciplinary graduate-level audience. ... Besides the elegance and the effectiveness in characterizing matrix-elements, a motivation for the algebraic approach also relies in the pedagogical expectation that beginners can be better driven to topics like coherent states and supersymmetric Quantum Mechanics. ... The book can be generally addressed to graduate students and young researchers in physics, theoretical chemistry, applied mathematics and electrical engineering ... ." (Giulio Landolfi, Zentralblatt MATH, Vol. 1130 (8), 2008)
"The book under review is an interesting and useful collection of results which fall under the headings of `factorization method´, `Darboux/Crum transformation´, `supersymmetric quantum mechanics´, `intertwining operator method´ and `shape invariance´. ... can be used by researchers in the field and by students of quantum mechanics. ... the overall impression of the book is that it is useful for a broad audience of physicists and mathematical physicists." (Marek Nowakowski, Mathematical Reviews, Issue 2008 k)
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 deillegalscription 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.
Audience:
Researchers and students of physics, mathematics, chemistry and electrical engineering.
METHOD.- THEORY.- LIE ALGEBRAS SU(2) AND SU(1, 1).- APPLICATIONS IN NON-RELATIVISTIC QUANTUM MECHANICS.- HARMONIC OSCILLATOR.- INFINITELY DEEP SQUARE-WELL POTENTIAL.- MORSE POTENTIAL.- PÖSCHL-TELLER POTENTIAL.- PSEUDOHARMONIC OSCILLATOR.- ALGEBRAIC APPROACH TO AN ELECTRON IN A UNIFORM MAGNETIC FIELD.- RING-SHAPED NON-SPHERICAL OSCILLATOR.- GENERALIZED LAGUERRE FUNCTIONS.- NEW NONCENTRAL RING-SHAPED POTENTIAL.- PÖSCHL-TELLER LIKE POTENTIAL.- POSITION-DEPENDENT MASS SCHRÖDINGER EQUATION FOR A SINGULAR OSCILLATOR.- APPLICATIONS IN RELATIVISTIC QUANTUM MECHANICS.- SUSYQM AND SWKB APPROACH TO THE DIRAC EQUATION WITH A COULOMB POTENTIAL IN 2+1 DIMENSIONS.- REALIZATION OF DYNAMIC GROUP FOR THE DIRAC HYDROGEN-LIKE ATOM IN 2+1 DIMENSIONS.- ALGEBRAIC APPROACH TO KLEIN-GORDON EQUATION WITH THE HYDROGEN-LIKE ATOM IN 2+1 DIMENSIONS.- SUSYQM AND SWKB APPROACHES TO RELATIVISTIC DIRAC AND KLEIN-GORDON EQUATIONS WITH HYPERBOLIC POTENTIAL.- QUANTUM CONTROL.- CONTROLLABILITY OF QUANTUM SYSTEMS FOR THE MORSE AND PT POTENTIALS WITH DYNAMIC GROUP SU(2).- CONTROLLABILITY OF QUANTUM SYSTEM FOR THE PT-LIKE POTENTIAL WITH DYNAMIC GROUP SU(1, 1).- CONCLUSIONS AND OUTLOOKS.- CONCLUSIONS AND OUTLOOKS.
From the reviews:
"An up-to-date organized account of material that can be addressed to an interdisciplinary graduate-level audience. ... Besides the elegance and the effectiveness in characterizing matrix-elements, a motivation for the algebraic approach also relies in the pedagogical expectation that beginners can be better driven to topics like coherent states and supersymmetric Quantum Mechanics. ... The book can be generally addressed to graduate students and young researchers in physics, theoretical chemistry, applied mathematics and electrical engineering ... ." (Giulio Landolfi, Zentralblatt MATH, Vol. 1130 (8), 2008)
"The book under review is an interesting and useful collection of results which fall under the headings of 'factorization method', 'Darboux/Crum transformation', 'supersymmetric quantum mechanics', 'intertwining operator method' and 'shape invariance'. ... can be used by researchers in the field and by students of quantum mechanics. ... the overall impression of the book is that it is useful for a broad audience of physicists and mathematical physicists." (Marek Nowakowski, Mathematical Reviews, Issue 2008 k)
Inhaltsverzeichnis
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: Pöschl-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: Pöschl-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 Schrödinger Equation for a Singular Oscillator. 14.1. Introduction. 14.2. Position-dependent effective mass Schrödinger 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
A fresh outlook and new ways of handling the important quantum systems in all branches of physics and chemistry
