Quantum Hamilton-Jacobi Formalism

Description

This book describes the Hamilton-Jacobi formalism of quantum mechanics, which allows computation of eigenvalues of quantum mechanical potential problems without solving for the wave function. The examples presented include exotic potentials such as quasi-exactly solvable models and Lame an dassociated Lame potentials. A careful application of boundary conditions offers an insight into the nature of solutions of several potential models. Advanced undergraduates having knowledge of complex variables and quantum mechanics will find this as an interesting method to obtain the eigenvalues and eigen-functions. The discussion on complex zeros of the wave function gives intriguing new results which are relevant for advanced students and young researchers. Moreover, a few open problems in research are discussed as well, which pose a challenge to the mathematically oriented readers. Preface Acknowledgements Contents Acronyms 1 Quantum Hamilton-Jacobi Formalism 1.1 Introduction 1.2 Quantum Hamilton-Jacobi Formalism 1.2.1 Working in the Complex Plane 1.2.2 Boundary Condition 1.2.3 Exact Quantization Condition 1.3 Plan of the Book References 2 Mathematical Preliminaries 2.1 Introduction 2.2 Results from Theory of Complex Variables 2.2.1 Laurent Expansion 2.2.2 Liuoville Theorem 2.2.3 Meromorphic Function 2.3 Results from Theory of Differential Equations 2.3.1 Solutions of Riccati Equation 2.3.2 Singular Points of Solution of Riccati Equation 2.3.3 Real and Complex Zeros of the Wave Function 2.3.4 Riccati Equation—Some General Results 2.3.5 Reduction to a Second-Order Linear Differential Equation 2.3.6 Most General Solution from Given Solution(s) 2.4 Some Frequently Used Results and Examples 2.4.1 Residue of QMF at a Moving Pole 2.4.2 Evaluating Action Integral 2.5 Harmonic Oscillator—Method-I 2.5.1 Behaviour of p Subscript clpcl for Large StartAbsoluteValue x EndAbsoluteValue|x| 2.5.2 Classical Momentum in the Complex Plane 2.5.3 Boundary Condition on QMF 2.5.4 Energy Eigenvalues 2.6 Harmonic Oscillator—Method-II 2.6.1 Using Square Integrability 2.6.2 General Form of QMF 2.6.3 Energy Eigenvalues 2.6.4 Energy Eigenfunctions References 3 Exactly Solvable Models 3.1 Introduction 3.2 Change of Variable 3.3 Morse Oscillator 3.4 Radial Oscillator 3.5 Particle in a Box 3.6 Exactness of SWKB Approximation 3.7 Concluding Remarks References 4 Exotic Potentials 4.1 Introduction 4.2 A Periodic Potential 4.3 A Potential with Two Phases of SUSY 4.3.1 QHJ Solution of Scarf-I Potential 4.3.2 Change of Variable 4.3.3 Meromorphic Form of the QMF chiχ 4.3.4 Computation of the Residues Using QHJ 4.3.5 Leacock-Padgett Boundary Condition 4.4 Quasi-Exactly Solvable Models 4.4.1 Introduction to QES Models 4.4.2 A List of QES Potentials 4.4.3 QHJ Analysis of Sextic Oscillator 4.4.4 The Residue at Infinity 4.4.5 Form of the QMF and Wave Function 4.4.6 Properties of the Solutions 4.5 Band Edges for a Periodic Potential 4.5.1 Explicit Solution for j equals 2j=2 4.6 A QES Periodic Potential 4.7 Some Observations References 5 Rational Extensions 5.1 Introduction 5.2 About ES Potentials and Orthogonal Polynomial Connection 5.3 Exceptional Orthogonal Polynomials 5.4 Extended Potentials 5.5 Supersymmetric Quantum Mechanics 5.6 Shape Invariance 5.7 Construction of New ES Models 5.7.1 Isospectral Deformation 5.7.2 Isospectral Shift Deformation 5.7.3 ISD and Shape Invariant Extensions 5.8 Radial Oscillator and Its Shape Invariant Extensions 5.9 Further Observations References 6 Complex Potentials and Optical Systems 6.1 Introduction 6.2 Complex script upper P script upper TmathcalPmathcalT-Symmetric Potentials 6.2.1 The Complex Scarf-II Potential 6.2.2 Variation of Parameters and Energy Spectrum 6.3 SUSY, script upper P script upper TmathcalPmathcalT-Symmetry and Optical Systems 6.4 Complex Scarf-II Potential: script upper P script upper TmathcalPmathcalT Symmetry and Supersymmetry 6.4.1 Broken and Unbroken Phases of script upper P script upper TmathcalPmathcalT-Symmetry 6.4.2 Unbroken script upper P script upper TmathcalPmathcalT and Broken SUSY Phase 6.4.3 Broken script upper P script upper TmathcalPmathcalT-Symmetric Phase References 7 Beyond One Dimension 7.1 Introduction 7.2 Classification of ES and QES Models 7.3 Open Questions and Concluding Remarks References Index