UID:
almafu_9960901281002883
Umfang:
1 online resource (122 pages)
ISBN:
3-031-10624-5
Serie:
SpringerBriefs in Physics
Anmerkung:
Intro -- 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.
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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.
Weitere Ausg.:
Print version: Kapoor, A. K. Quantum Hamilton-Jacobi Formalism Cham : Springer International Publishing AG,c2022 ISBN 9783031106231
Sprache:
Englisch
