This invaluable book provides an elementary description of supersymmetric quantum mechanics which complements the traditional coverage found in the existing quantum mechanics textbooks. It gives physicists a fresh outlook and new ways of handling quantum-mechanical problems, and also leads to improved approximation techniques for dealing with potentials of interest in all branches of physics. The algebraic approach to obtaining eigenstates is elegant and important, and all physicists should become familiar with this.The book has been written in such a way that it can be easily appreciated by students in advanced undergraduate quantum mechanics courses. Problems have been given at the end of each chapter, along with complete solutions to all the problems. The text also includes material of interest in current research not usually discussed in traditional courses on quantum mechanics, such as the connection between exact solutions to classical soliton problems and isospectral quantum Hamiltonians, and the relation to the inverse scattering problem.
Author(s): Fred Cooper; Avinash Khare; Uday Pandurang Sukhatme
Publisher: World Scientific Publishing Company Incorporated
Year: 2001
Language: English
Pages: 210
Supersymmetry in Quantum Mechanics
Preface
Contents
Chapter 1 Introduction
Chapter 2 The Schrödinger Equation in One Dimension
2.1 General Properties of Bound States
2.2 General Properties of Continuum States and Scattering
2.3 The Harmonic Oscillator in the Operator Formalism
Chapter 3 Factorization of a General Harniltonian
3.1 Broken Supersymmetry
3.2 SUSY Harmonic Oscillator
3.3 Factorization and the Hierarchy of Hamiltonians
Chapter 4 Shape Invariance and Solvable Potentials
4.1 General Formulas for Bound State Spectrum, Wave Functions and S-Matrix
4.2 Strategies for Categorizing Shape Invariant Potentials
4.2.1 Solutions Involving Traslation
4.2.2 Solutions Involving Scaling
4.2.3 Other Solutions
4.3 Shape Invariance and Noncentral Solvable Potentials
Chapter 5 Charged Particles in External Fields and Supersymmetry
5.1 Spinless Particles
5.2 Non-relativistic Electrons and the Pauli Equation
5.3 Relativistic Electrons and the Dirac Equation
5.4 SUSY and the Dirac Equation
5.5 Dirac Equation with a Lorentz Scalar Potential in 1 + 1 Dimensions
5.6 Supersymmetry and the Dirac Particle in a Coulomb Field
5.7 SUSY and the Dirac Particle in a Magnetic Field
Chapter 6 Isospectral Hamiltonians
6.1 One Parameter Family of Isospectral Potentials
6.2 Generalization to n-Parameter Isospectral Family
6.3 Inverse Scattering and Solitons
Chapter 7 New Periodic Potentials from Supersymmetry
7.1 Unbroken SUSY and the Value of the Witten Index
7.2 Lamé Potentials and Their Supersymmetric Partners
7.3 Associated Lamé Potentials and Their Supersymmetric Partners
7.3.1 a = b = Integer
Chapter 8 Supersymmetric WKB Approximation
8.1 Lowest Order WKB Quantization Condition
8.1.1 Simpler Approach for the Lowest Order Quantization Condition
8.2 Some General Comments on WKB Theory
8.3 Tunneling Probability in the WKB Approximation
8.4 SWKB Quantization Condition for Unbroken Supersymmetry
8.5 Exactness of the SWKB Condition for Shape Invariant Potentials
8.6 Comparison of the SWKB and WKB Approaches
8.7 SWKB Quantization Condition for Broken Supersymmetry
8.8 Tunneling Probability in the SWKB Approximation
Chapter 9 Perturbative Methods for Calculating Energy Spectra and Wave Functions
9.1 Variational Approach
9.2 SUSY d Expansion Method
9.3 Supersymmetry and Double Well Potentials
9.4 Supersymmetry and the Large-N Expansion
Appendix A Path Integrals and SUSY
A.1 Dirac Notation
A.2 Path Integral for the Evolution Operator
A.3 Path Integrals for Fermionic Degrees of Freedom
A.3.1 Hilbert Space for Fermionic Oscillator
A.4 Path Integral Formulation of SUSY Quantum Mechanics
A.5 Superspace Formulation of SUSY Quantum Mechanics
Appendix B Operator Transforms – New Solvable Potentials from Old
B.1 Natanzon Potentials
Appendix C Logarithmic Perturbation Theory
Appendix D Solutions to Problems
Index
