Supersymmetry In Quantum Mechanics

by Frederick M Cooper, Avinash Khare, Uday P Sukhatme

Overview

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.

Product Details

ISBN-13:	9789810246129
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	07/03/2001
Edition description:	New Edition
Pages:	224
Product dimensions:	6.08(w) x 8.60(h) x 0.48(d)

Table of Contents

	Preface	vii
Chapter 1	Introduction	1
Chapter 2	The Schrodinger Equation in One Dimension	7
2.1	General Properties of Bound States	8
2.2	General Properties of Continuum States and Scattering	9
2.3	The Harmonic Oscillator in the Operator Formalism	10
Chapter 3	Factorization of a General Hamiltonian	15
3.1	Broken Supersymmetry	23
3.2	SUSY Harmonic Oscillator	28
3.3	Factorization and the Hierarchy of Hamiltonians	30
Chapter 4	Shape Invariance and Solvable Potentials	35
4.1	General Formulas for Bound State Spectrum, Wave Functions and S-Matrix	36
4.2	Strategies for Categorizing Shape Invariant Potentials	38
4.2.1	Solutions Involving Translation	38
4.2.2	Solutions Involving Scaling	47
4.2.3	Other Solutions	53
4.3	Shape Invariance and Noncentral Solvable Potentials	56
Chapter 5	Charged Particles in External Fields and Supersymmetry	61
5.1	Spinless Particles	61
5.2	Non-relativistic Electrons and the Pauli Equation	62
5.3	Relativistic Electrons and the Dirac Equation	68
5.4	SUSY and the Dirac Equation	70
5.5	Dirac Equation with a Lorentz Scalar Potential in 1+1 Dimensions	72
5.6	Supersymmetry and the Dirac Particle in a Coulomb Field	75
5.7	SUSY and the Dirac Particle in a Magnetic Field	78
Chapter 6	Isospectral Hamiltonians	81
6.1	One Parameter Family of Isospectral Potentials	82
6.2	Generalization to n-Parameter Isospectral Family	84
6.3	Inverse Scattering and Solitons	88
Chapter 7	New Periodic Potentials from Supersymmetry	97
7.1	Unbroken SUSY and the Value of the Witten Index	97
7.2	Lame Potentials and Their Supersymmetric Partners	101
7.3	Associated Lame Potentials and Their Supersymmetric Partners	110
7.3.1	a = b = Integer	113
Chapter 8	Supersymmetric WKB Approximation	119
8.1	Lowest Order WKB Quantization Condition	120
8.1.1	Simpler Approach for the Lowest Order Quantization Condition	122
8.2	Some General Comments on WKB Theory	124
8.3	Tunneling Probability in the WKB Approximation	125
8.4	SWKB Quantization Condition for Unbroken Supersymmetry	126
8.5	Exactness of the SWKB Condition for Shape Invariant Potentials	128
8.6	Comparison of the SWKB and WKB Approaches	130
8.7	SWKB Quantization Condition for Broken Supersymmetry	131
8.8	Tunneling Probability in the SWKB Approximation	132
Chapter 9	Perturbative Methods for Calculating Energy Spectra and Wave Functions	137
9.1	Variational Approach	137
9.2	SUSY [delta] Expansion Method	141
9.3	Supersymmetry and Double Well Potentials	143
9.4	Supersymmetry and the Large-N Expansion	150
Appendix A	Path Integrals and SUSY	157
A.1	Dirac Notation	157
A.2	Path Integral for the Evolution Operator	158
A.3	Path Integrals for Fermionic Degrees of Freedom	162
A.3.1	Hilbert Space for Fermionic Oscillator	162
A.4	Path Integral Formulation of SUSY Quantum Mechanics	167
A.5	Superspace Formulation of SUSY Quantum Mechanics	174
Appendix B	Operator Transforms--New Solvable Potentials from Old	177
B.1	Natanzon Potentials	182
Appendix C	Logarithmic Perturbation Theory	185
Appendix D	Solutions to Problems	189
	Index	207