Supersymmetry in Quantum Mechanics

by Cooper, Fred; Khare, Avinash; Sukhatme, Uday

• ISBN13: 9789810246129
• ISBN10: 9810246129
• Format: Paperback
• Copyright: 2001-07-01
• Publisher: World Scientific Pub Co Inc

Table of Contents

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