Supersymmetry in Quantum Mechanics

The following sections are included:

• Preface

• Contents

Supersymmetry (SUSY) arose as a response to attempts by physicists to obtain a unified description of all basic interactions of nature. SUSY relates bosonic and fermionic degrees of freedom combining them into superfields which provides a more elegant description of nature. The algebra involved in SUSY is a graded Lie algebra which closes under a combination of commutation and anti-commutation relations. It may be noted here that so far there has been no experimental evidence of SUSY being realized in nature. Nevertheless, in the last fifteen years, the ideas of SUSY have stimulated new approaches to other branches of physics like atomic, molecular, nuclear, statistical and condensed matter physics as well as nonrelativistic quantum mechanics. Naively, unbroken SUSY leads to a degeneracy between the spectra of the fermions and bosons in a unified theory. Since this is not observed in nature one needs SUSY to be spontaneously broken. It was in the context of trying to understand the breakdown of SUSY in field theory that the whole subject of SUSY quantum mechanics was first studied…

The following section are include:

• General Properties of Bound States

• General Properties of Continuum States and Scattering

• The Harmonic Oscillator in the Operator Formalism

• References

• Problems

The following sections are included:

• Broken Supersymmetry

• SUSY Harmonic Oscillator

• Factorization and the Hierarchy of Hamiltonians

• References

• Problems

The following sections are included:

• General Formulas for Bound State Spectrum, Wave Functions and S-Matrix

• Strategies for Categorizing Shape Invariant Potentials
  • Solutions Involving Translation
  • Solutions Involving Scaling
  • Other Solutions

• Shape Invariance and Noncentral Solvable Potentials

• References

• Problems

The following sections are included:

• Spinless Particles

• Non-relativistic Electrons and the Pauli Equation

• Relativistic Electrons and the Dirac Equation

• SUSY and the Dirac Equation

• Dirac Equation with a Lorentz Scalar Potential in 1+1 Dimensions

• Supersymmetry and the Dirac Particle in a Coulomb Field

• SUSY and the Dirac Particle in a Magnetic Field

• References

• Problems

The following sections are included:

• One Parameter Family of Isospectral Potentials

• Generalization to n-Parameter Isospectral Family

• Inverse Scattering and Solitons

• References

• Problems

The following sections are included:

• Unbroken SUSY and the Value of the Witten Index

• Lamé Potentials and Their Supersymmetric Partners

• Associated Lamé Potentials and Their Supersymmetric Partners
  • a = b = Integer

• References

• Problems

The following sections are included:

• Lowest Order WKB Quantization Condition
  • Simpler Approach for the Lowest Order Quantization Condition

• Some General Comments on WKB Theory

• Tunneling Probability in the WKB Approximation

• SWKB Quantization Condition for Unbroken Supersymmetry

• Exactness of the SWKB Condition for Shape Invariant Potentials

• Comparison of the SWKB and WKB Approaches

• SWKB Quantization Condition for Broken Supersymmetry

• Tunneling Probability in the SWKB Approximation

• References

• Problems

The following sections are included:

• Variational Approach

• SUSY δ Expansion Method

• Supersymmetry and Double Well Potentials

• Supersymmetry and the Large-N Expansion

• References

• Problems

The following sections are included:

• Appendix A: Path Integrals and SUSY
  • Dirac Notation
  • Path Integral for the Evolution Operator
  • Path Integrals for Fermionic Degrees of Freedom
    • Hilbert Space for Fermionic Oscillator
  • Path Integral Formulation of SUSY Quantum Mechanics
  • Superspace Formulation of SUSY Quantum Mechanics
  • References

• Appendix B: Operator Transforms – New Solvable Potentials from Old
  • Natanzon Potentials
  • References

• Appendix C: Logarithmic Perturbation Theory
  • References
  • Problems

• Appendix D: Solutions to Problems

• Index