This book gives an introduction to supersymmetric quantum mechanics and a comprehensive review of its applications in quantum and statistical physics. The author discusses the classical and quantum versions of Witten's model and exact spectral properties of the model for the so-called shape invariant potentials. The quasi-classical quantization rules are derived and other topics include the supersymmetric structure of a classical stochastic dynamical system obeying the Langevin or the Fokker-Planck equation, Pauli's Hamiltonian and its application to the paragmagnetism of a non-interacting electron gas in two and three dimensions, and supersymmetry of Dirac's Hamiltonian. The book addresses graduate students as well as scientists.

Klappentext

The idea of supersymmetry was originally introduced in relativistic quantum field theories as a generalization of Poincare symmetry. In 1976 Nicolai sug­ gested an analogous generalization for non-relativistic quantum mechanics. With the one-dimensional model introduced by Witten in 1981, supersym­ metry became a major tool in quantum mechanics and mathematical, sta­ tistical, and condensed-IIll;l. tter physics. Supersymmetry is also a successful concept in nuclear and atomic physics. An underlying supersymmetry of a given quantum-mechanical system can be utilized to analyze the properties of the system in an elegant and effective way. It is even possible to obtain exact results thanks to supersymmetry. The purpose of this book is to give an introduction to supersymmet­ ric quantum mechanics and review some of the recent developments of vari­ ous supersymmetric methods in quantum and statistical physics. Thereby we will touch upon some topics related to mathematical and condensed-matter physics. A discussion of supersymmetry in atomic and nuclear physics is omit­ ted. However, the reader will find some references in Chap. 9. Similarly, super­ symmetric field theories and supergravity are not considered in this book. In fact, there exist already many excellent textbooks and monographs on these topics. A list may be found in Chap. 9. Yet, it is hoped that this book may be useful in preparing a footing for a study of supersymmetric theories in atomic, nuclear, and particle physics. The plan of the book is as follows.

Inhalt

1. Introduction.- 2. Supersymmetric Quantum Mechanics.- 2.1 Definition of SUSY Quantum Mechanics.- 2.1.1 The Pauh Hamiltonian (N = 1).- 2.1.2 Witten's SUSY Quantum Mechanics (N = 2).- 2.1.3 Nicolai's Supersymmetric Harmonic-Oscillator Chain.- 2.2 Properties of N = 2 SUSY Quantum Mechanics.- 2.2.1 The Witten Parity.- 2.2.2 SUSY Transformation.- 2.2.3 Ground-State Properties for Good SUSY.- 2.2.4 The Witten Index.- 2.2.5 SUSY and Gauge Transformations.- 3. The Witten Model.- 3.1 Witten's Model and Its Modification.- 3.2 Witten Parity and SUSY Transformation.- 3.3 The SUSY Potential and Zero-Energy States.- 3.3.1 Ground State for Good SUSY.- 3.3.2 An Additional Symmetry.- 3.3.3 Asymptotic Behavior of the SUSY Potential.- 3.4 Broken Versus Good SUSY.- 3.5 Examples.- 3.5.1 Systems on the Euclidean Line.- 3.5.2 Systems on the Half Line.- 3.5.3 Systems on a Finite Interval.- 4. Supersymmetric Classical Mechanics.- 4.1 Pseudoclassical Models.- 4.2 A Supersymmetric Classical Model.- 4.3 The Classical Dynamics.- 4.4 Discussion of the Fermionic Phase.- 4.5 Quantization.- 4.5.1 Canonical Approach.- 4.5.2 Path-Integral Approach.- 5. Exact Solution of Eigenvalue Problems.- 5.1 Supersymmetrization of One-Dimensional Systems.- 5.2 Shape-Invariance and Exact Solutions.- 5.2.1 An Explicit Example.- 5.2.2 Comparison with the Factorization Method.- 6. Quasi-Classical Path-Integral Approach.- 6.1 The Path-Integral FormaHsm.- 6.1.1 The WKB Approximation in the Path Integral.- 6.1.2 Quasi-Classical Modification for Witten's Model.- 6.2 Quasi-Classical Quantization Conditions..- 6.3 Quasi-Classical Eigenfunctions.- 6.4 Discussion of the Results.- 6.4.1 Exactly Soluble Examples.- 6.4.2 Numerical Investigations.- 7. Supersymmetry in Classical Stochastic Dynamics.- 7.1 Langevin and Fokker-Planck Equation.- 7.2 Supersymmetry of the Fokker-Planck Equation.- 7.3 Supersymmetry of the Langevin Equation.- 7.4 Implications of Supersymmetry.- 7.4.1 Good SUSY.- 7.4.2 Broken SUSY.- 8. Supersymmetry in the Pauli and Dirac Equation.- 8.1 Pauli's Hamiltonian in Two and Three Dimensions.- 8.2 Pauli Paramagnetism of Non-Interacting Electrons, Revisited..- 8.2.1 Two-Dimensional Electron Gas.- 8.2.2 Three-Dimensional Electron Gas.- 8.2.3 The Paramagnetic Conjecture and SUSY.- 8.3 The Dirac Hamiltonian and SUSY.- 8.3.1 Dirac Hamiltonian with a Scalar Potential.- 9. Concluding Remarks and Overview.- References.- Symbols.- Name Index.