Supersymmetric Methods in Quantum and Statistical Physics

ISBN-13: 9783642647420 / Angielski / Miękka / 2011 / 172 str.

Georg Junker

Opis

This book gives an introduction to supersymmetric quantum mechanics and a comprehensive review of its applications in quantum and statistical physics. The classical version and the quantum version of Witten's model are studied in detail. Exact spectral properties of the model for the so-called shape invariant potentials are discussed. The quasi-classical quantization rules are derived. The topics covered also 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, supersymmetry of Dirac's Hamiltonian, and others. The book addresses graduate students as well as scientists.

Spis treści

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.

Blurb

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.