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Supersymmetry in quantum mechanics. (English) Zbl 0988.81001
Singapore: World Scientific. 230 p. (2001).
This book makes available an organized description of supersymmetric quantum mechanics (SQM). The authors review the theoretical formulation of SQM and discuss how supersymmetry (SUSY) helps in finding exact and approximate solutions to quantum-mechanical problems. SUSY harmonic oscillator, Pauli equation, Dirac equation, Dirac equation with a Lorentz scalar potential in \(1+1\) dimensions, Dirac particle in a Coulomb field, Dirac particle in a magnetic field are considered. The connection between inverse scattering and isospectral potentials helps to construct multi-soliton solutions of the Korteweg-de Vries equation. It is shown that using the ideas of SUSY and shape invariance, enables one to solve a number of potential problems algebraically.
New types of approximations are developed to solve quantum-mechanical problems by the existence of a superpotential, partner potential, hierarchy of isospectral Hamiltonians. The authors focus on four new approximation methods, the \({1\over N}\) expansion within SQM, \(\delta\) expansion for the superpotential, a SUSY inspired WKB approximation and a variational method which utilizes the hierarchy of Hamiltonians related by SUSY and factorization. The many figures and exercises with answers make this book particularly suitable for students in advanced undergraduate and beginning graduate quantum-mechanical courses.

MSC:
81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
81Q60 Supersymmetry and quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81U40 Inverse scattering problems in quantum theory
35Q40 PDEs in connection with quantum mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
81R40 Symmetry breaking in quantum theory
81T60 Supersymmetric field theories in quantum mechanics
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