Supersymmetric approach to some problems in quantum mechanics and related topics

Marques, Fabricio

In this work we present an introduction to Supersymmetry in the context of 1-dimensional Quantum Mechanics. For that purpose we develop the concept of hamiltonians factorization using the simple harmonic oscillator as an example, we introduce the supersymmetric oscilator and, next, we generalize these concepts to introduce the fundamentals of Supersymmetric Quantum Mechanics. We also discuss useful tools to solve problems in Quantum Mechanics which are intrinsecally related to Supersymmetry as hierarchy of hamiltonians and shape invariance. We present two approximation methods which will be specially useful: the well known Variational Method and the Logarithmic Perturbation Theory, the latter being closely related to the concepts of superpotentials and hierarchy of hamiltonians. Finally, we present problems related to superpotentials which are monomials in even powers of the x coordinate multiplied by the sign function epsilon(x), which seems to be a new class of problems in Supersymmetric Quantum Mechanics.

Vladislav G. Bagrov, A. S. Vshivtsev

Leon A. Takhtajan

This book provides a comprehensive treatment of quantum mechanics from a mathematics perspective and is accessible to mathematicians starting with second-year graduate students. It addition to traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin, and it introduces the reader to functional methods in quantum mechanics. This includes the Feynman path integral approach to quantum mechanics, integration in functional spaces, the relation between Feynman and Wiener integrals, Gaussian integration and regularized determinants of differential operators, fermion systems and integration over anticommuting (Grassmann) variables, supersymmetry and localization in loop spaces, and supersymmetric derivation of the Atiyah-Singer formula for the index of the Dirac operator. Prior to this book, mathematicians could find these topics only in physics textbooks and in specialized literature. This book is written in a concise style with careful attention to precise mathematics formulation of methods and results.Numerous problems, from routine to advanced, help the reader to master the subject. In addition to providing a fundamental knowledge of quantum mechanics, this book could also serve as a bridge for studying more advanced topics in quantum physics, among them quantum field theory. Prerequisites include standard first-year graduate courses covering linear and abstract algebra, topology and geometry, and real and complex analysis.

ANURADHA LAHIRI, Papai Roy, Bijan Kumar Bagchi

A. D. Jannussis, I. Tsohantzis, D. Vavougios

Lu Huan, Wan Buyong, Liu Yao +1 more

The issues of the negative energy and the negative kinetic energy remain among the most profound and actively debated topics in physics. Based on the negative-kinetic-energy supersymmetric quantum mechanics (NKE-SUSYQM), this paper explores the supersymmetric algebra in depth. First, we begin with a concise overview of how the negative-kinetic-energy Schrödinger equation is derived and how NKE-SUSYQM is thereby established. Second, we analyze the fundamental quantum operators and elucidate the complete structure of the supersymmetric algebra within NKE-SUSYQM. Finally, we provide a detailed comparative study that contrasts the positive- and the negative-kinetic-energy SUSYQM, highlighting their key similarities and differences.