A Study about the Supersymmetry in the context of Quantum Mechanics

TL;DR

This paper introduces supersymmetry in one-dimensional quantum mechanics, covering Hamiltonian factorization, supersymmetric oscillators, problem-solving tools, approximation methods, and a new class of superpotential problems.

Contribution

It provides a comprehensive introduction to supersymmetric quantum mechanics, including new problem classes involving monomial superpotentials with sign functions.

Findings

Development of Hamiltonian factorization techniques

Introduction of supersymmetric oscillator models

Proposal of new superpotential problem class

Abstract

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.