A Perturbative Approach in the Minimal Length of Quantum Mechanics

Abstract

There are many pieces of evidence for a minimal lengthof the order of Planck length in the problems in quantum gravity, stringtheory, and black-hole physics etc. Existing of such a minimal lengthdescription modifies the traditional Heisenberg uncertainty principle. Thenovel form is called "the generalized uncertainty principle" in thejargon. Such a deformation in the uncertainty relation changes thecorresponding wave equation. The latter Schrodinger equation is now no more asecond-order differential equation. Consequently, this causes a greatdifficulty to obtain the analytic solutions. In this study, we propose a perturbativeapproach to the bound state solutions of the Woods-Saxon potential in theSchrodinger equation by adopting the minimal length. Here, we take the extraterm as a perturbative term to the Hamiltonian. Then, we calculate the firstorder corrections of the energy spectrum for a confined particle in a well by aWoods-Saxon potential energy.