Dirac δ δ -function potential in quasiposition representation of a minimal-length scenario

2017/04/30 by M. F. Gusson, A. Oakes O. Gonçalves, R. O. Francisco +4 · 9 citations
Physics and Astronomy · #Black Holes and Theoretical Physics #Bound state #Dirac (video compression format) #Dirac delta function #Dirac equation #Mathematical physics #Noncommutative and Quantum Gravity Theories #Physics #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum mechanics #Uncertainty principle #hep-th

paper · pdf · doi:10.1140/epjc/s10052-018-5659-6

published in The European Physical Journal C 78(3) (Springer Science+Business Media)

openalex created_date 2017/05/05 · openalex publication_date 2018/03/01 · arxiv created 2018/03/07 · arxiv updated 2018/03/08 · openalex updated_date 2026/08/05

Abstract

A minimal-length scenario can be considered as an effective description of quantum gravity effects. In quantum mechanics the introduction of a minimal length can be accomplished through a generalization of Heisenberg’s uncertainty principle. In this scenario, state eigenvectors of the position operator are no longer physical states and the representation in momentum space or a representation in a quasiposition space must be used. In this work, we solve the Schroedinger equation with a Dirac δ -function potential in quasiposition space. We calculate the bound state energy and the coefficients of reflection and transmission for the scattering states. We show that leading corrections are of order of the minimal length ( O(√(β ))) and the coefficients of reflection and transmission are no longer the same for the Dirac delta well and barrier as in ordinary quantum mechanics. Furthermore, assuming that the equivalence of the 1s state energy of the hydrogen atom and the bound state energy of the Dirac δ -function potential in the one-dimensional case is kept in a minimal-length scenario, we also find that the leading correction term for the ground state energy of the hydrogen atom is of the order of the minimal length and \varDelta x_min ≤ 10^-25 m.

Citations

• Calculating Vacuum Energies in Renormalizable Quantum Field Theories: A New Approach to the Casimir Problem
• Generalized virial theorem and pressure relation for a strongly correlated Fermi gas
• Some aspects of gravitational quantum mechanics
• Casimir effect in minimal length theories based on a generalized uncertainty principle
• The Casimir effect: Recent controversies and progress
• Universal properties of the ultracold Fermi gas
• Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations
• Exact Relations for a Strongly Interacting Fermi Gas from the Operator Product Expansion
• Minimal length uncertainty relation and the hydrogen atom
• Phenomenological Implications of the Generalized Uncertainty Principle