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

• Main Authors: M. F. Gusson, A. Oakes O. Gonçalves, R. O. Francisco, R. G. Furtado, J. C. Fabris, J. A. Nogueira
• Format: Article
• Language: English
• Published: SpringerOpen 2018-03-01
• Series: European Physical Journal C: Particles and Fields
• Online Access: http://link.springer.com/article/10.1140/epjc/s10052-018-5659-6
• ISSN: 1434-6044; 1434-6052

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 $$\delta $$ 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 $$({ O}(\sqrt{\beta }))$$ (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 $${{\delta }}$$ 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 $$\varDelta x_{\min } \le 10^{-25}$$ Δxmin≤10-25 m.