1D Schrodinger equations with Coulomb-type potentials

摘要

We employ Laplace and Fourier transforms in momentum space to find the bound states of the 1D Schrodinger equations with two different potentials; 1/x and 1/|x|. By performing inverse transforms we show that for the potential 1/|x| the solutions in real space reduce to those of the 1D hydrogen atom with eigenenergies proportional to 1~2 with n integer. Analogously, we find that for the potential 1/x the eigenenergies are proportional to 1/(n + 1/2)~2 and the eigenfunctions can be expressed in terms of fractional derivatives. Taking into account that both potentials are singular (the 1/x potential is analytical and the 1/|x| potential is not), we analyse the nature of their bound states.