One-Dimensional Hydrogen Atom

Abstract

The time-independent Schrödinger equation is solved for the bound state solutions of the one-dimensional Coulomb potential, −e2/|x|. The wave functions obtained are normalizable and continuous. The energy spectrum consists of a set of discrete levels with energies equal to the Bohr energies of hydrogen, and a set of continuum levels with energies lying strictly between the discrete levels. The odd wave functions associated with the discrete levels are differentiable everywhere, but the even wave functions associated with the continuum levels have a cusp with infinite slope at the origin. The energy levels are not degenerate. Following the method of Loudon, the bound state solutions of a truncated Coulomb potential −e2/(|x|+a), a>0, are also obtained. For small a, the discrete spectrum of this potential contains: (1) energy levels which approach arbitrarily close to the discrete levels of the true Coulomb potential as a→0, with odd wave functions identical to the odd wave functions of the true Coulomb potential at a = 0; (2) energy levels which approach arbitrarily close to the discrete levels of the true Coulomb potential as a→0, with, even wave functions which are not the even wave functions of the true Coulomb potential at a = 0; and (3) a ground state level belonging to an even wave function with an arbitrarily large binding energy as a→0. The ground state energy is quite sensitive to the actual value of a chosen. It is concluded that the solution to the one-dimensional hydrogen atom shows a critical dependence on the detailed behavior of the Coulomb potential at small distances.