Relation between Poisson and Schrodinger equations

A correspondence between the Poisson and Schrodinger equations is obtained through a simple transformation. This analogy between the two equations applies only to the restricted problem of quantum potentials consisting of a combination of attractive and repulsive delta functions in one dimension. The correspondence enables us to use elementary electrostatic results to find the ground state energy and wave function of the associated quantum problem. In particular, the result shows that it is possible to trap a single electron in a one dimensional ionic crystal modeled by repulsive and attractive delta functions. (C) 2012 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4722788]