Eigenvalues of the Square-Well Problem

Forrest I. Glick

D. Giordano, Pierluigi Amodio A, Felice Iavernaro

We address two questions regarding square-well potentials from a didactic perspective.The first question concerns whether or not the justification of the standard a priori omission of the potential's vertical segments in the analysis of the eigenvalue problem is licit.The detour we follow to find out the answer considers a trapezoidal potential, includes the solution, analytical and numerical, of the corresponding eigenvalue problem and then analyzes the behavior of that solution in the limit when the slope of the trapezoidal potential's ramps becomes vertical.The second question, obviously linked to the first one, pertains whether or not eigenfunction's and its first derivative's continuity at the potential's jump points is justified as a priori assumption to kick-off the solution process, as it is standardly accepted in textbook approaches to the potential's eigenvalue problem.We show that, by following the indicated detour, the irrelevance of the potential's vertical segments and the continuity of eigenfunctions and their first derivatives at the potential's jump points turn out to be proven results instead of initial assumptions.

W. Edward Gettys

The s-wave Schrödinger equation is solved for a square well potential with an attractive delta function at the well edge. This potential provides a number of soluble illustrations of the techniques of quantum mechanics. The bound state eigenvalue problem is solved as is the scattering problem for the cross section. The potential is capable of producing a sharp low energy resonance. The resonances are discussed using both R-matrix and S-matrix methods. Numerical examples are presented.

Osama M. Raisuddin, Suvranu De

Edward J. Robinson

Richard R. Chasman

The techniques of the Heisenberg matrix mechanics are extended to treat all potentials of the form Q2n, for n a positive integer. The square well (Q∞) and the potential Q4 are treated in detail.