Information entropy and bound quantum states

We explore a few advantages of studying the change in information entropy of a bound quantum state with energy. It is known that the property generally increases with the quantum number for stationary states, in spite of the concomitant gradual increase in the number of constraints for higher levels. A simple semiclassical proof of this observation is presented via the Wilson-Sommerfeld quantization scheme. In the small quantum number regime, we numerically demonstrate how far the semiclassical predictions are valid for a few systems, some of which are exactly solvable and some not so. Our findings appear to be significant in a number of ways. We observe that, for most problems, information entropy tends to a maximum as the quantum number tends to infinity. This sheds some light on the Bohr limit as a classical limit. Noting that the dependence of energy on the quantum number governs the rate of increase of information entropy with the degree of excitation, we extend our analysis to include the role of the kinetic energy. The endeavor yields a relation that possesses a universal character for any one-dimensional problem. Relevance of information entropy in studying the goodness of approximate stationary states obtained from finite-basis linear variational calculations is also delineated. Finally, we expound how this property behaves in situations where shape resonances show up. A typical variation is indeed observed in such cases when we proceed to detect Siegert states via the stabilization method.