Entropy and complexity analysis of Dirac-delta-like quantum potentials

The Dirac-delta-like quantum-mechanical potentials are frequently used to describe and interpret numerous phenomena in many scientific fields including atomic and molecular physics, condensed matter and quantum computation. The entropy and complexity properties of potentials with one and two Dirac-delta functions are here analytically calculated and numerically discussed in both position and momentum spaces. We have studied the information-theoretic lengths of Fisher, Rényi and Shannon types as well as the Cramér–Rao, Fisher–Shannon and LMC shape complexities of the lowest-lying stationary states of one-delta and twin-delta. They allow us to grasp and quantify different facets of the spreading of the charge and momentum of the system far beyond the celebrated standard deviation.