Crámer‐Rao complexity of the confined two‐dimensional hydrogen

The internal disorder of the confined two‐dimensional hydrogenic atom is numerically studied in terms of the confinement radius for the 1s, 2s, 2p, and 3d quantum states by means of the statistical Crámer‐Rao complexity measure. First, the confinement dependence of the variance and the Fisher information of the position and momentum spreading of its electron distribution are computed and discussed. Then, the Crámer‐Rao complexity measure (which quantifies the combined balance of the charge concentration around the mean value and the gradient content of the electron distribution) is investigated in position and momentum spaces. We found that confinement does distinguish complexity of the system for all quantum states by means of these two component measures. The electronic spreading of the confined two‐dimensional hydrogenic atom, which is the basic prototype of the general multidimensional confined quantum systems, is numerically studied for the 1s, 2s, 2p, and 3d stationary states by means of the Crámer‐Rao complexity measure in both position and momentum spaces. It is illustrated how and how much confinement is crucial not only for the energy spectrum of the system but also for its eigenfunction‐dependent electronic complexity.