Configuration Complexities of Hydrogenic Atoms

The Fisher-Shannon and Cramer-Rao information measures, and the LMC-like or shape complexity (i.e., the disequilibrium times the Shannon entropic power) of hydrogenic stationary states are investigated in both position and momentum spaces. First, it is shown that not only the Fisher information and the variance (then, the Cramer-Rao measure) but also the disequilibrium associated to the quantum-mechanical probability density can be explicitly expressed in terms of the three quantum numbers (n, l, m) of the corresponding state. Second, the three composite measures mentioned above are analytically, numerically and physically discussed for both ground and excited states. It is observed, in particular, that these configuration complexities do not depend on the nuclear charge Z. Moreover, the Fisher-Shannon measure is shown to quadratically depend on the principal quantum number n. Finally, sharp upper bounds to the Fisher-Shannon measure and the shape complexity of a general hydrogenic orbital are given in terms of the quantum numbers.Comment: 22 pages, 7 figures, accepted i

Inward matrix products, generalised density functions and Rayleigh-Schrödinger perturbation theory

A matrix product, the inward product, of two matrices is defined as an operation of internal composition involving two (m×n)-dimensional matrices and yielding another matrix of the same dimension. Such a product, known as Hadamard or Schur product in literature, presents typical properties and corresponds to a usual matrix product, within the isomorphic set of (mn)-dimensional diagonal matrices. It can be directly used to construct generalised density functions. A useful application to Rayleigh-Schrödinger perturbation theory is also discussed