Analytical Shannon information entropies for all discrete multidimensional hydrogenic states

Abstract

The entropic uncertainty measures of the multidimensional hydrogenic states quantify the multiple facets of the spatial delocalization of the electronic probability density of the system. The Shannon entropy is the most adequate uncertainty measure to quantify the electronic spreading and to mathematically formalize the Heisenberg uncertainty principle, partially because it does not depend on any specific point of their multidimensional domain of definition. In this work, the radial and angular parts of the Shannon entropies for all the discrete stationary states of the multidimensional hydrogenic systems are obtained from first principles; that is, they are given in terms of the states' principal and magnetic hyperquantum numbers (n, μ 1 , μ 2 , …, μ D−1 ), the system's dimensionality D and the nuclear charge Z in an analytical, compact form.Explicit expressions for the total Shannon entropies are given for the quasi-spherical states, which conform to a relevant class of specific states of the D-dimensional hydrogenic system characterized by the hyperquantum numbers μ 1 = μ 2 … = μ D −1 = n − 1, including the ground state. K E Y W O R D S dimensional dependence of Shannon entropy, entropic uncertainty, Shannon entropy of multidimensional hydrogenic states 1 | INTRODUCTIONThe entropic uncertainty measures have been shown to be much more adequate than the Heisenberg ones (ie, those based on the SD and its moment-type generalizations [1] ) not only to formulate the fundamental uncertainty principle of quantum mechanics [2][3][4][5] and for the development of quantum information and computation, [6][7][8][9][10][11] but also to describe the physical and chemical properties of multidimensional quantum systems. The latter is specially so in quantum chemistry, where D-dimensional notions and techniques play a very relevant role as the chemists Dudley R. Herschbach (Nobel Prize 1986) [12] and Gerhard Ertl (Nobel Prize 2007) and their collaborators have shown. For example, in chemistry, two dimensions are often better than three, since surface-bound reactions can be proved in greater detail than those in a liquid solution, as Ertl et al. illustrated in their pioneering contributions to the field of surface chemistry. [13][14][15] Most efforts have been focused on the entropic information entropies of Rényi and Shannon types and its variations, [16][17][18][19][20] basically because they have underpinned fundamental progress in a great diversity of scientific and technological fields such as applied mathematics, theoretical physics, quaand information technologies, including electronic correlations, complexity theory, statistical mechanics, quantum information theory, telecommunications, biochemical phenomena, and quantum information processing. [6,8,[21][22][23][24][25][26] The analytical determination of these entropic quantities is a serious task, [10] basically because the quantum probability density of these systems is not generally known, except for some extreme cases and/or some special states (high-dimensional and...

Shannon entropy as a predictor of avoided crossing in confined atoms

“…[28,31,42,52] In literature, the entropic uncertainty relation is proven to convey more information related to localization in comparison to the Heisenberg uncertainty relation. [31,36,42,52] Since the entropy is a logarithmic quantity, the entropic sum will be limited by inequality as in the case of uncertainty product. The entropic uncertainty relation in the confined H is given as…”

“…[30] Shannon information entropy has been applied to several complex atomic and molecular systems to investigate the electronic structure. [34][35][36][37][38][39][40][41][42][43][44] The application of information entropies on the atomic system reveals deeper knowledge of the electron correlations and delocalization of probability density. For example, a recent work by Dehesa and González [34] employed Shannon information entropy as an indicator of the avoided crossing by studying the dynamics of some excited states of hydrogen in the presence of parallel magnetic and electric fields.…”

“…[ 35 ] Electron correlation, a major problem in many‐body physics, is very well analyzed using the information entropies. The analytical determination of Shannon information entropies for all discrete stationary states of multidimensional hydrogenic system is reported by Toranzo et al [ 36 ] In a recent paper, Romera and Dehesa [ 37 ] introduced the concept of Fisher‐Shannon information plane as a specific electron correlation measure in two‐electron systems. The Fisher‐Shannon information product and plane for atoms were presented based on the Thomas‐Fermi‐Gáspár statistical model.…”

High Dimensional Atomic States of Hydrogenic Type: Heisenberg-like and Entropic Uncertainty Measures

“…The uncertainty measures of the Heisenberg (radial expectation values, variance) and entropy (Shannon, Rényi) types, which quantify the spreading properties of the electronic probability density, have recently been determined [24][25][26][27] for the D-dimensional hydrogenic system at all D from first principles; that is, in terms of the dimensionality D, the strength of the Coulomb potential (the nuclear charge) and the D hyperquantum numbers (η, µ 1 , µ 2 , . .…”

“…However, the theoretical expressions found for the uncertainty measures of general Ddimensional hydrogenic states [24][25][26][27] provide with algorithmic procedures to find their numerical values, but they are somewhat highbrow and not so handy for analytical manipulations. This is because they require the numerical evaluation at the unity of a generalized univariate hypergeometric functions of p+1 F p (z) type (Heisenberg-like measures), a multivariate Lauricella function of type A of s variables and 2s + 1 parameters F (s) A (x 1 , .…”