Entropic measures of Rydberg‐like harmonic states

In this work, we study the position and momentum information entropies of multiple quantum well systems in fractional Schrödinger equations, which, to the best of our knowledge, have not so far been studied. Through a confining potential, their shape and number of wells (NOW) can be controlled by using a few tuning parameters; we present some interesting quantum effects that only appear in the fractional Schrödinger equation systems. One of the parameters denoted by the Ld can affect the position and momentum probability densities if the system is fractional (1 < α < 2). We find that the position (momentum) probability density tends to be more severely localized (delocalized) in more fractional systems (ie, in smaller values of α). Affecting the Ld on the position and momentum probability densities is a quantum effect that only appears in the fractional Schrödinger equations. Finally, we show that the Beckner Bialynicki‐Birula‐Mycieslki (BBM) inequality in the fractional Schrödinger equation is still satisfied by changing the confining potential amplitude Vconf, the NOW, the fractional parameter α, and the confining potential parameter Ld.

Section: Introductionmentioning

confidence: 99%

Quantum information entropies of multiple quantum well systems in fractional Schrödinger equations

Solaimani

Dong

2019

“…Returning to asymptotics (36), we notice that for the expressions (18) and (19) of the harmonic systems, we have…”

Section: Shannon-like Integrals Of Laguerre Polynomials With Large mentioning

confidence: 99%

“…These results extend and complement various efforts about the information entropies of harmonic systems. [11,28,29,32,[36][37][38]63,69,70,73,76,77,[84][85][86][87][88][89]…”

Section: Shannon Entropy Of High-dimensional Harmonic Systemsmentioning

confidence: 99%

“…[25][26][27] The entropic properties of electronic systems (which crucially depend on the states' eigenfunctions) quantify the various facets of the spatial extension or multidimensional spreading of the electronic charge. Nowadays, there is an increasing interest on the dimensional dependence of the entropic properties for the stationary states of the multidimensional quantum systems [12,13,[28][29][30][31][32][33][34][35][36][37][38] in order to contribute to the emergent informational representation of the quantum systems which extends and complements the standard energetic representation. This is basically because of the increasing relevance of entropic properties in numerous biological, chemical, and physical phenomena [39][40][41][42][43][44] as well as in electronic correlations [45][46][47][48] and chemical reactions.…”

mentioning

confidence: 99%

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The Shannon entropy of high‐dimensional hydrogenic and harmonic systems

Dehesa

Belega

Toranzo

et al. 2019

Self Cite

There exists an increasing interest on the dimensionality dependence of the entropic properties for the stationary states of the multidimensional quantum systems in order to contribute to its emergent informational representation, which extends and complements the standard energetic representation. Nowadays, this is specially so for high-dimensional systems as they have been recently shown to be very useful in both scientific and technological fields. In this work, the Shannon entropy of the discrete stationary states of the high-dimensional harmonic (ie, oscillator-like) and hydrogenic systems is analytically determined in terms of the dimensionality, the potential strength, and the state's hyperquantum numbers. We have used an informationtheoretic methodology based on the asymptotics of some entropy-like integral functionals of the orthogonal polynomials and hyperspherical harmonics which control the wave functions of the quantum states, when the polynomial parameter is very large; this is basically because such a parameter is a linear function of the system's dimensionality. Finally, it is shown that the Shannon entropy of the D-dimensional harmonic and hydrogenic systems has a logarithmic growth rate of the type D log D when D ! ∞. K E Y W O R D S high-dimensional harmonic systems, high-dimensional hydrogenic systems, high-dimensional quantum systems, logarithmic integrals of orthogonal polynomials-asymptotics, Shannon entropy of high-dimensional quantum systems 1 | INTRODUCTION Multidimensional quantum systems consisting of many particles are a major challenge in Quantum Chemistry and Physics, as their behavior can be determined only with immense computational power. The physical solutions of their associated wave equations, and consequently all the chemical and physical properties of these systems, crucially depend on the space dimensionality D. Scientists are trying to discover elegant notions and techniques to simplify the problem (see, eg, ref. [1]). The 1986 Nobel Prize in Chemistry winner Dudley Robert Herschbach et al have developed [2][3][4][5] a very useful strategy to study the atomic and molecular systems, the D-dimensional scaling method, where the D is the basic variable. The pseudoclassical or highdimensional (D ! ∞) limit is the starting point of this strategy. Indeed, this method requires to solve a finite many-electron problem in the (D ! ∞)-limit, and then perturbation theory in 1D is used to have an approximate result for the standard dimension (D = 3), obtaining at times a quantitative accuracy comparable to the self-consistent Hartree-Fock calculations. Most important here is that the electronic structure for the (D ! ∞)-limit is beguilingly simple and exactly computable for any atom and molecule. For finite D but very large, the electrons are confined to harmonic oscillations about the fixed positions attained in the (D ! ∞)-limit. Indeed, at this high-dimensional limit, the electrons of a many-electron system assume fixed positions relative to

“…[45] In 2016, S and R were computed for highly excited quantum states within Laguerre asymptotic approximation [46] and some linearized method. [47] However, an elaborate information theoretic study for 3D CHO is still lacking as of now. Hence, in this communication our primary objective is to perform a systematic analysis of I in a CHO, for any arbitrary state characterized by quantum numbers n r , l, m, in both r and p spaces.…”

mentioning

confidence: 99%

Fisher information in confined isotropic harmonic oscillator

Mukherjee

Roy

2018

Fisher information (I) is investigated in a confined harmonic oscillator (CHO) enclosed in a spherical enclosure, in conjugate r and p spaces. A comparative study between CHO and a free quantum particle in spherical box (PISB), as well as CHO and respective free harmonic oscillator (FHO) is pursued with respect to energy spectrum and I. This reveals that, a CHO offers two exactly solvable limits, namely, a PISB and an FHO. Moreover, the dependence of I on quantum numbers n r, l, m in FHO and CHO are analogous. The role of force constant is discussed. Further, a thorough systematic analysis of I with respect to variation of confinement radius r c is presented, with particular attention on nonzero‐(l, m) states. Considerable new important observations are recorded. The results are quite accurate and most of these are presented for the first time.