Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems

Sobrino, Nahual; Puertas‐Centeno, David; Toranzo, I. V.; Dehesa, J. S.

doi:10.1088/1742-5468/aa7df4

Cited by 31 publications

(32 citation statements)

References 89 publications

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“…Extensive theoretical works have been published covering a broad variety of confining potentials. Two prototypical systems receiving maximum attention are harmonic oscillator (in 1D, 2D, 3D, D dimension) and confined hydrogen atom (CHA) inside a spherical enclosure . A CHA within an impenetrable (as well as penetrable) cavity was studied quite vigorously leading to a host of attractive properties—both from physical and mathematical point of view.…”

Section: Introductionmentioning

confidence: 99%

“…Besides, they have uses in cell-model of liquid, high-pressure physics, astrophysics, [8] study of impurities in semiconductor materials, matrix isolated molecules, endohedral complexes of fullerenes, zeolites cages, helium droplets, nanobubbles, [2] and so forth.Extensive theoretical works have been published covering a broad variety of confining potentials. Two prototypical systems receiving maximum attention are harmonic oscillator (in 1D, 2D, 3D, D dimension) [9][10][11][12][13][14] and confined hydrogen atom (CHA) inside a spherical enclosure. [3,10,[15][16][17][18][19][20][21][22][23][24] A CHA within an impenetrable (as well as penetrable) cavity was studied quite vigorously leading to a host of attractive properties-both from physical and mathematical point of view.…”

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confidence: 99%

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Information‐entropic measures in free and confined hydrogen atom

Mukherjee

Roy

2018

Int J of Quantum Chemistry

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Shannon entropy (S), Rényi entropy (R), Tsallis entropy (T), Fisher information (I), and Onicescu energy (E) have been explored extensively in both free H atom (FHA) and confined H atom (CHA). For a given quantum state, accurate results are presented by employing respective exact analytical wave functions in r space. The p‐space wave functions are generated from respective Fourier transforms—for FHA these can be expressed analytically in terms of Gegenbauer polynomials, whereas in CHA these are computed numerically. Exact mathematical expressions of Rrnormalα,Rpnormalβ, Trnormalα,Tpnormalβ,Er,Ep are derived for circular states of a FHA. Pilot calculations are done taking order of entropic moments (α, β) as (35,3) in r and p spaces. A detailed, systematic analysis is performed for both FHA and CHA with respect to state indices n, l, and with confinement radius (rc) for the latter. In a CHA, at small rc, kinetic energy increases, whereas Sboldr,Rboldrnormalα decrease with growth of n, signifying greater localization in high‐lying states. At moderate rc, there exists an interplay between two mutually opposing factors: (i) radial confinement (localization) and (ii) accumulation of radial nodes with growth of n (delocalization). Most of these results are reported here for the first time, revealing many new interesting features. Comparison with literature results, wherever possible, offers excellent agreement.

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Information‐entropic measures in free and confined hydrogen atom

Mukherjee

Roy

2018

Int J of Quantum Chemistry

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“…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.…”

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confidence: 99%

The Shannon entropy of high‐dimensional hydrogenic and harmonic systems

Dehesa

Belega

Toranzo

et al. 2019

Int J of Quantum Chemistry

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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

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“…Extensive theoretical calculations have been made in case of confined harmonic oscillator (CHO) (1D, 2D, 3D, d dimension) [8,[22][23][24][25][26] and confined hydrogen atom (CHA) inside an impenetrable cavity [3,23,[27][28][29][30][31][32][33][34][35][36]. They offer many extraordinary features, especially relating to simultaneous, incidental, inter-dimensional degeneracy [25] in their energy spectra.…”

Section: Introductionmentioning

confidence: 99%

Quantum mechanical virial-like theorem for confined quantum systems

Mukherjee

Roy

2019

Phys. Rev. A

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Confinement of atoms inside impenetrable (hard) and penetrable (soft) cavity has been studied for nearly eight decades. However, a uniform virial theorem for such systems has not yet been found. Here we provide a general virial-like equation in terms of mean square and expectation values of potential and kinetic energy operators. It appears to be applicable in both free and confined situations. Apart from that, a pair of equations has been derived using time independent Schrödinger equation, that can be treated as a sufficient condition for a given stationary quantum state. Change of boundary condition does not affect these virial equations. In hard confining condition, the perturbing (confining potential) does not affect the expression; it merely shifts the boundary from infinity to a finite region. In soft case, on the contrary, the final expression includes contributions from perturbing term. These are demonstrated numerically for several representative enclosed systems like harmonic oscillator (1D, 3D), hydrogen atom. The applicability in manyelectron systems has been discussed. In essence, a virial equation has been derived for free and confined quantum systems, from simple arguments.

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Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems

Cited by 31 publications

References 89 publications

Information‐entropic measures in free and confined hydrogen atom

Information‐entropic measures in free and confined hydrogen atom

The Shannon entropy of high‐dimensional hydrogenic and harmonic systems

Quantum mechanical virial-like theorem for confined quantum systems

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