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AIP conference proceedings

Shannon entropy information for nonrelativistic particles in doubled-ring shaped Coulomb oscillator potential

Miftahul Ma’arif, Suparmi Suparmi, C. Cari

Published Jan 1, 2018• 0 citations
Publisher

Abstract

The identification on probabilities of nonrelativistic particles moved in doubled-ring shaped Coulomb oscillator (DRSCO) potential has been analyzed by the definition on Shannon entropy information to get the quantify of the quantum entanglement and its correlations. We proposed the new modified potential called DRSCO potential that have been solved in recent research paper and the solution of ground state wave function in axial part was used to get its entropy information. The position-space and momentum-space information entropy were investigated by numerical results by using GNU Octave 4.2.2 and have been shown graphicaly.

DOI
10.1063/1.5054519

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The identification on probabilities of nonrelativistic particles moved in doubled-ring shaped Coulomb oscillator (DRSCO) potential has been analyzed by the definition on Shannon entropy information to get the quantify of the quantum entanglement and its correlations. We proposed the new modified potential called DRSCO potential that have been solved in recent research paper and the solution of ground state wave function in axial part was used to get its entropy information. The position-space and momentum-space information entropy were investigated by numerical results by using GNU Octave 4.2.2 and have been shown graphicaly.
There has been an interest in studies on informationtheoretic and complexity measures for quantum model systems in the recent years. The information entropy for the trigonometric Rosen-Morse potential, harmonic oscillator, hydrogenic atoms, Morse potential, Yukawa and Hulthen potentials have been investigated. The Kratzer potential plays an important role in atomic and molecular physics and quantum chemistry. Cheng and Dai proposed a new potential consisting of the modified Kratzer potential plus a new proposed ring-shaped potential which has possible applications to ring-shaped organic molecules like cyclic polyenes and benzene. In this work, we shall obtain the Fisher information, Fisher length, Shannon entropy, Shannon length, Renyi entropy, Renyi length, Tsallis entropy and the Fisher-Shannon complexity for the ring-shaped modified Kratzer potential. In this section, we present the definition of the information theoretical measures used in this work which are based on the 1-normalized electron density in the position and momentum space. The Shannon information entropy in the position space is given by Eq. (1). ( ) ( ) log ( )d S ρ ρ ρ = −∫ r r r ... (1)
We investigate the quantum-information properties of a particle confined by the double Morse potential in position and momentum spaces. The quasi-exact solvability of the model gives analytical expressions for the first two bound states, allowing the corresponding probability densities to be analyzed directly. Shannon entropy, Onicescu energy, Fisher information, statistical complexity, and Fisher-Shannon products are evaluated as functions of the parameter AAA, which controls the transition from a well-separated double well to a merged single-well profile. The position distribution is more delocalized and structurally complex when the double-well character is pronounced, whereas the momentum distribution exhibits the complementary trend. As the wells merge, the ground-state Fisher--Shannon product approaches its Gaussian reference value, whereas the excited state retains stronger non-Gaussian structure.
The study of quantum information entropies is significant in quantum computation and signal analysis. Much research work has been carried out in quantum information entropies under nonrelativistic wave equation. However, not much research has been reported in evaluating quantum information entropies using relativistic wave equation. In this work, we evaluate Shannon information entropy in one-dimensional space with generalised Hulthen plus Coulomb potential (GHPCP) in Klein-Gordon equation using parametric Nikiforov-Uvarov method. The numerical computations obtained both for position and momentum spaces as well as the Shannon sum obeys Bialynicki-Birula and Mycieslski (BBM) uncertainty inequality in order to ascertain the degree of accuracy of the analytical and numerical calculation. This research has relevance in atomic and molecular physics especially in localisation and delocalisation of electrons or particle in atomic orbitals.
Quantum information entropies have been properly studied within the confines of nonrelativistic quantum mechanics ( QM ) by solving Schrödinger wave equation for different considerable potential models to obtained the energy and total normalised wave function which are essential tools for determining various information entropies. However, not much research has been done on evaluation of information entropies within the relativistic quantum mechanics because of its complexities and existence of both multiple and quadratic eigenvalues for fixed quantum state ( n ) and orbital angular quantum number (ℓ)(\ell )(ℓ) . In this work, we investigate Fisher information and Shannon entropy in three dimensional space within the relativistic regime by solving Klein-Gordon ( KG ) equation for equal scalar and vector potential with Schioberg plus Yukawa potential ( SPYP ) using the Parametric Nikiforov-Uvarov ( PNU ) method and compare the results to its nonrelativistic counterpart. It is important to note that the numerical solutions for the three dimensional Fisher information and Shannon entropy obtained for both the relativistic and nonrelativistic wave equations using SPYP adequately satisfies Bialynicki-Birula and Mycieslski (BBM) uncertainty inequality. The radial wave function, radial probability density, momentum space wave function and momentum probability density graphical plots were all obtained for both cases respectively. The numerical computations for Fisher information and Shannon entropy in three dimensional space were carried out for ℓ=0\ell = 0ℓ=0 with different quantum states n=0n= 0n=0 and 1. The investigation of Fisher information and Shannon entropy are relevance in describing localisation and delocalisation mechanism of particles in atomic orbitals within a potential well, relativistic confinement effect and entropy-energy correlation.
The so-called Darboux III oscillator is an exactly solvable N-dimensional nonlinear oscillator defined on a radially symmetric space with non-constant negative curvature. This oscillator can be interpreted as a smooth (super)integrable deformation of the usual N-dimensional harmonic oscillator in terms of a non-negative parameter λ which is directly related to the curvature of the underlying space. In this paper, a detailed study of the Shannon information entropy for the quantum version of the Darboux III oscillator is presented, and the interplay between entropy and curvature is analysed. In particular, analytical results for the Shannon entropy in the position space can be found in the N-dimensional case, and the known results for the quantum states of the N-dimensional harmonic oscillator are recovered in the limit of vanishing curvature λ→0. However, the Fourier transform of the Darboux III wave functions cannot be computed in exact form, thus preventing the analytical study of the information entropy in momentum space. Nevertheless, we have computed the latter numerically both in the one and three-dimensional cases and we have found that by increasing the absolute value of the negative curvature (through a larger λ parameter) the information entropy in position space increases, while in momentum space it becomes smaller. This result is indeed consistent with the spreading properties of the wave functions of this quantum nonlinear oscillator, which are explicitly shown. The sum of the entropies in position and momentum spaces has been also analysed in terms of the curvature: for all excited states such total entropy decreases with λ, but for the ground state the total entropy is minimized when λ vanishes, and the corresponding uncertainty relation is always fulfilled.