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Physica Scripta

Quantum information entropy for a hyperbolical potential function

R Valencia-Torres, Guo-Hua Sun, Shi‐Hai Dong

Published Feb 5, 2015• 76 citations
Publisher

Abstract

The exact solutions to the Schrödinger equation with a hyperbolic potential are obtained. The position S x and momentum S p Shannon information entropies for the low-lying states are calculated. Some interesting features of the information entropy densities and as well as the probability densities and are demonstrated. We find that the choices of the values for those parameters have to satisfy the condition on . We also notice that the and are symmetric to the momentum p and the or is equal or greater than 1 at some positions r or momentum p . In addition, the Bialynicki-Birula–Mycielski inequality is tested from different cases and found to hold for these cases.

DOI
10.1088/0031-8949/90/3/035205

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The exact solutions to the Schrödinger equation with a hyperbolic potential are obtained. The position S x and momentum S p Shannon information entropies for the low-lying states are calculated. Some interesting features of the information entropy densities and as well as the probability densities and are demonstrated. We find that the choices of the values for those parameters have to satisfy the condition on . We also notice that the and are symmetric to the momentum p and the or is equal or greater than 1 at some positions r or momentum p . In addition, the Bialynicki-Birula–Mycielski inequality is tested from different cases and found to hold for these cases.
The Shannon information entropy for the Schrödinger equation with a nonuniform solitonic mass is evaluated for a hyperbolic-type potential. The number of nodes of the wave functions in the transformed space z are broken when recovered to original space x . The position S x and momentum S p information entropies for six low-lying states are calculated. We notice that the S x decreases with the increasing mass barrier width a and becomes negative beyond a particular width a , while the S p first increases with a and then decreases with it. The negative S x exists for the probability densities that are highly localized. We find that the probability density ρ ( x ) for n = 1, 3, 5 are greater than 1 at position x = 0. Some interesting features of the information entropy densities ρ s ( x ) and ρ s ( p ) are demonstrated. The Bialynicki–Birula–Mycielski (BBM) inequality is also tested for these states and found to hold.
Abstract The Shannon entropy (S) and the Fisher Information (I) entropies are investigated for a generalized hyperbolic potential in position and momentum spaces. First, the Schrodinger equation is solved exactly using the Nikiforov‐Uvarov‐Functional Analysis method to obtain the energy spectra and the corresponding wave function. By Fourier transforming the position space wave function, the corresponding momentum wave function was obtained for the low‐lying states corresponding to the ground and first excited states. The positions and momentum of Shannon entropy and Fisher Information entropies were calculated numerically. Finally, the Bialynicki‐Birula and Mycielski and the Stam‐Cramer‐Rao inequalities for the Shannon entropy and Fisher Information entropies, respectively, were tested and were found to be satisfied for all cases considered.
The Shannon information entropy for the Schrodinger equation with a nonuniform solitonic mass is evaluated for a hyperbolic-type potential. The number of nodes of the wave functions in the transformed space z are broken when recovered to original space x. The position Sx and momentum S p information entropies for six low-lying states are calculated. We notice that the Sx decreases with the increasing mass barrier width a and becomes negative beyond a particular width a,while the Sp first increases with a and then decreases with it. The negative Sx exists for the probability densities that are highly localized. We find that the probability density ρ(x) for n = 1, 3, 5 are greater than 1 at position x = 0. Some interesting features of the information entropy densities ρs(x) and ρs(p) are demonstrated. The Bialynicki–Birula–Mycielski(BBM)inequality is also tested for these states and found to hold.
We use the ansatz method to obtain the symmetric and antisymmetric solutions of a hyperbolic double‐well potential by solving the Heun differential equation. The Shannon entropy is studied. The positionSxand momentumSpinformation entropies for the low‐lying two statesN = 1, 2 are calculated. Some interesting features of the information entropy densitiesρs(x) andρs(p) as well as the probability densityρ(x) are demonstrated. We find thatρ(x) is equal or greater than 1 at positions for the allowed potential‐depth values ofU0 = 595.84 (symmetric case) andU0 = 1092.8 (antisymmetric case). This arises from the fact that most of the density is less than 1, the curve has to rise higher than 1 to have a total area of 1 as required for all probability distributions. We find that the position information entropySxdecreases with the potential strength but the momentum entropySpis contrary to theSx. The Bialynicki‐Birula–Mycielski inequality is also tested and found to hold for these cases. © 2015 Wiley Periodicals, Inc.
In this work we have studied the Shannon information entropy for two hyperbolic single-well potentials in the fractional Schrödinger equation (the fractional derivative number (0<n≤2) by calculating position and momentum entropy. We find that the wave function will move towards the origin as the fractional derivative number n decreases and the position entropy density becomes more severely localized in more fractional system, i.e., for smaller values of n, but the momentum probability density becomes more delocalized. And then we study the Beckner Bialynicki-Birula−Mycieslki (BBM) inequality and notice that the Shannon entropies still satisfy this inequality for different depth u even though this inequality decreases (or increases) gradually as the depth u of the hyperbolic potential U1 (or U2) increases. Finally, we also carry out the Fisher entropy and observe that the Fisher entropy increases as the depth u of the potential wells increases, while the fractional derivative number n decreases.
Shannon entropy for the position and momentum eigenstates of the symmetrically trigonometric Rosen–Morse potential for the lower states n = 1–4 is evaluated. The position information entropies S x for n = 1,2 are presented analytically. Some interesting features of the information entropy densities ρ s ( x ) and ρ s ( p ) are demonstrated graphically. We find that the ρ s ( p ) is inversely proportional to the range of potential a and the S x decreases with increasing the potential depth D . In particular, we note that the S x might become negative for some given parameters a and D . The Bialynicki-Birula–Mycielski inequality is also tested for a number of states and is found to generally hold well.