Bialynicki-Birula Mycielski Research Articles

Information-theoretic measures and Compton profile of H atom under finite oscillator potential

Information-theoretic measures for nl (2L) states of a H atom (with n=1−10 and l=0−2 , where n and l denote principal and angular momentum quantum numbers) have been investigated within a quantum dot by utilizing the Ritz variational principle, with the help of a Slater-type basis set. A well-established two-parameter (depth and width) model of finite oscillator potential is used to simulate the dot environment. The variationally optimized position (r)-space wave function is utilized to determine the momentum (p)-space wave function, leading to the generation of p-space radial density distribution. We explore the impact of cavity parameters on quantum information theoretic measures, such as Shannon (S) and Fisher information (I) entropy, in the ground as well as the excited state. The results of S were also used to test the Bialynicki–Birula–Mycielski inequality, related to the entropic uncertainty principle for the confined H atom. Some simple new fitting laws pertaining to S and I have been proposed. Furthermore, the p-space radial density is employed to derive the Compton profile of the confined H atom. Possible tunability of S,I and Compton profiles with respect to the parameters is noted.

Effects of magnetic and Aharonov–Bohm flux fields on information entropy with modified Kratzer plus screened Coulomb potential for selected diatomic molecules

The Shannon information entropy for scandium fluoride (ScF) and hydrogen ( H 2 ) molecules in the presence of magnetic and Aharanov–Bohm (AB) fields with modified Kratzer plus screened Coulomb potential in the position and momentum coordinate is investigated. The wavefunction and probability density are obtained by solving the two-dimensional (2D) Schrödinger wave equation using the Nikiforov–Uvarov functional analysis method and then applied to calculate the Shannon entropy. The graphical behaviour of the 2D wavefunction and probability density is showcased. Our results depict the effects of the external fields on Shannon entropy for ScF and ( H 2 ) , the negative values of the position Shannon entropies obtained imply higher localisation of the particles and molecules indicating more stability of the system. Another interesting part of this work is that Bialynicki–Birula–Mycielski inequality is tested and proven bounded for both diatomic molecules. A few to mention, this research will enhance the apprehension of the dynamics of quantum molecules and systems subjected to external fields, quantum information signal processing and communication.

Quantum Information Entropy for a Hyperbolic Double Well Potential in the Fractional Schrödinger Equation.

In this study, we investigate the position and momentum Shannon entropy, denoted as Sx and Sp, respectively, in the context of the fractional Schrödinger equation (FSE) for a hyperbolic double well potential (HDWP). We explore various values of the fractional derivative represented by k in our analysis. Our findings reveal intriguing behavior concerning the localization properties of the position entropy density, ρs(x), and the momentum entropy density, ρs(p), for low-lying states. Specifically, as the fractional derivative k decreases, ρs(x) becomes more localized, whereas ρs(p) becomes more delocalized. Moreover, we observe that as the derivative k decreases, the position entropy Sx decreases, while the momentum entropy Sp increases. In particular, the sum of these entropies consistently increases with decreasing fractional derivative k. It is noteworthy that, despite the increase in position Shannon entropy Sx and the decrease in momentum Shannon entropy Sp with an increase in the depth u of the HDWP, the Beckner-Bialynicki-Birula-Mycielski (BBM) inequality relation remains satisfied. Furthermore, we examine the Fisher entropy and its dependence on the depth u of the HDWP and the fractional derivative k. Our results indicate that the Fisher entropy increases as the depth u of the HDWP is increased and the fractional derivative k is decreased.

Thermal properties and quantum information theory with the shifted Morse potential

By employing the Nikiforov–Uvarov functional analysis (NUFA) method, we solved the radial Schrödinger equation with the shifted Morse potential model. The analytical expressions of the energy eigenvalues, eigenfunctions and numerical results were determined for selected values of the potential parameters. Variations of different thermodynamic functions with temperature were discussed extensively. Different quantum information-theoretic measures, including Shannon entropy, Fisher information and Fisher–Shannon product of the shifted Morse potential, were investigated numerically and graphically in position and momentum spaces for the ground and the first excited states. The quantum information theories considered satisfied their corresponding inequalities, including Bialynicki–Birula–Mycielski, Stam–Cramer–Rao inequalities and the Fisher–Shannon product relation.

Quantum information entropy and squeezing of 𝒫𝒯-symmetric potential

The information theoretic concepts are crucial to study the quantum mechanical systems. In this paper, the information densities of [Formula: see text]-symmetric potential have been demonstrated and their properties deeply analyzed. The position space and momentum space information entropy is obtained and Bialynicki-Birula–Mycielski inequality is saturated for different parameters of the potential. Some interesting features of information entropy have been discussed. The variation in these entropies is described which gets saturated for specific values of the parameter. These have also been analyzed for the [Formula: see text]-symmetry breaking case. Further, the entropy squeezing phenomenon has been investigated in position space as well as momentum space. Interestingly, [Formula: see text] phase transition conjectures the entropy squeezing in position space and momentum space.

Quantum information-entropic measures for exponential-type potential

The approximate solutions of the Schrödinger equation with the central generalized Hulthén and Yukawa potential are investigated within the framework of the functional method. The obtained wave function and the energy levels are used to study the Shannon entropy, Renyi entropy, Fisher information, Shannon Fisher complexity, Shannon power and Fisher-Shannon product in both position and momentum spaces for the ground and first excited states. The theoretic information theories such as Shannon entropies Sr,Sp, Fisher information Ip,Ip and Renyi entropies Rβ as well as the information entropy probability densities ρ(r),ϕ(p) are illustrated graphically. The Bialynicki-Birula – Mycielski (BBM) inequality for the Shannon entropies and the Cramer-Rao inequality for the Fisher information are satisfied for the potential model.

Shannon entropy and Fisher information-theoretic measures for Mobius square potential

We solve the Schrodinger equation for the Mobius square potential using the newly proposed Nikiforov–Uvarov functional analysis method. The approximate energy spectrum and unnormalized wave function are obtained in a closed form. The Shannon entropy, Fisher information, Fisher–Shannon product and the expectation values for the Mobius square are investigated in position and momentum space for the low lying states corresponding to $$ n = 0 $$ and 1. All the theoretic information theories investigated satisfied their corresponding inequality such as Bialynicki–Birula–Mycielski, Stam–Cramer–Rao inequalities and the Fisher–Shannon product relation $$ P = P_{r} P_{p} > 1 $$ .

Effect of confining potential on information entropy measures in hydrogen atom: extensive and non-extensive entropy

A spherically confined hydrogen atom has been considered using two different confined potentials such as modified Kratzer and non-spherical oscillator potentials. First, the Schrodinger equation in $$ r $$ -space has been solved and the exact analytical wave functions and energy levels have been obtained. Then, the Renyi entropy and Shannon entropy in $$ r $$ -space have been calculated. In addition, we have numerically calculated the wave functions in $$ p $$ -space by performing Fourier transform. The calculations have been done for different quantum numbers $$ n $$ and $$ l $$ and also quantum size $$ r_{0} $$ . The Bialynicki–Birula–Mycielski inequality is also tested. It is found that this inequality depends on non-extensive parameter $$ q $$ .

Information theoretic measures of uncertainty of a noncommutative anisotropic oscillator in a homogeneous magnetic field

A noncommutative anisotropic oscillator in the presence of a homogeneous magnetic field is studied and the Schrödinger equation is solved analytically. We evaluated various information theoretic measures of uncertainty e.g., Heisenberg uncertainty relation, Shannon area and Fisher area for several states, including the ground state. It has been shown that the ground state is coherent for different magnetic fields. The optimum values of the Shannon area have found in position and momentum spaces. We observe that the Fisher area in momentum space has a maximum value without a magnetic field. Bialynicki–Birula–Mycielski (BBM) and Cramer–Rao uncertainty relations have been verified.

Quantum information measures of infinite spherical well

In this work, we study the Shannon information entropies [Formula: see text] and [Formula: see text] of an infinite spherical well. The Shannon entropy [Formula: see text] is calculated numerically in terms of the analytical result of the wave function in momentum space. Some typical features of the position and momentum probability densities [Formula: see text] and [Formula: see text] as well as the information entropy densities [Formula: see text] and [Formula: see text] are demonstrated. We find that the position entropy [Formula: see text] increases with the radius a of the spherical well for given quantum numbers l, m and n. It is interesting to note that the position entropy [Formula: see text] decreases with the quantum numbers l and n for a fixed radius a and quantum number m. The position entropy [Formula: see text] is almost independent of the quantum numbers l, m and n. The momentum entropy [Formula: see text] first increases and then decreases with respect to the radius a. We also note that the [Formula: see text] increases with the radius a and finally arrives at a constant. In addition, the Bialynicki–Birula–Mycielski (BBM) inequality is verified and also hold for this confined system.

Shannon entropy and Fisher information of the one-dimensional Klein–Gordon oscillator with energy-dependent potential

In this paper, we studied, at first, the influence of the energy-dependent potentials on the one-dimensionless Klein–Gordon oscillator. Then, the Shannon entropy and Fisher information of this system are investigated. The position and momentum information entropies for the low-lying states n = 0, 1, 2 are calculated. Some interesting features of both Fisher and Shannon densities, as well as the probability densities, are demonstrated. Finally, the Stam, Cramer–Rao and Bialynicki–Birula–Mycielski (BBM) inequalities have been checked, and their comparison with the regarding results have been reported. We showed that the BBM inequality is still valid in the form [Formula: see text], as well as in ordinary quantum mechanics.

Theoretical information measurement in nonrelativistic time-dependent approach

The information-theoretic measures of time-dependent Schrodinger equation are investigated via the Shannon information entropy, variance and local Fisher quantities. In our calculations, we consider the two first states $$n = 0,1$$ and obtain the position $$S_{x} (t)$$ and momentum $$S_{p} (t)$$ Shannon entropies as well as Fisher information $$I_{x} (t)$$ in position and momentum $$I_{p} (t)$$ spaces. Using the Fourier transformed wave function, we obtain the results in momentum space. Some interesting features of the information entropy densities $$\rho_{s} (x,t)$$ and $$\gamma_{s} (p,t)$$ , as well as the probability densities $$\rho (x,t)$$ and $$\gamma (p,t)$$ for time-dependent states are demonstrated. We establish a general relation between variance and Fisher’s information. The Bialynicki-Birula–Mycielski inequality is tested and verified for the states $$n = 0,1$$ .

Information Theoretic Global Measures of Dirac Equation With Morse and Trigonometric Rosen–Morse Potentials

In this study, the information-theoretic measures of (1+1)-dimensional Dirac equation in both position and momentum spaces are investigated for the trigonometric Rosen–Morse and the Morse potentials. The solutions of the corresponding Dirac equation are obtained in an exact analytical manner in the first step. Next, using the Fourier transformation, the position and momentum Shannon information entropies are obtained and some features of the probability densities are analyzed. The consistency with Bialynicki-Birula–Mycielski inequality and Heisenberg uncertainty is checked.

Entropy and Information of a harmonic oscillator in a time-varying electric field in 2D and 3D noncommutative spaces

We analyze the noncommutativity effects on the Fisher information (Frˆ,pˆ) and Shannon entropies (Srˆ,pˆ) of a harmonic oscillator immersed in a time-varying electric field in two and three dimensions. We find the exact solutions of the respective time-dependent Schrödinger equation and use them to calculate the Fisher information and the Shannon entropy for the simplest case corresponding to the lowest-lying state of each system. While there is no problem in defining the Shannon entropy for noncommutating spaces, the definition of the Fisher information have to be modified to satisfy the Cramer–Rao inequalities. For both systems we observe how the Fisher information and Shannon entropy in position and momentum change due to the noncommutativity of the space. We verify that the Bialynicki-Birula–Mycielski (BBM) entropic uncertainty relation still holds in the systems considered.

Nonrelativistic Shannon information entropy for Killingbeck potential

The Shannon entropy of a nonrelativistic quantum system is investigated with the Killingbeck potential, which includes Coulomb, linear, and harmonic terms. The position and momentum information entropies for the low-lying states n = 0, 1 are calculated. Some interesting features of the information entropy densities as well as the probability densities are demonstrated. The Bialynicki-Birula–Mycielski inequality is checked regarding the results.

Nonrelativistic Shannon information entropy for Kratzer potential

The Shannon information entropy is investigated within the nonrelativistic framework. The Kratzer potential is considered as the interaction and the problem is solved in a quasi-exact analytical manner to discuss the ground and first excited states. Some interesting features of the information entropy densities as well as the probability densities are demonstrated. The Bialynicki–Birula–Mycielski inequality is also tested and found to hold for these cases.

Shannon information entropies for position-dependent mass Schrödinger problem with a hyperbolic well

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 Sx and momentum Sp 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.

Quantum information entropy for a hyperbolical potential function

The exact solutions to the Schrödinger equation with a hyperbolic potential are obtained. The position Sx and momentum Sp 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.

Quantum information entropies for position-dependent mass Schrödinger problem

The Shannon entropy for the position-dependent Schrödinger equation for a particle with a nonuniform solitonic mass density is evaluated in the case of a trivial null potential. The position Sx and momentum Sp information entropies for the three lowest-lying states are calculated. In particular, for these states, we are able to derive analytical solutions for the Sx entropy as well as for the Fourier transformed wave functions, while the Sp quantity is calculated numerically. We notice the behavior of the Sx entropy, namely, it decreases as the mass barrier width narrows and becomes negative beyond a particular width. The negative Shannon entropy exists for the probability densities that are highly localized. The mass barrier determines the stability of the system. The dependence of Sp on the width is contrary to the one for Sx. Some interesting features of the information entropy densities ρs(x) and ρs(p) are demonstrated. In addition, the Bialynicki-Birula–Mycielski (BBM) inequality is tested for a number of states and found to hold for all the cases.

Quantum information entropies for a squared tangent potential well

The particle in a symmetrical squared tangent potential well is studied by examining its Shannon information entropy and standard deviations. The position and momentum information entropy densities ρs(x), ρs(p) and probability densities ρ(x), ρ(p) are illustrated with different potential range L and potential depth U. We present analytical position information entropies Sx for the lowest two states. We observe that the sum of position and momentum entropies Sx and Sp expressed by Bialynicki-Birula–Mycielski (BBM) inequality is satisfied. Some eigenstates exhibit entropy squeezing in the position. The entropy squeezing in position will be compensated by an increase in momentum entropy. We also note that the Sx increases with the potential range L, while decreases with the potential depth U. The variation of Sp is contrary to that of Sx.