Fisher information in confined isotropic harmonic oscillator

This paper revisits the generalized pseudospectral (GPS) method on the calculation of various radial expectation values of atomic systems, especially on the spatially confined hydrogen atom and harmonic oscillator. As one of the collocation methods based on global functions, the powerfulness and robustness of the GPS method has been well established in solving the radial Schrödinger equation with high accuracy.However, in our recent work, it was found that the previous calculations based on the GPS method for the radial expectation values of confined systems show significant discrepancies with other theoretical methods. In this work we have tackled such a problem by tracing its source to the GPS method and found that the method itself may not be able to obtain the system wave function at the origin. Combined with an extrapolation method developed here, the GPS method can fully reproduce the radial quantities obtained by other theoretical methods, but with more flexibility, efficiency, and accuracy. We apply the GPS-extrapolation method to investigate the relatievistic fine structure and hyperfine splitting of confined hydrogen atom in s-wave states where the zero-point wave function dominates. Good agreement with previous predictions is obtained for confined hydrogen in low-lying states, and benchmark results are obtained for high-lying excited states. The perturbation treatment of the fine and hyperfine interactions is validated in the confining environment. K E Y W O R D Sconfined atom, fine structure, generalized pseudospectral method, hyperfine splitting, radial expectation value | INTRODUCTIONIn the past few years, the generalized pseudospectral (GPS) method, which is also known as the semispectral method or collocation method, has been successfully applied to investigate the structural properties of atomic and molecular systems, [1][2][3][4][5] the quantum scattering processes in nuclear and atomic physics, [6,7] and the laser-atom interactions. [8][9][10] Its usefulness has been continuously revealed in recent years in accurately and efficiently solving both the time-independent and time-dependent Schrödinger and Dirac equations. Being a type of discrete variable representation (DVR) method, the GPS method shows its special superiority over other generalizations of DVR, such as the finite difference and finite element methods. For example, the Numerov and Runge-Kutta methods, as in the latter case, are local approaches to the unknown function by a sequence of overlapping low-order polynomials in a small subset of user-defined grid points. [11,12] The GPS method and its variants in different forms are generally global approaches to the unknown function using global basis functions with a high degree, for example, the trigonometric

Section: Introductionmentioning

confidence: 99%

Revisiting the generalized pseudospectral method: Radial expectation values, fine structure, and hyperfine splitting of confined atom

Zhu

Jiao

et al. 2020

“…Therefore, intense theoretical studies on information‐theoretical measures for different quantum systems have been performed. In this way, different potential profiles in the Schrödinger equation have been assumed, for example, Dirac‐delta‐like potentials, hyperbolical potential, power‐type potentials, D ‐dimensional harmonic oscillator and hydrogen atom, for Morse and Pöschl‐Teller potentials, the Rydberg‐like harmonic states, infinite potential well, double square well potential, infinite circular and spherical wells, an electron in one‐dimensional nonuniform systems, one‐dimensional Anderson model, two‐electron atoms, hydrogen atom under soft spherical confinement, the information‐entropic measures in free and confined hydrogen atom, information entropy for Eckart potential, modified Hylleraas plus exponential Rosen‐Morse potential, a squared tangent potential well, a parity‐restricted harmonic oscillator, the Fisher entropy for infinite circular and spherical wells, and so on. The quantum information theory plays an important role in the measurement of uncertainty and other related parameters of an assumed quantum system.…”

Section: Introductionmentioning

confidence: 99%

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

Solaimani

Dong

2019

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.

“…Most efforts have been centered around the spectroscopic properties and some density‐functional descriptors of physical and chemical quantities for the ground state of spherically confined atoms . However, not so much is known about the information‐theoretical measures of the multidimensional confined systems except for a few recent entropy‐like and complexity‐like results of the three‐dimensional confined hydrogenic atom. The aim of this work is to cover this lack of information by determining the confinement dependence of some entropy (Shannon, Fisher) and complexity (Fisher‐Shannon, lopezruiz‐mancini‐alvet (LMC) and LMC‐Rényi) measures for the 1s, 2s, 2p, and 3d quantum states of the two‐dimensional confined hydrogenic atom (2D‐CHA, in short) in both position and momentum spaces.…”

Section: Introductionmentioning

confidence: 99%

Two‐dimensional confined hydrogen: An entropy and complexity approach

Estañón

Aquino

Puertas‐Centeno

et al. 2020

The position and momentum spreading of the electron distribution of the two-dimensional confined hydrogenic atom, which is a basic prototype of the general multidimensional confined quantum systems, is numerically studied in terms of the confinement radius for the 1s, 2s, 2p, and 3d quantum states by means of the main entropy and complexity information-theoretical measures. First, the Shannon entropy and the Fisher information, as well as the associated uncertainty relations, are computed and discussed. Then, the Fisher-Shannon, lopezruiz-mancini-alvet, and LMC-Rényi complexity measures are examined and mutually compared. We have found that these entropy and complexity quantities reflect the rich properties of the electron confinement extent in the two conjugated spaces. K E Y W O R D SFisher information of the two-dimensional confined hydrogen atom, Fisher-Shannon complexity measure of the two-dimensional confined hydrogen atom, LMC-Rényi complexity measure of the two-dimensional confined hydrogen atom, Shannon entropy of the two-dimensional confined hydrogen atom, two-dimensional confined hydrogen atom | INTRODUCTIONThe fundamental and practical relevance of the spherically confined quantum systems has been manifested from the early days of quantum physics [1,2] up until now. [3][4][5][6][7] They have been used as prototypes to explain numerous phenomena and systems not only in the three-dimensional world but also for nonrelativistic and relativistic D-dimensional (D ≥ 2) chemistry and physics. [8][9][10][11][12][13][14] For example, pioneered by Gerhard Ertl, the 2007 Nobel Laureate in Chemistry, and his collaborators, surface chemistry has been extensively studied and has become a key branch in chemistry. [11][12][13] In physics of materials, two-dimensional systems have been used to gain insight into the properties of semiconductors (see, eg, refs. [15, 16]), and they start to play a fundamental role in bringing strong and new forms of control to the atomic-scale limit over the dynamics of matter in the extreme confinement of electromagnetic energy by phonon polaritonics. [10] Moreover, the properties of the real fluids can be studied by means of crystalline fluids (ie, with a symmetry) with nonstandard dimensions (see, eg, ref. [17]).The idea of two-and multidimensional spherical confinement of atoms has been used not only to simulate the effect of high pressure on the static dipole polarizability in hydrogen [18] but also to model a great deal of nanotechnological objects such as quantum dots; quantum wells and quantum wires; [19,20] atoms and molecules embedded in nanocavities, for example, in fullerenes, zeolites cages, and helium droplets; [4,6,7,[21][22][23] dilute bosonic and fermionic systems in magnetic traps of extremely low temperatures; [24][25][26] and a variety of quantum-information elements. [27,28] This has provoked a fast development of a density functional theory of independent particles moving in multidimensional central potentials with various analytical forms (see, eg, refs. [5-7, 29-...