Relative Fisher information in some central potentials

Abstract

Relative Fisher information (IR), which is a measure of correlative fluctuation between two probability densities, has been pursued for a number of quantum systems, such as, 1D quantum harmonic oscillator (QHO) and a few central potentials namely, 3D isotropic QHO, hydrogen atom and pseudoharmonic potential (PHP) in both position (r) and momentum (p) spaces. In the 1D case, the n = 0 state is chosen as reference, whereas for a central potential, the respective circular or node-less (corresponding to lowest radial quantum number n r ) state of a given l quantum number, is selected. Starting from their exact wave functions, expressions of IR in both r and p spaces are obtained in closed analytical forms in all these systems. A careful analysis reveals that, for the 1D QHO, IR in both coordinate spaces increase linearly with quantum number n. Likewise, for 3D QHO and PHP, it varies with single power of radial quantum number n r in both spaces. But, in H atom they depend on both principal (n) and azimuthal (l) quantum numbers. However, at a fixed l, IR (in conjugate spaces) initially advance with rise of n and then falls off; also for a given n, it always decreases with l.

Crámer‐Raocomplexity of the confined two‐dimensional hydrogen

“…The present results, together with some recent efforts on information entropies of global (Shannon, Rényi, Tsallis) [37,40,41] and local (Fisher, relative Fisher) [37,42,44,62,63] character for numerous excited states of three-dimensional free and confined hydrogen-like systems, illustrate how and how much confinement is crucial not only for the energy spectrum of multidimensional hydrogen [20,64] but also for its eigenfunctiondependent information-theoretical properties that control all the chemical and physical properties. The mean expectation value hpi of the momentum probability density of the free d-dimensional hydrogenic atom is not yet explicitly known for a generic stationary state (n, l, {μ}) (n, l μ 1 , μ 2 , …, μ d − 1 ) despite the efforts of various authors.…”

“…The present results, together with some recent efforts on information entropies of global (Shannon, Rényi, Tsallis) [ 37,40,41 ] and local (Fisher, relative Fisher) [ 37,42,44,62,63 ] character for numerous excited states of three‐dimensional free and confined hydrogen‐like systems, illustrate how and how much confinement is crucial not only for the energy spectrum of multidimensional hydrogen [ 20,64 ] but also for its eigenfunction‐dependent information‐theoretical properties that control all the chemical and physical properties.…”

Fisher Information of Free-Electron Landau States

“…The relative Fisher information can be a suitable measure of similarity or dissimilarity between two states when one uses Fisher information contained in systems. It is recently applied to study atomic shell structures [23], atomic ionization processes and isoelectronic series [24], sinusoidal and gamma-like densities [25], one-particle densities of (non-)relativistic hydrogenic systems [12,26], the Morse potential and isotropic quantum harmonic oscillators [27,28], diatomic molecules with the pseudoharmonic potential [27], the deviation of the Pauli and Weizsäcker kinetic energy densities from the local density approximation [29], and the derivation of the Euler equation of the orbital-free excited-state density functional theory as a variational problem [30].…”

“…Here, we measure the relative information between the excited Landau level specified by n ρ and the lowest Landau level n ρ = 0 having the common |m|. This choice is natural, since we have set the ground state of a system as a reference state in the studies of the radial wavefunctions of hydrogen-like atoms in position space [12], quantum harmonic oscillators [27,28], and under some central potentials in both position and momentum spaces [27]. Note that the lowest Landau level wavefunction is widely used in the literature of the quantum Hall effect and various material sciences.…”

Entropic Kullback‐Leibler type distance measures for quantum distributions

“…KL has also been used in molecular dynamics simulations . Relative information, based on other measures such as Fisher information, or Rényi entropy, has also been examined in varied contexts.…”