Relative Fisher information for Morse potential and isotropic quantum oscillators

Abstract

The relative Fisher information associated with quantum-mechanical states for two prototypical models is presented; the radial density functions that characterize the Morse oscillator and the d-dimensional isotropic quantum oscillators are analyzed in terms of this relative information by specifying the ground state as the reference. We find that the information increases as the eigenvalue (i.e., the degree of the associated Laguerre polynomials) increases for the Morse density function, but for the isotropic quantum harmonic models, it decreases with the dimension and increases almost linearly with the increasing quantum numbers. We present also the explicit expression for the information associated with the angular density as a function of the orbital quantum number and the magnetic quantum number for the three-dimensional isotropic quantum oscillator.

Fisher Information of Free-Electron Landau States

“…In contrast to the constant slope of 4 derived in Equation ( 13) when one sets the reference density as η 0,m , the linearity depends on both quantum numbers. This dependency is remarkably analogous to the case of the D-dimensional isotropic quantum oscillator, in which the relative Fisher information of the radial wavefunction depends on both the principal quantum number n and the orbital quantum number l [28]. We observe in Figure A1 that for a fixed n ρ , the Landau states with a higher azimuthal quantum number get closer to the exponentially decreasing density.…”

“…We evaluate these integrals below. Note that similar integrals also appear in calculating relative Fisher information for the D-dimensional isotropic quantum oscillator [28]. There would be several ways to obtain these integrals.…”

“…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.…”

Fisher information of Landau states and relative information against the lowest level

“…We here focus on Fisher information associated with these Landau states specified by the two quantum numbers. Fisher information provides a useful measure of the electronic structure in quantum systems such as hydrogen-like atoms [1,2] and molecules under Morse potentials [3]. We numerically evaluate the generalized Laguerre polynomials contained in the radial wave functions.…”