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.
An electron in a constant magnetic field has the energy levels known as the Landau levels. One can obtain the corresponding radial wave function in cylindrical polar coordinates (e.g., textbook of Landau & Lifshitz). This system is not explored so far in terms of information-theoretical point of view. 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. We report that Fisher information increases linearly with the quantum number n that specifies energy levels, but decreases monotonically with the quantum number m (i.e., the index of the generalized Laguerre polynomial). Also, we present relative Fisher information of the Landau states by setting the lowest Landau state as a reference density. The analytical form is just 4n, which does not depend on the other quantum number m. References: 1. T. Yamano, Relative Fisher information of hydrogen-like atoms, Chem. Phys. Lett. 691 (2018) 1962. T. Yamano, Fisher information of radial wavefunctions for relativistic hydrogenic atoms, Chem. Phys. Lett. 731 (2019) 1366183. T. Yamano, Relative Fisher information for Morse potential and isotropic quantum oscillators, J. Phys. Commun. 2 (2018) 085018
We investigate the quantum-information properties of a particle confined by the double Morse potential in position and momentum spaces. The quasi-exact solvability of the model gives analytical expressions for the first two bound states, allowing the corresponding probability densities to be analyzed directly. Shannon entropy, Onicescu energy, Fisher information, statistical complexity, and Fisher-Shannon products are evaluated as functions of the parameter
A, which controls the transition from a well-separated double well to a merged single-well profile. The position distribution is more delocalized and structurally complex when the double-well character is pronounced, whereas the momentum distribution exhibits the complementary trend. As the wells merge, the ground-state Fisher--Shannon product approaches its Gaussian reference value, whereas the excited state retains stronger non-Gaussian structure.