Relative Fisher information of hydrogen-like atoms

Liangyong Wu, Sheng‐Qi Zhang, Bowen Li

Neetik Mukherjee, Sangita Majumdar, Amlan K. Roy

Takuya Yamano

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

Takuya Yamano

An electron in a constant magnetic field has energy levels, known as the Landau levels. One can obtain the corresponding radial wavefunction of free-electron Landau states in cylindrical polar coordinates. However, this system has not been explored so far in terms of an information-theoretical viewpoint. Here, we 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 and under some potentials. By numerically evaluating the generalized Laguerre polynomials in the radial densities, we report that Fisher information increases linearly with the principal quantum number that specifies energy levels, but decreases monotonically with the azimuthal quantum number m. We also present relative Fisher information of the Landau states against the reference density with m=0, which is proportional to the principal quantum number. We compare it with the case when the lowest Landau level state is set as the reference.

Shu Hotta