Fisher Information of Free-Electron Landau States

Abstract

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.

Skewed Jensen—Fisher Divergence and Its Bounds

“…Relative Fisher information is potentially useful in many areas of science. In quantum mechanical applications, the excited states of density functions (the square of the wave functions) are compared with its ground state one, and the information shows significant changes according to the eigen states in respective quantum systems under potentials (e.g., [17,[32][33][34][35][36][37][38]). We mention two more examples of the role of relative Fisher information in statistical physics.…”

Distance in Information and Statistical Physics III

“…Fisher information has also been used to quantify the information of wave functions for quantum mechanical systems, such as hydrogen-like atoms. Paper [7] investigates the Fisher information and relative Fisher information of the radial wave functions of the free-electron Landau states realized by electrons in a uniform magnetic field. With numerical evaluation of the generalized Laguerre polynomials, the study reveals that these measures of information change monotonically with quantum numbers.…”