# Information-entropic measures for non-zero l states of confined hydrogen-like ions

## TL;DR

This paper derives analytical expressions for the relative Fisher information in position and momentum spaces for various quantum systems, revealing how it varies with quantum numbers and system types.

## Contribution

It provides exact closed-form formulas for IR in different quantum systems and analyzes their dependence on quantum numbers, extending previous studies to confined hydrogen-like ions.

## Key findings

- IR increases linearly with quantum number n in 1D QHO.
- IR varies as a power of n_r in 3D QHO and PHP.
- In hydrogen atom, IR depends on both n and l, showing initial increase then decrease with n at fixed l.

## 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 $l$.