Quantum information and statistical complexity of hydrogen-like ions in Dunkl-Schrödinger system

TL;DR

This work extends the hydrogenic problem to the Dunkl-Schrödinger system by providing exact solutions for the Coulomb potential, yielding analytic expressions for eigenvalues, eigenfunctions, and densities with Dunkl parameters $\mu_x,\mu_y,\mu_z$. It goes beyond standard quantum mechanics by deriving linear information measures—standard deviation, Shannon entropy, and Rényi entropy—and a set of statistical complexities (LMC, SRC, GRC, RCR) within this deformed framework. The study reveals how Dunkl reflections modify density distributions, degeneracies, and information quantities, with explicit $Z$-scaling and parity-dependence, and reports first-time results for these quantities in DS hydrogen-like ions ($Z=1-3$). Overall, the paper provides a comprehensive information-theoretic analysis of Coulomb systems in the Dunkl setting, offering new insights into how symmetry deformations influence localization and complexity measures in quantum states.

Abstract

In this work, we present analytical solutions of Schrödinger equation for Coulomb potential in presence of a Dunkl reflection operator. Expressions are offered for eigenvalues, eigenfunctions and radial densities for H-isoelectronic series (Z=1-3). The degeneracy in energy in absence and presence of the reflection has been discussed. The standard deviation, Shannon entropy, Rényi entropy in position space have been derived for arbitrary quantum states. Then several important complexity measures like López-Ruiz-Mancini-Calbet (LMC), Shape-Rényi complexity (SRC), Generalized Rényi complexity (GRC), Rényi complexity ratio (RCR) are considered in the analytical framework. Representative results are given for three one-electron atomic ions in tabular and graphical format. Changes in these measures with respect to parity and Dunkl parameter have been given in detail. Most of these results are offered here for the first time.

Figures (6)

• Figure 1: Probability of finding the electron in between $r$ to $r+dr$. The first, second and third rows, from top, are given for H, He$^{+}$ and Li$^{2+}$. The first, second, third and fourth columns from left are represented for $(n_r,\ell,m)=(0,0,0), (1,0,0), (2,0,0)$ and $(3,0,0)$ states. Solid lines correspond to "no reflection" ($\mu_x=\mu_y=\mu_z=0$), whereas dashed lines refer to reflections with Dunkl parameters $(\mu_x,\mu_y,\mu_z)=(0.4,0.5,0.3)$.
• Figure 2: Probability densities of H, He$^+$ and Li$^{2+}$, in $y=1$ plane. The quantum numbers $(n_r, \ell,m)$ are (A) $(0,0,0)$, (B) $(1,1,0)$, (C) $(1,2,0)$, (D) $(1,3,0)$. (E) $(0,0,0)$ (F) $(1,1,1/2)$ (G) $(1,1,1/2)$ (H) $(1,1,1)$ (I) $(1,1/2,0)$ (K) $(1,1/2,1/2)$ (J) $(1,1/2,1/2)$ (L) $(2,3/2,1)$. For (A)-(D), $(\mu_x,\mu_y,\mu_z)=(0,0,0)$ and for (E)-(L) the parities are $+++,++-,+-+,+--,-++,-+-,--+,---$ respectively with Dunkl parameters $(\mu_x,\mu_y,\mu_z)=(0.1,-0.2,0.1)$.
• Figure 3: The degeneracy of first excited state of H atom for $\ell=m=0$, with reflections having parities $+++$ (blue surface) and without reflections (red plane). Panel (A) corresponds to that with respect to $\mu_x$, $\mu_y$; (B) with respect to $\mu_y$, $\mu_z$; (C) with respect to $\mu_z$, $\mu_x$. The red plane has same energy value of $E_{1,0,0}^{(OD)}=-3.3995731\,eV$ of H atom, without reflection, in the usual Schrödinger system.
• Figure 4: Plot of the ratio, $\left[\left(\Delta r\right)_{0,\ell,m}/\left\langle r\right\rangle_{0,\ell,m}\right]^{(DD)}$ of H atom in Dunkl plane $\mu_x+\mu_y+\mu_z=0$, with reflection, having parities $+++$ (star), $-++$ (circle), and without reflection (square) $\left[\left(\Delta r\right)_{0,\ell,m}/\left\langle r\right\rangle_{0,\ell,m}\right]^{(OD)}$. The red circle and red square indicate minimum values of corresponding ratios.
• Figure 5: Shannon and Rényi entropies of H atom with respect to $m$ in DS system. The connected lines join the points with different colors for ease of understanding. All quantities in a.u. The parameters are $(\mu_x,\mu_y,\mu_z)=(0.01,-0.02,0.01)$ and for Rényi entropy order $\alpha=1.5$.