Shannon Entropy of a Hydrogenic Impurity on a Conical Surface: Confinement and Aharonov–Bohm Effects

In this work, we solve the Schrödinger equation for a hydrogenic impurity located at the apex of a right circular cone, with the electron constrained to move on the conical surface of semi-aperture angle θ0 and subjected to an Aharonov-Bohm magnetic flux along the symmetry axis. Analytical expressions for the energy eigenvalues and normalized radial wave functions are obtained in terms of the principal quantum number <i>n</i> and the angular quantum number m, the magnetic flux ν, and the cone angle. The Shannon entropy is evaluated in both configuration and momentum spaces for several low-lying states, and its variation with ν and θ0 is analyzed in detail. When the magnetic flux vanishes, pairs of states n, m and n, -m share the same entropic behavior; for finite flux, this degeneracy is lifted and the entropies depend explicitly on the state, the cone geometry, and the flux strength. Finally, we verify that the entropic sum Sr+Sp fulfills the Bialynicki-Birula-Mycielski bound, providing an information-theoretic consistency check for the model.