Shannon information entropies of the Hulthén potential and their critical behavior near the system bound limit

Abstract The Shannon entropies in both coordinate and momentum spaces for some low-lying eigenstates of the Hulthén potential (HP) are investigated over a wide range of screening parameters where bound states exist. The system wave functions are expanded in terms of multiple groups of Slater-type orbitals and the Hamiltonian is solved by employing the Rayleigh–Ritz variational method. The accuracy of the obtained bound-state eigenenergies, Shannon entropies, and radial mean values are validated by comparing with the analytical solutions of the HP in s -wave states. The Bialynicki–Birula and Mycielski inequalities for Shannon entropies in both the coordinate and momentum spaces as well as their sum are examined in both weak and strong screening situations. It is shown that the variation of Shannon entropy sum is governed by its upper bound for the system in a well-defined bound state. In the critical bound region where the system undergoes a bound-continuum transition, the Shannon entropies and entropy sum differ in states by their orbital angular momenta and their asymptotic behavior can be properly shaped by the rigorous upper and lower bounds in terms of the radial and momentum expectation values <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>r</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mi>p</mml:mi> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We finally extended the calculation to a modified HP (MHP) that has been widely discussed in the literature and compared our results with previous predictions. Benchmark Shannon entropies are provided for further investigation of the information measures of MHPs.