Critical screening parameters and critical behaviors of one-electron systems with screened Coulomb potentials

Abstract

The critical screening parameters for one-electron systems screened by Hulthén, Debye–Hückel, and exponential cosine screened Coulomb potentials are calculated with an accuracy close to the precision of numerical arithmetic. The results for a H atom with an infinitely heavy nucleus are reported from the ground to high-lying excited states, and those for arbitrary two-body charged systems are derived from the Zm-scaling law. A thorough comparison of the critical screening parameters for the ground and the first p-wave excited states with previous predictions is made to demonstrate the accuracy of our calculations. The critical behaviors of system-bound and pseudo-continuum eigenenergies for s- and non-s-wave states are shown to follow the quadratic and linear laws, respectively. The variation of the corresponding wave functions is analyzed in detail. For systems with non-zero orbital angular momenta, the bound states convert into shape-type resonances when the screening parameter exceeds the critical value. The resonance energy shares the same linear law as the pseudo-continuum state, while the resonance width varies by an l-dependent power law. It is further shown that the different asymptotic behaviors of the resonance energy and width are consistent with the complex analog of the Hellmann–Feynman theorem.

Multipole polarizabilities of spherically confined hydrogen atom and positronium in screened Coulomb potential

“…As a result, the eigenenergies of the H atom in ECSCP, with or without spatial confinement, are overall larger than the eigenenergies in SCP. At r max = ∞, the critical screening parameter for the ground state of H atom in ECSCP is located at about λ c (1s) = 0.720524085881953… [41,44], which is much smaller than the corresponding value in SCP.…”

“…It is worth noting that, when there is no confinement, the second scaling law shown in Equation ( 9) is also valid for the critical screening parameter λ c which is defined as the point beyond which the bound state ceases to exist (either absorbed into the continuum or transformed into a shape resonance state, depending on the orbital angular momentum) [41]. Therefore, the scaling law for the multipole polarizabilities is applicable in the entire range of screening parameters where the stable bound state exists.…”

“…show perfect agreement with the predictions of Kang et al [24] for all digits listed in the table, but have quite large discrepancies with the former two works at relatively large screening parameters. At r max = ∞, for example, the earliest calculation of Saha et al [22] shows that the ground state is not bound at λ = 1 and the more recent work of Saha et al [23] gave an estimate of λ c = 1.1158, while the most accurate prediction of the critical screening parameter for the ground state of H atom in SCP predicted by the GPS method is located at about λ c (1s) = 1.190612421060617… [41,44]. It is worth noting that the definition of such a critical screening parameter for spherically confined atoms is of no practical significance or is even misleading, because the electron would always be bound by the impenetrable sphere box with any large positive eigenenergies.…”

Critical screening parameters of one-electron systems with screened Coulomb potentials: circular Rydberg states

“…Investigation of the critical stability of few-body systems under external potentials provides us with a unique opportunity to understand the novel phenomena in quantum systems such as the quantum phase transition and configuration symmetry breaking [1][2][3][4][5][6][7][8][9][10]. By fine tuning the potential parameters, the system can take bound-continuum, bound-virtual, or bound-quasibound (resonance) transitions near the critical binding region, and in some circumstances, the quantum system reproduces classical behavior [10][11][12][13][14][15][16]. The short-range screened Coulomb potentials (SCPs) are a class of parametric potentials that attract wide interest in atomic and molecular physics as well as nuclear and plasma physics.…”

Revisiting the generalized pseudospectral method: Energy degeneracies, radial expectation values, and Schwarz‐like inequalities of the shell‐confined atom

“…In our previous work [29], we made great effort in developing the generalized pseudospectral (GPS) method to investigate various physical properties of the spherically confined hydrogen-like atoms. The GPS method, which is a numerical method in discrete variable representation and can also be viewed as a finite-difference method with infinite order of accuracy, has shown its high flexibility and fast convergence in solving the time-independent (and time-dependent) Schrödinger (and Dirac) equation for free atomic and molecular systems [30][31][32][33][34][35]. For spherically confined atoms, the spatial restriction of the radial variable can conveniently be taken into account through a proper nonlinear mapping function, and the boundary conditions for the system wave functions can be satisfied by reducing the dimension of the derived eigenvalue problem.…”