The noncommutative quadrupole field effect for the H-atom

Abstract

In this paper, we obtained the expression for the noncommutative quadrupole potential and computed the second-order noncommutative quadrupole effect on the ground-state energy of the hydrogen atom. This computation was done using both the second-order perturbation theory and the exact computation using the Dalgarno–Lewis method. As a result, a sum rule was obtained and shown to be consistent with the quadrupole sum rule derived by Bell.

On certain sum rules for the hydrogen atom

“…is the zeta function. We appreciate that S < 6.889304238 is considerably smaller than the result (6) given by Chair et al [2]. These authors compared their sum rule with one of those derived earlier by Bell [4].…”

“…We have proved that the sum rules derived by Chair and Dalabeeh [1] and Chair et al [2] are incorrect because the authors did not take into account the continuous spectrum of the hydrogen-atom Hamiltonian H 0 . Although those authors may have calculated the result of the sum exactly by means of the method of Dalgarno and Lewis [3] they omitted the contribution of the continuous spectrum in the sum over intermediate states (5).…”

“…In order to obtain their sum rules Chair and Dalabeeh [1] and Chair et al [2] proved that 0| H ′ |0 = 0 and managed to calculate the matrix elements n| H ′ |0 and 0| H ′ F |0 that appear in equation (5). However, since they omitted the continuous spectrum of hydrogen the bound states in their sums do not span the whole state space (that is to say: do not satisfy equation ( 3)) and, consequently, their sum rules cannot be exact.…”

“…However, Bell obtained the results of the sums and did not calculate the terms in the series explicitly. Chair et al [2] did it but only for the discrete states; for this reason their explicit sum rule is incorrect.…”

“…Chair and Dalabeeh [1] and more recently Chair et al [2] derived sum rules for the hydrogen atom by means of the Rayleigh-Schrödinger perturbation theory. To this end, they compared the explicit sum over intermediate states with the exact result provided by the method of Dalgarno and Lewis [3].…”