Higher-order corrections of the hydrogen-like atoms: the effect of Darwin term - Mahmood Jwailes; Imad Barghouthi; Qusay Atawnah (Arab Journal of Basic and Applied Sciences, 2025)
The article extensively explores higher-order corrections of the time-independent non-degenerate perturbation theory, providing a thorough analysis of its fundamental principles. We derive the general formula for both energy levels and wave function corrections, spanning all orders. Then, we use them to determine exactly the higher-order, specifically second- and third-order corrections, for all quantum states of the hydrogen-like atoms, when we include the effect of the Darwin term in the calculations. Additionally, we present numerical results that show how the wave function and its higher-order corrections change at different states, under the influence of the Darwin term. Finally, we discuss the validity of non-degenerate perturbation theory within the same context, showing that it should provide a reasonable approximation for atomic numbers up to 60.