Answer
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Work Step by Step
In high-angular-momentum ($ l $) states, the electron follows a nearly circular orbit that stays mostly outside the inner electron shells. Because these inner electrons act as a shield, the outer electron effectively experiences a reduced nuclear charge of $ +e $, where $ Z-1 $ of the $ Z $ protons are masked by the surrounding electrons. This shielding makes the force on the high-$ l $ electron comparable to the force on an electron in a hydrogen atom. Consequently, the energy levels for high-$ l $ states are nearly the same as those for a hydrogen atom with the same principal quantum number ($ n $).
On the other hand, low-$ l $ states involve more elliptical orbits that allow the electron to get much closer to the nucleus. During these close passes, the electron experiences the full nuclear charge of $ +Ze $, as it is no longer shielded by the inner electrons. The stronger attractive interaction between the positively charged nucleus and the negatively charged electron reduces the electron's energy, making it harder for the electron to escape from the atom. Although the electron may also face increased repulsive forces from other electrons due to the proximity, this repulsion is relatively weak compared to the enhanced attraction toward the nucleus.
Overall, the energy of a low-$ l $ state is lower than that of a hydrogen atom's corresponding energy level with the same $ n $, as the additional attraction to the nucleus outweighs the effects of electron-electron repulsion.
