Energy levels and the Aufbau Principle

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The energy levels of the hydrogen atom, described by the formula En=-me4/(32π2εohbarn2), are independent of the orbital angular momentum quantum number l, creating a paradox with the Aufbau principle, which suggests varying energies for s, p, and d orbitals. This discrepancy can be explained through the concepts of shielding and penetration, as well as the relativistic effects that are often neglected in simpler models. A more accurate treatment of electron behavior, including the fine structure of hydrogen, reveals that these factors influence energy levels. Experimental evidence supports this by showing differences in energy levels for multi-electron atoms, where electron interactions and relativistic effects become significant. Understanding these complexities resolves the apparent contradiction between the hydrogen atom's energy levels and the Aufbau principle.
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The energy levels of the hydrogen atom are given by En=-me4/(32\pi2\epsilonohbarn2, and are independant of the orbital angular momentum quantum number l, yet the Aufbau principle clealry implies that the energies of s, p, and d orbitals are different. Explain this apparent paradox and provide an example of experimental evidence that supports your conclusion.

I feel like I'm on the right track if I go about it from a shielding/penetration standpoint. But I could use the additional help! Thanks!
 
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There are certain approximations - things you neglected - that a more careful treatment will not neglect. Consider: the electron is actually a relativistic particle, yet you used the nonrelatavistic schrodinger equation to derive those energy levels.

Look up the fine structure of the hydrogen atom.
 
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