I came across the following link: Which orbitals of the hydrogen atom are degenerate for $n=3$

And all the answers said that for the hydrogen atom, the energy of an orbital depends only on n. Is this really true? And also, isn't it true that for other atoms (at least according to those who answered the other question) the energy depends upon both $l$ and $n$?

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    $\begingroup$ Yes, that's true. In a hydrogen atom, orbitals with same $n$ and different $l$ are degenerate, that is, have the same energy; in multi-electron atoms, they are different. Also, welcome to Chem.SE. $\endgroup$ – Ivan Neretin Aug 15 '16 at 7:45
  • $\begingroup$ Oh, I didn't know about this!(Hmph. Why can't they mention this exception on textbooks?!) Do u know why this is so? Also, thanks! :) $\endgroup$ – user33789 Aug 15 '16 at 7:47
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    $\begingroup$ "Why" is a tricky question. What kind of explanation would you be satisfied with? We just solve the Schrödinger equation (exactly for H, approximately for other atoms), and that's it. Some people say the degeneracy is due to certain (and quite unobvious) higher symmetry. $\endgroup$ – Ivan Neretin Aug 15 '16 at 8:02
  • $\begingroup$ Well, OK, although that's not much of an explanation as to "why", I don't think I'd be able to understand anyway, since I only just graduated high school. Thanks for ur help! $\endgroup$ – user33789 Aug 15 '16 at 13:25

It is true that if you solve the Schroedinger equation the solutions depend only on the principal quantum number $n$ and the energies (often called eigenenergies or eigenstates) have the values predicted by the Bohr formula, $E=-hcR_\infty /n^2$ and energy differences as predicted by the Balmer formula. The calculated energy does not depend on $l$; this degeneracy of wavefunctions with different $l$ is a peculiar and special feature of the symmetry of the Coulomb potential.

However, nature cares little for our equations and experiments show that in fact the energy of the $l$ levels are different in energy to one another. This is called 'fine structure' and the energy shifts are very small as the name suggests. The first cause is explained as being due to the fact that the electron has an intrinsic quantum property called 'spin', which does not mean that it is literally spinning, but crudely speaking that it can be described by the same equations as angular momentum. As the electron is charged its spin produces a magnetic field which interacts with the magnetic field of its orbital motion, which in turn depends on $l$. This interaction is called spin-orbit coupling, and technically it is a relativistic effect.
The second effect is called the Darwin term, for electrons with $l=0$, and the third effect is the Lamb Shift, which needs explanation beyond relativistic effects and requires Quantum Electrodynamics.

The fine structure due to spin-orbit coupling can be observed easily in many atoms so the $l>0$ levels are split for each $n$ and so are no longer degenerate. The common example is the two lines of Sodium 'D' emission. The difference is small, the two transitions are 589.59 nm to 588.99 nm. This is the yellow colour associated with sodium easily seen if you sprinkle salt into a gas cooker's flame and in some types of street lights. The splitting can be measured with a simple spectrometer and is a common teaching lab experiment. The other effects mentions are far,far more difficult to measure and in a Chemistry course you are unlikely ever to meet them.

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    $\begingroup$ I would clarify that energy differences for "fine structure" are very small. Avoid this clarification may result in a misinterpretation of your nice details. $\endgroup$ – user1420303 Aug 15 '16 at 21:43
  • $\begingroup$ @ user1420303, thanks, have added a line to emphasise this. $\endgroup$ – porphyrin Aug 16 '16 at 8:35
  • $\begingroup$ You are right, nature is more important than our equations (and moreover, Schrödinger equation is not even the "ultimate equation of everything", for it does not include relativistic things). I stand corrected. You might want to add Wikipedia links to Darwin term and Lamb shift, and maybe also to the thing I mentioned earlier which explains that the said degeneracy is, in fact, not accidental. $\endgroup$ – Ivan Neretin Aug 16 '16 at 8:52
  • $\begingroup$ 'accidental', I take the point and I have reworded this part of the text. $\endgroup$ – porphyrin Aug 16 '16 at 10:07
  • $\begingroup$ I took the liberty of formatting the links in a different way. $\endgroup$ – Ivan Neretin Aug 16 '16 at 10:32

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