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I'm working on the fine structure correction to the Hydrogen atom. I have more of a conceptal, maybe historical question, why is this correction called this way? and why is the fine structure constant called this way?

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We use the term "fine structure" to refer to the three terms in the Hamiltonian that contribute to the splitting of energy levels (the relativistic kinetic energy correction, the spin-orbit coupling term, and the Darwin term). As you may end up showing, these perturbative corrections are of order $(Z\alpha)^2$ times the unperturbed energy, where $\alpha$ is the fine structure constant. Given that - for the case of the hydrogen atom, where $Z=1$ - this product is $\sim5\times10^{-5}$, we see that the correction is indeed small compared to the unperturbed energy, and thus fine. We therefore call the resulting splitting fine structure to distinguish it from the energy levels obtained without the perturbations (the gross strucuture), which are comparatively large.

As the fine structure constant gives us a way of quantifying the strength of the splitting, it seems natural to name it after the fine structure it describes. It appears that Arnold Sommerfeld introduced the term in 1916.

This of course then leads to the natural name of the hyperfine structure, which arises because of interactions between the magnetic moments of the proton and electron. The energy splitting is of the same order as the fine structure perturbations by multiplied by the ratio of the particles' masses, $(m_e/m_p)\sim5\times10^{-4}$, making it over three orders of magnitude smaller than the fine structure contributions, and thus hyperfine.

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  • $\begingroup$ I haven't seen the links and this sounds like a plausible reason as to why it should be called fine structure constant. But was it named so historically because of the same reason? $\endgroup$ – Dvij Mankad May 20 at 3:04
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    $\begingroup$ @dvij... They saw that the lines were not just "thick" but had a tiny split, there were two lines, so it was a fine distinction: merriam-webster.com/dictionary/fine .the entry 3. of definition of "fine" $\endgroup$ – anna v May 20 at 3:45
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    $\begingroup$ It might be worthy to mention that the energy levels obtained from the non-relativistic Schrodinger equation are sometimes referred to as the gross structure of the atom. $\endgroup$ – eranreches May 20 at 18:55
  • $\begingroup$ @eranreches Good point; done. $\endgroup$ – HDE 226868 May 20 at 19:07

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