Talk:Fine structure

Untitled

Moved from the article:

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To add:

• Spin-orbit hamiltonian and effect on levels (choice of eigenstates for perturbation calculation)
• Relativistic kinetic energy correction
• Unusual hydrogenic atoms eg Muonium
• Fine structure in many-electron atoms

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--Smack (talk) 01:14, 18 May 2005 (UTC)

Kinetic energy relativistic correction

$E_{n}=-{\frac {e^{2}}{2a_{0}n^{2}}}$ looks wrong shouldn't it be $E_{n}=-{\frac {e^{2}}{8\pi \epsilon _{0}a_{0}n^{2}}}$ Found the reason: CGS vs. SI, maybe there should be a hint... Maxiantor (talk) 13:11, 14 July 2013 (UTC)

What is e?

The variable e is not defined. Please define it. — Preceding unsigned comment added by 99.224.212.49 (talk) 22:00, 2 April 2012 (UTC)

origin of the term

Could anyone add something about the origin of the term?

Relativistic Correction

Can someone comment why p^4 is deemed nonhermitian (comment on the bottom of that section). Looks hermitian to me. — Preceding unsigned comment added by 129.67.66.131 (talk) 11:07, 2 February 2012 (UTC)

• Given a wavefunction $\psi$, while it makes sense to use $p^{2}\psi$, the latter may not be a normalizable wavefunction. I assume you're making the argument that $\langle \psi _{1}|p^{4}\psi _{2}\rangle =\langle p^{2}\psi _{1}|p^{2}\psi _{2}\rangle =\langle p^{4}\psi _{1}|\psi _{2}\rangle$. Neither of those equalities makes sense, however, unless $p^{2}\psi$ (for the relevant $\psi$) is a normalizable wavefunction, as the hermiticity of p^2 depends on the normalizability of $\psi$. Griffiths gives an example where $\langle \psi _{n00}|p^{4}\psi _{m00}\rangle ={\frac {8\hbar ^{4}}{a^{4}}}{\frac {n-m}{(nm)^{\frac {5}{2}}}}+\langle p^{4}\psi _{n00}|\psi _{m00}\rangle$--Justplainuncool (talk) 06:07, 18 April 2012 (UTC)

P^4 is Hermit! Griffiths has corrected it on his web. http://academic.reed.edu/physics/faculty/griffiths/QM2b.pdf

I have corrected the article accordingly.KoreanApple (talk) 09:04, 12 December 2014 (UTC)

Spin Orbit

Added detail. Removed section on spin of bosons and fermions as it seemed fairly irrelevant? —The preceding unsigned comment was added by 80.192.82.128 (talk) 14:19, 7 April 2007 (UTC).

Variables

It would be nice to have something explaining what the variables are, for a layman such as myself it is impossible to understand what the equations mean without that. 82.216.248.178 08:57, 15 July 2007 (UTC)

Spin-orbit coupling

The article currently reads:

The spin-orbit correction arises when we shift from the standard frame of reference (where the electron orbits the nucleus) into one where the electron is stationary and the nucleus instead orbits it.

This doesn't seem precise as it could be — the spin-orbit interaction is always there. (Clearly we can't have an physical phenomenon that depends on our reference frame!) It's just that it becomes much easier to model when we consider the electron as our reference point — then we can just consider a moving nucleus. Maybe this should be reworded? —Preceding unsigned comment added by 125.238.124.128 (talk) 06:26, 15 October 2007 (UTC)

I reworded it, but I would still prefer a direct argument. 89.217.29.25 (talk) 16:07, 17 April 2015 (UTC)

Pictures

Would it not be worth having some relevant pictures of spectra? Or is that not meaningful? PJTraill (talk) 13:42, 3 October 2008 (UTC)

Image:Hydrogen fine structure.svg might be useful. /Pieter Kuiper (talk) 15:08, 3 October 2008 (UTC)

How about File:Fabry Perot Etalon Rings Fringes.png? — Ti89TProgrammer (talk) 00:50, 17 February 2009 (UTC)

This picture has nothing to do with fine structure of atomic energy levels. It shows wave interference.

Define variables

The cardinal sin of writing physics articles is to not define the variables being used. This problem needs to be corrected before the article is of much use to anyone. —Preceding unsigned comment added by 129.78.64.100 (talk) 04:50, 4 June 2010 (UTC)

First order corrections to theories falsified a century ago cannot be the cause of physical phenomena.

From the introduction: "the fine structure describes the splitting of the spectral lines of atoms due to first order relativistic corrections".

This is putting the theoretical cart before the empirical horse. The observed splitting of spectral lines is certainly not caused by approximate corrections to falsified archaic theories. The ad-hoc corrections are in fact due to the observed splitting of the spectral lines.

<snark>Theorists run amok these days. Statements of the sort "the observed phenomena is due to the theory of the observed phenomena" are not merely meaningless, but strikingly disingenuous.</snark>

NOrbeck (talk) 09:17, 6 November 2010 (UTC)

Since this page is mainly about the theoretical explanation rather than the experimental description of the finestructure splitting, I guess there's nothing wrong with this sentence. In fact, since there are many reasons for the splitting of the hydrogen lines, it would be plain wrong to speak of _the_ fine structure splitting; the finestructure corrections are only one of several reasons (perhaps some yet to be discovered) for the non-degenerate energy levels. After all, IMHO it's just common to say this and that phenomenon is due to some theory which explains it; of course nobody would assume that nature works this way because of our theories! --Jan Krieg (talk) 23:06, 9 June 2011 (UTC)

Cleanup requested

This article has non-standarized formatting, inline footnotes (beginning with "Remark:" or "Note:"), and has equations listed without the context of a sentence or paragraph. Please consider nominating it for cleanup. — Preceding unsigned comment added by 152.3.174.236 (talk) 00:39, 16 December 2013 (UTC)

Improved Darwin-Term section and definitions of variables

I tried to improve the article by defining the variables and improving the section about the Darwin term by a short calculation justifying the "quantum fluctuations" interpretation. Please let me know if there is anything else that is unclear. 35.13.164.198 (talk) 16:42, 30 April 2014 (UTC)

P^4 is Hermit, the author of the book has corrected on his web.

http://academic.reed.edu/physics/faculty/griffiths/QM2b.pdf — Preceding unsigned comment added by 2606:A000:95C1:7200:9DD4:B180:F151:FB9D (talk) 16:39, 1 June 2014 (UTC)

A step was skipped in "Kinetic energy relativistic correction"

When you correct the equation to first order, you have to correct both E_n and ψ_n. I believe the latter contribution is zero because the relevant expression is quadratic in ψ. But this should be argued for.

Also, one should write the index n on ψ^0, i.e. write ψ^0_n, just as we write E^0_n. 89.217.29.25 (talk) 15:26, 17 April 2015 (UTC)

Fine structure constant is forgotten after the introduction

The fine structure constant is mentioned in the introduction, and we are told that the corrections are on the order of (Zα)^2. But this is not borne out in the text; none of the three corrections are expressed this way. They really should be. 89.217.29.25 (talk) 15:47, 17 April 2015 (UTC)

What does this paragraph mean?

From the article:

Remark: On the (n,l,s)=(n,0,1/2) and (n,l,s)=(n,1,-1/2) energy level, which the fine structure said their level are the same. If we take the g-factor to be 2.0031904622, then, the calculated energy level will be different by using 2 as g-factor. Only using 2 as the g-factor, we can match the energy level in the 1st order approximation of the relativistic correction. When using the higher order approximation for the relativistic term, the 2.0031904622 g-factor may agree with each other. However, if we use the g-factor as 2.0031904622, the result does not agree with the formula, which included every effect.

This is badly written to the point of incomprehensibility, but seems to be making interesting comparisons. Could someone rewrite this? 89.217.29.25 (talk) 16:49, 17 April 2015 (UTC)

The article could be more systematic in its discussion of the 3 correction terms

From Griffiths, paraphrased: The exact fine-structure formula for hydrogen (obtained from the Dirac equation without recourse to perturbation theory) is

$E_{nj}=mc^{2}\left\{\left[1+\left({\frac {\alpha }{n-(j+1/2)+{\sqrt {(j+1/2)^{2}-\alpha ^{2}}}}}\right)^{2}\right]^{-1/2}-1\right\}$

where α is the fine structure constant.

So the 3 derivations in the article are not the full essence of fine structure, but just an approximation to a more exact theoretical value (coming from Dirac and/or QED). This should be stated in the article.

The starting formula for the spin-orbit coupling correction, and especially the derivation of the Darwin term, seems to be pulled out of a hat.

How are the 3 correction terms related to each other? Evidently they are nonoverlapping and can therefore be summed. But they appear to come from 3 disparate, ad-hoc motivations. So how can we be sure that they are really all the corrections that are needed? How close do they come to the "gold standard" of either (1) the Dirac equation, (2) QED, or (3) observation? How close do these "gold standards" come to each other?

To what extent was the Dirac equation and/or QED used in motivating or computing the corrections? I see that the Darwin term seems to come from Dirac, and that something called the "g-factor" already slips a bit of QED into the spin-orbit coupling, but I can't get an overview of what comes from what or how complete it all is.

More from Griffiths (in the introduction to his book, emphasis mine): Finally the treatment of the Dirac equation in the last chapter (20) is intended to show that several things such as electron spin, its magnetic moment, the spin-orbit interaction, etc. which were introduced in an ad hoc fashion in earlier chapters, emerge as a coherent whole from the Dirac equation, and also to give students a glimpse of what lies ahead. It would be great to have a bit of this coherent whole in the current article. 89.217.29.25 (talk) 20:00, 17 April 2015 (UTC)

The introduction of the paragraph on the Dirac equation is confusing:

“The total effect can also be obtained by using the Dirac equation. In this case, the electron is treated as non-relativistic. The exact energies are given by [6]”

... electron treated as non-relativistic? -- well, it's just the opposite with the Dirac equation.

... cite [6] = Sommerfeld from 1919 for the Dirac eigenvalues? -- that's impossible, better quote Dirac's seminal paper (although his precision is not the same as the quoted formula) or a standard textbook like Messiah vol II. DieHenkels (talk) 10:39, 6 January 2023 (UTC)

Introduction

"In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom by Albert A. Michelson and Edward W. Morley in 1887,[1][2] laying the basis for the theoretical treatment by Arnold Sommerfeld, introducing the fine-structure constant.[3]"

It would be helpful if the introduction quoted above could be clarified. What is fine structure? It says it is "the splitting of the spectral lines." Does that mean the location of the emission or absorption lines in the spectrum? That is not clear. The fine structure is discussed in terms of quantum mechanical and relativistic processes, but it was observed and measured before quantum mechanics and relativistic physics were discovered. That's confusing. A better description of the phenomenon itself would be very useful. Perhaps a graphic would help. Jpipersson (talk) 22:41, 29 April 2020 (UTC)