## Question - fine structure

Recently I read about element 137 (feynmanium / Untriseptium ), the element which would theoretically have its electrons moving faster than light (if I understood it the right way). I know that if you involve relativity in the analysis, the problem occurs in element 139, but thats kind of irrelevant to my question.

Here is my first question : The article said the speed of electrons in the 1s electron orbital could be obtained with this equation : v = Z *alpha* c = (Z*c)/(137.036) . Why is that so?

This leads me to my second question. I did a bit of research myself, and found that the fraction 1/137.036 was the fine structure constant, and thats where I hit a roadblock. What is the fine structure? I know it's the splitting of the spectral lines in atoms, but I'm having trouble understanding this concept. And I need help, please!
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 Quote by Luidge Recently I read about element 137 (feynmanium / Untriseptium ), the element which would theoretically have its electrons moving faster than light (if I understood it the right way). I know that if you involve relativity in the analysis, the problem occurs in element 139, but thats kind of irrelevant to my question. Here is my first question : The article said the speed of electrons in the 1s electron orbital could be obtained with this equation : v = Z *alpha* c = (Z*c)/(137.036) . Why is that so?
The electrons will not be moving FTL!

Here is a nice background - http://physics.nist.gov/cuu/Constants/alpha.html
 I realize I didn't express the situation correctily. I'm not one implying that electrons would be movint FTL. Instead of saying hte electrons in element 137 would theoretically move FTL (which makes no sense when you do the math), I should've said it would be the last element in which elctrons on th 1s electron orbital would not move FTL. This raises interesting questions, because we know that electrons can'T move FTL. Is element 137 the last possible element? or if it's not, what happens in the following elements? Sorry I didn'T express the situation correctly in my first post, by I guess my questions still stand.

## Question - fine structure

The fine structure constant alpha = e^2/(hbar c 4 pi epsilon_0), e being the electron charge (I hope everything else is self-explanatory).
I would check out that NIST page, it looks pretty good. The fsc originated as a perturbation parameter, 0th order quantum effects have alpha^0 (= 1) coefficients, fine structure has alpha^2, hyperfine has alpha^4, etc. Later, it was found that this constant determines the interaction between electrons and photons. It kept popping up in places, and since it's dimensionless, people would try to assign all kinds of theoretical importance to it, and make physical interpretations.

As for an electron moving FTL, don't worry about it. It's not. That expression you gave for velocity (I'm guessing) treats an electron as being in a circular orbit, and goes from there. You can find all sorts of contradictions if you treat QM situations classically. Another example is a single electron, with spin hbar/2. Spin is angular momentum, and treating an electron as a rotating sphere, you find that
L = hb/2 = m_e r_e^2 w -> v_m = r_e w = hb/(2 * m_e * r_e)
v_m being the velocity at the equator of the electron. The classical electron radius
(http://en.wikipedia.org/wiki/Classical_electron_radius) is about 3 picometers, which will give you 2e10 m/s, or ~ 60 c.

The point I'm trying to make is that one can't always apply classical logic to quantum mechanics. In fact, one often can't. An electron can't be treated as a point particle at a fundamental level. That formula for velocity you gave can be arrived at in the following way:
For a charged particle of charge e (electron) orbiting at distance r a body of charge Z*e(nucleus), the CLASSICAL velocity would be
sqrt( (z k e^2) /(m_e r) ), where k is the k in F = ke^2/r^2, Coulomb constant in whatever your system of units is.
The AVERAGE (not constant) distance for an electron in a 1s orbital is a_b/Z, a_b being the Bohr radius = hb/( m_e c alpha). Work that all through, and you'll get the velocity as Z alpha c.

I tried finding the expectation value of (p/m)^2= v^2, and square rooting it, and you actually get the same thing. So I guess that's where it came from first. But that was dealing with the non-relativistic Schrodinger equation, it wouldn't be valid for any velocity close to c. Although it would be an expectation value, meaning one would make measurements both higher and lower than it, it comes from an equation which is not relativistically correct.

Hope that helps.
 The fine structure constant was first introduced to us by Sommerfeld, and its name is a little confusing. It is really just the velocity of an electron in Bohr's n=1 quantum orbit divided by the speed of light. $$\alpha = \frac{v}{c}$$ And while there is the other equation related above, the first definition is just as definitive. Fine structure relates to the subtle differences between electron states in the spectra of the elements. Wikpedia is a good place to start, or you could try Hyperphysics. A word of caution about the fine structure constant, it pops up alot. There are many crackpot attempts to reproduce 1/137... You should try and avoid these. Bethe worked on the fine structure of the elements- again the spectra.