Fine structure, exact formula. Dirac.

[andrewr]

Hello all,

I'm still plugging away at the meaning of spin, and spin orbital coupling. I am at the stage where I am testing out various formulations of corrections to Schrodinger's equation and beginning to test my ideas against data.

Right now I am looking at Hydrogen spectra because being a single proton and electron system it may be handled precisely using a reduced mass formulation; there are no electron-electron repulsions to mess it up. There is also excellent data available on the Balmer series that I am looking at.

I have an equation obtained from the Dirac equation, that I am told is "exact". (Bethe and Salpeter, p. 238 "Quantum Mechanics of one- and two-electron atoms")[tex]
E_{nj}=mc^{2}\left\{ \left[1+\left(\frac{\alpha}{n-(j+1/2)+\sqrt{(j+1/2)^{2}-\alpha^{2}}}\right)^{2}\right]^{-\frac{1}{2}}-1\right\}
[/tex]

Alpha is the fine structure constant.
The mass in this equation is the reduced mass of the single proton and electron in hydrogen for the Balmer series; j is the total angular momentum and n is the quantum number.
Spin is not conserved separate from orbital angular momentum, so the sum of these is j;
j is therefore 1/2 for an S orbital, 1/2 *again* for a P orbital since spin and orbital momentum will partially cancel (the preferred lower energy state). etc.

Now, when I check the above equation against NIST data:
http://physics.nist.gov/PhysRefData/ASD/lines_form.html
656.4522566nm :: n=3,2 j=1.5,0.5 (3D->2P)
656.4664650nm :: n=3,2 j=2.5,1.5 (3D->2P)
656.4564685nm :: n=3,2 j=0.5,0.5 (3S->2P)
656.4722362nm :: n=3,2 j=0.5,1.5 (3S->2P)
656.4537698nm :: n=3,2 j=1.5,0.5 (3P->2S)
656.4584417nm :: n=3,2 j=0.5,0.5 (3P->2S)
As you can see there 8+ significant digits. All of my constants, c, h bar, etc. are codata values that have at least 8 digits of accuracy. And, indeed, when I plug some values of the above table into the Dirac derived equation -- I get answers with that many digits of precision; precise matches; However, approximately half of these values DO NOT come out anywhere near the accuracy they ought.

eg:
656.4537698nm :: n=3,2 j=1.5,0.5 (3P->2S) will yield a ~5 digit accurate result which is so low as to be nearly useless qualitatively.

I would expect, that if hyperfine corrections (the only ones I haven't been explicitly told are included in the Dirac equation derivation -- but they might be..!) were the culprit they would disturb all answers significant digits. However, that's not the case -- so I don't think it is an equation difference for hyperfine corrections.

Am I correct in assuming that hyperfine corrections are too small to account for the magnitude error I am seeing in fine structure? (Worst errors being particularly S->P orbital transitions)?
I don't have hyperfine data to know what it's magnitude is, and I haven't learned the hyperfine mathematics yet; But I really do need an exact formula for Hydrogen/Balmer series to compare my new models against ---

Does anyone know why the Dirac equation derivative that I posted would work in roughly half the cases but loose accuracy on the other half?

Is there a more correct version of the equation that I posted somewhere else?

Thanks.

 

[Bill_K]

Did you remember to take into account the radiative corrections, particularly the Lamb shift? This has its largest effect on the S states.

 

[ytuab]

Andrewr,
Reduced mass is not a real mass.
And I think it is very difficult and complicated to couple the reduced mass and the relativistic effect.

As you know, the effect of the reduced mass is larger than the relativistic effect.
So it is very important to couple the reduced mass with the relativistic mass.

And the one important thing is that Dirac equation (which is same as Sommerfeld model) uses the approximation.
So it is not complete.
The approximation means the nonrelativistic approximation. (Because the electron's velocity is much slower thant the light speed in the hydrogen.)
For example, relativistic Sommerfeld model uses the nonrelativistic approximation, but it is completely consistent with Dirac equation, which means the Dirac equation is one of the nonrelativistic approximation, I think.
(And they both use the Coulomb potential (Coulomb gauge).)

So can we consider the 2S1/2 and 2P1/2 (which name difference exists only in Dirac equation (not Sommerfeld) ) as the same energy states based only on the approximate Dirac equation ?

And the Lamb shift in the hydrogen is very small like hyperfine levels, so the relation between the reduced mass and the relativistic effect is much influential, I think.

 

[andrewr]

Did you remember to take into account the radiative corrections, particularly the Lamb shift? This has its largest effect on the S states.

No, as I didn't derive the equation but was told it is "exact".
All I am really aware of is that the only permitted radiation/adsorbtion is going to happen when 'l' changes by +-1 as a boundary condition. (j itself may stay the same).

In reading my texts, none of them speak about radiative corrections -- so the mathematics for the radiative corrections isn't something I have any knowledge of.

is there a formula for how it would disturb the wavelength of emitted light?.

EG: Given this case alone:
Hydrogen 3P to 2S, spin 3/2 to 1/2,
When I plug this into the Dirac energy formula, I come up with a wavelength of :656.452272013
656.4537698nm is the exact frequency from NIST.
error=0.001497...nm.
So I calculate a mass/energy error of around 4.309μeV

Is there a general formula that would tell me how much correction is needed for radiation compensation in hydrogen/Balmer series?

 

[andrewr]

Andrewr,
Reduced mass is not a real mass.
And I think it is very difficult and complicated to couple the reduced mass and the relativistic effect.

As you know, the effect of the reduced mass is larger than the relativistic effect.
So it is very important to couple the reduced mass with the relativistic mass.

And the one important thing is that Dirac equation (which is same as Sommerfeld model) uses the approximation.
So it is not complete.
The approximation means the nonrelativistic approximation. (Because the electron's velocity is much slower thant the light speed in the hydrogen.)
For example, relativistic Sommerfeld model uses the nonrelativistic approximation, but it is completely consistent with Dirac equation, which means the Dirac equation is one of the nonrelativistic approximation, I think.
(And they both use the Coulomb potential (Coulomb gauge).)

So can we consider the 2S1/2 and 2P1/2 (which name difference exists only in Dirac equation (not Sommerfeld) ) as the same energy states based only on the approximate Dirac equation ?

And the Lamb shift in the hydrogen is very small like hyperfine levels, so the relation between the reduced mass and the relativistic effect is much influential, I think.

Yes, a 2S 1/2 and 2P 1/2 would be the same since n=2 in both of them according to the equation I was given. They have no difference in energy.

I didn't have the background on the derivation of either equation; and I have absolutely no idea of the relationship between Sommerfeld and Dirac. I don't think the similar outcome necessarily means that because one approximation equals the other that they are both non-relativistic... It does suggest it.

I do agree with you that the error is *probably* related to the difference in mass that a difference in kinetic energy represents. My own derivations converge on that perspective; but I want to compare my new formulas with whatever is standard in the field to see how it tracks qualitatively. I only have a fixed number of real data points and that is anemic at best when trying to understand the correlations I am seeing.

In mentioning reduced mass, I have run the above dirac derived equation with normal masses and with reduced masses. The answer is only close when reduce mass is used -- that's why I said mass is the reduced mass. The inferences between relativity and reduced mass weren't a consideration -- just which gets the closer to correct answer.

 

[dextercioby]

The formula in Bethe & Salpeter is <exact> only under the specified assumptions, one of these being the infinite mass of nucleus. <Exact> means that in reaching it no perturbation theory was used.

The formula by Sommerfeld of 1915 coincides with the formula obtained from Dirac's equation, if the electron's spin effects are neglected.

 

[andrewr]

The formula in Bethe & Salpeter is <exact> only under the specified assumptions, one of these being the infinite mass of nucleus. <Exact> means that in reaching it no perturbation theory was used.

The formula by Sommerfeld of 1915 coincides with the formula obtained from Dirac's equation, if the electron's spin effects are neglected.

Thanks Dexter, that helps. Since the hydrogen atom has mostly low velocities, I am getting away with reduced mass (classical approximation) to several significant digits (8+) in some calculations and 5 in others. I am working on a correction factor using relativistic transformations of kinetic energy to see if I can come up with a figure which compensates for the reduced mass of the nucleus based on the orbital energy (and perhaps shape) as well; but it will be a few days and it is just a guess (order of magnitude) type correction.

Do you know of an already existing, but better equation than the one I posted for getting the Balmer series fine constants correct in hydrogen?

 

[andrewr]

Well, I have some progress...
I was able to come up with a partial correction to the center of mass issue (infinte mass of nucleus), but it turns out not to be the major error I am seeing by itself.

When I compare the error energies between Dirac's and the measured light, the errors are on the order of 4.4[μeV] or less (n=3 to 2 , j=0.5 to 0.5 to ) for the Balmer series, The one Lyman series error for 121.566...nm increases the energy offset by over factor of 10 to 47.5[μeV]

For some reason, that I don't understand, there is an alternate measured frequency of the Lyman 2S->1S which is anomalous in that the dirac equation predicts that frequency almost exactly -- but not the others. Does anyone have an idea of why?

So it seems to me, the energy of the error between actual light and predicted light (Δhf) correlates with orbitals, but it varies enough to be highly nonlinear / makes the correlation at least a second order correction of some kind between levels of n.

The overall mass of the system is a much more predictable/well behaved error. When reduced mass is computed for the electron, (which is the *only* scale variable available to the Dirac derived eqn.) a change of mass of -278.1...[eV] is computed for proton/electron alone. That is an overall error correction of around 0.05% in mass; eg: 4 digits down -- and it brings all of Dirac's solutions to within 5 digit accuracy of the NIST measured values.

I came up with an additional correction based on assuming point masses of nucleus and electron; The correction has two effects, one of which I can't add yet -- and that is a correction to the scale of the Coloumb energy vs. radius.

The other correction is the effective mass change of the electron on the reduced scale. I was able to compute that. I simply disturb the entire system, proton and electron, with additional mass energy (ΔEm) -- and then compute the effect on reduced mass, and on wavelength in Dirac's equation -- to figure out how much overall energy needs to be added to the system to make the answers come out right with NIST.

The energy corrections are on the order of 12[meV] to 2.3[eV]; a tiny fraction of the initial change of 278[eV] -- BUT They are also much too large to be caused by the recoil energy of the emitted photon changing the momentum of the atom.

More importantly, the error is roughly proportional to the amount of the probability of the electron which lies closest to the nucleus -- and the error change appears to be cubic in nature; which re-affirms the classical correction for a magnetic disturbance based on a slight change in radius. So I am pretty sure it is the angular momentum near the nucleus which is correlated with most of this error.

My total energy offset error is as follows: (Energy change to total mass/reduced mass, and Energy change to light)

Balmer series
λ= 656.45226[nm] ΔEm= 11.977[meV] Δhf= 44.292[neV] 3p/d to 2s/p
λ= 656.46646[nm] ΔEm= -12.124[meV] Δhf= -44.834[neV] 3d to 2p/d

λ= 656.45647[nm] ΔEm= 361.36[meV] Δhf= 1.3363[μeV] 3s/p to 2s/p ritz
λ= 656.47224[nm] ΔEm= 333.07[meV] Δhf= 1.2317[μeV] 3s/p to 2p/d ritz

λ= 656.45377[nm] ΔEm= -1.1653[ eV] Δhf= -4.3094[μeV] 3p/d to 2s/p
λ= 656.45844[nm] ΔEm= -1.1738[ eV] Δhf= -4.3407[μeV] 3s/p to 2s/p

Lyman series
λ= 121.56699[nm] ΔEm= -2.3846[ eV] Δhf= -47.620[μeV] 2p to 1s/p
λ= 121.56731[nm] ΔEm= -1.4723[ eV] Δhf= -29.401[μeV] 2s/p to 1s/p
λ= 121.56699[nm] ΔEm= -118.22[meV] Δhf= -2.3607[μeV] Alternate 2s/p to 1s/p

I still haven't figured out what the lamb shift correction would be.

I saw this equation of Wikipedia...
[tex]Elamb=\alpha^{5}m_{e}c^{2}\frac{1}{4n^{3}}\left[k(n,l)\pm\frac{1}{\pi(j+0.5)(l+0.5)}\right][/tex]

The constant k(n,l) is <0.05, so can be ignored for scaling the general correction size.
Using a non-reduced mass electron, as the mass in the equation; (maximize energy change).

And that puts the correction energy in the range of: 560[neV] for n=1 orbitals;
and 70[neV] for n=2 orbitals.
and don't bother for n=3 orbitals.

That makes the energy correction way too small even for correcting the frequency of light (E=Δhf) based on micron electron volts of energy error. I don't know QED, so I haven't any way to check my answer -- Does anyone spot an obvious mistake on the lamb shift equation?

 

[Bill_K]

Does anyone spot an obvious mistake on the lamb shift equation?

The Lamb shift for the 2S state is usually quoted as 1000 MHz. If I can multiply (not guaranteed) that corresponds to about 4 μeV.

 

[andrewr]

The Lamb shift for the 2S state is usually quoted as 1000 MHz. If I can multiply (not guaranteed) that corresponds to about 4 μeV.

Thanks for the figure, it helped me find the mistake.

I missed a note on a second formula, lamb shift is largest on the S type orbitals. The formula I gave was for P,D,F,... which are tiny in comparison. The K for s orbitals is much larger ~13.xx
The energy shift is by crude estimate:

34.4[ueV] 1 s
4.30[ueV] 2 s
1.27[ueV] 3 s

It requires a difference in levels to account for the exact figure, but this is close enough for what I want to do. The order of magnitude of the error is right.

 

[Qubix]

I have an equation obtained from the Dirac equation, that I am told is "exact". (Bethe and Salpeter, p. 238 "Quantum Mechanics of one- and two-electron atoms")


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

It's on page 83, the very last formula on the page.

 

[andrewr]

It's on page 83, the very last formula on the page.

oops.


Thanks for the correction; I quoted the footnote to the footnote page in my secondhand text... The actual formula is on P83 in the original.