- #1
andrewr
- 263
- 0
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.
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.
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