Hydrogen 1S-2S transition frequency

[Aether]

I'm reading some papers on recent measurements of the hydrogen 1S-2S transition frequency, and they report 2466.061 THz (with about nine additional digits of precision ). However, when I calculate this frequency from the "exact fine-structure formula for hydrogen" (using 1S(n=1,j=0) and 2S(n=2,j=0) I get 2467.554 THz. Has anyone here computed a theoretical value for this transition frequency, and if so what number did you get?

 

[jtbell]

Usually when physicists measure something precisely, it's to compare it against some theory. Don't those papers do that, or at least have references to theoretical calculations?

 

[Aether]

Usually when physicists measure something precisely, it's to compare it against some theory. Don't those papers do that, or at least have references to theoretical calculations?

"This paper describes the theoretical model used to analyze the experimental hydrogen spectra." -- A. Huber et al., High-resolution spectroscopy of the 1S-2S transition in atomic hydrogen, Physical Review A 59(3), 1844 (1999), but it seems to be an analysis of the spectrometer per se rather than a prediction of the transition frequency. The only guidance that I have seen in these papers wrt to predicting the transition frequency are general references to QM texts. I'm now predicting 2467.401 THz using a QM text by D.J. Griffiths as a guide, and this page http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html#c4 (Z=1, n_2=2, n_1=1) has an online calculator that gives a wavelength of 121.502231393543 nm which translates to 2467.382 THz. My calculation agrees with the one generated by this online calculator much better than either one of them agrees with the experimental value.

This paper describes an experimental determination of the 1S-2S transition frequency: M. Niering et al., Measurement of the Hydrogen 1S-2S Transition Frequency by Phase Coherent Comparison with a Microwave Cesium Fountain Clock, Physical Review Letters 84(24), 5496 (2000)), and reports a measured value of the transition frequency of: 2 466 061 413 187 104(46) Hz.

 

[Creator]

I'm now predicting 2467.401 THz using a QM text by D.J. Griffiths as a guide, and this page http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html#c4 (Z=1, n_2=2, n_1=1) has an online calculator that gives a wavelength of 121.502231393543 nm which translates to 2467.382 THz.
,,,,,Physical Review Letters 84(24), 5496 (2000)), and reports a measured value of the transition frequency of: 2 466 061 413 187 104(46) Hz.

Did you take into account fine structure; i.e.,the energy difference of the spin -orbit coupling ?

Creator

 

[Creator]

Oops.:zzz:

Apparently, fine structure won't be enough to cover the discrepancy;...
...try re-calculating using the 'reduced mass'.
That gives a corrected difference from the Bohr/Schrödinger formula which may account for the discrepancy.

Creator

 

[jtbell]

There are also corrections due to hyperfine structure, and higher-order QED effects such as the Lamb shift. Atomic physics isn't my field, so I don't know how big these effects are for this transition, offhand, or where to look for detailed calculations.

At any rate, in order to get really accurate values for transition energies, you have to go beyond undergraduate-level quantum physics.

 

[Physics Monkey]

Hi Aether,

I assume by "exact fine structure" you mean the Dirac energies. You will definitely have to include the hyperfine structure (the finite size of the nucleus has a relatively strong effect on the 1s state for example) and also the Lamb shift. If you have a library about I suggest you pick up the book "Physics of Atoms and Molecules" by Bransden and Joachim for further information.

I find it amusing that even good ol' hydrogen is insanely complicated; physics is fun.

 

[Aether]

Oops.:zzz:

Apparently, fine structure won't be enough to cover the discrepancy;...
...try re-calculating using the 'reduced mass'.
That gives a corrected difference from the Bohr/Schrödinger formula which may account for the discrepancy.

Creator

Using the reduced mass I get 2466.058 THz vs. the experimentally determined value of 2466.061 THz. Haha...the online calculator is wrong. Thanks.

There are also corrections due to hyperfine structure, and higher-order QED effects such as the Lamb shift. Atomic physics isn't my field, so I don't know how big these effects are for this transition, offhand, or where to look for detailed calculations.

At any rate, in order to get really accurate values for transition energies, you have to go beyond undergraduate-level quantum physics.

Yes, the paper (M. Niering et al., 2000) includes a correction for hyperfine splitting of f_hf=310 712 233(13) Hz. A different paper (Th. Udem et al., PRL 79(14), 2646 (1997)) gives the 1S-Lamb shift as 8172.876(29) MHz. Thanks.

Hi Aether,

I assume by "exact fine structure" you mean the Dirac energies. You will definitely have to include the hyperfine structure (the finite size of the nucleus has a relatively strong effect on the 1s state for example) and also the Lamb shift. If you have a library about I suggest you pick up the book "Physics of Atoms and Molecules" by Bransden and Joachim for further information.

I find it amusing that even good ol' hydrogen is insanely complicated; physics is fun.

Hi Physics Monkey,
Yes, that is what I (D.J. Griffiths actually) mean by "exact fine structure". I'll keep that book in mind. Thanks.