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The energy levels calculated from the Dirac equation with the electron in a Coulomb potential is given by: where Z is the number of positive charges, m is the mass of the electron, and is the fine structure constant. The formula predicts that the energy levels of the hydrogen atom should depend only on the principal quantum number n and the total angular momentum j = L + S, so that the energy levels of the 2S1/2 (n=2, j=1/2) and 2P1/2 (n=2, j=1/2) states should be degenerate (the same). In 1947, Lamb and Retherford demonstrated that this is not the case. The 2P1/2 and 2S1/2 energy levels are separated by more than 1000 MHz (see Figure 03a). | |
Figure 03a Lamb Shift |
overlapped more with the proton charge cloud and shifted more accordingly. By summing the contributions from : fine structure, hyperfine splittings, radiative corrections, recoil and proton size, ..., the total computed 2S1/2F=1 - 2P3/2F=2 energy difference (the dominate transition) is : E = 209.9779(49) - 5.2262(rp)2 + 0.0347(rp)3 where rp is the root-mean-square charge radius of the proton, numbers in parenthesis indicate uncertainty, and the energy is expressed in unit of | ||
Figure 03b Muonic Lamb Shift [view large image] |
Figure 03c Laser Resonance [view large image] |
mev = 10-3ev. The first term in the formula is dominated by vacuum polarization, which causes the 2S states to be more tightly bound than the 2P states (Figure 03b,c). |
A report in 2013 indicates that the proton size as measured from the muonic hydrogen atoms is indeed smaller (see Fiigure 03d). The accuracy of the new data is two times better than the previous attempt. Various new forces are proposed to resolve the discrepancy. They are constrained by the requirements that it should not deviated too much from the Standard Model, especially the property of neutrinos cannot be altered beyond the current observation. Such constraints can be bypassed with the "dark photon" (associate with dark matter) or gravity leaking to another dimension. All the remedies look rather contrived. The "tidal force" of the electron or muon on the proton has also been measured to be too small to explain the | |
Figure 03d Proton Size Discrepancy [view large image] |
discrepancy. It seems that the inherent difficulty of defining the proton size is at the root of the problem as explained below. |
The traditional assumption for proton mass distribution is about 75% concentrated in a central core (size rp) with the other 25% lying outside in the halo up to 1.4 fm. An August 2010 calculation suggests that the discrepancy between the original proton size and the latest (2010) | |
Figure 03e Latest Proton Size |
experimental data can be reconciled if the halo band extends 4.7 times as far as the previous definition (Figure 03e). |