Erratum to: The hydrogen atom without external fields
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(18.5A) 20{j. Corrections for nuclear structure. In the text we have shown that there is a contribution to the level shift of 5-states, due to the finite size of a nucleus, which is proportional to the mean square radius (r2 ) of the charge distribution inside the nucleus. We have stated that a similar but smaller spread of charge of the proton cannot yet be calculated from meson theory. This is still true but 1 1
J. W. DuMoND: Phys. Rev. 106, 501 (1957). Phys. Rev. 107, 328 (1957).
R. L. SHACKLETT and
C.
M. SOMMERFIELD:
Bethe and Salpeter, Quantum Mechanics.
H. A. Bethe et al., Quantum Mechanics of One- and Two-Electron Atoms © Springer-Verlag Berlin Heidelberg 1957
23
352
BETHE and SALPETER: Quantum Mechanics of One- and Two-Electron Systems.
experiments on high energy scattering of electrons by protons do give an experimental value for the mean square radius (r 2 ) of the proton charge distribution. This value is somewhat larger than one might have suspected on purely theoretical grounds and is 1 (r 2 )
= (0.77 ± 0.10) 2 x w- 26 cm2 ~ 2.1 x w-10 a 2
{20-JA) for the proton, while the equivalent quantity (r 2 ) for the neutron is almost exactly zero. This spread of the proton charge contributes an amount 2 of (+0.12±0.03) Me to the shift of the 25-state in hydrogen. We quoted in our text a contribution of + 0.73 M c to the shift of the 2 5state in deuterium, due to the charge spread inside the deuteron (made up of one proton and one neutron). This calculation also had not taken into account the charge spread of the proton and recent experiments 3 on electron-deuteron scattering indicate that this contribution of + 0.73 M c for deuterium should also be increased by approximately 0.12 M c. 21. Fine structure and the LAMB shift. The theoretical expression (21.5) for the fine structure separation 2J1- 2.11 in deuterium can be written in the form eRn 2 [1+ 2g + -cx 5 2 -m rx J F=--cx 1 16 8
Mv :rr: '
(21.5A)
where g1 is the anomalous magnetic moment of the electron. If we accept the change in the theoretical value of g1 from expression (18.5) in the text to expression (18.5 A) in these Addenda, we have to modify the value in (21.6) for the fine structure constant to ...!._ = 137.0390 ± 0.0012. {21.6A) (X The change of the fourth order moment contribution to g1 from -2.973 cx 2/n2 to -0.328 cx 2 jn2 affects the LAMB shift marketlly: The fourth order moment contributes about -0.94 M c to the n = 2 LAMB shift in H and in D if we use the old value of g1 , but only - 0.10 M c if we use the new one. Further, the change in the fine structure constant from (21.6) to (21.6A) decreases the LAMB constant L in (21.8) by about 43 ppm and decreases 5~ by about 0.05 M c. If we accept these changes and add the contributions due to nuclear size, discussed in Sect. 20{J of these Addenda, the theoretical values in Table 3 for the LAMB shift (in M cjsec) are changed to H
D
1058.03±0.15
He+
1059-38±0.15114055±3
Comparison with Table 3 shows that these changes decrease the magnitude and change the sign of the discrepanc
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