According to the Dirac equation with static Coulomb source the energy levels with same quantum numbers nand j are degenerate. Lamb and Retherford have measured in 1947 an energy splitting (called Lamb shift) of about 1 GHz between the 2S1/2 and the 2P1/2
energy levels. This energy difference is not the tiny l dependence in Eq. (A.12) which accounts for 2 kHz for the 2S1/2−2P1/2 energy splitting. This discovery together with the
110 Theory of hydrogen energy levels
discovery of the electron’s anomalous magnetic moment has triggered the development of quantum electrodynamics (QED).
Generally speaking the “Lamb shift” is any deviation from the energy level predicted by the Dirac equation (partially corrected for the finite nucleus mass). In some application [19] the Lamb shiftLnjl is defined as the sum of all contributions to the energy levels beyond the first four terms in Eq. (A.12) averaging over (or without considering) the hyperfine splitting 1 contributions EnjlFhfs EnjlFtot =mc2+M c2+ (f(n, j)−1)mrc2 −(f(n, j)−1)2 m 2 rc2 2(m+M) +Lnjl+E hfs njlF (A.13) whereEtot
njlF is the total binding energy of the level characterized by then, j, l, F quantum numbers, F being the total angular momentum. The first two terms always cancel out when energy differences between two levels are measured. The third term contributes about 2.5×103 THz to the 1S−2S energy splitting in hydrogen, while the forth term
(recoil) contributes about 22 MHz. The difference of 1Sand 2Slevel Lamb shifts is about
L1S−L2S '8−1 = 7 GHz. The hyperfine splitting of the 1Sstate isEtot1S(F=1)−E1totS(F=0)' 1420 MHz. In muonic hydrogen the hyperfine structure is considered as a part of the Lamb shift.
This gives an overview of the size of the various contributions. Noticeable is the size of the main finite mass correction (mrinstead ofmin the third term) of about 1 THz relative to the size of the Lamb shift of about 7 GHz. It is desirable that as much as possible of the correct mass dependence is included in the effective Dirac equation in order to avoid treating all effects of the motion perturbatively.
The Lamb shift does not arise in the framework of quantum mechanics with a potential but it requires field–theoretical methods which can be written in terms of Feynman graphs. This quantum electrodynamic (QED) corrections to the energy levels can be written in the form of a power series expansion in three small dimensionless parameters: α, Zα, and m/M. These parameters, in particular α and Zα, enter the wave function and the particle propagator (via the energy, see Eq. (A.18) in the next section) in a nonperturbative way. Although bound–state QED is nonperturbative, it is possible to make use of these small parameters to develop expressions in increasing order of smallness. However the nonperturbative nature of this expansion shows up in the coefficients of the power series. Some of these coefficients are not constants but slowly varying functions (e.g., ln (Zα)−2
in Eq. (A.22)) of the expansion parameters. Hence the energy shift caused by a given kernel can not be estimated by simply counting the powers in α and Zαit contains.
It is convenient to classify the various contributions to the Lamb shift in accordance with the small parameters they depend on. They may be classified in four groups which are listed in decreasing size: radiative, recoil, radiative–recoil, and finite nuclear size and structure corrections.
Radiative corrections : They take into account pure QED effects like self–energy and vacuum polarization for an electron in a Coulomb potential of an infinitely heavy and point–like nucleus. These contributions depend only onα and Zα.
1The Dirac equation with a Coulomb source takes into account the electron spin but does not include
the magnetic moment of the heavy nucleus, and hence the hyperfine contributions has to be taken into account separately.
A.4 111
Table A.1: Theoretical expectation (not updated) of the 1S and 2S Lamb shifts taken from
M. Niering’s thesis [120]. Note that uncertainty related to proton radius extracted from scattering experiment [121] has been underestimated. The first uncertainty of the summed contributions refers to the pure theoretical uncertainty and the second to the uncertainty caused by the uncertainty of the proton radius.
Contribution 1S state Uncertainty 2S state Uncertainty
[kHz] [kHz] [kHz] [kHz]
radiative (one–loop) 8 181 285.5 0.2 1 046 061.43 4×10−2 radiative (two–loop) 721 24 90.2 3×10−1
recoil 2 402.118 0.002 340.3646 3×10−4 radiative–recoil −12 777.97 0.04 −1.635.437 5×10−3
finite nuclear size 1167 32 145.8 4
sum 8172 798(32)(24) 1045 002.4(4.0)(0.3)
Recoil corrections : They describe corrections due to the finite mass of the nucleus, apart for the two larger finite mass contributions which are already included in the third and forth term of Eq. (A.13), without considering any QED correction. These contributions arise from the fact that the introduction of the reduced mass can not account for all recoil corrections in a relativistic two–body system. They depend simultaneously only on Zαand m/M.
Radiative–recoil corrections : They take into account mixed radiative and recoil con- tributions, e.g., recoil contributions with one or more photon loops on the fermion line. They depend simultaneously on α, Zα, and m/M.
Finite nuclear size corrections : The assumption that the nucleus is point–like is re- leased. The extension of the nuclear charge causes a deviation of the (1/r) Coulomb potential. An additional contribution comes from the nucleus polarization which arises from the interaction of the electron with the nucleus excited in a virtual state. It is fortunate that the different physics involved can be largely compartmentalized, e.g.,
the recoil and nuclear structure corrections can be neglected when treating the main ra- diative corrections and vice versa. Nevertheless, the actual level of precision (experimental and theoretical) has reached a point where some corrections have to be treated together as the recoil–radiative corrections. Table A.1 summarizes the contributions to the hydrogen Lamb shift for the 1S and 2S state according to the groups defined above. Note that the reported values refer to the status in the year 1999. As will be explained later the theoretical uncertainty has been reduced in the past five years by an order of magnitude, whereas the uncertainty related to the proton radius from electron scattering experiments has been underestimated (cf. Eq. (A.38)). In the following, for brevity, it was chosen to report some details only for the radiative and the nuclear finite size corrections. This is justified since the radiative corrections are the largest ones and their uncertainty domi- nates the theoretical uncertainty of the Lamb shift besides the finite size effect. The finite size corrections contributes only 1.4 ×10−4 to the Lamb shift, but contain the largest uncertainty caused by the uncertainty of the rms proton radius extracted from electron scattering experiments.
112 Theory of hydrogen energy levels
p
k q =p−k
Figure A.1: Self–energy graph for a free (left) and a bound (right) electron. Single and double straight lines represent the free electron propagator and the bound electron propagator respectively.
= + + + + . . . .
Figure A.2: Expansion of the bound self–energy in terms of Coulomb fields. The “X” represents the nucleus.