Hydrogen energy levels

The kinetic energy operator in the Schrodinger equation corresponds to the quadratic term in this nonrelativistic expansion, and thus the Schrodinger equation describes only the leading nonrelativistic approximation to the hydrogen energy levels. 

The contributions that have been considered in order to obtain precise theoretical expressions for hydrogenic energy levels are as follows the Dirac eigenvalue with reduced mass, relativistic recoil, nuclear polarization, self energy, vacuum polarization, two-photon corrections, three-photon corrections, finite nuclear size, nuclear size correction to self energy and vacuum polarization, radiative-recoil corrections, and nucleus self energy. 

QED contributions to the Lamb shift consist of electron self-energy and vacuum polarization terms. In one-electron atoms the former is both the larger and the more difficult to calculate and has been the focus of much recent theoretical work. Up to Feynman diagrams including two-loops the self-energy contribution to a hydrogenic energy level can be written as [32]... 

Abstract. The calculation of the last unknown contribution to hydrogen energy levels at order ma7, due to the three loop slope of the Dirac form factor, is described. The resulting shift of the nS energy level is found to be 3.16/n3 kHz. Adding this result to many known contributions to the 1S Lamb shift and comparing with experimental value, we derive the value of the proton charge radius rp = 0.883 0.014 fm. 

The contribution due to the three-loop slope of the Dirac form factor was the last unknown contribution to the hydrogen energy levels at order a3(Za)4. The two other contributions come from the three-loop electron anomalous magnetic moment and the three-loop vacuum polarization correction to the Coulomb propagator. These contributions can be extracted from the literature [10,13]. 

This paper describes the progress of a laser resonance experiment which aims to measure the Lamb shift in hydrogenic silicon with an accuracy that will allow it to test the two-loop binding corrections mentioned above. This in turn should allow the viability of calculable frequency standards, based on transitions in lower-Z one-electron systems such as hydrogen and He+, to be assessed. Following a review of some theoretical contributions to hydrogenic energy levels, the details of the laser resonance experiment are outlined. 

Table 2. The values of the relativistic recoil correction to hydrogen energy levels beyond the Salpeter contribution, in kHz. The values given in the second and third rows include the (aZ)6m2/M contribution and all the contributions of higher orders in aZ. In the last row the sum of the (otZ f m2/M and (otZ)7 og2 (aZ)m2/M contributions is given...

Fig. 2.25. Schematic diagram of the hydrogen energy levels in the plasma and in a-Si H.

It is inevitable that work of this kind faces limits as far as determination of numerical values for the constants and also as precision tests of fundamental theory. These limits may be seen to arise in two ways. First of all, the hydrogenic energy levels and their splittings are sensitive to some extent to the inner structure of proton. Secondly, since interesting electrodynamic effects scale with some power of Za, measurements of great precision are required. In the case of the former problem, one is attracted to the notion of the spectroscopy of purely leptonic systems, while the latter invites consideration of the single electron spectra of high Z ions. 

From the results contained in Table 4,2 the transition probabilities and radiative lifetimes of hydrogenic energy levels can be calculated using equations (4,30) and (4.26)... 

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