Level Shift

Level Shift


the slight deviation of the fine structure of the energy levels of hydrogen-like atoms from the predictions of re-lativistic quantum mechanics based on the Dirac equation.

According to the exact solution of the Dirac equation, the atomic energy levels are doubly degenerate: the energies of states with the same principal quantum number n = 1, 2, 3,… and the same total angular momentum quantum number j= 1/2, 3/2, … should coincide regardless of the two possible values of the azimuthal quantum number l = j ± 1/2 ≤ n – 1 (excluding j + 1/2 = n when l = j –1/2 = n –1). In 1947, however, W. Lamb and R. Retherford measured by radio-frequency spectroscopy the splitting of the degenerate levels 2S1/2 (n = 2, l = 0, j = 1/2) and 2P1/2 (n = 2, l = l, j = 1/2) in the hydrogen atom. This displacement of the levels from the positions predicted by the Dirac theory is called the Lamb shift. The most recent experimental value of this quantity is LexpH= (1058.90 ± 0.06) megahertz (MHz).

The Lamb shift has been theoretically explained and calculated within the framework of quantum electrodynamics. The principal contribution to the shift is made by two radiative effects (seeRADIATIVE CORRECTION) and is approximately equal to α3R, where α is the fine structure constant and R is the Ryd-berg constant. The first effect is the emission and absorption of virtual photons by a bound electron (seeVIRTUAL PARTICLES); as a result, the effective mass of the electron is changed, and the electron acquires an anomalous magnetic moment. The second effect is the possibility of the virtual production and annihilation of electron-positron pairs in the vacuum, or what is known as vacuum polarization; this effect distorts the Coulomb potential of the nucleus at distances of the order of the Compton wavelength of the electron (∼4 × 10-11 cm). The contribution of the effects of the motion and structure of the hydrogen nucleus (a proton) has also been found.

The most recent theoretical value of the Lamb shift in hydrogen, LtheoryH = (1058.911 ± 0.012) MHz, is in complete accord with the experimental value. This is a brilliant confirmation of the fundamental principles of quantum electrodynamics. The measured and calculated shifts of other hydrogen levels and of levels in other hydrogen-like atoms (such as D and He+) are also in good agreement.


Sdvig urovnei atomnykh elektronov i dopolnitel’nyi magnitnyi moment elektrona soglasno noveishei kvantovoi elektrodinamike: Sb. statei. Edited by D. D. Ivanenko. Moscow, 1950.
Faustov, R. N. “Urovni energii i elektromagnitnye svoistva vodorodopodobnykh atomov.” Fizika elementarnykh chastits i atomnogo iadra, 1972, vol.3, issue 1, p.238.


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