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Physical Review A

Nonrelativistic Lamb shift for a nonadiabatic many-particle system: An application to thedtμ ion

Robert Bukowski, Bogumił Jeziorski

Published Nov 1, 1992• 1 citation
Publisher

Abstract

The soft-photon contribution to the Lamb shift of a bound state of a nonrelativistic many-particle system has been expressed by a general formula, which allows for the fully nonadiabatic treatment of the particles. This formula has been applied to estimate the radiative corrections to the binding energies of the S and P states of the dt\ensuremath{\mu} molecule. The correction for the weakly bound P state has been found equal to 0.048 meV, and for the ground state the corresponding value is -1.184 meV. Simple adiabatic models for dt\ensuremath{\mu} are examined and found to provide fairly accurate estimates of the Lamb shift for the two lowest states of this molecule.

DOI
10.1103/physreva.46.5437

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The soft-photon contribution to the Lamb shift of a bound state of a nonrelativistic many-particle system has been expressed by a general formula, which allows for the fully nonadiabatic treatment of the particles. This formula has been applied to estimate the radiative corrections to the binding energies of the S and P states of the dt\ensuremath{\mu} molecule. The correction for the weakly bound P state has been found equal to 0.048 meV, and for the ground state the corresponding value is -1.184 meV. Simple adiabatic models for dt\ensuremath{\mu} are examined and found to provide fairly accurate estimates of the Lamb shift for the two lowest states of this molecule.
Reports a natural all-order decorrelation procedure for the interaction of a single two-level atom with all modes of the radiation field which replaces the semiclassical boson approximation for the atom by an exact fermion treatment. In consequence, the spontaneous emission of the semiclassical theory is extended to include stimulated emission, a semiclassical Lamb-type shift is explicitly replaced by a shift satisfying the Bethe formula manifestly due to vacuum fluctuations, and there can be two intensity-dependent terms, one of which generalizes the Bethe formula.
The radiation damping terms of classical electrodynamics are introduced into a set of stochastic equations of nonrelativistic quantum mechanics to study their effects on the energy levels of a stationary system to first order in perturbation theory. In the potential case there appear correction terms to Schr\"odinger's equation, one of which is calculated by solving a subsidiary equation. In particular, we obtain for the Lamb shift an expression of essentially stochastic origin, whose structure is in complete agreement with known results. According to this calculation, which has the advantage of being devoid of divergence problems, the non-relativistic spinless contribution to the Lamb shift of the 2s2s2s level amounts of 60% of the total effect.
The influence of the modification of the free vacuum on the quantum Zeno and anti-Zeno effects, as well as the Lamb (energy) shift of a multilevel atom, is studied. A unitary transformation method has been used to include the counter-rotating terms and to obtain the ground state of the whole Hamiltonian. The anti-Zeno effect is recovered if the modified density of states (DOS) has a dip near the transition frequency. The modification of DOS by a dip or a peak will change the Lamb shift compared with that in the free vacuum. Together with our unitary transformation method, there are three different methods to obtain the change in the Lamb shift. Using our method we investigate the change in the Lamb shift and obtain a formula for the change, which is the same as obtained with the method of Louisell [Quantum Statistical Properties of Radiation (Wiley, New York, 1973)] but different from the method of Cohen-Tannoudji et al. [Atom-Photon Interactions: Basic Processes and Applications (Wiley, New York, 1992), pp. 317 and 417]. The correctness of the Lamb shift formulas obtained by the three methods can be easily testified experimentally.
An alternative approach is introduced for adiabatic and nonadiabatic calculations in the ABA Coulomb systems from H2+{\mathrm{H}}_{2}^{+}H2+​ to the positronium negative ion ${\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}$ (${\mathit{e}}^{\mathrm{\ensuremath{-}}}$e+{\mathit{e}}^{+}e+${\mathit{e}}^{\mathrm{\ensuremath{-}}}$) so that they are treated as molecular systems. The adiabatic vibrational solutions point out the most relevant ``nuclear'' configurations that should appear in the construction of the nonadiabatic wave function and reveal the importance of the AA vibrations at this stage. Applications provide evidence for the separability of the ground state of ${\mathrm{Ps}}^{\mathrm{\ensuremath{-}}}$ and give energy values for six S bound states of ${\mathrm{\ensuremath{\mu}}}^{+}$${\mathit{e}}^{\mathrm{\ensuremath{-}}}$${\mathrm{\ensuremath{\mu}}}^{+}$ that seem not to be reported so far.
It is shown that algebraic techniques based on the Lie algebra so(4,2) provide efficient tools for evaluating Lamb shifts and radiative decay rates for hydrogenic energy eigenstates as they systematically exploit the intrinsic symmetry of the hydrogenic Hamiltonian. As a main result in lowest order perturbation theory with respect to the fine-structure constant integral representations are derived for the complex-valued energy shifts of hydrogen-like ions from which Lamb shifts and radiative decay rates can be evaluated in a unified way, thus generalizing a recently discussed algebraic approach of Maclay. In order to exemplify the usefulness of this algebraic approach numerical results are presented for Lamb shifts and radiative decay rates which transcend the dipole approximation and contain the dipole approximation as a limiting case.