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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Self-Interaction Corrections in a Nonrelativistic Stochastic Theory of Quantum Mechanics

L. de la Peña-Auerbach, Ana Marı́a Cetto

Published Feb 15, 1971• 31 citations
Publisher

Abstract

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 $2s$ level amounts of 60% of the total effect.

DOI
10.1103/physrevd.3.795

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Physical review. B, Condensed matter•Feb 1, 1991•12 citations
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.
A nonlinear equation for the description of self-interaction corrections is developed by means of stochastic mechanics. It is derived within a nonrelativistic framework in which fields do not appear. The linearized approximation of this equation is also examined, and an alternative, more conventional, interpretation of the resulting effective interaction is given. Calculations involving effects which have a possible interpretation as the Lamb shift and anomalous magnetic moment of the electron, and which are free of cutoffs, are made; these give semiquantitative agreement, offering some encouragement for a possible attempt at a relativistic generalization.
A set of basic evolution equations for the mean values of dynamical variables is obtained from the Fokker–Planck equation applied to the general problem of a particle subject to a random force. The specific case of stochastic electrodynamics is then considered, in which the random force is due to the zero-point radiation field. Elsewhere it has been shown that when this system reaches a state of energy balance, it becomes controlled by an equation identical to Schrödinger’s, if the radiationless approximation is made. The Fokker–Planck equation was shown to lead to the Ehrenfest theorem under such an approximation. Here we show that when the radiative terms are not neglected, an extended form of the Ehrenfest equation is obtained, from which follow, among others, the correct formulas for the atomic lifetimes and the (nonrelativistic) Lamb shift.
We apply the theory of stochastic electrodynamics to the study of the (nonrelativistic) harmonic oscillator by using the Fokker–Planck method. It is demonstrated that the equilibrium distribution in phase space is exactly equal to that given by quantum statistical mechanics, i.e., the corresponding Wigner distribution, and that analysis of this distribution by means of a decomposition in terms of canonical densities leads automatically to the usual description of quantum mechanics in terms of excited states. All fundamental equations of quantum mechanics are recovered as aprproximations to zero order in the radiation terms; the first-order terms lead to the radiative corrections predicted by quantum electrodynamics, namely, the decay of states and the Lamb shift of the energy levels. The necessary differences between both treatments of the oscillator and their implications are briefly discussed.
In the literature of calculating atomic and molecular structures, most Schrodinger equations are described by Coulomb potential. However, there are also a few literatures that discuss some magnetic correction methods, such as Pauli and Shortley's early work. But in fact, the calculation accuracy of these Schrodinger equations is not consistent with Lamb shift. Therefore, in the traditional ab initio calculation of quantum mechanics, it is common and necessary to use Dirac theory or quantum electrodynamics (QED) to correct the energy level of Schrodinger equation. However, the calculation of Feynman diagram is a daunting problem, including the application of self-consistent field in relativity and density functional theory. So recently, we have noticed the simplicity of the modified Newtonian mechanics, and we think that quantum mechanics will have similar properties. Here, we state this and improve the correction function in our previous action potential. In addition, through the demonstration of hydrogen-like and helium-like systems here, it can be proved that this conclusion is a potential application, that is, the energy level structure of our modified Schrodinger equation is consistent with Lamb shift.
A natural all-order decorrelation procedure is described for the interaction of a single two-level atom with all modes of the radiation field which replaces the semi-classical boson approximation for the atom by an exact fermion treatment. In consequence the vacuum Lamb shift appears as twice the Bethe shift for each level separately instead of the limiting neo-classical shift obtained in the Bose theory. Stimulated emission and absorption modify the Einstein A-coefficient and the vacuum shift is extended to include a field dependent shift. The all-order decorrelated perturbation theory yields an equation of motion for the single-atom dipole moment which includes those reported from operator radiation reaction theory recently; but it also generalizes these by including the effects of stimulated processes. The method is extended to a many-atom case in which it becomes necessary to modify the procedure slightly.
Hydrodynamical equations for the probability density \ensuremath{\rho} and the local velocity v of a non-Markovian stochastic process have been recently obtained. The corresponding complex equation in terms of the wave function \ensuremath{\psi} coincides with the Schr\"odinger equation if the first-order terms only are retained. The second-order terms vanish and the third-order terms could be considered as QED-type corrections. To obtain the coefficient D4{\mathit{D}}_{4}D4​ of the third-order term the probability density \ensuremath{\rho} for a free particle is obtained by stochastic electrodynamics. This solution is substituted in the new complex equation, thus obtaining D4{\mathit{D}}_{4}D4​. By this coefficient the calculation of the Lyman-\ensuremath{\alpha} wavelength is in agreement with the best direct measurements for the transition 2P\ensuremath{\rightarrow}1S. The 1S displacement is 1% of the QED value for the Lamb shift. It is suggested how the new negative contribution could be the difference between the (1+0.01) QED value for the 3S state and the corresponding experimental value, which is 20% less than that given by QED.