Unstable states in a model of nonrelativistic quantum electrodynamics: corrections to the Lorentzian distribution

TL;DR

This paper analyzes the decay behavior of atomic resonances in a quantum electrodynamics model, showing that the return probability amplitude combines exponential decay with universal corrections, improving understanding of resonance stability.

Contribution

It provides a new approximation method for the resolvent matrix element, avoiding complex energies, and offers a physical interpretation of decay corrections and lifetime characterization.

Findings

Return probability amplitude includes exponential decay and universal correction.

Approximation method avoids complex energies and unphysical Riemann sheets.

Characterization of sojourn time as the average lifetime of the decaying state.

Abstract

We review the Lee-Friedrichs model as a model of atomic resonances in the hydrogen atom, using the dipole-moment matrix-element functions which have been exactly computed by Nussenzveig. The Hamiltonian $H$ of the model is positive and has absolutely continuous spectrum. Although the return probability amplitude $R_{\Psi}(t)=(\Psi,{t\exp(-iHt)\Psi)$ of the initial state $\Psi$, taken as the so-called Weisskopf-Wigner (W.W.) state, cannot be computed exactly, we show that it equals the sum of an exponentially decaying term and a universal correction $O(\beta^{2}\frac{1}{t})$ for large positive times $t$ and small coupling constants $\beta$, improving on some results of C. King. The remaining, non-universal part of the correction is also shown to be of the same qualitative type. The method consists in approximating the matrix element of the resolvent operator in the W.W. state by a Lorentzian distribution. No use is made of complex energies associated to analytic continuations of the resolvent operator to "unphysical" Riemann sheets. Other new results are presented, in particular a physical interpretation of the corrections and the characterization of the so-called sojourn time as the average lifetime of the decaying state, a standard quantity in (quantum) probability.