Natural Line Width: Wigner-Weisskopf Treatment

Abstract

Because we consider a world made of atomic systems and electromagnetic fields, i.e.. atomic systems couplcd to the photon fields, we must consider not only the atomic bound states with zero photons present, i.e.. states such as |A n0›, but also atomic bound states with one photon present, states such as \( |{A_{n'}},\vec k\mu \rangle \). Because the coupling between atoms and photons is weak, states with two photons, such as \( |{A_{n''}},\vec k\mu ,\vec k'\mu '{\text{ }}\rangle \), may be negligible in dominant order of perturbation theory, although they may also play a role, particularly for systems for which selection rules eliminate the one photon states. Fig. 68.1 shows the discrete spectrum of an atomic system with no photons present, with ground state, A 0, and excited states, A n. with n = 1.2, … The figure also shows the continuous spectra of states with one photon present. The atomic cigenstatc, A. e.g., is seen to be degenerate with the system where the atom is in the ground stale and the photon has energy,\( \hbar\omega=E_{{A_3}}^{\left(0\right)}-E_{{A_0}}^{(0)} \). It is also degenerate with the system where the atom is in the first excited state and the photon has energy \( \hbar\omega=E_{{A_3}}^{\left( 0 \right)}-E_{{A_1}}^{(0)} \) and with the system composed of the atom in the second excited state and with photon energy, \( \hbar\omega=E_{{A_3}}^{\left(0\right)}-E_{{A_2}}^{(0)} \). More precisely, the discrete excited state |A n 〉 of the bare atom is coupled to states, \( |{A_{n'}},\vec k\mu \rangle , \), in the continuum, where the photon energy lies between the above \( \hbar\omega \) and\( \hbar\omega+d{E_{photon}} \), wit\( \hbar \omega = E_n^{\left( 0 \right)} - E_{n'}^{(0)} \), where n < n. In other words, the discrete state is coupled to the photon continuum with the weighting factor \( p({E_{photon}})d{E_{photon}} = \frac{{Vol.}}{{{{(2\pi )}^3}}}\frac{{{\omega ^2}d\Omega}}{{\hbar{c^3}}}d{E_{photon}}. \)