Aspects of the Theory and Computation of Resonances, with Applications to Field-Free and Field-Dressed Atomic States

Abstract

A review of certain aspects of the theory and computation of resonance states is presented, from the point of view of the work by the author and his colleagues in atomic physics. Two issues are mainly discussed: one is the understanding and ab initio calculation of resonance states of real systems from a time-dependent point of view. The other is the derivation and application of the complex eigenvalue Schrödinger equation from a superposition of the localized wave packet Ψ₀ with the orthogonal to it scattering wave functions ϕ(E), when outgoing-wave boundary conditions are imposed. It is shown how two complex adjoint solutions, the hallmark of resonance state theory, correspond to the Fano solution for a resonance state on the real energy axis, obtained from the application of Hermitian quantum mechanics. The forms of the complex eigenfunctions are used for non-Hermitian calculations of resonance states in polyelectronic atoms. The question of time-asymmetry at the quantum level is tackled by observing that the time-evolution has to be considered with boundary conditions t ≥ 0 and ∞ > E > 0 and a complex energy distribution given by the diagonal matrix element of the Green's function with respect to Ψ₀. Using a model whereby the self-energy of the decaying state, A(z), is approximated by A(z) ≈ A(E ₀), where E ₀ = 〈Ψ₀|H|Ψ₀〉, it is shown that time-asymmetry, if present as defined in this work, should have an effect on the as yet unobserved long-time deviation from exponential decay. Although not described explicitly, it is indicated, via the forms of the trial wave functions and via the references, how poly-electronic calculations have been carried out, for field-free resonance states as well as for resonance states that are created by the presence of an external electromagnetic field.