Time dependence, complex scaling, and the calculation of resonances in many‐electron systems

Abstract

The theory of this paper deals with certain aspects of the formal properties of atomic and molecular highly excited nonstationary states and the problem of calculating their wave functions, energies, and widths. The conceptual framework is a decay theory based on the consistent definition and calculation of the t = 0 localized state, |Ψ. Given this framework, the following topics are treated: (a) The variational calculation of Ψ0 and E0 using a previously published theory that generalized the projection operator approach to many‐electron systems. (b) The exact definition of the resonance energy. (c) The possibility of bound states in the continuum. (d) The relation of Ψ0 to the resonance (Gamow) function Ψ and of the Hamiltonian to the rotated Hamiltonian H(θ) based on the notion of perturbation of boundary conditions in the asymptotic region. (e) The variational calculation of real and complex energies employing matrix elements of H and H2 with square‐integrable and resonance functions. (f) The mathematical structure of the time evolution of |Ψ0〉 and the possibility of observing nonexponential decays in certain autoionizing states that are very close to the ionization threshold. (g) A many‐body theory of atomic and molecular resonances that employs the coordinate rotation method.

Non-dispersive wave packets in periodically driven quantum systems

“…Using Sturmian functions, the properties of the resonances can be calculated directly using the complex rotation technique [119][120][121][122][123][124][125]. The price to pay is to diagonalize complex symmetric matrices, instead of real symmetric ones.…”

“…By inserting equations (121,122) in eq. (164), after appropriate account for the projection of the body-fixed frame (x ′ , y ′ , z ′ ) onto the laboratory frame (x, y, z), eq.…”

Reply to the Comment by Cleanthes A. Nicolaides

“…Moreover, for long times, Nicolaides work [4,5] predicts non-exponential decay to be an inverse power law proportional to t −2 . Again, later work has shown that this long-time non-exponential decay is actually proportional to t −3/2 [2].…”

“…In his earlier work, he also derived the expression for P (t) given by equation (1) in his Comment which he claims holds for all t. Numerical evaluation of this function as shown in [4,5] does indeed show an exponential region followed by non-exponential decay after many lifetimes for large R (= E r /Γ).…”

“…In particular, he uses a formal solution to the time dependent Schrödinger equation which was derived in [4,5] and plausible assumptions relevant to certain physical situations, together with experimental estimates of the two parameters E r and Γ, to argue that non-exponential decay for 14 C would only appear after hundreds of lifetimes, which would again be beyond experimental observation. In his earlier work, he also derived the expression for P (t) given by equation (1) in his Comment which he claims holds for all t. Numerical evaluation of this function as shown in [4,5] does indeed show an exponential region followed by non-exponential decay after many lifetimes for large R (= E r /Γ).In response, we consider the form of non-exponential decay at short and long times as predicted by Nicolaides' model. In the earlier papers [4,5], the complex function G(t) was derived where the survival probability P (t) = |G(t)| 2 .…”