Time asymmetry, nonexponential decay, and complex eigenvalues in the theory and computation of resonance states

Abstract

Stationary-state quantum mechanics presents no difficulties in defining and computing discrete excited states because they obey the rules established in the properties of Hilbert space. However, when this idealization has to be abandoned to formulate a theory of excited states dissipating into a continuous spectrum, the problem acquires additional interest in many fields of physics. In this article, the theory of resonances in the continuous spectrum is formulated as a problem of decaying states, whose treatment can entail time-dependent as well as energy-dependent theories. The author focuses on certain formal and computational issues and discusses their application to polyelectronic atomic states. It is argued that crucial to the theory is the understanding and computation of a multiparticle localized wavepacket, ⌿ 0 , at t ϭ 0, having a real energy E 0 . Assuming this as the origin, without memory of the excitation process, the author discusses aspects of time-dependent dynamics, for t Ϸ 0 as well as for t 3 ϱ, and the possible significance of nonexponential decay in the understanding of time asymmetry. Also discussed is the origin of the complex eigenvalue Schrödinger equation (CESE) satisfied by resonance states and the state-specific methodology for its solution. The complex eigenvalue drives the decay exponentially, with a rate ⌫, to a good approximation. It is connected to E 0 via analytic continuation of the complex selfenergy function, A(z), (z is complex), into the second Riemann sheet, or, via the imposition of outgoing wave boundary conditions on the stationary state Schrödinger equation satisfied by the Fano standing wave superposition in the vicinity of E 0 . If the nondecay amplitude, G(t), is evaluated by inserting the unit operator I ϭ ͐dE|EϾϽE| G(t) ϭ Ͻ ⌿ 0 |e ϪiHt |⌿ 0 Ͼ, then the resulting spectral function is real, g(E) ϭ|Ͻ⌿ 0 |EϾ| 2 , and does not differentiate between positive and negative times. The introduction of time asymmetry, which is associated with irreversibility, is achieved by starting from Ͻ ⌿ 0 |(t)e ϪiHt |⌿ 0 Ͼ, where (t) is the step function at the discontinuity point t ϭ 0. In this case, the spectral function is complex. Within the range of validity of exponential decay, the complex spectral function is the same as the coefficient of ⌿ 0 in the theory of the CESE. A calculation of G(t) using the simple pole approximation and the constraints that t Ͼ 0 and E Ͼ 0 results in a nonexponential decay (NED) correction for t ӷ 1/⌫ that is different than when a real g(E) is used, representing the contribution of both "in" and "out" states. Earlier formal and computational work has shown that resonance states close to threshold are good candidates for NED to acquire nonnegligible magnitude. In this context, a pump-probe laser experiment in atomic physics is proposed, using as a paradigm the He Ϫ 1s2p

Time-dependent tunnelling via path integrals. Connection to results of the quantum mechanics of decaying states

“…The motivation for this work came from the general interest in comparing the results of quantum and of semiclassical mechanics and from the existence of two such results on the long-time nonexponential decay (NED) of the nondecay (survival) probability, P(t), of an isolated nonstationary state. Specifically, there is agreement for this quantity between the analytic result that was obtained in a formal way by Nicolaides and Beck [1][2][3] via quantum mechanics, and that obtained later by Holstein [4], who obtained P(t) for tunnelling from a square well, using semiclassical path integral mechanics. Although Holstein did not discuss it, this agreement is significant because, as explained below, the result on NED has to do with the fundamentals of irreversible quantum dynamics connected to the transient formation of decaying (resonance) states.…”

“…The present paper is a contribution to the theory and calculation of nonstationary states from a time-dependent point of view. Specifically, among other things, we show that semiclassical mechanics of tunnelling, implemented via the path integral formalism on a one-dimensional model of generic character, produces the same results for the existence of complex eigenvalues (with energy shift and width), and for the time-dependent probability beyond the exponential decay (ED) regime, as the ones derived from the quantum mechanics of decaying states, subject to two physical constraints: the lower energy bound of the continuous spectrum, and the arrow of time (t > 0) characterizing irreversible dissipation [1][2][3].…”

Mean lifetime of a false vacuum in terms of the Krylov-Fock nonescape probability

“…(3) is not new; for instance, one can find a similar discussion in [19], but it is not common knowledge. The mean lifetime is often defined in the literature as (see [20][21][22], possibly one can add many other references)…”

Mean lifetime of a false vacuum in terms of the Krylov-Fock non-escape probability

“…( 3) is not new, for instance one can find similar discussion in [19], but is not a common knowledge. The mean-lifetime is often defined in the literature as (see [20][21][22], possibly one can add many other references)…”