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

Abstract

The probability, P(t), of the irreversible dissipation into a continuous spectrum of an initially (t = 0) localized (Ψo) nonstationary state acquires, as time increases, ‘memory’ due to the lower energy bound of the spectrum, and eventually follows a nonexponential decay (NED). Regardless of the degree of dependence on energy, the magnitude of this deviation from exponential decay depends on the degree of proximity to threshold, and on whether the theory employs a real energy distribution, one form of which is g(E) ≡ ⟨Ψo |δ(H − E)|Ψo ⟩, or a complex energy distribution, G(E) ≡ ⟨Ψo |(H − E + i0)|Ψo ⟩. It is the latter that is physically consistent, since it originates from the singularity at t = 0, which breaks the S-matrix unitarity, in accordance with the non-Hermitian character of decaying states. In order to test the quantum mechanical theory, we carried out semiclassical path integral calculations of the P(t) for an isolated narrow tunnelling state, whereby the truncated Fourier transform of a semiclassical Green function, Gsc(E), is obtained. The results are in agreement with the analytic results of quantum mechanics when energy and time asymmetry are taken into account. It is shown that the analytic structure of Gsc(E) is [Dregular + Dpole], where Dpole is a finite sum over complex poles, which are the complex eigenvalues, Wn, that the potential can support. The Wn are given by En + Δn − (i/2)Γn, where En are the real eigenvalues of the corresponding bound potential, Γn are the energy widths and Δn are the energy shifts, both expressed in terms of computable semiclassical quantities. The spherical harmonic oscillator (SHO) with and without angular momentum, and unstable ground states of diatomic molecules, are treated as particular cases. The exact spectrum of the SHO is recovered only when the Kramers–Langer semiclassical expression for the centrifugal potential is used, thereby bypassing the difficulty of the singularity at r = 0. The spectrum from the use of the quantal form l(l + 1) reduces to that of l(l + 1/2)2 in the limit of large l, i.e., for orbits far from r = 0. Using previously computed energies and widths for the vibrational levels of He22+1σ g2 1Σ g+, the application of two formulae for P(t), one derived from a Lorentzian real energy distribution and the other from the corresponding complex energy distribution, shows that, for the lowest level, in the former case NED starts after about 193 lifetimes, and in the latter after about 102 lifetimes. The fact that this difference is large should have consequences for the deeper understanding of irreversibility at the quantum level.

Semiclassical calculation of the pendulum period

“…) and is related to the semiclassical 'bounce time' t bounce the particle spends under the barrier [41]. Recalling our past research [28,42], it was related to the knowledge of the particle's position as well as to the energy indeterminacies during tunneling. On the other hand, T rev is the period for fractional revivals of wavepacket reconstruction and is given by the following relation [18] ( )…”