Flux projections onto channel-specific transit time eigenspaces Randall S. Dumont * Department of Chemistry, McMaster University, 1280 Main St. W., Hamilton, Ont., Canada L8S 4M1 Received 19 March 2002; revised 3 April 2002; accepted 3 April 2002 Abstract Time-evolved flux expectations are shown to provide channel-specific transit time probability distributions, including the distribution of nonadiabatic tunneling time, for one-dimensional scattering of an effective particle with n internal quantum levels, e.g. the close coupled equations description of bimolecular scattering. Results presented here strengthen and generalize the results of Dumont and Marchioro [Phys. Rev. A 47 (1993) 85]. The analysis hinges on the demonstration that the time- dependent flux operator can be interpreted as a projector onto a channel-specific (i.e. transmission or reflection in a specific adiabatic level) transit time eigenspace. q 2002 Elsevier Science B.V. All rights reserved. Keywords: Tunneling; Transit time; Nonadiabatic transition 1. Introduction There is a long-standing debate over how best to characterize the time elapsed when a particle tunnels through a barrier [1–3]. The controversy arises from the standard approach of quantum mechanics wherein one adopts a Hermitian operator to represent an observable of interest. Tunneling time constitutes simultaneous observations that the particle tunneled, and is observed at some reference point at a particular time. However, operators representing the two observables in question—the ‘tunneling flag’ and ‘transit time’ operators—do not commute, and the desired joint distribution function does not exist. Nevertheless, we will see that by restricting the space of initial wave packets, a tunneling time probability distribution can in fact be constructed. Some authors treat the problem of tunneling time in the energy domain, and emphasize the importance of a physical clock [4,5]. In this framework, the imaginary time [7–9] spent under the barrier (note that the effective momentum in the classically unallowed portion of the barrier is imaginary) plays an important role. Wave packet propagation, i.e. the time domain approach has also been considered [10–13]. 1 A key feature, made clear in the time domain, is the quantum speedup or Hartman effect [10]. The effective speedup of a tunneling particle results because the barrier acts as a filter for the high- energy components of the incident wave packet. One can associate a distribution of tunneling times to a propagating wave packet if a tunneling time operator is known. According to the standard approach of quantum mechanics [15], the distribution is given by the expectation values (with respect to the initial wave packet) of the projection operators onto the tunneling time operator eigenspaces. Sojourn or transit time operators have been proposed for this 0166-1280/02/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S0166-1280(02)00247-6 Journal of Molecular Structure (Theochem) 591 (2002) 267–276 www.elsevier.com/locate/theochem * Tel.: þ1-905-525-9140; fax: þ 1-905-522-2509. E-mail address: dumontr@mcmaster.ca (R.S. Dumont). 1 Wave packet propagation with coupling to a Larmor clock is treated in Ref. [6].