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The Journal of Chemical Physics

The significance of imaginary time in quantal reactive scattering

Eli Pollak

Published Aug 1, 1985• 51 citations
Publisher

Abstract

The concept of time in quantal tunneling processes is reexamined. We find that the Wigner–Eisenbud definition of real time and the definition of imaginary time may be understood in terms of stationary phase analysis in complex time of the microcanonical flux–flux correlation function. This analysis explains why the real time should not be used to justify adiabatic approximations for perpendicular degrees of freedom at tunneling energies. A semiclassical analysis shows that, as suggested by Buettiker and Landauer, the imaginary time should be used to determine the validity of the adiabatic approximation for tunneling processes. Numerical examples for the hydrogen exchange reaction are provided. The implications on adiabatic and sudden approximations in reactive scattering are discussed. A theory unifying the two approaches is outlined.

DOI
10.1063/1.449473

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The concept of time in quantal tunneling processes is reexamined. We find that the Wigner–Eisenbud definition of real time and the definition of imaginary time may be understood in terms of stationary phase analysis in complex time of the microcanonical flux–flux correlation function. This analysis explains why the real time should not be used to justify adiabatic approximations for perpendicular degrees of freedom at tunneling energies. A semiclassical analysis shows that, as suggested by Buettiker and Landauer, the imaginary time should be used to determine the validity of the adiabatic approximation for tunneling processes. Numerical examples for the hydrogen exchange reaction are provided. The implications on adiabatic and sudden approximations in reactive scattering are discussed. A theory unifying the two approaches is outlined.
It is widely believed that the WKB approximation to quantum barrier penetration is described in the path-integral formalism by a saddle-point approximation based on an expansion about an imaginary-time semiclassical path. This article examines the relationship between tunneling amplitudes and imaginary-time paths. It is shown, using a simple example, that for some tunneling amplitudes the dominant semiclassical path cannot be chosen to have purely imaginary time if all other variables are chosen to be real.
Asymptotic time evolution of a wave packet describing a non-relativistic particle incident on a potential barrier is considered, using the Wigner phase-space distribution. The distortion of the trasmitted wave packet is determined by two time-like parameters, given by the energy derivative of the complex transmission amplitude. The result is consistent with various definitions of the tunneling time (e.g. the Büttiker-Landauer time, the complex time and Wigner's phase time). The speed-up effect and the negative dispersion are discussed, and new experimental implications are considered.
We study tunneling in one-dimensional quantum mechanics using the path integral in real time, where solutions of the classical equation of motion live in the complex plane. Analyzing solutions with small (complex) energy, relevant for constructing the wave function after a long time, we unravel the analytic structure of the action, and show explicitly how the imaginary time bounce arises as a parameterization of the lowest order term in the energy expansion. The real time calculation naturally extends to describe the wave function in the free region of the potential, reproducing the usual WKB approximation. The extension of our analysis to the semiclassical correction due to fluctuations on the saddle is left for future work.
The concept of ‘tunnelling time’ in the context of quantum particle tunnelling is reviewed. Various suggestions of linking the tunnelling dynamics with a characteristic time (real or complex) like the phase time, barrier interaction time (bounce time), presence time, etc. are analysed. A simple but fully quantal method of defining and estimating a real tunnelling time is examined in a variety of situations. The recently proposed idea of interpreting ‘tunnelling time’ as the cavity lifetime of a particle is also explored. We emphasize that proton or H-atom transfer reactions in double or triple wells offer systems in which the signature of the tunnelling time should be recognizable not just indirectly through the tunnelling splitting of spectral lines, but by following the relaxation dynamics of the subsystem that the proton or H atom leaves by tunnelling.
How much time does it take for a particle to tunnel through a barrier? This question continues to baffle up to this very day, one of the reasons being the non-existence of an energy–time commutator analogous to the position–momentum commutation relation. Instead of worrying about the definition of an operator one may consider time as a parameter in the time-dependent Schrödinger equation and study the time evolution of a wavepacket appropriately scattered on a potential barrier. Using this approach, one may formally define a tunnelling flight time that may be measured in a time-of-flight experiment. The resulting tunnelling flight time either vanishes or is very small. The implications of this observation with respect to recent hydrogen and helium atom attosecond photoionization experiments are discussed. The vanishing or small flight time does not contradict the finite time measured in Larmor clock experiments, where the tunnelling particle affects an external degree of freedom whose dynamics induced by the tunnelling may be interpreted in terms of a time scale. Flight times are also relevant to quantum reflection where coherences due to resonance scattering are not well accounted for by Wigner's phase time delay.