Phase-Space Approach to the Tunnel Effect: A New Semiclassical Traversal Time

Abstract

We determine the semiclassical coherent-state propagator for a particle going through onedimensional evolution in a simple barrier potential. The described semiclassical method makes use of complex trajectories which, by its turn, enables the definition of (real) traversal times in the complexified phase space. We then discuss the behavior of this time for a wave packet whose average energy is below the barrier height. [S0031-9007(97)04310-X]

The semiclassical coherent state propagator for systems with spin

“…(17) and (19) define the critical path of the Feynman integral (10). Nevertheless, these trajectories must satisfy the boundary conditions z(0) = z ′ , s(0) = s ′ , z * (T ) = z ′′ * and s * (T ) = s ′′ * , as can be seen from Eqs.…”

“…In spite of this, a first numerical application of the semiclassical formula was performed by Adachi [8] for an two-dimensional chaotic map, obtaining good agreement with exact quantum results. The correct evaluation of the second order fluctuations appeared in the works of Baranger and Aguiar [9], Xavier and Aguiar [10,11,12] and, independently, by Kochetov [13]. However, a detailed derivation of the semiclassical coherent state propagator for one-dimensional systems has appeared only later in [14].…”

Semiclassical tunnelling of wavepackets with real trajectories

“…Next we describe the evolution of a Gaussian through a square potential barrier in its three separate regions: before, inside and after the barrier. Finally in section IV we discuss the calculation of tunneling times, as proposed in [2]. We find that the barrier slows down the wavepacket at high energies, but that it speeds it up at energies comparable to the barrier height.…”

“…With this addition the semiclassical approximation becomes again very accurate inside the barrier. In the next section we shall briefly discuss the possibility of using our results to calculate the tunneling time as defined in [2].…”

“…In the case of plane waves, the tunneling and reflection coefficients can be easily calculated in the semiclassical limit, giving the well known WKB expressions [1]. For wavepackets, however, the problem is more complicated and few works have addressed the question from a dynamical point of view [2,3,4]. The time evolution of a general wavefunction with initial condition ψ(x, 0) = ψ 0 (x) can be written as ψ(x, T ) =< x|K(T )|ψ 0 >= < x|K(T )|x i > dx i < x i |ψ 0 >,…”