Quantum Traversal Time and Path Integrals

Abstract

To find out how much time a classical particle spends in a given region of space one only has to use a stopwatch. The same question posed in the context of quantum mechanics has caused controversy for several decades and remains controversial to date. As early as in 1932 McColl [1] noted that tunneling must be characterised not only by the transmission rate but also by the speed of transmission. The problem attracted renewed attention with the progress in nano-technology [3,4] and photonic tunneling experiments [2]. It has been hoped that a properly defined “traversal time” τ (i.e., the time a tunneling particle spends in the barrier) would, among other things, describe the response of a tunneling device to a time modulation of the barrier, provide an insight into the nature of the image forces affecting the tunneling rate and explain why a transmitted wavepacket appears to arrive at a detector ahead of the one that propagates freely. However, anyone interested in the subject soon discovers that standard texts on quantum theory offer neither a clear definition nor a unique recipe for determining the duration τ. Numerous attempts have been made to obtain a suitable quantum mechanical generalisation of the classical traversal time (for reviews see [5]) with approaches ranging from using specially designed “clocks” to invoking non-standard interpretations of the quantum mechanics, such as that of Bohm [6].