The “Tunneling-Time Problem” for Electrons

Abstract

The very concept of the motion of an electron during the time interval between its preparation in a given state and the subsequent “collapse” of the time-evolved state during a measurement is widely regarded as a meaningless one within conventional quantum mechanics. Despite this, a large number of theoretical papers, including several reviews (e.g., Hauge and Stovneng 1989; Landauer and Martin 1994), have been devoted to various characteristic times associated with the motion of a particle interacting with a potential barrier. Most of the approaches involve, at least implicitly, some relatively straightforward extension of the fundamental postulates of standard quantum theory. These will be referred to as “conventional” approaches to distinguish them from those based on alternatives to quantum theory, such as Bohmian mechanics. In this paper, the approach based on Bohm’s theory is compared to several “conventional” methods that do not involve such radical departures from orthodoxy. In section 2, the characteristic times of interest are defined in words and the underlying reason for the difficulty in translating these words into unique, universally accepted mathematical expressions discussed. In Section 3, Bohmian mechanics is applied to the derivation of expressions for the mean dwell, transmission, reflection and arrival times. The systematic projector approach of Brouard et al. (1994) is considered in Section 4 where it is shown that none of the infinite number of possibilities for the mean transmission time that are generated by this method can be equal to the (unique) result from Bohmian mechanics. In Section 5, “conventional” probability current approaches are criticized from the point of view of Bohm’s theory. In Section 6, the quantum clock of Salecker and Wigner (1958) is applied to the calculation of the mean and mean-square dwell time. The results for the free particle case (no barrier) are particularly illuminating. Concluding remarks are made in Section 7.