Quantum Tunneling Time:relativistic Extensions

Quantum tunneling is not only one of the most amazing phenomena in quantum mechanics, but also one of the most basic and important process in nature. It has been the focus of debates for decades since it was first pointed out by MacColl in1932. The analysis of tunneling time is complicated because time is different from that of quantities like position energy, momentum and so on. It is not usually treated as an operator; rather it is a parameter. To this end, several methods have been proposed, leading to different or controversial results. Salecker and Wigner illustrated a very simple way to calculate the time of quantum tunneling to avoid the use of measuring rods which are essentially macrophysical objects. The idea is to construct a clock coupled very weakly to the tunneling particle. Peres conducted in-depth research on the basis and indicated that we define the tunneling time in terms of the change in the clock variable from the time that the particle reaches the barrier to the time it emerges. And the Hamiltonian is time independent.Davies took Pere’s old prescription and illustrated a very simple way to calculate the time of quantum tunneling. The idea is to construct a clock coupled very weakly to the tunneling particle and then measure the time difference between two events, i.e. the particle’s entering and exiting the barrier. The clock measures only time differences between two events, not the absolute time of either event. Basing on this idea, Davies argued that for a stationary state Φ=φ(x)e-iEi the particle’s traveling time can be evaluated from the phase difference δ(E) of wave function between two points as T=δ’(E). Davies’ calculation is performed in non-relativistic quantum mechanics with Schrodinger equation.It is natural to extend his treatment to relativistic quantum mechanics, utilizing the Klein-Gordon equation or the Dirac equation. This is our mission in the present paper. Section3and section4are successful relativistic extensions of Davies’results with Klein-Gordon equation and Dirac equation respectively. In each case, both the potential step and the potential hill are considered, and we focus on the energy regime with negligible particle creation/annihilation effects. However, in relativistic quantum mechanics, particle and antiparticle creation/annihilation is a common phenomenon, sometimes playing a crucial role.An outstanding example is the Klein paradox. We will study this example and a similar one for potential step in section5. In both examples, the sojourn time turns out to be non-positive. This indicates the failure of Davies’method in the case of copious particle/antiparticle creation and annihilation. Section6, treating with similar examples for a potential hill, provides more evidence on this failure. In section7we summarize our results and discuss some open problems.