Quantum-mechanical calculation of travel time across a potential barrier and the problem of classical limit

Summary

Conditions are examined under which the quantum-mechanical travel time across a potential barrier goes over to the classical result. It is found that the limiting process is quite complex and sensitive to many details such as the nature of the incident state vector and the analytic properties of the potential barrier. An interesting result is that in the present problem the classical limit is reached for a wave packet that is sharply localised in momentum space only and is insensitive to the spatial extension. This is in sharp contrast to the results we obtained for scattering. Another significant conclusion is that the classical limit may converge either to that of a single classical particle or of a classical statistical ensemble depending on the nature of the initial wave packet.