Quantum tunneling time and tunneling in time-dependent potentials: A general formulation and some exactly solvable models

The article reviews the basic physics involved in quantum tunneling time and tunneling in time-dependent potentials. A generalization of classical transit time is employed to define the quantum tunneling time and it is expressed in terms of the asymptotic values of the wave function and its derivative with respect to the energy and position. This enables us to calculate the tunneling time in terms of the derivatives of the reflection and transmission phase shifts with respect to the particle energy. Tunneling in time-dependent potentials is analyzed in terms of various multi-channel processes. Some exactly solvable models of time-dependent potentials are discussed.