The tunneling solutions of the time-dependent Schroumldinger equation for a square-potential barrier

The exact tunneling solutions of the time-dependent Schroumldinger equation with a square-potential barrier are derived using the continuous symmetry group G(S) for the partial differential equation. The infinitesimal generators and the elements for G(S) are represented and derived in the jet space. There exist six classes of wave functions. The representative (canonical) wave functions for the classes are labeled by the eigenvalue sets, whose elements arise partially from the reducibility of a Lie subgroup G(LS) of G(S) and partially from the separation of variables. Each eigenvalue set provides two or more time scales for the wave function. The ratio of two time scales can act as the duration of an intrinsic clock for the particle motion. The exact solutions of the time-dependent Schroumldinger equation presented here can produce tunneling currents that are orders of magnitude larger than those produced by the energy eigenfunctions. The exact solutions show that tunneling current can be quantized under appropriate boundary conditions and tunneling probability can be affected by a transverse acceleration.