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

The exact tunneling solutions of the time-dependent Schrodinger equation with a square-potential barrier are derived using the continuous symmetry group GS for the partial differential equation. The infinitesimal generators and the elements for GS 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 GLS of GS 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 Schrodinger 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 bo...