Quantum Reflection in Step-Exponential Barrier and Parabolic Barrier

We calculate transition and reflection coefficients of arbitrary barrier by analytical transfer matrix method(ATMM),and then apply the method to the quantum reflection above the barrier.The results in step-exponential barrier and parabolic barrier by ATMM are more accurate than those by WKB approximation.Furthermore,when the energy of the particle is greater than the barrier maximum,only ATMM rather than WKB can get precise results.The results show that reflection coefficients gradually change to zero,where quantum reflection happens.These problems can be solved by ATMM exactly.