Instantaneous Reflection and Transmission Coefficients and a Special Method to Solve Wave Equation

摘要

People are familiar with quantum mechanical reflection and transmissioncoefficient. In all those cases corresponding potentials are usually assumed asof constant height and depth. For the cases of varying potential, correspondingreflection and transmission coefficients can be found out using WKBapproximation method. But due to change of barrier height, reflection andtransmission coefficients should be changed from point to point. Here we showthe analytical expressions of the instantaneous reflection and transmissioncoefficients. Here as if we apply the WKB approximation at each point, so wecall it as Instantaneous WKB method or IWKB method. Once we know the forwardand backward wave amplitudes we can find out corresponding wave function bycalculating Eiconal. For the case of analytically complicated potentialcorresponding differential equation seems to be unsolvable analytically. If thepotentials are well behaved then obviously these could be replaced by functionsof simple expression of same behaviour which can be integrated analytically andthe equation is now possible to solve analytically.