This is another common problem in quantum mechanics which involves scattering of gaussian wave packet from a potential step. We assume the initial incident packet to be gaussian. The problem is to find out the probability of the reflected and transmitted waves as well as reflection and transmission coeffecients. The approach is pretty straight forward.

The supposed gaussian wave packet looks like this;

We solve SWE as usual and find solutions which respect the boundry conditions and are well behaved. The solutions (Psi) can can be used to find probability density of packet in various points in space and time. The trajectory is :

An over simplified equation (setting many terms as unity) is useful in my case to give a beautiful overview of the fate of the packet.