A. Propagation of initial wavepacket into barrier region
B. Auto correlation function of initial wavepacket
C.
Calculation of reflection and Transmission Coefficients via Wavepacket Auto-Correlation Functions. The movie below shows a wavepacket incident from the left on an asymmetric one dimensional potential. The average energy of the wavepacket is slightly below the energy of the barrier height, According to classical mechanics an incident particle would be reflected with unit probability but quantum mechanically, the wavepacket bifurcates, with one portion reflected and one portion transmitted. When the bifurcation is complete, we can define the amplitude on the product side as
and that on the reactant side as and write
The bottom part of the movie shows the wavepacket in the momentum representation, at each of the three stages - before, during and after the collision. The bifurcation process in the momentum representation as can be represented:
D. Propagation of transmitted & reflected wavepacket out of barrier region
E. Reflected & transmitted auto correlation functions
F.
H.
I.
We now consider the spectrum of the transmitted and reflected wavepackets. Defining we write:
Where we have used the fact that the correlation function is invariant with respect to the zero of time and that the cross-terms of the form
vanish. Taking the time-energy Fourier transform of both sides of eq. (3) we find that
where we have defined the spectrum of the incident, the reflected and the transmitted wavepacket. Dividing both sides of equation (4) by σ(E) we find
The statement that R(E) and T(E) sum to 1 is simply a statement that at asymptotic times all particles must be either reflected or transmitted.
and that on the reactant side as 
and write
we write:
Where we have used the fact that the correlation function is invariant with respect to the zero of time and that the cross-terms of the form 
vanish. Taking the time-energy Fourier transform of both sides of eq. (3) we find that
where we have defined the spectrum of the incident, the reflected and the transmitted wavepacket. Dividing both sides of equation (4) by σ(E) we find
The statement that R(E) and T(E) sum to 1 is simply a statement that at asymptotic times all particles must be either reflected or transmitted.