Calculation of Reflection and Transmission Coefficients via Wavepacket Cross-Correlation Functions.
We define four fundamental wavepackets,
and
indicates the region to the left, of the barrier,
β
to the right of the barrier. The superscript + indicates incoming to the barrier at negative times; the superscript - indicates outgoing from the barrier at positive times. We may define a 2 x 2 matrix of correlation functions:
Note that we are free to choose
and
in which case the matrix is symmetric, i.e.
We now define the scattering- or
S-matrix,
S(E), which is a matrix of transition amplitudes from initial incoming state α to final outgoing state β at total energy
E:
The elements of this matrix are proportional to the Fourier transforms from time to energy of the elements of the correlation function matrix, normalized by the coefficients of the free particle energy eigenstates in t he incoming or outgoing wavepackcts,
or
respectively. The reflection and transmission coefficients arc given by:
wavepacket incident from left
wavepacket incident from right
