I am doing a scattering simulation of a Gaussian wave packet on a finite square well. I have solved numerically the Schroedinger equation and I know the values of the wave function after the scattering process before and after the square well.
To be more clear, this is the situation after the scattering process: (the wave function is not normalized, the biggest packet is moving towards
left, the smallest towards right)
I know that the tasmission coefficient is defined as:
$T=\dfrac{\vec{j}_{trasmitted}}{\vec{j}_{incident}}$
where the flux $\vec{j}$ is defined as:
$\vec{j}(x)=\psi^*(x)\frac{\hat{p}}{2m}\psi(x)-\psi(x)\frac{\hat{p}}{2m}\psi(x)^*$
Now, for an unnormalizable function like $Ce^{ikx}$ I am able to determine $T$, it is easy because the flux is constant. But in this case the flux is not constant before and after the square well.
Maybe the probability to find the particle after the square well (and so the area under the part of the normalized $|\psi|^2$ that crossed the barrier) is the correct value of the transmission coefficient?
If my hypotesis is correct, could you give me some references where it is explained?