I have simulated the scattering of a gaussian wave packet with a potential barrier (Crank-Nicolson), and through many simulations I have determined the dependence of the transmission coefficient with the height of the potential barrier. However, when trying to compare it to the theoretical function it fails.
I suppose that that is mainly because the derivation of the coefficient is done for wavefunctions of the form $\sim e^{ikx}$. The theoretical coefficient that I've been using is [Sakurai, Modern Quantum Mechanics, Appendix B3]:
$$\begin{cases} T=\{ 1+[V_0^2/4E(V_0-E)]\sinh ^2(2a\sqrt{2m(V_0-E)/\hbar^2}) \} ^{-1}\quad \mathrm{if} \quad E<V_0\\ T=\{ 1+[V_0^2/4E(E-V_0)]\sin ^2 (2a\sqrt{2m(E-V_0)/\hbar^2}) \} ^{-1}\quad \mathrm{if} \quad E>V_0\\ \end{cases}$$
And I get what you can see in the image, which is quite off.
There have been some changes of units, so that everything is easier to treat numerically, $m=1/2, \ \hbar=1, \ V_0=\lambda k^2, \ E=\hbar k^2/2m=k^2$ and $\lambda$ would be the x-axis. My parameters are: $a=200$ and $k=0.4398$.
I think that the main problem has to do with the expression being only useful for a wave that is totally determined in the momenta-space, as I said, $\sim e^{ikx}$, but I've been researching for an expression for gaussians and failed.