Transmission coefficient of a Gaussian wave packet through a potential barrier

Question

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.

Answer 1

If your incident wavefunction is $$\psi_i=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^\infty{A(k)e^{i(kx-\omega_kt)}dk},$$ the transmitted wavefunction is $$\psi_t=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^\infty{\alpha(k)A(k)e^{i(kx-\omega_kt)}dk}$$ where the transmission coefficient for an incident wave $e^{ikx}$ is $T(k)=|\alpha(k)|^2$. The transmission coefficient for the wave packet would be $$T'=\frac{\int\limits_{-\infty}^\infty|\psi_t|^2dx}{\int\limits_{-\infty}^\infty|\psi_i|^2dx}=\frac{\int\limits_{-\infty}^\infty T(k)|A(k)|^2dk}{\int\limits_{-\infty}^\infty|A(k)|^2dk},$$ where Parseval's relation has been used. In other words, the transmission probability is the average of $T(k)$ weighted by $|A(k)|^2$. The denominator would be just $1$ if your initial wavefunction is normalized.

Try expressing your wavefunction in momentum space (i.e. computing $A(k)$) and computing $T'$ as above to verify your results.