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.

Theoretical transmission coefficient and experimental results.

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.

  • $\begingroup$ Please label your axes. How are you defining/computing the transmission coefficient for a Gaussian wave packet? $\endgroup$ – Puk Jun 22 '20 at 1:57
  • $\begingroup$ The y axis is T and as I've said, the x axis is lambda (related to the height of the potential barrier V_0). The definition for T is already in the text, just substituting the parameters after the figure, to make an easier expression. $\endgroup$ – Adrián David Jun 22 '20 at 2:06

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.

  • $\begingroup$ Great, is $T(k)$ just the same expression that I've used before (but converting E to k) or is it another one that I don't know of? $\endgroup$ – Adrián David Jun 22 '20 at 2:51
  • $\begingroup$ It's the same one. $\endgroup$ – Puk Jun 22 '20 at 2:53

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