A wave packet and a Delta potential

[Poirot]

Homework Statement

I want to plot using Mathematica a wave packet entering a delta potential $V(x) = s\delta(x)$ (s is the strength) but I need to get the physics right first and I'm having trouble with a a few parts. I need to compute the integral $\int_{0}^{\infty} e^{-i\omega t}\phi_{\omega} f(\omega) d\omega$ to get the linear superpositions and get the wave packet but Mathematica can't handle it and I can't compute it by hand.
I'm taking $f(\omega) = e^{-\omega^2/\alpha^2}$ (I assume to make this a gaussian wave packet).

Homework Equations

$\phi_{\omega} (x) = e^{ikx} + R_{\omega}e^{-ikx}, x<0 \\ \phi_{\omega}(x) =T_{\omega}e^{ikx}, x>0 \\ k^2 = 2mE/\hbar \\ \omega = E/\hbar$

The Attempt at a Solution

I found the Reflection and Transmission coefficients:
$R_{\omega} = \frac{1}{\frac{\hbar^2ik}{sm}-1} \\ T_{\omega} = \frac{1}{1-\frac{sm}{\hbar^2ik}}$
from the continuity conditions and also from integrating the Schrodinger equation.

From what I understand I think I then need to compute the integral as stated above and plot this against time, but I can't get Mathematica to solve it and I was told it could be computed exactly so I must be messing something up.

Any help would be greatly appreciated thank you.

 

[mark726]

Thank you for your question. It sounds like you are on the right track with your approach to solving this problem. However, there are a few things that you may want to consider in order to get the correct physics and make your computation easier.

First, when computing the integral for the linear superposition, you may want to use the Fourier transform of your Gaussian wave packet instead of directly integrating. The Fourier transform of a Gaussian function is also a Gaussian, which may make your computation easier and more manageable for Mathematica.

Secondly, when using the reflection and transmission coefficients, make sure you are using the correct wave vector k for each region. In your equations, you have used k for both regions, but they should be different as they correspond to different energies. This may be causing issues with your computation.

Lastly, it may be helpful to use dimensionless variables in your computation, such as scaled position and momentum, to make the equations simpler and more manageable. This can also help with numerical stability in your calculations.

I hope this helps and good luck with your plot! If you have any further questions, please don't hesitate to ask.