I'm trying to use the Animate function on Mathematica to show a gaussian wave packet passing through a delta potential. I'm quite new to Mathematica and this is by far the hardest thing I've had to do so please bear with me.(adsbygoogle = window.adsbygoogle || []).push({});

I effectively want to solve the integral:

##

\phi_k(x) = \left\{

\begin{array}{ll}

e^{ikx} + R_k e^{-ikx} & \quad x < 0 \\

T_k e^{ikx} & \quad x > 0

\end{array}

\right.

##

I have found the reflection and transmission coefficients:

##

R_k =\frac{1}{\frac{\hbar^2ik}{sm} -1} \\

T_k =\frac{1}{1-\frac{sm}{\hbar^2ik}}

##

where s is the strength of the delta

## V(x) = s\delta(x) ##

I then need to plug my phi into the integral

##\int_{-\infty}^{\infty} \phi_k(x) e^{-(k-k_0)^2/\alpha^2} e^{-i\hbar k^2 t/2m} dk##

I've been told the integrals are exactly solvable but not to worry about that just yet so I've been plugging in the limits k-> -100 and 100 in the mean time. I am able to solve the first part of the x<0 by hand but the thing that's causing me the most issues is the transmission and reflection coefficients as the denominator goes to infinity when k=1 (I have set all my constants to 1 in the mean time for simplicity except k0 the initial momentum which i've toggled with to see what happens). Every time I put my piecewise function into Mathematica to integrate it spits out the integral unsolved. I am not really sure how I'm supposed to animate this as I can only solve 1 part of the integral.

So really the first issue I'm having is with the integral, I am *hoping* once I can do that it should all fall into place but I've been stuck for a long time now.

Thanks for any help.

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# Mathematica QM Animation of delta potential

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