# Is this analytically solvable?

1. Feb 26, 2017

### Poirot

1. The problem statement, all variables and given/known data
I am trying to solve integrals of the form:
$\int_{-\infty}^{\infty} \frac{k}{k-1} e^{itk^2}e^{ikx}e^{-(k-k_0)^2/\alpha^2} dk \\ \int_{-\infty}^{\infty} \frac{1}{k-1} e^{itk^2}e^{ikx}e^{-(k-k_0)^2/\alpha^2} dk$

A bit of background:
I'm trying to create an animation in Mathematica to show a gaussian wave packet passing through a delta potential. The fractions at the start of the integrals come from the transmission and reflection coeffecients such that
$\phi_k(x) = \left\{ \begin{array}{ll} e^{ikx} + R_k e^{-ikx} & \quad x < 0 \\ T_k e^{ikx} & \quad x > 0 \end{array} \right.$
so Rk and Tk are the fractions at the start then we multiply the phi(x) by a time dependent part and then the gaussian part and integrate for the linear superposition.

2. Relevant equations
The overall integral is
$\int_{-\infty}^{\infty} \phi_k(x) e^{-(k-k_0)^2/\alpha^2} e^{-i\hbar k^2 t/2m} dk \\ R_k =\frac{1}{\frac{\hbar^2ik}{sm} -1} \\ T_k =\frac{1}{1-\frac{sm}{\hbar^2ik}}$

where s is the strength of the delta potential such that:

$V(x) = s\delta(x)$
and I've set virtually every constant to 1 (except k0 and alpha which I plan to toggle once I get something).
3. The attempt at a solution
I'm told these integrals should be possible analytically but I can't see how. I've also tried using mathematica to compute it but it's having trouble with the function not converging on the interval. I've tried using NIntegrate too but I think the fact there's factors of x and t in the integrand is causing issues when trying to get a numerical approximation.

It might be worth mentioning that I've managed to solve the integral for the the e^(ikx) part for x<0 using completing the square but I can't see how to do it with the fraction.

Any help or guidance would be greatly appreciated thank you!

2. Feb 28, 2017

### Khashishi

It seems that Mathematica can calculate
FourierTransform[ Exp[I t k^2] Exp[-(k - k0)^2/\[Alpha]^2] , k, x]
and
FourierTransform[k/(k - 1) , k, x]
but it chokes on the whole thing.
Maybe you can do something with the convolution theorem to combine them?

3. Feb 28, 2017

### Ray Vickson

Let $f(k) = \exp(i t k^2 + i x k - (k-k_0)^2/ \alpha^2)$. If you can get $\int_{\mathbb{R}} f(k) \, dk$ and $\int_{\mathbb{R}} k f /(k-1) \, dk$, then you an get $\int_{\mathbb{R}} f/(k-1) \, dk$ because $1/(k-1) = (1-k + k)/(k-1) = k/(k-1) - 1.$

Last edited: Feb 28, 2017