Find the wave function of a Gaussian wave packet

[serverxeon]

In particular, i am solving part b.
I pulled off a couple of formulas from a textbook, but I'm quite sure they are incorrect to apply here.
Can anyone guide me?

Below is my attempt.

 

[BruceW]

I'm pretty sure that you are fine up to the line
\psi(x,t)=\frac{1}{\sqrt{2\pi}} \cdot \frac{\sqrt{\sigma}}{\pi^{1/4}} \int_{-\infty}^\infty e^{-i\hbar tk^2/2m+ixk} dk
But then after that you apply the 'general form' of the Gaussian integral. But this 'general form' is only true when alpha, beta and gamma are real numbers. But in this case, they are imaginary numbers (well, gamma is zero, but the others are imaginary).

 

[BruceW]

huh, hold on. that 'general form' should work for imaginary numbers too. So I would agree with your answer. (except in the last line I think you have not written the square root around all the things that you are meant to put it around). The answer seems really strange though. I would not expect that at all...

 

[mfb]

I would be very surprised if the k in your initial wavefunction is related to momentum. How would you interpret this? I think it is some fixed constant, which changes every calculation afterwards.

 

[BruceW]

yep, that's where the problem is, I think. You should define the Fourier transform as something like $\phi(k')$ i.e. use a different variable to the $k$ that is in the equation for $\psi(x,t)$. Since you generally want $k$ and $k'$ to be two different variables.

edit: or you can just re-name the $k$ in the equation for $\psi(x,t)$ as some other letter, $\kappa$ (kappa) for example. And then use $\phi(k)$ as your Fourier transform. This will save you from having to write out the prime a lot of times, which can be annoying. But go with whichever way you prefer. Main thing to remember is that they are different variables. I'm guessing you know this, but just forgot (as I did to begin with).