QM: psi(x,t) for Gaussian Wave Packet

[Gumbercules]

Homework Statement

For a free particle, Given psi(x,0) = Aexp(-ax^2), find psi(x,t)

Homework Equations

phi(k) = 1/(sqrt(2pi)) times integral from -inf to +inf (psi(x,0)exp(-ikx))dx
psi(x,t) = 1/(sqrt(2pi)) times integral from -inf to +inf (phi(k)exp(i(kx - (hk^2)t/2m)))dk
my apologies for the messy notation

The Attempt at a Solution

I have normalized psi(x,0) to get A = (pi/a)^-1/4 and have my psi(k) = (1/(sqrt(2pi))) ((pi/a)^-1/4) times integral from -inf to +inf (exp(-ax^2) exp(-ikx)) dx.

regrettably, my math is quite out of practice, and I am unsure how to proceed. the text says something about 'completing the square' which gives y = (sqrt(a))[x + (b/2a)], then ((ax^2) + bx) = (y^2) - (b^2)/4a. After this, integration by parts doesn't seem to help (or I'm missing something, which is quite likely). Any help is greatly appreciated!

 

[turin]

Why do you need integration by parts? Maybe you've just been staring at QM too long. If A and B are c-numbers, then e^AB=e^Ae^B. One of these factors will come out of the integral.

 

[Gumbercules]

Perhaps you are right Turin, I do feel a little braindead at the moment. Do you mean exp(a+b) = exp(a)exp(b)? In that case, I would take the exp((b^2)/4a) out of the integral, which would leave the integral from -inf to + inf (exp(-y^2)), which I can solve. My apologies if I have this wrong, maybe I should come back to it later.