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

  1. Jul 23, 2009 #1
    1. The problem statement, all variables and given/known data
    For a free particle, Given psi(x,0) = Aexp(-ax^2), find psi(x,t)

    2. Relevant 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

    3. 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 doesnt seem to help (or I'm missing something, which is quite likely). Any help is greatly appreciated!
  2. jcsd
  3. Jul 23, 2009 #2


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    Homework Helper

    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 eAB=eAeB. One of these factors will come out of the integral.
  4. Jul 23, 2009 #3
    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.
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