1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    turin

    User Avatar
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: QM: psi(x,t) for Gaussian Wave Packet
  1. Gaussian Wave Packet (Replies: 2)

  2. Gaussian Wave packet (Replies: 8)

Loading...