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Find the wavefunction for a 1 dimensional wave packet

  1. May 16, 2008 #1
    Hi,

    I'm teaching myself quantum mechanics (so this isn't homework). I came across the following question:

    [tex]\Phi(p) = A\Theta\left[\frac{\hbar}{d}-|p-p_{0}|\right][/tex]

    I have to find the constant of normalization, [itex]\psi(x)[/itex], and the coordinate space wave function [itex]\psi(x,t)[/itex] in the limit [itex]\frac{h/d}{p_{0}} << 1[/itex].

    I started by finding [itex]A[/itex]:

    [tex]\int_{-\infty}^{\infty}\frac{|\Phi(p)|^2}{2\pi\hbar}dp = 1[/tex]

    This gives [itex]A = \sqrt{\pi d}[/itex].

    Now,

    [tex]\psi(x) = \frac{1}{2\pi\hbar}\int_{-\infty}^{\infty}dp'\Phi(p')e^{-ip'x/\hbar}[/tex]

    which gives

    [tex]\psi(x) = \sqrt{\frac{d}{\pi}}e^{ip_{0}x/\hbar}\frac{\sin(x/d)}{x}[/tex]

    (There may be an algebraic error here..)

    My problem is: how do I find [itex]\psi(x,t)[/itex]? I am not sure how to proceed here.
     
  2. jcsd
  3. May 17, 2008 #2
  4. May 18, 2008 #3
    Solved

    Got it.

    [tex]\Phi(p,t) = \Phi(p)e^{-\frac{p^{2}t}{2m\hbar}}[/tex]

    and

    [tex]\psi(x,t) = \int \Phi(p,t)e^{-ipx/\hbar}dp[/tex]
     
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