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A one-dimensional Gaussian Wave Packet

  1. Sep 24, 2011 #1
    1. The problem statement, all variables and given/known data
    I have been given the function
    [tex]
    \varphi_{G}(z,t) = \frac{2}{\Delta k_0\sqrt{\pi}} \int_{-\infty}^{\infty}dk\ e^{-\frac{4(k-k_0)^2}{\Delta k_0^2}}e^{i(kz-\omega t)}
    [/tex]
    and been told to do the integration and then to specify the phase and group velocity of the wave package. I also have to decide if there is dispersion or not.


    2. Relevant equations
    [tex]
    \omega = ck
    [/tex]
    [tex]
    \int_{-\infty}^{\infty}dx e^{-ax^2} = \sqrt{\frac{\pi}{a}},\quad a > 0
    [/tex]

    3. The attempt at a solution
    Using the method of completing the square I get
    [tex]
    \begin{align*}
    \varphi_{G}(z,t)
    &= \frac{2}{\Delta k_0\sqrt{\pi}} \int_{-\infty}^{\infty}dk\ \exp{\left(-\frac{4(k-k_{0})^2}{\Delta k_{0}^{2}} + i(kz - kct)\right)} \\
    &= \frac{2}{\Delta k_0\sqrt{\pi}} \int_{-\infty}^{\infty}dk\ \exp{\left(-\frac{4}{\Delta k_{0}^{2}}(k^2 - 2k(k_0 + i\frac{\Delta k_{0}^{2}}{8}(z-ct)) + k_{0}^{2})\right)} \\
    &= \frac{2}{\Delta k_0\sqrt{\pi}} \int_{-\infty}^{\infty}dk\ \exp{\left(-\frac{4}{\Delta k_0^2}( (k-(k_0 + i\frac{\Delta k_0^2}{8}(z-ct)))^2 - 2k_0 i\frac{\Delta k_0^2}{8}(z-ct) + \frac{\Delta k_0^4}{64}(z-ct)^2)\right)} \\
    &= \frac{2}{\Delta k_0\sqrt{\pi}} \exp{\left(ik_0(z-ct) - \frac{\Delta k_0}{16}(z-ct)^2\right)} \int_{-\infty}^{\infty}dk\ \exp{\left(-\frac{4}{\Delta k_0^2}(k-(k_0 + i\frac{\Delta k_0^2}{8}(z-ct)))^2\right)} \\
    &= \frac{2}{\Delta k_0\sqrt{\pi}} \exp{\left(ik_0(z-ct) - \frac{\Delta k_0}{16}(z-ct)^2\right)} \frac{\sqrt{\pi}\Delta k_0}{2} \\
    &= \exp{\left(-\frac{\Delta k_0^2}{16}(z-ct)^2 + ik_0 (z-ct)\right)}
    \end{align*}
    [/tex]

    Now, I don't really have a lot of experience doing things like this (in fact this is the first time). Is the result I got correct or have I done an error somewhere? Also, is there maybe an easier way to calculate the integral?
     
  2. jcsd
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