1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: 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
    \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)}
    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
    \omega = ck
    \int_{-\infty}^{\infty}dx e^{-ax^2} = \sqrt{\frac{\pi}{a}},\quad a > 0

    3. The attempt at a solution
    Using the method of completing the square I get
    &= \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)}

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted