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Wavepackets [k-space to z-space]

  1. Jan 10, 2007 #1
    Hi, this type of question has been confusing my slightly as of late, an a pointer in the right direction would be greatly appreciated

    1. The problem statement, all variables and given/known data
    The wavefunction associated with a Gaussian wavepacket propagating in free space can be shown to be [included as attachment - it's too complicated for here] where delta k is withe width of the wavepacket in k space and v is the velocity of the wavepacket.

    Deduce an expression for the width of the wavepacket in real space (z-space)as a function of time


    2. Relevant equations

    again, as attached

    3. The attempt at a solution

    I'm suspecting it has something to do with Fourier Transforms, but I'm really stumped. it's probably straightforward, but i'm a bit blind to it at the moment

    Thanks in advance
     

    Attached Files:

  2. jcsd
  3. Jan 11, 2007 #2

    dextercioby

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    I have a hunch that [itex] \Delta z\Delta p =\frac{\hbar}{2} [/itex], since a gaussian wavepacket is minimizing the uncertainty relations.

    Daniel.
     
  4. Jan 19, 2007 #3
    To find the witdth of the wave packet you should consider the form of
    [tex] |\psi|^2 [/tex] .
    This will have the form
    [tex] \psi \propto \exp \left\{- \frac{(z - vt)^2}{A(t)} \right\} [/tex]

    This has the form of a Gaussian curve. The maximum occurs where [tex] z = vt [/tex] where the exponens takes on the value 1.
    The width is given by the lenght between the points where the exponent is [tex] 1/2 [/tex]. So the expression used to find the widht is
    [tex] \exp \left\{ - \frac{(z-vt)^2}{A(t)} \right\} = \frac{1}{2} [/tex].
    Solving this gives two solutions [tex] z_1(t) [/tex] and [tex] z_2 (t) [/tex] and the difference between these are the width of the wave packet.

    You can expect that the width is increasing with time, since the Schrödinger equation has a dispersive term (a term that causes different Fourier components of the wave to propagate with different velocities).
     
    Last edited: Jan 19, 2007
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