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Find equation obeyed following Fourier transform

  1. Feb 18, 2017 #1
    1. The problem statement, all variables and given/known data
    I have a potential V(x,t) = scos(ωt)δ(x) where s is the strength of the potential. I need to find the equations obeyed by φn given that
    ##
    \psi_E (x,t) =\phi_E exp[\frac{-iEt}{\hbar}] \\

    \phi_E (x, t + T) = \phi_E (x,t)\\

    \phi_E = \sum_{-\infty}^{\infty}\phi_{nE}exp[in\omega t]
    ##


    Do the same again for the potential V(x,t) = ħ/2m sδ(x-acos(ωt))
    2. Relevant equations
    Time-Dependent Schrodinger Equation:
    ##
    \frac{-\hbar^2}{2m} \frac{\partial^2 \psi(x,t)}{\partial x^2} + V(x,t)\psi(x,t) = i\hbar \frac{\partial \psi(x,t)}{\partial t}
    ##

    3. The attempt at a solution
    At first I simply plugged in the periodic wave function into the Schrodinger equation along with my potential and got out an answer that was too simple I believe and I'd ignored a crucial point: as there is only 1 Fourier harmonic we have that V1(x) = sδ(x) and V-1(x) = V1(x) (this was advice given) the notation comes from
    ##
    V(x,t+T) =V(x,t) = \sum_{-\infty}^{\infty} V_n(x) exp[in\omega t]
    ##
    I may be wrong but I think this condition comes from a reality condition Vn = V-n.
    I've been told the second one is going to be very tricky, but I don't even understand the first one. I don't understand the importance of the Fourier harmonic or what it is/means really.

    Any help would be greatly appreciated, and I hope my question makes sense as it's now homework per se but will help with a project.

    Thanks in advance!

    Sorry for the equations not being in LaTeX, I can't figure out what's wrong so if someone can see my blunder that would also be great, thank you.
     
    Last edited: Feb 18, 2017
  2. jcsd
  3. Feb 23, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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