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Solve time-dependent Schrodinger equation for V=V(x,t)

  1. Feb 19, 2017 #1
    1. The problem statement, all variables and given/known data
    For the potential
    ##V(x,t) = scos(\omega t)\delta (x) ## where s is the strength of the potential, find the equations obeyed by ##\phi_n(x)##
    And again for ##V(x,t) = \frac{\hbar^2}{2m} s \delta(x - acos(\omega t))##
    2. Relevant equations
    Time-Dependent Schro:
    ## \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} ##
    from floquet theorem:
    ## \psi_E(x,t) = \phi_E(x,t) exp[-iEt/\hbar]##
    with
    ## \phi_E(x,t + T) = \phi_E(x,t) = \sum_{n=-\infty}^{\infty} \phi_{En}(x) exp[in\omega t] ##
    and
    ## V(x,t) = V(x,t+T) = \sum_{n=-\infty}^{\infty} V_n (x) exp[in\omega t]##

    3. The attempt at a solution
    I tried simply plugging in the period ψ and V(x,t) into the Schrodinger equation and ended up with an expression with no summations that seems far too simple. I was given a hint that I needed to think about how there's only 1 Fourier harmonic: ## V_{1}(x) = s\delta (x)## and ##V_{-1} = V_{1} ## but I don't really know what this means and as for second potential it should be very tricky but by my method it would be very simple. I don't really know how to use the Fourier transform of the potential here which I think is the issue.

    Thanks in advance, any help would be greatly appreciated!
     
  2. jcsd
  3. Feb 24, 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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