Find equation obeyed following Fourier transform

1. Feb 18, 2017

Poirot

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.

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. Feb 23, 2017