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

1. Feb 19, 2017

### Poirot

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. Feb 24, 2017

### PF_Help_Bot

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