I'm trying to solve a 1D time-dependent Schrodinger equation:

$$ i\,\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} \frac{\partial^2}{\partial x^2}+V(x)+F(t)\,x\right]\!\psi(x,t) $$

where $V(x)$ is a periodic Dirac delta potential (or the Dirac comb):

$$ V(x)=\sum_{k=-\infty}^\infty\delta(x-ka) $$

and $a$ is some constant.

The driving force $F(t)$ is some arbitrary function of time, but for simplicity can be assumed to be simple trigonometric functions here:

$$ F(t)=F_0\cos(\omega_0t) $$

where $F_0$ and $\omega_0$ are constants ($F_0$ is not weak so that the perturbation theory may not apply).

Or for case of potential is a simple trigonometric function

$$ V(x)=\cos(a x). $$

How can I solve this 1D TDSE analytically?

  • $\begingroup$ I guess that you may need to start with easier problems. Set $F(t)=0$ and solve for time independent equation for $\psi(x,t_0)$. For delta potential $V(x)=\delta(x)$ the solution is like $\psi(x,t_0)=C \exp^{-\lambda |x|}$. You may then try to solve for $\psi(x,t_0)$ with the Dirac-comb potential. Finally you may add $F(t)x$ term in the potential. $\endgroup$ – mike Jun 9 '14 at 10:01
  • $\begingroup$ @mike Thanks, but I guess the difficult part is the time-dependence. The time-independent problem can be solved using ODE technics. $\endgroup$ – xslittlegrass Jun 10 '14 at 0:21
  • $\begingroup$ You may do Fourier transform for variable $x$, so that the Schrodinger equation becomes $\left(i\partial_t -(1/2)p^2 -i F(t)\partial_p+()\right)\psi(p,t)=0$. You may check arxiv.org for Schrodinger equation for time dependent mass $m(t)$. $\endgroup$ – mike Jun 10 '14 at 15:34
  • $\begingroup$ With $f=0$ you get the Dirac Comb which is solvable. A reason it can be solved is that the potential is both time-independent and periodic. This last fact allows us to use Bloch’s theorem to propagate a solution in one well to another. In your problem you have lost both of these symmetries and because of this it seems unlikely that analytical solutions exist (beyond perturbation theory). I think you will be better off searching through books or research articles in this field than asking it here. Either this problem has been solved before or if not: such a solution would be publishable. $\endgroup$ – Winther Jun 15 '14 at 21:58
  • $\begingroup$ @Winther The Bloch's theorem is essentially the Floquet theory. By adding a harmonic time-dependent term, the system is both periodic in space and time, and seems to be more symmetric. I was wondering if there was some techniques to combine the Floquet theory and the Bloch theory to deal with this equation. $\endgroup$ – xslittlegrass Jun 16 '14 at 19:21

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