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

[Poirot]

Homework Statement

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 $V(x,t) = \frac{\hbar^2}{2m} s \delta(x - acos(\omega t))$

Homework 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
$V(x,t) = V(x,t+T) = \sum_{n=-\infty}^{\infty} V_n (x) exp[in\omega t]$

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!

 

[mark726]

Dear fellow scientist,

Thank you for your post. It seems like you are on the right track with your approach. However, you are correct in thinking that the Fourier transform of the potential will be important in solving this problem.

Let's start with the first potential, $V(x,t) = scos(\omega t)\delta (x)$. As you mentioned, there is only one Fourier harmonic, $V_1(x) = s\delta(x)$, which means that the Fourier transform of the potential is simply $V_1(x)$. This will be important when we use the Floquet theorem to solve for $\phi_n(x)$.

As a reminder, the Floquet theorem tells us that the time-dependent wavefunction can be written as a sum of stationary states, $\psi_E(x,t) = \phi_E(x,t) exp[-iEt/\hbar]$. We can use this to solve for $\phi_n(x)$ by plugging in the Fourier transform of the potential and using the fact that $V(x,t) = V(x,t+T) = \sum_{n=-\infty}^{\infty} V_n (x) exp[in\omega t]$.

For the first potential, we can write $V(x,t) = s\delta(x)$ and plug this into the Schrodinger equation, which will give us an expression for $\phi_n(x)$. Then, using the fact that $V_1(x) = s\delta(x)$, we can solve for $\phi_n(x)$ and find the equations that it obeys.

For the second potential, $V(x,t) = \frac{\hbar^2}{2m} s \delta(x - acos(\omega t))$, we have a more complicated Fourier transform. In this case, we will have to use the fact that $V(x,t) = \sum_{n=-\infty}^{\infty} V_n (x) exp[in\omega t]$ and solve for $\phi_n(x)$ using the Fourier transform of the potential.

I hope this helps guide you in the right direction. Don't hesitate to reach out if you have any further questions.