Find equation obeyed following Fourier transform

[Poirot]

Homework Statement

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 φ_n 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))

Homework Equations

Time-Dependent Schrodinger Equation:
$\frac{-\hbar^2}{2m} \frac{\partial^2 \psi(x,t)}{\partial x^2} + V(x,t)\psi(x,t) = i\hbar \frac{\partial \psi(x,t)}{\partial t}$

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.

Thanks in advance!

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.

 

[mark726]

I am a scientist and I would be happy to assist you with your questions. Firstly, let me address your first problem regarding the potential V(x,t) = scos(ωt)δ(x). The importance of the Fourier harmonic in this case is that it represents the periodicity of the potential. In other words, it tells us that the potential is repeating itself at regular intervals of time, in this case, the period is T = 2π/ω.

To solve for the equations obeyed by φn, we can use the time-dependent Schrodinger equation. We plug in the given periodic wave function and the potential into the equation and solve for φn. However, as you mentioned, there is a crucial point to consider. Since there is only one Fourier harmonic in this potential, we have V1(x) = sδ(x) and V-1(x) = V1(x). This means that the potential is symmetric with respect to the Fourier index n. This also leads to the condition that Vn = V-n, which comes from the reality condition.

Moving on to the second potential, V(x,t) = ħ/2m sδ(x-acos(ωt)), this is a more complicated case. The potential is now dependent on both space and time, and it is also not symmetric with respect to the Fourier index n. This means that Vn ≠ V-n and we cannot use the same approach as before.

To solve for the equations obeyed by φn in this case, we can use the time-dependent Schrodinger equation again. However, we will need to consider the potential at different time intervals, specifically at t = 0 and t = T = 2π/ω. This will give us two equations, and by solving them simultaneously, we can obtain the equations obeyed by φn. This method is known as the Floquet theory.

I hope this helps to clarify your understanding of the problem. If you have any further questions, please do not hesitate to ask.