# Can a potential act in a small region of spacetime?

1. Feb 26, 2012

### Spinnor

I would like to perturb the wave-function of a localized charged particle with a potential that is close to a delta function in space and time. Do Maxwell's equations prevent such a potential in theory if not in practice?

If so can I in a very loose sense think of the potential as giving the wave-function a localized sharp "kick"?

Thanks for any help!

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2. Feb 28, 2012

### PhilDSP

3. Feb 28, 2012

### TriTertButoxy

Also, I don't know how to interpret your sketch; can you tell me what's on that graph? I'd love to know :)

4. Feb 29, 2012

### Spinnor

Thank you for your help! My question might not be clear? My first concern is if the following potential might be realized or in a similar form for the purposes of perturbing the wave-function of a localized charged particle,

V(X,t) = positive constant*δ(X)*δ(t)

where the delta functions were smeared out, peaked and finite, not infinite, like a very sharp Gaussian in both space and time, and X coincides with some small part of the localized particle.

Thanks for any help!

5. Feb 29, 2012

### Spinnor

I will include a better sketch below. I think I messed up the signs on the sketch? I should probably refer to time dependent perturbation theory but if we use a delta function like potential we might cut some corners but still get a feeling for what is "going on"?

Consider the ground state of particle of mass m and charge e constrained to a one dimensional distance L. The wave-function is like,

ψ(x,t) ≈ sin(∏x/L)*exp(-iE*t/hbar)

Let there be a potential that acts in a small region of space-time (smeared out delta functions) that coincides with where the particle is likely to be found. (I'm not sure such a potential can be realized?)

Let V(x,t) = ε*δ(x°)*δ(t°).

where ε is a small constant. Then consider how ψ(x,t) changes when x = x° and t = t°.

Hψ(x,t) = [H + V(x,t)]ψ(x,t) =-i∂ψ(x,t)/∂t so ?

Δψ(x°,t°) = iΔt[H + V(x°,t°)]ψ(x°,t°)

where Δt is the small time the potential acts.

So depending on the sign of the potential the, the potential gives the wave-function a little push forwards or backwards in the direction of Δψ?

Thanks for any help!

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