Can a potential act in a small region of spacetime?

[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!

 

[PhilDSP]

Hi Spinnor,

While the Maxwell equations don't provide you with any means of determining a unique electric potential or magnetic potential there are several ways of determining retarded potentials such as Liénard–Wiechert potentials and Jefimenko's equation (which is more advanced in conception and potentially more useful)

http://en.wikipedia.org/wiki/Retarded_potential
http://en.wikipedia.org/wiki/Classical_electromagnetism#General_field_equations

 

[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 :)

 

[Spinnor]

Hi Spinnor,

While the Maxwell equations don't provide you with any means of determining a unique electric potential or magnetic potential there are several ways of determining retarded potentials such as Liénard–Wiechert potentials and Jefimenko's equation (which is more advanced in conception and potentially more useful)...

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!

 

[Spinnor]

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

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!