Electron Scattering: Resolve Distance Scale Below 10^-15 m

[UnIssued]

Homework Statement

Assuming that in electron scattering off a target proton you need to resolve a distance
scale below R = 10^-15 m (that is, the uncertainty in the proton’s position is no larger than
about 10^-15 m), determine a condition on the electron momentum, and justify it in one or
two sentences. Decide if this corresponds to a relativistic or a non-relativistic situation

Homework Equations

($$\Delta$$p)($$\Delta$$x) $$\geq$$ hbar/2

The Attempt at a Solution

$$\Delta$$x = 10^-15
$$\Delta$$p $$\geq$$ hbar/(2*$$\Delta$$x) = 5.273*10^-20I'm not sure I completely understand the question. If I got this right, it's because I saw what my book did and took a guess; I'd rather understand what's going on.

I understand the uncertainty principle for the target proton, but I don't understand how those uncertainties translate to the uncertainty of the electron.
Also, a simple point of confusion related to the topic: when is $$\Delta$$p = p or $$\Delta$$x = x? I've seen these down in solutions in what appears to be a haphazard fashion. I'd be very gracious for any insight into that.

Thank you for any help.

 

[zhermes]

I understand the uncertainty principle for the target proton, but I don't understand how those uncertainties translate to the uncertainty of the electron.

Good question. This is actually a little weird: the $$\Delta x$$ isn't really referring to the proton (per se) at all. Think about it more like: 'in general, to resolve anything with $$\Delta x$$ positional accuracy...'
You're using the electron itself as a measurement tool.
This certainly isn't an obvious answer, and the details get even uglier: e.g. what the given accuracy can actually tell you about the proton... etc.

Also, a simple point of confusion related to the topic: when is $$\Delta$$p = p or $$\Delta$$x = x? I've seen these down in solutions in what appears to be a haphazard fashion.

In the context of the uncertainty principle, they should always be deltas (i.e. $$\Delta x$$). Often, however, with something like momentum people often assume that the uncertainty is comparable to the value, i.e. $$\Delta p \approx p$$, but this is an approximation and a generalization.
The other explanation is people just being lazy---and this happens to.