# A Minimum uncertainty in electron position

#### FrederikPhysics

I am wondering about the minimum possible uncertainty (standard deviation) in an electron's position (Δx). How precise can one know the electron's whereabouts without creating other sorts of particles and phenomenons.
I know of the localization energy interpretation of the energy uncertainty (ΔE). Using a Heisenberg relation we estimate Δx as
ΔEΔx≈ħc/2 ⇒ Δx≈ħ/2mc,​
since E=mc2 for an isolated electron (m is the rest mass). If Δx is smaller than this, extra localization energy manifests itself through other particles.

Now the questions are simple. Is this interpretation valid? What is the actual minimum possible Δx for an isolated electron? Have this been measured?

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#### BvU

Homework Helper
Wasn't it $\Delta p\Delta x \ge \displaystyle {\hbar\over 2}$ ?

#### PeterDonis

Mentor
How precise can one know the electron's whereabouts without creating other sorts of particles and phenomenons.
This is much too vague. What "other sorts of particles and phenomenons" are you thinking of?

What is the actual minimum possible Δx for an isolated electron?
Theoretically, there is none; you can make $\Delta x$ as small as you like (as long as it's not zero), at the cost of making $\Delta p$ larger.

I'm not sure what the smallest $\Delta x$ is that has been achieved experimentally, but I think it's somewhere around the size of an atomic nucleus, about $10^{-15}$ meters. I'm basing that on the deep inelastic scattering experiments that first provided evidence for quarks; in these experiments, high energy electrons were fired into nuclei and scattered off quarks inside the nuclei, meaning that the electrons' positions had to be within the size of the nucleus, roughly, for the scattering to occur.

#### dextercioby

Homework Helper
Just as a side remark to post # 3: in whatever deep inelastic scattering of (fundamental or not) particles which are highly accelerated (energies of many GeV), there's no practical/theoretical way to test if the (non-relativistic) HUP is valid or not, for you can never measure the position of a particle in a scattering event, nor can you calculate its exact state vector. Remember that $\Delta x$ (the standard deviation from the mean) is theoretically computed as the statistical spread of the expectation values of x and x2 in the scattering state call it $\psi(t)$. You cannot determine the state, because, due to the interaction term, the SE is not solvable, therefore you cannot calculate any expectation value.

"Minimum uncertainty in electron position"

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