QM Particles and their space boundaries

[philipp2020]

Hi

On Wikipedia some author wrote: According to Quantum Mechanics, particles can't inhabit a place smaller than their wavelength.

I googled around a little bit but couldn't find any formula which is consistent with this sentence.

Anybody knows by which formula this sentence makes sense?

Thanks for an answer

Philipp

 

[dextercioby]

That's wrong. Don't read unverified statements on Wiki. They are the downside of this great website. Any moron can write there...

 

[philipp2020]

So there are no boundaries at all? Any formula which relates to that?

 

[DrewD]

In an infinite well the ground state is in space the size of half the wavelength.
The boundaries depend on the specific system, which is probably why you didn't find what you are looking for through Google. I think the only relation that would hold in all situations is the uncertainty principle.

 

[philipp2020]

In an infinite well the ground state is in space the size of half the wavelength.

From which formula can this explanation be derived from?

Also, the Heisenberg Uncertainty principle only says something about the place a particle could be found, and nothing about that the surrounding space could be smaller than the particle's actual wavelength.

 

[DrewD]

Here is an online derivation from the Schrödinger equation. I haven't read through it to be sure that its correct, but I would be surprised if there are mistakes.

http://panda.unm.edu/Courses/Fields/Phys491/Notes/TISEInfiniteSquare.pdf

I am only an undergrad, so others should point out if the following interpretation is wrong.

I said that the uncertainty principle related to the spatial boundary of a particle because it relates the space in which a particle is confined to the possible momentum and therefore to the wavelength by $\Delta p=\frac{h}{\Delta\lambda}$. Since the particle doesn't have doesn't have a physical size (as far as I know from QM; I don't really know if QFT would change this interpretation), the space that it occupies is the space that it has non-zero probability of occupying during the given time frame. Therefore a particle confined to a certain space has an uncertainty in position of ΔX and therefore the possible wavelengths will have range of $\ \Delta\lambda\geq 4\pi*\Delta X$ (DO NOT QUOTE THIS! I did it quick in my head and very well may have made a mistake but the idea is all that matters). In that way, the uncertainty does relate the space in which a particle exists and the possible wavelength. This is the best "answer" I can think of for your question.

 

[philipp2020]

This explanation is great. Thank you. I searched a bit more on the internet to find other reasonable solution to the author's claim, but couldn't find anything which could cover his sentence.

Still I would like to ask the author what properties of QM he had in mind when he was writing this on Wikipedia. But unfortunately, its not possible to contact the author as far as I know.