Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,846 answers , 20,597 comments
1,470 users with positive rep
501 active unimported users
More ...

The minimum uncertainty of a particle's position

+ 0 like - 0 dislike
93 views

It is stated that the minimum uncertainty $\Delta x$ of the position of some particle with the mass $m$ at rest is
$$
\Delta x \sim \frac{\hbar}{mc}
$$
For the particle with energy $E$ corresponding statement is
$$
\Delta x \sim \frac{\hbar c}{E}
$$
But I don't understand how to obtain these results (except, of course, thinking in the spirit of dimensional analysis). 

Could you please explain how to obtain them quantitatively, from the ground of relativistic physics?

asked Aug 30 in Theoretical Physics by NAME_XXX (1,010 points) [ revision history ]
edited Aug 31 by NAME_XXX

The particle position uncertainty depends on the state in which you prepare the particle. A plane wave with energy $E$ may have a huge wave-packet size. What you wrote looks like the De Broglie wave-length estimation rather than the position uncertainty. The latter is determined with the wave-packet size.  Note that $mc$ and $E$ have different dimensions.

@VladimirKalitvianski : in the source (Landau, Vol. 4, paragraph 1) where I've seen these results there is a comparison with DeBroglie wavelength only after obtaining the results. I'll also correct the misprint.

Relativistic quantum theory/experiment demonstrates a multi-particle states after collisions. So if you measure the particle position with help of collisions, then you can encounter a situation with many particles and anti-particles in the final state, so you may not ascribe any of them to your initial particle. The threshold of the transferred energy for pair creation is $2mc^2$. If the transferred energy is too huge, you can obtain a "shower" of particles in the final state, so your second formula is not about the particle position uncertainty.

@VladimirKalitvianski : but Landau tells about the second example as about the position uncertainty.

Look, the first relationship is a Compton wave-length. I arises naturally in the relativistic particle treatment. The second relationship is a Lorentz-contracted Compton wave-length. By its definition, it is a longitudinal length. But there is a transversal Compton wave-length too, which is not Lorentz-contracted.

Also, the De Broglie wave-length says nothing about the wave packet size including many-many wave-lengths. Thus, what is written in Landau textbook has a very limited meaning.

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysic$\varnothing$Overflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...