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

PO is now at the Physics Department of Bielefeld University!

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

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,353 answers , 22,789 comments
1,470 users with positive rep
820 active unimported users
More ...

  What's the definition of a short range potential and why is it defined this way?

+ 2 like - 0 dislike
5632 views

This question is related to this one on SE. However, I'm still not satisfied with the answers. My question can be divided as these 2:

  1. Is this just a convention (as Claudius said) or is there any profound reason? 
  2. Why is $r^{-2}$ also long range? (As an example, on the P.5 of Quantum Mechanics: Selected Topics, "The centrifugal potential cannot be regarded as a short-range potential." and $V(r)=\frac{l(l+1)}{r^2}$)
asked May 18, 2016 in Theoretical Physics by forthebest (25 points) [ revision history ]
Most voted comments show all comments

@Arnold Neumaier Sorry for my late response. You've said that depend on the specific cases we study, we may have different definitions of short or long range potential. Could you give a concrete example? Or for the second part of my OP, why centrifugal potential is treated as long range?

@forthebest : The centrifugal potential can well be "short-range" if $r$ is within $(a,b)$ with finite $a$ and $b$. In order to tell what potential is short- or long-range, one has to compare the potential term with the others at $r\to\infty$. With kinetic term, for example.

@Vladimir Kalitvianski. I think your criterion about the centrifugal potential is wrong. Since for any non-singular potential it will definetely finite, even for exponentially growing potential. This is absurd. As to compare with kinetic term, in Schordinger equation, there are no r dependent kinematic term at all, then how to compare them?

@forthebest : I gave a reason why tan exponentially falling potential is considered short range. jjscale gave on SE a reason why a potential $O(r^{-2}$ is considered short-range in the context of a scattering calculation. Your reference on the centrifugal potential probably gives a reason why in its context it cannot be considered as short range. I can't check since you didn't give acomplete reference.

@forthebest : You have to compare the term with a given solution $\psi$, not just $d^2/dr^2$. And a given $\psi$ is what Arnold Neumaier means when speaking of particular problem. If the potential term prevails the kinetic one at finite $r$ and when $r\to\infty$, then the potential in this problem is short-range. If it starts gradually to prevail only when $r\to\infty$, then it is a long-range potential at the given circumstances.

Most recent comments show all comments

As jjcale mentioned in the answer on SE, what counts as short-scale depends on the application. In relativistic quantum field theory, massive particle generate short-range forces with an exponentially decaying potential, while massless particles generate (in the absense of confinement) long range forces with a modified Coulomb potential, essntially $1/r$ for large $r$. The two cases behave very differently in many respects.
 

@ArnoldNeumaier I think what @Forthebest is asking for is why we call it short- or long-range potentials. So why is it that a exponentially dying potential results in "short-range" and 1/r potential results in "long-range." I would have to think more about it, but my first guess would be to look at the power spectrum. In my mind I think of a long-range force as one who can emit radiation as you go towards infinity (dipole, quadrupole, etc.), while short-range forces are always screened due to the exponentially decaying potential.

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:
$\varnothing\hbar$ysicsOverflow
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).
Please complete the anti-spam verification




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

Your rights
...