Yukawa potential

From Wikipedia, the free encyclopedia

Jump to: navigation, search

A Yukawa potential (also called a screened Coulomb
potential) is a potential of the form

Hideki Yukawa showed in
the 1930s that such a potential arises from the exchange of a massive scalar
field such as the field of the pion whose mass is m. Since the field mediator is massive the
corresponding force has a certain range due to its decay, where the range is
inversely proportional to the mass. If the mass is zero, then the Yukawa
potential becomes equivalent to a Coulomb potential, and the range is said to be
infinite.

In the above equation, the potential is negative, denoting that the force is
attractive. The constant g is a real number; it is equal to the coupling constant
between the meson field and the fermion field with which it
interacts. In the case of the nuclear force, the
fermions would be a proton and another proton or a neutron.

[edit]
Fourier transform

The easiest way to understand that the Yukawa potential is associated with a
massive field is by examining its Fourier transform. One
has

where the integral is performed over all possible values of the 3-vector
momentum k. In this form, the fraction 4π /
(k2 + m2) is seen to be the propagator
or Green's function of
the Klein-Gordon
equation.

[edit]
Feynman amplitude

The Yukawa potential can be derived as the lowest order amplitude of the
interaction of a pair of fermions. The Yukawa interaction
couples the fermion field ψ(x) to the meson
field φ(x) with the coupling term

The scattering
amplitude for two fermions, one with initial momentum p1 and the other with momentum p2, exchanging a meson with momentum
k, is given by the Feynman diagram on the
right.

The Feynman rules for each vertex associate a factor of g with the
amplitude; since this diagram has two vertices, the total amplitude will have a
factor of g2. The line in the
middle, connecting the two fermion lines, represents the exchange of a meson.
The Feynman rule for a particle exchange is to use the propagator; the
propagator for a massive meson is − 4π /
(k2 + m2). Thus, we see that the
Feynman amplitude for this graph is nothing more than

From the previous section, this is seen to be the Fourier transform of the
Yukawa potential.

[edit]
References

Gerald Edward
Brown and A. D. Jackson, The Nucleon-Nucleon Interaction, (1976)
North-Holland Publishing, Amsterdam ISBN
0-7204-0335-9

Retrieved from "http://en.wikipedia.org/wiki/Yukawa_potential"

Categories: Nuclear
physics | Quantum field
theory | Potential

Personal tools

Log
in / create account

Namespaces

Article

Discussion

Variants

Views

Read

Edit

View
history

Actions

Search

Navigation

Main page

Contents

Featured
content

Current events

Random article

Donate
to Wikipedia

Interaction

Help

About Wikipedia

Community
portal

Recent
changes

Contact Wikipedia

Toolbox

What
links here

Related
changes

Upload file

Special pages

Permanent
link

Cite
this page

Print/export

Create a book

Download as PDF

Printable
version

Languages

العربية

Česky

Deutsch

Español

Français

한국어

Italiano

Nederlands

日本語

Русский

Українська

This page was last modified on 28 December 2010 at
11:24.

Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. See Terms of Use for
details.
Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a
non-profit organization.

Contact us

Privacy policy

About Wikipedia

Disclaimers