Lecture 7
39
Nov 13, 2002
6
Forces in Nature
The forces in nature are Gravity, the weak interaction, the electromagnetic interaction (the latter two
now unified as the electroweak interaction, the cornerstone of the Standard Model), and the strong or
color interaction. All modern interaction models are based on the exchange of elementary quanta, in
the simplest case tensor, vector, or scalar bosons. Feynman graphs picture such exchange processes,
but more than that, they associate precise terms for the amplitude describing the process with the vari
ous items in the graph: vertices (where lines meet) correspond to precisely prescribed vertex factors,
internal lines to propagator terms, and in and outgoing particle lines to phasespace factors dependent
on particle type.
6.1
The Yukawa Interaction
In 1934 Hideki Yukawa proposed the existence of a light mass boson (bosonic field) that would be re
sponsible for (the mediator of) the shortrange nuclear force.
2
The mass of the boson would be com
mensurate with the observed mean range of about 2 fm. The conjugate of the range would be the
boson’s mass or energy:
m
=1/
r =
ħ
c/r
= 197 MeV
⋅
fm / 2 fm
≈
100 MeV. Powell
et al.
much later
(1947) observed the Yukawa particle, the pion (140 MeV) or pimeson, in photographic emulsions ex
posed to cosmic rays at highaltitude (mountain tops and balloons).
3
Earlier, the muon (105 MeV) had
been discovered (at sea level),
4
but was quickly recognized for what it was: a heavy version of the
electron (a charged lepton), which did NOT have the strong interaction, and thus could be hardly its
messenger!
An appropriate way of deriving this is to start with the relativistic massenergy relationship, and “quan
tify” it by making the momentum and energy into operators:
2
2
2
2
2
2
2
;
using
, and
,
this becomes:
( , )
0
p
m
E
E
i
i
m
t
t
t
ψ
∂
∂
∂
+
=
→
→ −
−
+
+
=
∂
∂
∂
p
r
r
∇
(I.94)
This equation is the famous
KleinGordon equation
, the relativistic version of the Schrödinger equa
tion or the massive version of the equation of motion for the electromagnetic field. For spherically
symmetric solutions
ψ
(
r
,
t
) =
ψ
(
r
,
t
) the threedimensional space derivative can be written purely in
terms of radius
r
(the derivatives with respect to polar and azimuthal angles
θ
and
φ
have nothing to
operate on!):
2
2
2
2
2
2
2
2
2
2
2
2
2
( )
2
1
1
1
( )
( )
cot
( )
sin
r
r
r
r
r
r
r
r
r
r
r
ψ
ψ
ψ
θ
ψ
θ
θ
θ
φ
∂
∂
∂
∂
∂
∂
∂
∂
≡
=
+
+
+
+
=
∂
∂
∂
∂
∂
∂
∂
∂
r
∇
r
(I.95)
In the stationary case, the time derivative also vanishes. For a pointlike nucleon at rest, equation (I.94)
has a stationary, spherically symmetric solution
( )
4
mr
e
r
r
α
ψ
π
−
=
, the
Yukawa potential
. Here
r
=
r
;
α
is
a proportionality “constant” (which could be a function of something other than
r
, and is in fact a func
tion dependent on the nucleon spin as well); and the exponential has the particle (boson) mass
m
in it
(in the proper units!).
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 Fall '01
 Rijssenbeek
 Physics, Angular Momentum, Force, Gravity, Schrodinger Equation, wave function, Quark, expectation value

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