Intro to Particles and Fields - ex3
April 2, 2017
due date is: 23.04.2017
Contents
1 Yukawa potential and the Klein Gordon equation
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2 Yukawa potential and the weak force
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3 Yukawa potential of weak and EM forces
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1
Yukawa potential and the Klein Gordon equation
In this assignment we will expand on the connection between the Yukawa potential and the Klein Gordon equation.
1) Calculate by direct differentiation that −∇2 + m2 φ = 0 everywhere,
where φ =
ge−mr
.
r
2) Calculate explicitly the divergence of
r̂
r2 .
3) However using the divergence theorem one finds:
Z
r̂
∇ · 2 dV = 4π
r
V
Explain this discrepancy.
4) You now have the tools necessary to prove that:
−∇2 + m2
• The divergence theorem:
Z
ge−mr
= gδ(r)
r
I
~
F~ · dS
div F~ dV =
V
S
• Divergence in spherical coordinates:
1
∂
1
∂
~ · f~ = 1 ∂ r2 fr +
∇
(sin(θ)fθ ) +
(fφ )
r2 ∂r
r sin(θ) ∂θ
r sin(θ) ∂φ
• Gradient in spherical coordinates:
1
∂F
~ (F ) = ∂F r̂ + 1 ∂F θ̂ +
∇
φ̂
∂r
r ∂θ
r sin(θ) ∂φ
• Reminder: ∇2 f =div(grad f )
A bit more about the so-called Gauss’ theorem - simply as Green’s theorem
applied to 3D → 2D: Green’s Theorem
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Yukawa potential and the weak force
Yukawa potential is widely used as a model for interactions mediated by Bosons,
it is also a model for a ”masked” or ”screened” coulomb potential. In this question we will aim to find the characteristic ranges of the weak and electromagnetic
forces.
1) Find the masses of the following Bosons:
– W ± Boson.
– Z 0 Boson.
– H Boson (Higgs Boson).
– Aµ Boson (electromagnetic Boson - i.e. photon)
– π ± ,and π 0
2) Think about a working definition for effective range. Explain why you
chose that definition.
3) Reintroduce natural constants into the dimensionless Yukawa potential
−mr
φ = Ae r , to arrive at a physically correct version. Perform the dimensional analysis explicitly.
4) Choose one massive particle from the above list, and find by explicit calculation the characteristic range of the interaction it mediates. Do the
same for the rest of the particles - no need for explicit calculation.
5) Look-up and write the charge radius of the proton.
6) Compare the effective range of a charged weak Boson to the charge radius
of the proton.
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Yukawa potential of weak and EM forces
In this assignment we will calculate some values for weak interactions and compare to EM, and gravitational interactions, assuming a Yukawa potential of the
−mr
form ge r for the weak interaction and setting g = 1:
1) What is the weak force between two protons at a range of 1f m?
2) What is the range at which the electric force and the weak force will
balance each other, for two protons?
3) What is the value of the weak interaction for two protons, in two separate
◦
Hydrogen atoms, where the protons can be said to be a distance of 1A
Compare to the gravitational force between the two protons.
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