7. Relativity and Cosmology

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Setup

\(Clear["Global`*"]; 
Off[General::spell]; 
Off[General::spell1]; 
    \)

Needs["Calculus`VectorAnalysis`"]

Needs["Graphics`PlotField3D`"]

<< "D:\Mma\Inits\ch7init.m"

Clear["Global`*"];

7.2 Problems

7.2.1  Special Relativity Problems

Problem 3:  Compton  Scattering

A photon with wavelength λ0 collides with an electron of mass m at rest and scatters with a new wavelength λ1.  Let the incident photon move along the x axisand let the scattered photon make an angle θ1 with the x axis.  The scattered photon moves in a direction making an angle θ2 with the x axis.  Let p1 and p2 be the initial four-momenta fo the photon and electron, respectively.

Part (a)  Write the equations that follow from the conservation of four-momentum.

p1 = (h {1, 0, 0, 1})/λ0

p2 = m {1, 0, 0, 0}

p3 = (h*{1, Sin[θ1], 0, Cos[θ1]})/λ1

p4 = m*γf*{1, -βf*Sin[θ2], 0, βf*Cos[θ2]}

eq1 = Thread[p1 + p2 == p3 + p4]

ColumnForm[eq1]

Part (b)  Consider the scalar invariant m^2 = (p1+p2-p3).η. (p1+p2-p3) and solve for the wavelength λ1 of the final photon.

eq2 = ExpandAll[Simplify[m^2 == (p1 + p2 - p3) . η . (p1 + p2 - p3)]]

λRule = ExpandAll[Solve[eq2, λ1][[1]]]

Part (c)  Find the kinetic energy of the recoil electron.

eq3 = eq1[[1]] /. λRule

γfRule = Simplify[Solve[eq3, γf][[1]]]

Simplify[m*γf - m /. γfRule]

Part (d)  Find a relation between θ1, θ2, and λ0.

eq4 = Simplify[eq1[[{2, 4}]]]

sol1 = Solve[eq4, {γf, Sin[θ1]}][[1]]

eq5 = Sin[θ1] == (Sin[θ1] /. sol1)

sol2 = Simplify[Solve[eq5, {Tan[θ2]}][[1]] /. λRule]