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Right now, I am working on a hw problem asking me to derive the Compton effect, which is given by [tex]\lambda\prime-\lambda=\frac{h}{m_ec}(1-cos\theta)[/tex]

A diagram of the situation can be found here: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/compton.html#c1 (However, in my diagram, the recoil electron and scattered photon are switched with the electron and [tex]\phi[/tex] below the x axis and the photon and [tex]\theta[/tex] above the x axis.)

To derive the equation, I use the momentum conservation equations and energy conservation equation.

There are two equations for the momentum, one for the x component and one for the y component:

x-component: [tex]\frac{h}{\lambda}=\frac{h}{\lambda\prime}cos\theta+\gamma m u cos\phi[/tex]

y-component: [tex]0=\frac{h}{\lambda\prime}sin\theta-\gamma mu sin\phi[/tex]

where [tex]\gamma=\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}[/tex]

m=electron mass

u=electron velocity after the collision

My equation for energy conservation is:

[tex]h\frac{c}{\lambda}+mc^2=h\frac{c}{\lambda\prime}+\gamma m c^2[/tex]

To start off, I solved the x component of the momentum for cos(phi) and the y component for sin(phi). I then added the equations together and got (since sin^2 + cos^2 = 1):

[tex]\frac{h^2}{\lambda\prime^2\gamma^2m^2u^2}+\frac{h^2}{\lambda^2\gamma^2m^2u^2}-\frac{2h^2cos\theta}{\lambda\prime\lambda\gamma^2m^2u^2}=1[/tex]

At this point, I am to now eliminate u using the resulting equation and the equation for the energy conservation as given above. However, I am stuck on this next step because it seems as though any algabraic manipulation I try makes the equation extremely complicated. Could anyone please give me some hints as to what step should be next?

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# Homework Help: Deriving the Compton effect

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