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# hwsol3 - Physics 31 Spring 2007 Solution to HW#3 Square...

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Physics 31 Spring, 2007 Solution to HW #3 Problem A When we considered the classical analog of Compton scattering, we wrote the following conservation equations for the x and y components of momentum and for the energy: mv = mv ± cos φ + Mu ± cos θ (1) 0= mv sin φ ± sin θ (2) 1 2 mv 2 = 1 2 mv ± 2 + 1 2 ± 2 (3) (a) Compare Eqs. (1) and (2) with the corresponding equa- tions in the text for Compton scattering [Eqs. (2.16) and (2.17)]. How do they diFer? (b) Write the relativistic energy conservation equation for Compton scattering, that is, the case of a photon (fre- quency ν ) incident on an electron (mass M ) at rest. You will need Eq. 1.24 from page 76 of the text. Hint: If the mass of the electron is M , and its momentum after the collision is p ± e , then its energy after the collision is E = p M 2 c 4 +( p ± e ) 2 c 2 . See if you can massage your energy equation to get Eq. (2.19) in the text. (Note that the text is a little sloppy about using the symbol “ p ” for several diFerent things. In Eq. (2.19), p is the electron momentum after the collision, the same quantity denoted by p ± e here.) (c) (Optional) Solve Eqs. (1)–(3) to verify the result pre- sented in class, 1 E ± 1 E 4 1 cos φ Mv 2 . You should ±nd the solution to an exact quadratic equa- tion for v , and then look for simplications using results valid for m/M ± 1. ²or part (a): the equations in the text use the momentum p e of the electron, rather than mv . In that formulation, you don’t have to worry about the factors of γ . ²or part (b), equate the initial energy to the ±nal: + Mc 2 = ± + q ( 2 ) 2 p ± e c ) 2 Rearrange the terms: ± + 2 = q ( 2 ) 2 p ± e c ) 2 .

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hwsol3 - Physics 31 Spring 2007 Solution to HW#3 Square...

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