How to Translate Cross-Sections between Center-of-Mass and Lab Frames

In quantum physics, once you relate the angles of the scattered particles in the lab frame and the center-of-mass frame, you can translate the differential cross section — the bull's eye when you're aiming to scatter the particles at a particular angle — between the lab and center-of-mass frames.

The differential area

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is infinitesimal in size, and it stays the same between the two frames. But the angles that make up

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the scattering angle, vary when you translate between frames. You get to take a look at how that works now, relating the lab differential cross section:

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to the center-of-mass differential cross section:

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In the lab frame,

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And in the center-of-mass frame,

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Because

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the following equation is true:

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Putting that equation with the equations for the lab frame and the center-of-mass frame, you have

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Because you have cylindrical symmetry here,

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You've already seen that

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You can also show that

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