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
![image0.png](https://www.dummies.com/wp-content/uploads/398216.image0.png)
is infinitesimal in size, and it stays the same between the two frames. But the angles that make up
![image1.png](https://www.dummies.com/wp-content/uploads/398217.image1.png)
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:
![image2.png](https://www.dummies.com/wp-content/uploads/398218.image2.png)
to the center-of-mass differential cross section:
![image3.png](https://www.dummies.com/wp-content/uploads/398219.image3.png)
In the lab frame,
![image4.png](https://www.dummies.com/wp-content/uploads/398220.image4.png)
And in the center-of-mass frame,
![image5.png](https://www.dummies.com/wp-content/uploads/398221.image5.png)
Because
![image6.png](https://www.dummies.com/wp-content/uploads/398222.image6.png)
the following equation is true:
![image7.png](https://www.dummies.com/wp-content/uploads/398223.image7.png)
Putting that equation with the equations for the lab frame and the center-of-mass frame, you have
![image8.png](https://www.dummies.com/wp-content/uploads/398224.image8.png)
Because you have cylindrical symmetry here,
![image9.png](https://www.dummies.com/wp-content/uploads/398225.image9.png)
You've already seen that
![image10.png](https://www.dummies.com/wp-content/uploads/398226.image10.png)
You can also show that
![image11.png](https://www.dummies.com/wp-content/uploads/398227.image11.png)