How to Relate the Angles between Center-of-Mass and Lab Frames

Quantum physics experiments take place in the lab frame, but you do scattering calculations in the center-of-mass frame, so you have to know how to relate the angle between the two frames.

Here's how this works: The following figure shows scattering in the lab frame.

Scattering in the lab frame.
Scattering in the lab frame.

One particle, traveling at

image1.png

is incident on another particle that's at rest

image2.png

and hits it. After the collision, the first particle is scattered at

image3.png

and the other particle is scattered at

image4.png

Now in the center-of-mass frame, the center of mass is stationary and the particles head toward each other. After they collide, they head away from each other at angles

image5.png

You have to move back and forth between these two frames — the lab frame and the center-of-mass frame — so you need to relate the velocities and angles (in a nonrelativistic way).

To relate the angles

image6.png

you start by noting that you can connect

image7.png

using the velocity of the center of mass,

image8.png

In addition, here's what you can say about the velocity of particle 1 after it collides with particle 2:

image9.png

Now you can find the components of these velocities:

image10.png

Dividing the equation in the second bullet by the one in the first gives you

image11.png

But wouldn't it be easier if you could relate

image12.png

by something that didn't involve the velocities, only the masses, such as the following?

image13.png

Well, you can. To see that, start with

image14.png

And you can show that

image15.png

You can also use the conservation of momentum to say what happens after the collision. In fact, because the center of mass is stationary in the center-of-mass frame, the total momentum before and after the collision is zero in that frame, like this:

image16.png

Therefore

image17.png

And after the collision,

image18.png

which means that

image19.png

Also, if the collision is elastic, kinetic energy is conserved in addition to momentum, so that means the following is true:

image20.png

Substituting

image21.png

into this equation gives you

image22.png

Given these two equations, you can redo

image23.png

Dividing the magnitude of each side of

image24.png

by the magnitude of the above equation gives you

image25.png

And because you saw earlier that

image26.png

substituting

image27.png

into this equation gives you at last

image28.png

Okay, that relates

image29.png

which is what you were trying to do. Using the relation

image30.png

you can rewrite

image31.png

as the following:

image32.png

You can also relate

image33.png

You can show that

image34.png

which, using a little trig, means that

image35.png

You've now related the angles between the lab and center-of-mass frames.

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