Casimir's Trick: Summing vs. Averaging Spin Configurations

[Silviu]

Hello! I am reading (from Griffiths' book) about Casimir's trick in calculating the amplitude of an interaction. At a point it says that we need to average over all initial spin configurations and sum over all final spin configurations. Why is it a sum for the final states and not an average, too? Thank you!

 

[mfb]

If we don't measure the spin, all possible configurations will contribute to the result. It's like throwing 2 dice and not caring about the result of die 1: You have to add the probabilities of all possible results of die 1.

 

[Silviu]

If we don't measure the spin, all possible configurations will contribute to the result. It's like throwing 2 dice and not caring about the result of die 1: You have to add the probabilities of all possible results of die 1.

But then, why we average on the initial spin configuration. We also don't know the initial spin (this is why we need to average, right?)?

 

[mfb]

We don't know it, but the particles have to be in one specific initial spin value.

There is no time symmetry in this experiment.

 

[ChrisVer]

Think of it this way: you shoot 1000 A's on 1000 B's...
A can be in states [1,2] and B can be in states [1,2]
The final result of those collisions can be C and D each in state [1,2].
In the end of the experiment you can measure the numbers of C[1],C[2],D[1],D[2] (4 possible outcomes). But you cannot keep track of what were the Ai,Bi (so you average them out).

 

[Silviu]

Thank you for your explanations. It made sense, but I encountered a problem about the amplitude of the electron positron interaction in a positronium. Here they say that we can't average over the spins as system is either in a singlet configuration (spins antiparallel) or triplet (spins parallel). Now I am a bit confused again. Isn't this always the case? When you collide 2 particles, the z components of the spin are parallel or antiparallel (there is no 3rd option for a spin 1/2 particle). So why in a normal collision you have to average, but in this case you can't?

 

[mfb]

For positronium, you want to study both cases separately. Their relative formation rate depends on the initial conditions, for example. If you know the formation rates, and don't care about things like lifetime, you can average.