Casimir's Trick: Summing vs. Averaging Spin Configurations

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Discussion Overview

The discussion revolves around the application of Casimir's trick in quantum mechanics, specifically regarding the treatment of spin configurations in interactions. Participants explore the reasoning behind summing over final spin states while averaging over initial spin states, and the implications of these choices in different contexts, such as electron-positron interactions in positronium.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions why the final states are summed while the initial states are averaged, suggesting that all configurations contribute when not measuring spin.
  • Another participant draws an analogy to rolling dice to illustrate the need to sum probabilities for final states while averaging initial states.
  • A further contribution emphasizes that while the initial spin is unknown, particles must have a specific initial spin value, indicating a lack of time symmetry in the experiment.
  • One participant presents a scenario involving multiple collisions to clarify the distinction between averaging initial states and summing final outcomes.
  • A later reply introduces confusion regarding the treatment of spins in positronium, noting that spins can only be in singlet or triplet configurations, questioning why averaging is permissible in normal collisions but not in this specific case.
  • Another participant responds by suggesting that the formation rates of the singlet and triplet states depend on initial conditions, which may allow for averaging under certain circumstances.

Areas of Agreement / Disagreement

Participants express differing views on the treatment of initial and final spin configurations, with some agreeing on the necessity of averaging initial states while others question the consistency of this approach in specific scenarios like positronium interactions. The discussion remains unresolved regarding the implications of these differing treatments.

Contextual Notes

The discussion highlights limitations in understanding the role of initial conditions and the specific configurations of spins in quantum interactions, as well as the dependence on the context of the experiment.

Silviu
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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!
 
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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.
 
mfb said:
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?)?
 
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.
 
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).
 
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?
 
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.
 

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