Question about Dr. Chinese's example of Bell's Theorem

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Discussion Overview

The discussion revolves around Bell's Theorem, specifically addressing the probabilities associated with detector outcomes in quantum mechanics versus local hidden variable theories. Participants explore the implications of these probabilities in the context of different experimental setups, including Stern-Gerlach analyzers and the interpretations of quantum mechanics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the interpretation of a probability of 1/4 for complete matches between detectors, suggesting it should be at least 1/3 based on the provided table.
  • Another participant asserts that Bell's Inequality is violated, implying that local interpretations cannot account for the observed probabilities.
  • Some participants discuss the implications of Einstein's objections to local hidden variables, noting that EPR believed in local hidden variables while Bell's response acknowledged the need for non-locality.
  • A participant introduces the Consistent Histories approach, suggesting that classical definitions of objective reality may be inadequate for quantum mechanics.
  • Questions arise regarding the quantum mechanical probability formula, specifically whether it should be cos²(θ/2) and how it applies to different experimental setups.
  • Another participant raises questions about the rationale for using Stern-Gerlach analyzers with fixed orientations and the implications of extending this to an infinite number of positions.
  • Concerns are expressed about how probabilities are calculated in these setups, particularly regarding integration over angles.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of probabilities in quantum mechanics versus local hidden variable theories. There is no consensus on the correct interpretation or the implications of the probabilities discussed.

Contextual Notes

Some participants note that the quantum prediction is not fully represented in the initial table, and there are unresolved questions about the assumptions underlying the calculations and interpretations presented.

Who May Find This Useful

This discussion may be of interest to those studying quantum mechanics, Bell's Theorem, or the philosophical implications of quantum interpretations, particularly in relation to experimental setups involving Stern-Gerlach analyzers.

prajor
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Hello, this could be a basic question. I saw the link http://www.drchinese.com/David/Bell_Theorem_Easy_Math.htm" on one of the threads.

Following the table there the probability of complete match between two detectors happens twice out of 8 times. i.e. 1/4.

How how can we say that it happens at least 1/3 times, where the table clearly shows 1/4.
 
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We can't. Bell's Inequality is violated.
 


Luke, But the local interpretation (EPR) (the Table with 8 possibilities) also gives 1/4 only !
 


prajor said:
Luke, But the local interpretation (EPR) (the Table with 8 possibilities) also gives 1/4 only !
Yes, this was the source of Einstein's objection. EPR had essentially believed in local hidden variables and complained that Quantum Mechanics doesn't make any sense unless their hidden variables were non-local. Bell responded by saying "yes, that's quite right".

There are ways out of this however. The usual way of doing this is to say that hidden variables do not exist and not assume the existence of anything that cannot be constructed from the rules of Quantum Mechanics.
Many Bohmians would rather allow their descriptions even though they are non-local.

I, myself, having learned Consistent Histories from Robert Griffiths (not to be confused with any other Welshman with the name Griffiths), am of the view that a classical definition of objective reality is simply too limited to talk about Quantum Mechanics. Griffiths, by the way, constructed his ideas firmly in the mathematics of Quantum theory. However, and I'm trying to work this out myself, Consistent Histories seems to be able to use very similar tools and the same formulas as the Bohmian approach.
 
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prajor said:
Hello, this could be a basic question. I saw the link http://www.drchinese.com/David/Bell_Theorem_Easy_Math.htm" on one of the threads.

Following the table there the probability of complete match between two detectors happens twice out of 8 times. i.e. 1/4.

How how can we say that it happens at least 1/3 times, where the table clearly shows 1/4.

Welcome to PhysicsForums, prajor!

I think if you look again, the thing you say happens 1/4 of the time actually happens 1/2 of the time. When you consider the lowest possible scenario, from the table, you can get as low as 1/3 but no lower. That is for the classical analysis.

On the other hand: the Quantum prediction is not presented in the table. It is per the cos^2 theta formula, and is .25 or 1/4.

Does this help?
 
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Thanks DrC for the welcome and clarification.

1. Should the QM probability be cos2(\theta/2) or is it true only if we take Stern-Gerlach devices instead of photons with polarizers ?

2. Obviously 3 polarisers at 120 degrees is a special case. Isn't it right to have a generic case for QM probability ?
\int cos2\theta integrated over 0 and 2\pi
 
ok. I will try and explain why I have the below questions. May be someone can throw some light on this.

I am following the argument of Bell's theorem with Stern Gerlach Analyzers (SGA) and silver atoms (Daniel Styer). In this first probablilty that is calculated is for a simple setup where we know the direction of SGA, so a given single atom coming through has a 1/2 probability of going to output +mb (and same probability for -mb).

Now we introduce a SGA mounted on a pivot with three specific orientations and then do the same experiment. We get probability 1/2 again for the output to be +mb.

The questions:
1. Why complicate the setup by introducing SGA which can take one of the 3 fixed positions ? What is the rationale behind this ?
2. If we extend the above to infinite number of positions around the circle also, the probability of output being +mb should be 1/2 ?
3. How is the probability in above (#2) calclulated ? Is it not by integrating cos2(\theta/2) around the circle ?
 
Folks, any thoughts on my questions below ?
 
Guess these are not right questions..
 

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