Minus sign in Bell derivation

In summary, the conversation revolves around reproducing Bell's calculation for the expectation value of paired spin measurements on particles in the singlet state. The calculation involves calculating P(a,b) via the commutation and anticommutation relations for the Pauli matrices, resulting in a final answer of P(a,b)=(\hat{a}\cdot\hat{b})=\cos(\theta). However, the commonly quoted answer is -\cos(\theta), leading to confusion about where the minus sign is coming from. The possibility of inserting the information about anti-parallel spins again via the operator \vec{\sigma} is also discussed.
  • #1
Adam Lewis
16
0
Hello,

I am trying to reproduce Bell's calculation for the expectation value of paired spin measurements on particles in the singlet state. For unit vectors [itex] \hat{a} [/itex] and [itex] \hat{b} [/itex] we want to calculate

[tex] P(a,b)=<\psi|(\hat{a}\cdot\vec{\sigma})(\hat{b} \cdot \vec{\sigma})|\psi>[/tex]

where [itex]|\psi>[/itex] is the singlet state.

Via the commutation and anticommutation relations for the Pauli matrices the enclosed operator is simply

[tex](\hat{a}\cdot\hat{b})I + \imath\vec{\sigma}\cdot(\hat{a}\times\hat{b}).[/tex]

As a scalar the dot product can be pulled from the bra-ket, leaving [itex](\hat{a}\cdot\hat{b})<\psi|I|\psi>=(\hat{a}\cdot \hat{b}) [/itex] since the singlet state is normalized. The cross product's expectation value turns out to vanish. Thus the final answer is

[tex] P(a,b)=(\hat{a}\cdot\hat{b})=\cos(\theta). [/tex]

The answer usually quoted, however, is [itex] -\cos(\theta) [/itex], and I can't figure out where the minus sign is coming from. Any ideas?
 
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  • #2
Well, that would get you the minus sign. But I had thought the fact the spins were anti-parallel to be already encoded by the singlet state. It seems odd to me that you should have to insert this information again via the operator. Maybe I'm misunderstanding how [itex]\vec{\sigma}[/itex] is supposed to work?
 

What is the significance of the minus sign in Bell derivation?

The minus sign in Bell derivation represents the correlation between measurements of entangled quantum particles. It is a crucial aspect in understanding the phenomenon of quantum entanglement and its effects on the measurement outcomes.

Why is the minus sign often referred to as the "quantum sign"?

The minus sign in Bell derivation is commonly referred to as the "quantum sign" because it is a unique characteristic of quantum mechanics. It reflects the non-classical behavior of entangled particles and is not present in classical physics.

How does the minus sign affect the probabilities in Bell's inequality?

The minus sign in Bell's inequality alters the probabilities of measurement outcomes by introducing a negative correlation between entangled particles. This leads to violations of the inequality, demonstrating the non-local nature of quantum mechanics.

Can the minus sign be explained by classical physics?

No, the minus sign in Bell derivation cannot be explained by classical physics. It is a purely quantum phenomenon that arises from the superposition and entanglement of particles, which have no classical analog.

What implications does the minus sign have on our understanding of reality?

The minus sign in Bell derivation challenges our classical notion of reality by demonstrating the non-local and non-causal nature of quantum mechanics. It suggests that there are fundamental aspects of reality that are beyond our classical understanding and may require further exploration and study.

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