Why Does Bell's Calculation Yield a Negative Cosine?

[Adam Lewis]

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 \hat{a} and \hat{b} we want to calculate

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

where |\psi> is the singlet state.

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

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

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

P(a,b)=(\hat{a}\cdot\hat{b})=\cos(\theta).

The answer usually quoted, however, is -\cos(\theta), and I can't figure out where the minus sign is coming from. Any ideas?

 

[Adam Lewis]

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 \vec{\sigma} is supposed to work?