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# 30 - Physics 6210/Spring 2007/Lecture 30 Lecture 30...

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Physics 6210/Spring 2007/Lecture 30 Lecture 30 Relevant sections in text: § 3.9, 5.1 Bell’s theorem (cont.) Assuming suitable hidden variables coupled with an assumption of locality to determine the spin observables with certainty we found that correlation functions must satisfy |h A n 1 ) B n 2 ) i - h A n 1 ) B n 3 ) i| ≤ 1 + h A n 2 ) B n 3 ) i . We now show that quantum mechanics is not compatible with this inequality. We compute the expectation value of the product of the two observers’ measurements (in units of ¯ h/ 2) using quantum mechanics: h A n 1 ) B n 2 ) i = 1 ¯ h/ 2 2 h S 1 S 2 i = 4 ¯ h 2 h ψ | n 1 · ~ S 1 )(ˆ n 2 · ~ S 2 ) | ψ i . (Again, this quantity is the correlation function of the two spins.) With z chosen along ˆ n 1 , this quantity is easily computed (exercise): h ψ | n 1 · ~ S 1 )(ˆ n 2 · ~ S 2 ) | ψ i = ¯ h 4 ( h + - | - h- + | n 2 · ~ S 2 ( | + -i + | - + i ) = - ¯ h 2 4 cos θ, where θ is the angle between ˆ n 1 and ˆ n 2 . To get the last equality we assume that ˆ n 1 and ˆ n 2 are in the x - z plane with z along and ˆ n 1 . Using θ to denote the angle between ˆ n 1 and ˆ n 2 we have ( h + -|-h- + | n 2 · ~ S 2 ( | + -i + |- + i ) = ( h + -|-h- + | )[cos θS 2 z +sin θS 2 x ]( | + -i + |- + i ) = - cos θ Of course the result is geometric and does not depend upon the choice of coordinates. Thus, defining θ ij = ˆ n i · ˆ n j , Bell’s inequality – if it applied in quantum mechanics would imply | cos θ 13 - cos θ 12 | ≤ 1 - cos θ 23 , which is not true. Thus quantum mechanics is not consistent with all observables having local definite values based upon some (unknown) “hidden variables”. On the other hand, if reality is such that all observables for the individual particles are compatible and locally defined (with QM just giving an incomplete statistical description), then this To see this, just let ˆ n 1 point along y , let ˆ n 3 point along x , and let ˆ n 2 lie at 45 from x (or y ) in the x - y plane, so that θ 12 = π/ 4 = θ 23 , θ 13 = π/ 2. 1

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Physics 6210/Spring 2007/Lecture 30 inequality should be valid, experimentally speaking. (Assuming of course that the correct description can be obtained using some hidden variables λ as described above.) Experiments to check the Bell inequality have been performed since the 1960’s. Many regard the “Aspect experiment” of the early 1980’s as definitive. It clearly showed that the Bell inequality was violated, while being consistent with quantum mechanical predictions.
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