# 29 - Physics 6210/Spring 2007/Lecture 29 Lecture 29...

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Physics 6210/Spring 2007/Lecture 29 Lecture 29 Relevant sections in text: § 3.9 Spin correlations and quantum weirdness: Spin 1/2 systems Consider a pair of spin 1/2 particles created in a spin singlet state. (Experimentally speaking, this can be done in a variety of ways; see your text.) Thus the state of the system is defined by the state vector | ψ i = 1 2 ( | + -i - | - + i ) . Let us suppose that the particles propagate – undisturbed and not interacting – until they are well-separated. Particle 1 has a component of spin measured by Observer 1 and particle 2 has a component of spin measured by Observer 2. To begin, suppose both observers measures spin along the z axis. If observer 1 sees spin up, what does observer 2 see? You probably will correctly guess: spin down. But can you prove it? Well, the reason for this result is that the state of the system is an eigenvector of S z with eigenvalue zero. So, the two particles are known with certainty to have opposite values for their z -components of spin. Alternatively, you can see from the expansion of | ψ i in the product basis that the only states that occur (with equal probability) are states with opposite spins. Let us see how to prove this systematically; it’s a good exercise. We can ask the question as the following sequence of simple questions. What is the probability P ( S 1 z = ¯ h 2 ) that observer 1 gets spin up? That’s easy:* P ( S 1 z = ¯ h 2 ) = |h + + | ψ i| 2 + |h + - | ψ i| 2 = 1 2 . Of course, there is nothing special about particle 1 compared to particle 2; the same result applies to particle 2. What’s the probability for getting particle 1 with spin up and particle 2 with spin down? We have P ( S 1 z = ¯ h 2 , S 2 z = - ¯ h 2 ) = |h + - | ψ i| 2 = 1 2 . * Well, it’s “easy” if you realize that, in a state | ψ i , the probability for getting an eigenvalue a of an operator A is (exercise) P ( a ) = d X i =1 |h i | ψ i| 2 , where | i i are a basis for the d -dimensional subspace of vectors with eigenvalue a : A | i i = a | i i , i = 1 , 2 , . . . d. 1

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Physics 6210/Spring 2007/Lecture 29 You can now easily infer that – in the singlet state – when particle 1 is determined to have spin up along z , then particle 2 will have spin down along z with certainty. We say that the S z variables for particles 1 and 2 are “completely correlated”. Another way to state this is the following. If – in the singlet state – particle 1 is determined to have spin up along z , then the state vector of the system for the purposes of all subsequent measurements is given by | + -i . A subsequent measurement of the z component of spin for particle 2 is then going to give spin-down with certainty. The foregoing discussion will work for any
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