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Appendix 14 THE EPR PARADOX In their  paper “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” Einstein, Podolsky, and Rosen (EPR) pointed out that, in judging the success of a theory, we have to ask two questions: () Is the theory correct? and () Is the theory complete? EPR define completeness as meaning that “every element of the physical reality must have a counterpart in the physical theory.” Something is an element of reality “if, without disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of the physical reality corresponding to this physical quantity.” Now, according to Heisenberg’s uncertainty principle, when the momentum of a particle is known exactly, the position has no physical reality. Applying the definitions given above, a thought experiment can be designed to test whether the quantum mechanical view of Heisenberg ’s uncertainty is correct, or the quantum mechanical description of reality incomplete. The exercise was prompted by Einstein’s deep dissatisfaction with the strangeness of quantum theory. While he appreciated the mathematical consistency and the utility of this theory, he also thought that it should be replaced by something that allowed a more reasonable view of reality. Thus, when it was impossible to doubt that quantum mechanics was correct, it was possible to prove that it was deficient because it was incomplete. To perform the EPR thought experiment, consider two particles: because of Heisenberg’s uncertainty, the position and momentum of particle , x1 and p1, are incompatible observables. Similarly, for particle , x2 and p2 are incompatible observables. They cannot be known exactly at the same time. However, the relative positions of the two particles on an axis, say the x-axis, or the distance between them, given by the difference, x1 - x2, between their coordinates, and their total momentum (the sum p1 + p2) are not subject to Heisenberg’s uncertainty. That is, it is in principle possible that ∆(x1 - x2) ∆(p1 + p2) = 0.  1SCHÄFER_PAGES:SCHÄFER PAGES 4/29/10 11:14 AM Page 189 Assume that the two particles at one time interact and then are separated so that information on the distance between them and on their total momentum is not disturbed. Then at a certain time, t, an observer can measure x1, and will know x2 because the difference x1 - x2 is known; or he can choose to measure p1, and then will know p2 because the sum p1 + p2 is known. Since either x2 or p2 is exactly predictable in this way, both must correspond to an element of reality. EPR: “Since quantum mechanics forbids this, it must be incomplete.” Taken from a different point of view: At a certain time t, one can decide to measure either the position or the momentum of a distant particle, using a local particle as the measuring device. This means that immediately prior to the chosen measurement, the distant particle must actually have owned both a definite position and a definite momentum, or something I do here, now, has an instantaneous effect a long distance away. Einstein, Podolsky, and Rosen concluded that such a violation of the locality principle was unacceptable : “No reasonable definition of reality could be expected to permit this.” In  Bohm formulated a version of the EPR paradox involving the spin components of two spin-½ particles (such as electrons or protons, with s = ½) in what is called a singlet state. When two electrons form a singlet state, their angular momenta are counter-aligned in such a way that they cancel and the total angular momentum is zero. In terms of quantum mechanics, the total angular momentum quantum number is zero. The condition that the individual angular momenta of the two particles cancel in the singlet state implies that, when the spin components of the two particles are measured along the z-direction and when the orientation of one of them is found up, the orientation of the other one must be down. If s(z,) = z+, s(z,) = zIf s(z,) = z-, s(z,) = z+ According to EPR, s(z,) is an element of reality. However, instead of measuring s(z,), one could have measured s(x,). This would have predicted s(x,) exactly; that is, s(x,) is also an element of reality. However, in quantum mechanics, s(x) and s(z) are subject to Heisenberg ’s uncertainty, and when one is known the other one is not...

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