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Interpretation of Quantum Mechanics Lectures Quantum Mechanics - January 2009 Lectures-QM-08/09 – p. Critique of Einstein, Podolski and Rosen 1935 paper by Einstein, Podolski and Rosen Can Quantum-Mechanical Description of Physical Reality Be Considered Complete ? Question posed: Is the description given by quantum mechanics complete ? What is meant by complete ? Herefore two propositions: A. Every element of physical reality must have a counterpart in the physical theory B. If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity Lectures-QM-08/09 – p. EPR critique EPR invented Gedankenexperiments, each of which can determine the eigenvalues of one member of a group of incompatible observables without disturbing the system to which the observable pertains. This they did by by considering a system composed of two parts, I and II, which interacted in the past, but which subsequently separate to an extent where a measurement performed on one part is no longer able to produce an inﬂuence on the other. Examples: Two-particle decay of a system 1 Spin singlet state decays into two spin 2 particles 1 |0 >= √ (|+ >1 |− >2 −|− >1 |+ >2 ) (1) 2 - Perfect correlations between the two spin projections Classical physics of decaying particles would also exhibit correlations, but no inﬂuence of the measurement of one particle on the other if they are spatially separated. Lectures-QM-08/09 – p. EPR critique What causality requires is that there be no inﬂuence on an object due to any action taken in a region that is at a space-like position with respect to the object. It is the combination of this requirement with the existence of incompatible observables that creates the enigmatic situations that EPR pointed to. The paradox: Let A be an observable for system I, with eigenvalues aα and eigenfunctions uα (x1 ) and the total state is X Ψ(x1 , x2 ) = ψα (x2 )uα (x1 ) (2) α B is another observable belonging to I, which does not commute with A, with the eigenvalues bβ and eigenfunctions vβ (x1 ) and X Ψ(x1 , x2 ) = φβ (x2 )vβ (x1 ) (3) α Lectures-QM-08/09 – p. EPR critique ...continued: I and II are spacelike separated: No real change in I or II due to an action in II or I, respectively From a measurement of the observable A in I with outcome uν we know from the wave function that the distant state of system II is ψν . Alternatively, by measuring B, and ﬁnding I to be in the state vλ , we know from the wave function that II is in the state φλ Suppose further: ψα (x2 ) are the eigenfunctions of an observable P belonging to system II, with eigenvalues pα , while the φβ (x2 ) are the eigenfunctions of another observable Q of system II with eigenvalues qβ , and that P and Q do not commute. As no interaction between I and II is possible when they have a space-like separation this has the following consequences: C. By measuring either A or B we are in a position to predict with certainty, and without in any way disturbing the second system, either the value of the quantity P (that is pα ) or the value of the quantity Q (that is qβ ). In accordance with our criterion of reality, in the ﬁrst case we must consider the quantity P as being an element of reality, while in the second case the quantity Q is an element of reality. Lectures-QM-08/09 – p. EPR critique However, quantum mechanics tells us D. When two operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous physical reality. and furthermore that any attempt to determine Q experimentally will alter the state of the system in such a way as to destroy the knowledge of P. C. and D. contradict each other ! Since the Gedankenexperiment satisﬁes proposition B, EPR concluded that quantum mechanics is a theory that fails to comply with proposation A i.e. the quantum mechanical description of physical reality given by the wave function is not complete ! =⇒ EPR: Not every element of physical reality has a counterpart in quantum mechanics. Bohr argued subsequently: The EPR conception of reality is in conﬂict with how factual statements must be made in the quantum domain. Lectures-QM-08/09 – p. EPR critique Bohr further argues: Distinct experiments measure mutually incompatible observables of one particle (Spin projections along different axes as being examples for the two observable A and B) EPR were mislead by their deﬁnition of elements of physical reality The procedure of measurement has an essential inﬂuence on which the very deﬁnition of the physical quantities rests Nowadays typical attitude: EPR does not pose a paradox, because it is impossible to devise any experiment that simultaneously measures precise values of incompatible observables. Arguments for EPR paradox typical employ counterfactual arguments: Contradictions between an actual experiment that does measure P, and another that could, were it done instead, measure the incompatible observable Q. Lectures-QM-08/09 – p. EPR critique Nevertheless, entangled states have properties that appear to be due to non-local interventions, as is made strikingly clear by the spin singlet state. For if a long string of measurements of teh distant spins σ1 , σ2 are made along the same direction, then in each and every individual case an observer who knows the outcomes v1 = ± at end I can, without fail, tell the distant observer at II the outcome v2 = ∓, and vice versa. Thsi does not allow superluminal signaling between I and II because observer I cannot control which of the outcomes ± will appear in any one measurement at I, nor for taht reason the correlated outcomes at II. The data have to be brought together before the ’prediction’ can be conﬁrmed. Nevertheless, the correlations provoke the thought that a conspiracy is afoot, with the members of each pair carrying instructions that tell them what to do when they are questioned. Lectures-QM-08/09 – p. Hidden variables and Bell theorem The EPR argument raises the question: Is there a description of nature that is more reﬁned than that of quantum mechanics, and which explains the the spooky action at a distance that quantum mechanics appears to imply Request for a ’local realistic’ theory satisfying EPRs deﬁnitions of realism and completeness, while leaving the established consequences of quantum mechanics intact =⇒ Introduce additional ’hidden variables’ that provide a ’complete’ description of individual systems by assigning values to all the observables of such an individual prior to measurement, with an average over the hidden variables giving the quantum state as a description of ensembles of individuals. (Note: Quantum mechanics assigns no value to the observable prior to measurement) Bell-Kochen-Specker (BKS) theorem: It is not possible to consistently assign values to all the observables in any Hilbert space larger than H2 . Quantum mechanics is not a statistical rendition of an underlying hidden variable theory. Lectures-QM-08/09 – p. Hidden variables and Bell theorem Bell’s theorem and its generalizations establish that hidden variable theories can only reproduce the statistical predictions of quantum mechanics if the hidden variables in one region can be affected by measurements carried out in a space-like separated region (as they are in the de Broglie-Bohm theory). Such theories therefore violate the relativistic causality principle, surely an ironic outcome from the perspective of EPR ! The experiments motivated by Bell’s theorem give results in excellent agreement with quantum mechanics, hence only non-local hidden variable theories are viable options, which, one could say, offers a cure that is worse than the malaise ..... Lectures-QM-08/09 – p.1