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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Complementarity in the double-slit experiment: Quantum nonseparability and a quantitative statement of Bohr's principle

William K. Wootters, Wojciech Hubert Zurek

Published Jan 15, 1979• 617 citations
Publisher

Abstract

A detailed analysis of Einstein's version of the double-slit experiment, in which one tries to observe both wave and particle properties of light, is performed. Quantum nonseparability appears in the derivation of the interference pattern, which proves to be surprisingly sharp even when the trajectories of the photons have been determined with fairly high accuracy. An information-theoretic approach to this problem leads to a quantitative formulation of Bohr's complementarity principle for the case of the double-slit experiment. A practically realizable version of this experiment, to which the above analysis applies, is proposed.

DOI
10.1103/physrevd.19.473

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A detailed analysis of Einstein's version of the double-slit experiment, in which one tries to observe both wave and particle properties of light, is performed. Quantum nonseparability appears in the derivation of the interference pattern, which proves to be surprisingly sharp even when the trajectories of the photons have been determined with fairly high accuracy. An information-theoretic approach to this problem leads to a quantitative formulation of Bohr's complementarity principle for the case of the double-slit experiment. A practically realizable version of this experiment, to which the above analysis applies, is proposed.
As per Einstein's design, particles are introduced into the double-slit experiment through a small hole in a plate which can either move up and down (and its momentum can be measured) or be stopped (and its position can be measured). Suppose one measures the position of the plate and this act verifies the statement that the interference pattern is observed in the experiment. However, if it is possible to think about the outcome that one would have obtained if one had measured plate's momentum instead of its position, then it is possible to consider, together with the aforesaid statement, another statement that each particle passes through either slit of the double-slit screen. Hence, the proposition affirming the wave-like behavior and the proposition affirming the particle-like behavior might be true together, which would imply that Bohr's complementarity principle is incorrect. The analysis of Einstein's design and ways to refute it based on an approach that uses exclusively assignments of the truth values to experimental propositions is presented in this paper.
Einstein's version of the double-slit interferometer is reexamined by taking into account possible degrees of freedom of the detector. This can be considered as a natural generalization of an earlier attempt by Wooters and Zurek, however, it is found that in this approach a new element which is directly related to the uncertainties of physical variables appears and that it affects considerably the visibility of the interference pattern. Thi fact turns out to be a quantitative realization of the complementarity principle in quantum mechanics.
Young’s double-slit experiment played a crucial role in establishing the wave nature of light. In this chapter, the shocking result that incident electrons yield a similar interference pattern as that formed by light waves is described. It is shown that the only way the experimental results could be explained is via a wave function description of electrons. It is also shown that, in the same experiment, the interference fringes disappear if the which-path information becomes available. This is the essence of wave–particle duality. The first of the Einstein–Bohr debates on wave-particle duality and Bohr’s principle of complementarity in the double-slit experiment is also discussed. Also presented are the counterintuitive notions of delayed choice and quantum eraser effects showing how the availability or erasure of information generated in the future can affect how the data in the present can be interpreted.
Some modified two-slit interference experiments claim to demonstrate a violation of Bohr’s complementarity principle. A typical such experiment is theoretically analyzed using wave-packet dynamics. The flaw in the analysis of such experiments is pointed out and it is demonstrated that they do not violate complementarity. In addition, it is quite generally proved that if the state of a particle is such that the modulus square of the wave-function yields an interference pattern, then it necessarily loses which-path information.
Abstract It is argued that two distinct types of complementarity are implied in Bohr's complementarity principle. While in the case of complementary variables it is the quantum mechanical uncertainty relation which is at work, the collapse hypothesis ensures this exclusiveness in the so-called wave-particle complementarity experiments. In particular it is shown that the conventional analysis of the double slit experiment which invokes the uncertainty principle to explain the absence of the simultaneous knowledge of the which-slit information and the interference pattern is incorrect and implies consequences that are quantum mechanically inconsistent.