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arXiv:0705.2764 · American Journal of Physics

Complementarity in the Einstein–Bohr photon box

Dennis Dieks, Sander Lam

Published Sep 1, 2008• 7 citations
arXivHTMLPDF

Abstract

The Bohr–Einstein photon box thought experiment is a forerunner of the EPR experiment: a packet of radiation escapes from a box, and the box-plus-radiation state remains entangled. Hence, a measurement on the box makes a difference for the state of the far-away radiation long after its escape. This consequence is analogous to the EPR experiment, but an additional aspect of the Bohr–Einstein box is that complementary magnitudes of the escaped radiation seem correlated to jointly measurable box quantities, which appears to imply a violation of the uncertainty relations. We show how the details of the dynamics make this latter paradox disappear. This calculation makes Bohr’s qualitative arguments in defense of the uncertainty relations precise and reinforces the EPR-like behavior that Einstein later claimed to have had in mind. The calculation reveals that the quantum description of the far-away radiation depends on exactly when the box measurement is performed.

DOI
10.1119/1.2919740

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The Bohr–Einstein photon box thought experiment is a forerunner of the EPR experiment: a packet of radiation escapes from a box, and the box-plus-radiation state remains entangled. Hence, a measurement on the box makes a difference for the state of the far-away radiation long after its escape. This consequence is analogous to the EPR experiment, but an additional aspect of the Bohr–Einstein box is that complementary magnitudes of the escaped radiation seem correlated to jointly measurable box quantities, which appears to imply a violation of the uncertainty relations. We show how the details of the dynamics make this latter paradox disappear. This calculation makes Bohr’s qualitative arguments in defense of the uncertainty relations precise and reinforces the EPR-like behavior that Einstein later claimed to have had in mind. The calculation reveals that the quantum description of the far-away radiation depends on exactly when the box measurement is performed.
The photon box thought experiment can be considered a forerunner of the EPR-experiment: by performing suitable measurements on the box it is possible to ``prepare'' the photon, long after it has escaped, in either of two complementary states. Consistency requires that the corresponding box measurements be complementary as well. At first sight it seems, however, that these measurements can be jointly performed with arbitrary precision: they pertain to different systems (the center of mass of the box and an internal clock, respectively). But this is deceptive. As we show by explicit calculation, although the relevant quantities are simultaneously measurable, they develop non-vanishing commutators when calculated back to the time of escape of the photon. This justifies Bohr's qualitative arguments in a precise way; and it illustrates how the details of the dynamics conspire to guarantee the requirements of complementarity. In addition, our calculations exhibit a ``fine structure'' in the distribution of the uncertainties over the complementary quantities: depending on \textit{when} the box measurement is performed, the resulting quantum description of the photon differs. This brings us close to the argumentation of the later EPR thought experiment.
In 1930, Einstein argued against the consistency of the time–energy uncertainty relation by discussing a thought experiment involving a measurement of the mass of the box which emitted a photon. Bohr seemingly prevailed over Einstein by arguing that Einstein's own general theory of relativity saves the consistency of quantum mechanics. We revisit this thought experiment from a modern point of view at a level suitable for an undergraduate readership and find that neither Einstein nor Bohr was correct. Instead, this thought experiment should be thought of as an early example of a system demonstrating nonlocal 'EPR' quantum correlations, five years before the famous Einstein–Podolsky–Rosen paper.
Abstract ‘At the next meeting with Einstein at the Solvay Conference in 1930 our discussions took quite a dramatic turn’, Bohr wrote. Bohr turned Einstein’s ‘photon box’ experiment on its head to defend the uncertainty relations. But Einstein’s purpose was more subtle, arguing that quantum theory implied that the properties of a photon are governed by the choice made by an experimenter, no matter how distant. This argument resurfaced in his epoch-making paper with Boris Podolsky and Nathan Rosen. The experimenter chooses what kind of measurement to make on one of a pair of correlated particles, instantly determining the properties of the distant other. Schrödinger called it ‘entanglement’, and exposed the theory’s absurdities in his paradox of the cat. As the founders were rewarded with Nobel prizes, Pauli resolved the problem of nuclear beta-decay by proposing the neutrino, inspiring a witty parody of Goethe’s Faust performed in Bohr’s institute in 1932.
The EPR paradox is shown to be a result of the wrong definitions in quantum mechanics of such physical quantities as momentum and position. The corrected definitions show that momentum and position of particles can be precisely defined simultaneously and uncertainty relations do not prevent it. Therefore, with correct definition of physical quantities in quantum mechanics, the paradox disappears. The Bohm Aharonov version of the EPR paradox is illustrated by the example of measuring the polarization of photon pairs. The common view of the two photon state radiated by a source is presented. The common Bell’s inequality is discussed and the simplest version of it for a specific hidden variable is derived. A possibility of an experimental study of whether a photon and its polarization are preexistent before measurements is considered.
Abstract We present a reformulation of Bohr’s analysis of Einstein’s thought experiment with the photon box, which hopefully settles some unclear issues. The box is weighed by two position measurements, the first immediately before the emission of the photon and the second after a quarter period of oscillation of the box, or alternatively after a half period. These times are precisely defined with respect to the moving clock installed in the box, but vary statistically when measured with a stationary clock outside the box, due to gravitational time dilation in accordance with Bohr’s original idea. Moreover, we will discuss two other proposals to solve the apparent inconsistency without invoking the theory of general relativity.
The two-slit experiment with quantum particles provides many insights into the behavior of quantum mechanics, including Bohr’s complementarity principle. Here, we analyze Einstein’s recoiling slit version of the experiment and show how the inevitable entanglement between the particle and the recoiling slit as a which-way detector is responsible for complementarity. We derive the Englert–Greenberger–Yasin duality from this entanglement, which can also be thought of as a consequence of sum-uncertainty relations between certain complementary observables of the recoiling slit. Thus, entanglement is an integral part of the which-way detection process, and so is uncertainty, though in a completely different way from that envisaged by Bohr and Einstein.