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arXiv:2112.14659 · International Journal of Theoretical Physics

Einstein’s photon box revisited

Heinz–Jürgen Schmidt

Published Jul 21, 2022• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/s10773-022-05154-2

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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 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.
In 1930, Einstein proposed the Gedankenexperiment that seems to show the invalidity of the time-energy uncertainty relation. Bohr gave an answer to this Einstein's challenge by resorting to the Einstein's general theory of relativity. However, many scientists and philosophers have critiqued his answer, and proposed new ones. The important proposals in these are ones given by Treder and Busch. In this paper, I survey and critique these answers. The point is how to interpret the time-energy uncertainty relation appeared in the Einstein's Photon Box. Next, I suggest a new answer to the Photon Box Gedankenexperment. My suggestion is that measurement of energy of phton is done by measuring the mass of box before and after the emission of photon.
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.
In a previous paper, “On the Applicability of the Lorentz Transformations”, the issue of the physical vs. apparent velocity of a photon, c vs. (c±v), was examined in the cases of the emitter moving relative to the detector and vice versa. In the present paper, the case is examined of both the emitter and detector moving relative to the origin of a third coordinate system, the case used by Einstein for his derivation of the Lorentz transformations. It is shown that both Einstein’s derivation and a more general derivation are inconsistent with the behavior of the photon, and that the physical (not apparent) velocity of the photon as measured in the stationary coordinate system in this case must be (c±v). It is also noted that because the velocity of the photon is independent of its momentum and energy, the dispersion relations between conjugate Fourier variables of energy and time, ∇E∇t ≥ ħ/2, and momentum and position, ∇p∇x ≥ ħ/2, do not affect its velocity. As a result, unlike for non-zero rest mass quanta, the position of a photon as a function of time can in principle be determined with an arbitrary accuracy.
The meaning and scope of a recent type of uncertainty relation of a very general character are elucidated using the notions of time-indicating dynamical variables (clock variables) and place-indicating dynamical variables (position variables). It is shown that if the total energy (momentum) of a system is certain, all time-indicating (place-indicating) dynamical variables are completely uncertain. The quantum clock is discussed as an illustration of the energy–time uncertainty relation. The relations can be successfully applied to the thought experiments that Einstein introduced into his debate with Bohr about the uncertainty principle(s) and, in particular, to the famous photon-box experiment. It is shown that due to this general relation the photon box can never serve its purpose, independent of the details of the experiment.