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arXiv:quant-ph/0309184

Uncertainty relations on the slit and Fisher information

Z. Hradil, Jaroslav Řeháček

Published Sep 25, 2003• 0 citations
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Abstract

Diffraction on the slit can be interpreted in accordance with the Heisenberg uncertainty principle. This elementary example hints at the importance of the information theory for the quantum physics. The role played by one particularly interesting measure of information -- the Fisher information -- in quantum measurements is further discussed in the context of quantum interferometry.

DOI
10.48550/arxiv.quant-ph/0309184

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Diffraction on the slit can be interpreted in accordance with the Heisenberg uncertainty principle. This elementary example hints at the importance of the information theory for the quantum physics. The role played by one particularly interesting measure of information -- the Fisher information -- in quantum measurements is further discussed in the context of quantum interferometry.
In this paper the following information interpretation of uncertainty relation is proposed: if one bit of information was extracted from the system as a result of the measurement process, then the measurement itself adds an additional uncertainty (chaos) into the system equaled to one bit. This formulation is developed by calculating of the Shannon information entropy for the classical N-slit interference experiment. This approach allows looking differently at several quantum phenomena. Particularly, the information interpretation is used for explanation of entangled photons diffraction picture compression.
R. P. Feynman, by far, physics’ most successful communicator, said: The uncertainty principle protects quantum mechanics. W. Heisenberg recognized that if it were possible to measure the momentum and the position simultaneously with a greater accuracy, quantum mechanics would collapse. Quantum mechanics maintains its perilous but correct existence”. In his book on Quantum Mechanics and Path Integrals , Feynman makes use of the two-slit experiment to provide an approximate description of the physics behind the uncertainty principle. Feynman’s description is outlined using a slightly different notation. The uncertainty principle is widely applied in optics. It applies to interferometry, linewidth measurements, and beam divergence measurements. The chapter focuses on the uncertainty principle and beam divergence. An important application of the uncertainty principle manifests itself in calculations of the angular resolution limit of telescopes used in astronomical observations. The Heisenberg uncertainty principle can be used to derive some useful identities in optics and interferometry.
Abstract 1. In the general synthesis of classical dynamics with the quantum theory, the Uncertainty Principle plays a most useful part. It is of course only one aspect of the new mechanics, but it is a very helpful one since by its means it becomes easy to see where the old classical ideas broke down. The state of affairs in the quantum theory is not unlike that of the early days of relativity, when most of those who studied the subject felt the need of supporting the formal theory by seeing how the old ideas failed in specific cases. Here the formal theory is very abstract and is not easy to follow intuitively, and the Uncertainty Principle plays much the same rôle as did the examples of clocks and rods in relativity theory. For this reason it is more appropriate for illustrative examples, than for any extreme generality, and though a number of examples have been already given by Heisenberg and Bohr, it may not be amiss to have some more. There are probably some who will have shared my experience that it is often by no means easy to detect how the uncertainty enters into a given experiment, though once detected the arguments are usually very simple. In a recent conversation Professor Bohr criticised some rather careless remarks that I had made, and the present work was undertaken to clear matters up. It may be shortly described as a study of the Uncertainty Principle in connection with electrometers and magnetometers. It may be of interest to point out that the Uncertainty Principle can be regarded from a rather different aspect. The "resolving power" of optical instruments was discussed very fully by Rayleigh on wave principles, but as long as matter was regarded in the classical manner, a mechanical instrument could be considered as capable of measuring quantities with absolute accuracy. Now that we know that matter also has wave properties, there is need of a theory of the resolving power of mechanical instruments, and this is exactly what the Uncertainty Principle supplies.
We show that the quantum mechanical interpretation of the diffraction of light on a slit, when a wave function is assigned to a photon, can be used for a direct experimental study of Heisenberg’s position-momentum and equivalent positionwave vector uncertainty relation for the photon. Results of an experimental test of the position-wave vector uncertainty relation, where the wavelength is used as the input parameter, are given and they very well confirm our approach. The same experimental results can also be used for a test of the position-momentum uncertainty relation when the momentum p0 of a photon is known as the input parameter. We show that a measurement of p0, independent of the knowledge of the value of the Planck’s constant, is possible. Using that value of p0, a test of the position-momentum uncertainty relation could be regarded as a method for a direct measurement of the Planck’s constant. This is discussed, since the diffraction pattern is also well described by classical electrodynamics in the considered experimental conditions. This approach for testing the Heisenberg’s uncertainty relations is very simple and consequently suitable as a quantitative exercise in undergraduate experimental courses, as well as a visual and attractive demonstration of the Heisenberg’s uncertainty principle in courses of quantum mechanics.
The information gained by performing a measurement on a physical system is most appropriately assessed by the Fisher information, which in fact establishes lower bounds on estimation errors for an arbitrary unbiased estimator. We revisit the basic properties of the Fisher information and demonstrate its potential to quantify the resolution of optical systems. We illustrate this with some conceptually important examples, such as single-slit diffraction, spectroscopy and superresolution techniques.