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European Journal Of Physics Education

Gamma-Ray Telescope and Uncertainty Principle.

T Shivalingaswamy, B. A. Kagali

Published Jan 1, 2012• 0 citations
Publisher

Abstract

Heisenberg’s Uncertainty Principle is one of the important basic principles of quantum mechanics. In most of the books on quantum mechanics, this uncertainty principle is generally illustrated with the help of a gamma ray microscope, wherein neither the image formation criterion nor the lens properties are taken into account. Thus a better illustration is required for the clear understanding of this principle. In this article we illustrate Heisenberg’s uncertainty principle with a new thought experiment that could be called as gamma ray telescope with the inclusion of properties of the lens and quantum nature of light.

DOI
10.20308/ejpe.21003

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Heisenberg’s Uncertainty Principle is one of the important basic principles of quantum mechanics. In most of the books on quantum mechanics, this uncertainty principle is generally illustrated with the help of a gamma ray microscope, wherein neither the image formation criterion nor the lens properties are taken into account. Thus a better illustration is required for the clear understanding of this principle. In this article we illustrate Heisenberg’s uncertainty principle with a new thought experiment that could be called as gamma ray telescope with the inclusion of properties of the lens and quantum nature of light.
The two most important areas in modern physics are quantum mechanics and the theory of relativity. Unlike classical physics, where Newton's mechanics dominates, these two areas change human beings' fundamental view of the universe. One of the theories that build up the base of quantum mechanics is Heisenberg's Uncertainty Principle. Starting from a thought experiment, Heisenberg's microscope in the setting of classical physics, Werner Heisenberg built a bridge between classical and quantum physics by presenting a counterintuitive outcome in the thought experiment. Since then, the observer of a physics phenomenon is no longer a bystander. The behavior of observation became a part of the physical experiment. To come up with a mathematical expression that can describe such a new discovery, Heisenberg came up with matrix mechanics and the concept of complementary variables. There is a trade-off between a pair of complementary variables. When one of them is measured precisely, meaning the information of that variable is known on a large scale, the other variable can not be measured precisely, meaning there is no way to know enough information about the other variable. The principle indicates a fundamental limit on what human beings can know about the unknown variables. The discoveries of other complementary variables help physicists know the new image of the physics world under new rules.
The principle of uncertainty is one of the essential principles of quantum mechanics as it outlines that there is a reflective limit to the measurement of microscopic particle motion. However it is impossible to measure at once and accurately both the position and momentum data of any microscopic particle no matter what means are used to do so. This paper starts by discussing the notion and mathematical derivation of the uncertainty principle as put forth by Heisenberg explaining the physical relevance of the principle. It introduces the natural uncertainty the principle introduces to the dynamics of particles in the microscopic world, and the significant part it plays in the theory of quantum mechanics. The article overviews the explanations of the principle on phenomena including atomic stability and scanning tunneling microscopes, and the interference of measurement actions on the states of particles and the shortcomings of measurement precision. According to the wave-particle duality of quantum mechanics, this article gives a required theoretical basis of microscopic phenomena. Likewise, the use of the research theory on the uncertainty principle by distinguished scholars in the physics fraternity helps the readers in achieving the knowledge in related subjects more easily.
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 Heisenberg introduced his famous uncertainty relations in a seminal 1927 paper entitled The Physical Content of Quantum Kinematics and Mechanics . He motivated his arguments with a gedanken experiment, a gamma ray microscope to measure the position of a particle. A primary result was that, due to the quantum nature of light, there is an inherent uncertainty in the determinations of the particle’s position and momentum dictated by an indeterminacy relation, δ q δ p ∼ h . Heisenberg offered this demonstration as ‘a direct physical interpretation of the [quantum mechanical] equation pq − qp = − i ℏ ’ but considered the indeterminacy relation to be much more than this. He also argued that it implies limitations on the very meanings of position and momentum and emphasised that these limitations are the source of the statistical character of quantum mechanics. In addition, Heisenberg hoped but was unable to demonstrate that the laws of quantum mechanics could be derived directly from the uncertainty relation. In this paper, we revisit Heisenberg’s microscope and argue that the Schrödinger equation for a free particle does indeed follow from the indeterminacy relation together with reasonable statistical assumptions.
Heisenberg's uncertainty principle has been understood to set a limitation on measurements; however, the long-standing mathematical formulation established by Heisenberg, Kennard, and Robertson does not allow such an interpretation. Recently, a new relation was found to give a universally valid relation between noise and disturbance in general quantum measurements, and it has become clear that the new relation plays a role of the first principle to derive various quantum limits on measurement and information processing in a unified treatment. This paper examines the above development on the noise–disturbance uncertainty principle in the model-independent approach based on the measurement operator formalism, which is widely accepted to describe a class of generalized measurements in the field of quantum information. We obtain explicit formulae for the noise and disturbance of measurements given by measurement operators, and show that projective measurements do not satisfy the Heisenberg-type noise–disturbance relation that is typical in the gamma-ray microscope thought experiments. We also show that the disturbance on a Pauli operator of a projective measurement of another Pauli operator constantly equals , and examine how this measurement violates the Heisenberg-type relation but satisfies the new noise–disturbance relation.
Heisenberg's uncertainty principle has been understood to set a limitation on measurements; however, the long standing mathematical formulation established by Heisenberg, Kennard, and Robertson does not allow such an interpretation. Recently, a new relation was found to give a universally valid relation between noise and disturbance in general quantum measurements, and it has become clear that the new relation plays a role of the first principle to derive various quantum limits on measurement and information processing in a unified treatment. This paper examines the above development on the noise-disturbance uncertainty principle in the model independent approach based on measurement operator formalism, which is widely accepted to describe a class of generalized measurements in the field of quantum information. We obtain explicit formulas for noise and disturbance of measurements given by measurement operators, and show that projective measurements do not satisfy the Heisenberg-type noise-disturbance relation that is typical in the gamma-ray microscope thought experiments. We also show that the disturbance on a Pauli operator of a projective measurement of another Pauli operator constantly equals to the square root of 2, and examine how this measurement violates the Heisenberg-type relation but satisfies the new noise-disturbance relation.