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CRC Press eBooks

The Uncertainty Principle

F. J. Duarte

Published Jan 15, 2024• 2 citations
Publisher

Abstract

Heisenberg’s uncertainty principle, Δ p Δ x ≈ h , is derived from the generalized quantum N -slit interferometric probability. Various versions of the uncertainty principle utilized in quantum optics are introduced in the chapter. Applications to beam divergence Δθ, astronomical telescopes, and the tuning of laser microcavities are discussed.

DOI
10.1201/9781003398707-3

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Heisenberg’s uncertainty principle, Δ p Δ x ≈ h , is derived from the generalized quantum N -slit interferometric probability. Various versions of the uncertainty principle utilized in quantum optics are introduced in the chapter. Applications to beam divergence Δθ, astronomical telescopes, and the tuning of laser microcavities are discussed.
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.
Heisenberg's uncertainty principle is one of the unique and iconic features of quantum mechanics that distinctly separates this physics, based on indeterminacy, from any and all classical deterministic theories. Besides its succinct utter beauty, https://www.w3.org/1998/Math/MathML" display="inline"> Δ x Δ p ≈ h , as articulated by Feynman ‘it protects quantum mechanics.’ In addition to its foundational importance and ‘protection’ function, the uncertainty principle plays crucial experimental roles in the measurement of coherent emission linewidth https://www.w3.org/1998/Math/MathML" display="inline"> Δ ν and in determining the nonlocality of the photon via measurements of coherence length https://www.w3.org/1998/Math/MathML" display="inline"> Δ x . Physically, it can be shown that https://www.w3.org/1998/Math/MathML" display="inline"> Δ x Δ p ≈ h has its origin in the physics of measurable interferometric quantum probability distributions. This insight teaches us that, unlike probability amplitudes, https://www.w3.org/1998/Math/MathML" display="inline"> Δ x Δ p ≈ h firmly resides in the measurable realm.
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.
Since the publication of Heisenberg’s paper1 on the “anschaulichen Inhalt” of quantum mechanics, discussions of the fundamental limitations on the accuracy of physical measurements have been much in the foreground. According to Heisenberg, the quantum mechanics implies that it is impossible to measure simultaneous values of a coordinate and its conjugate momentum with unlimited precision. Instead, if Δp be the estimated error or uncertainty in a momentum and Δq that in the associated coordinate one must have the inequality, ΔpΔq>h2π \Delta p\Delta q>\frac{h}{{2\pi}} ΔpΔq>2πh​ (1) This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.