The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited

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The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited Bởi: OpenStaxCollege Particle physics as we know it today began with the ideas of Hideki Yukawa in 1935 Physicists had long been concerned with how forces are transmitted, finding the concept of fields, such as electric and magnetic fields to be very useful A field surrounds an object and carries the force exerted by the object through space Yukawa was interested in the strong nuclear force in particular and found an ingenious way to explain its short range His idea is a blend of particles, forces, relativity, and quantum mechanics that is applicable to all forces Yukawa proposed that force is transmitted by the exchange of particles (called carrier particles) The field consists of these carrier particles The strong nuclear force is transmitted between a proton and neutron by the creation and exchange of a pion The pion is created through a temporary violation of conservation of massenergy and travels from the proton to the neutron and is recaptured It is not directly observable and is called a virtual particle Note that the proton and neutron change identity in the process The range of the force is limited by the fact that the pion can only exist for the short time allowed by the Heisenberg uncertainty principle Yukawa used the finite range of the strong nuclear force to estimate the mass of the pion; the shorter the range, the larger the mass of the carrier particle Specifically for the strong nuclear force, Yukawa proposed that a previously unknown particle, now called a pion, is exchanged between nucleons, transmitting the force between them [link] illustrates how a pion would carry a force between a proton and 1/4 The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited a neutron The pion has mass and can only be created by violating the conservation of mass-energy This is allowed by the Heisenberg uncertainty principle if it occurs for a sufficiently short period of time As discussed in Probability: The Heisenberg Uncertainty Principle the Heisenberg uncertainty principle relates the uncertainties ΔE in energy and Δt in time by ΔEΔt ≥ h 4π , where h is Planck’s constant Therefore, conservation of mass-energy can be violated by h an amount ΔE for a time Δt ≈ 4πΔE in which time no process can detect the violation This allows the temporary creation of a particle of mass m, where ΔE = mc2 The larger the mass and the greater the ΔE, the shorter is the time it can exist This means the range of the force is limited, because the particle can only travel a limited distance in a finite amount of time In fact, the maximum distance is d ≈ cΔt, where c is the speed of light The pion must then be captured and, thus, cannot be directly observed because that would amount to a permanent violation of mass-energy conservation Such particles (like the pion above) are called virtual particles, because they cannot be directly observed but their effects can be directly observed Realizing all this, Yukawa used the information on the range of the strong nuclear force to estimate the mass of the pion, the particle that carries it The steps of his reasoning are approximately retraced in the following worked example: Calculating the Mass of a Pion Taking the range of the strong nuclear force to be about fermi (10 −15 m), calculate the approximate mass of the pion carrying the force, assuming it moves at nearly the speed of light Strategy The calculation is approximate because of the assumptions made about the range of the force and the speed of the pion, but also because a more accurate calculation would require the sophisticated mathematics of quantum mechanics Here, we use the Heisenberg uncertainty principle in the simple form stated above, as developed in Probability: The Heisenberg Uncertainty Principle First, we must calculate the time Δt that the pion exists, given that the distance it travels at nearly the speed of light is about fermi Then, the Heisenberg uncertainty principle can be solved for the energy ΔE, and from that the mass of the pion can be determined We will use the units of MeV / c2 for mass, which are convenient since we are often considering converting mass to energy and vice versa Solution 2/4 The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited The distance the pion travels is d ≈ cΔt, and so the time during which it exists is approximately Δt 10 −15 m ≈ d c ≈ 3.3 × 10 −24 s = 3.0 × 108 m/s Now, solving the Heisenberg uncertainty principle for ΔE gives ΔE ≈ h 4πΔt ≈ 6.63 × 10 −34 J⋅s ( 4π 3.3 × 10 −24 s ) Solving this and converting the energy to MeV gives ΔE ≈ (1.6 × 10 −11 J) MeV 1.6 × 10 −13 J = 100 MeV Mass is related to energy by ΔE = mc2, so that the mass of the pion is m = ΔE / c2, or m ≈ 100 MeV/c2 Discussion This is about 200 times the mass of an electron and about one-tenth the mass of a nucleon No such particles were ...Quantum theory, the Church-Turing principle and the universal quantum computer DAVID DEUTSCH Appeared in Proceedings of the Royal Society of London A 400, pp. 97-117 (1985) (Communicated by R. Penrose, F.R.S. — Received 13 July 1984) Abstract It is argued that underlying the Church-Turing hypothesis there is an implicit physical assertion. Here, this assertion is presented explicitly as a physical prin- ciple: ‘every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means’. Classical physics and the universal Turing machine, because the former is continuous and the latter discrete, do not obey the principle, at least in the strong form above. A class of model computing machines that is the quantum generalization of the class of Tur- ing machines is described, and it is shown that quantum theory and the ‘universal quantum computer’ are compatible with the principle. Computing machines re- sembling the universal quantum computer could, in principle, be built and would have many remarkable properties not reproducible by any Turing machine. These do not include the computation of non-recursive functions, but they do include ‘quantum parallelism’, a method by which certain probabilistic tasks can be per- formed faster by a universal quantum computer than by any classical restriction of it. The intuitive explanation of these properties places an intolerable strain on all interpretations of quantum theory other than Everett’s. Some of the numerous connections between the quantum theory of computation and the rest of physics are explored. Quantum complexity theory allows a physically more reasonable definition of the ‘complexity’ or ‘knowledge’ in a physical system than does clas- sical complexity theory. Current address: Centre for Quantum Computation, Clarendon Laboratory, Department of Physics, Parks Road, OX1 3PU Oxford, United Kingdom. Email: david.deutsch@qubit.org This version (Summer 1999) was edited and converted to L A T E X by Wim van Dam at the Centre for Quantum Compu- tation. Email: wimvdam@qubit.org 1 Computing machines and the Church-Turing principle The theory of computing machines has been extensively developed during the last few decades. In- tuitively, a computing machine is any physical system whose dynamical evolution takes it from one of a set of ‘input’ states to one of a set of ‘output’ states. The states are labelled in some canonical way, the machine is prepared in a state with a given input label and then, following some motion, the output state is measured. For a classical deterministic system the measured output label is a definite function of the prepared input label; moreover the value of that label can in principle be measured by an outside observer (the ‘user’) and the machine is said to ‘compute’ the function . Two classical deterministic computing machines are ‘computationally equivalent’ under given la- bellings of their input and output states if they compute the same function under those labellings. But quantum computing machines, and indeed classical stochastic computing machines, do not ‘compute functions’ in the above sense: the output state of a stochastic machine is random with only the prob- ability distribution function for the possible outputs depending on the input state. The output state of a quantum machine, although fully determined Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 32585, 24 pages doi:10.1155/2007/32585 Research Article A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem Jingbo Dou, Pengcheng Niu, and Zixia Yuan Received 22 March 2007; Revised 26 May 2007; Accepted 20 October 2007 Recommended by L ´ aszl ´ oLosonczi Based on properties of vector fields, we prove Hardy inequalities with remainder terms in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The best constants in Hardy inequalities are determined. Then we discuss the existence of solutions for the nonlinear eigenvalue problems in the Heisenberg group with weights for the p- sub-Laplacian. The asymptotic behaviour, simplicity, and isolation of the first eigenvalue are also considered. Copyright © 2007 Jingbo Dou et al. This is an open access article distributed under the Creative Commons Attribution License, w hich permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let L p,μ u =−Δ H,p u −μψ p |u| p−2 u d p ,0≤ μ ≤  Q − p p  p , (1.1) be the Hardy operator on the Heisenberg group. We consider the following weighted eigenvalue problem with a singular weight: L p,μ u = λf(ξ)|u| p−2 u,inΩ ⊂ H n , u = 0, on ∂Ω, (1.2) where 1 <p<Q = 2n +2, λ ∈ R, f (ξ) ∈ Ᏺ p :={f : Ω→R + | lim d(ξ)→0 (d p (ξ) f (ξ)/ (ψ p (ξ)) = 0, f (ξ) ∈ L ∞ loc (Ω \{0})}, Ω is a bounded domain in the Heisenberg group, and the definitions of d(ξ)andψ p (ξ); see below. We investigate the weak solution of (1.2)and the asymptotic behavior of the first eigenvalue for different singular weights as μ increases to ((Q − p)/p) p . Furthermore, we show that the first eigenvalue is simple and isolated, as 2 Journal of Inequalities and Applications well as the eigenfunctions corresponding to other eigenvalues change sign. Our proof is mainly based on a Hardy inequality with remainder terms. It is established by the vec- tor field method and an elementary integral inequality. In addition, we show that the constants appearing in Hardy inequality are the best. Then we conclude a compact em- bedding in the weighted Sobole v space. Themaindifficulty to study the properties of the first eigenvalue is the lack of regu- larit y of the weak solutions of the p-sub-Laplacian in the Heisenberg group. Let us note that the C α regularity for the weak solutions of the p-subelliptic operators formed by the vector field satisfy ing H ¨ ormander’s condition was given in [1] and the C 1,α regularity of the weak solutions of the p-sub-Laplacian Δ H,p in the Heisenberg group for p near 2 was proved in [2]. To obtain results here, we employ the Picone identity and Harnack inequality to avoid effectively the use of the regularity. The eigenvalue problems in the Euclidean space have been studied by many authors. We refer to [3–11]. These results depend usually on Hardy inequalities or improved Hardy inequalities (see [4, 12–14]). Let us recall some elementary facts on the Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 32585, 24 pages doi:10.1155/2007/32585 Research Article A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem Jingbo Dou, Pengcheng Niu, and Zixia Yuan Received 22 March 2007; Revised 26 May 2007; Accepted 20 October 2007 Recommended by L ´ aszl ´ oLosonczi Based on properties of vector fields, we prove Hardy inequalities with remainder terms in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The best constants in Hardy inequalities are determined. Then we discuss the existence of solutions for the nonlinear eigenvalue problems in the Heisenberg group with weights for the p- sub-Laplacian. The asymptotic behaviour, simplicity, and isolation of the first eigenvalue are also considered. Copyright © 2007 Jingbo Dou et al. This is an open access article distributed under the Creative Commons Attribution License, w hich permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let L p,μ u =−Δ H,p u −μψ p |u| p−2 u d p ,0≤ μ ≤  Q − p p  p , (1.1) be the Hardy operator on the Heisenberg group. We consider the following weighted eigenvalue problem with a singular weight: L p,μ u = λf(ξ)|u| p−2 u,inΩ ⊂ H n , u = 0, on ∂Ω, (1.2) where 1 <p<Q = 2n +2, λ ∈ R, f (ξ) ∈ Ᏺ p :={f : Ω→R + | lim d(ξ)→0 (d p (ξ) f (ξ)/ (ψ p (ξ)) = 0, f (ξ) ∈ L ∞ loc (Ω \{0})}, Ω is a bounded domain in the Heisenberg group, and the definitions of d(ξ)andψ p (ξ); see below. We investigate the weak solution of (1.2)and the asymptotic behavior of the first eigenvalue for different singular weights as μ increases to ((Q − p)/p) p . Furthermore, we show that the first eigenvalue is simple and isolated, as 2 Journal of Inequalities and Applications well as the eigenfunctions corresponding to other eigenvalues change sign. Our proof is mainly based on a Hardy inequality with remainder terms. It is established by the vec- tor field method and an elementary integral inequality. In addition, we show that the constants appearing in Hardy inequality are the best. Then we conclude a compact em- bedding in the weighted Sobole v space. Themaindifficulty to study the properties of the first eigenvalue is the lack of regu- larit y of the weak solutions of the p-sub-Laplacian in the Heisenberg group. Let us note that the C α regularity for the weak solutions of the p-subelliptic operators formed by the vector field satisfy ing H ¨ ormander’s condition was given in [1] and the C 1,α regularity of the weak solutions of the p-sub-Laplacian Δ H,p in the Heisenberg group for p near 2 was proved in [2]. To obtain results here, we employ the Picone identity and Harnack inequality to avoid effectively the use of the regularity. The eigenvalue problems in the Euclidean space have been studied by many authors. We refer to [3–11]. These results depend usually on Hardy inequalities or improved Hardy inequalities (see [4, 12–14]). Let us recall some elementary facts on the The MBR Book To Sam and Oliver (again) The MBR Book: Principles and Applications of Membrane Bioreactors in Water and Wastewater Treatment Simon Judd With Claire Judd AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Elsevier Ltd, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB UK Elsevier BV, Radarweg 29, PO Box 211, 1000 AE, Amsterdam, The Netherlands Elsevier Inc., 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA Elsevier Ltd, 84 Theobald’s Road, London, WC1Z 8RR, UK © 2006 Elsevier Ltd All rights reserved This work is protected under copyright by Elsevier Ltd, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use Permissions may be sought directly from Elsevier’s Rights Department in Oxford, UK; phone: (ϩ44) (0) 1865 843830; fax: (ϩ44) (0) 1865 853333; e-mail: permissions@elsevier.com Requests may also be completed on-line via the Elsevier homepage (http://www.elsevier.com/locate/permissions) In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (ϩ1) (978) 7508400, fax: (ϩ1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (ϩ44) 20 7631 5555; fax: (ϩ44) 20 7631 5500 Other countries may have a local reprographic rights agency for payments Derivative Works Tables of contents may be reproduced for internal circulation, but permission of the Publisher is required for external resale or distribution of such material Permission of the Publisher is required for all other derivative works, including compilations and translations Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher Address permissions requests to: Elsevier’s Rights Department, at the fax and e-mail addresses noted above Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made First edition 2006 Library of Congress Control Number: 2006927679 ISBN-13: 978-1-85-617481-7 ISBN-10: 1-85-617481-6 ϱ The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper) Typeset by Charon Tec Ltd, Chennai, India www.charontec.com Printed in Great Britain 06 07 08 09 10 10 Contents Preface ix Contributors Chapter Chapter xiii Introduction 1.1 Introduction 1.2 Current MBR market size and growth projections 1.3 Barriers to MBR technology implementation 1.4 Drivers for MBR technology implementation 1.4.1 Legislation 1.4.2 Incentives and funding 1.4.3 Investment costs 1.4.4 Water scarcity 1.4.5 Greater confidence in MBR technology 1.5 Historical perspective 1.5.1 The early days of the MBR: the roots of the Kubota and Zenon systems 1.5.2 Development of other MBR products 1.5.3 ... converting mass to energy and vice versa Solution 2/4 The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited The distance the pion travels is d ≈ cΔt, and so the time during which... class of particles, as we shall see in Particles, Patterns, and Conservation Laws) 3/4 The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited The pions, or π-mesons as they are.. .The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited a neutron The pion has mass and can only be created by violating the conservation of mass-energy This is allowed by the