The Double Slit Experiment and Quantum Mechanics∗
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Classical and Modern Diffraction Theory
Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3701993/frontmatter.pdf by guest on 29 September 2021 Classical and Modern Diffraction Theory Edited by Kamill Klem-Musatov Henning C. Hoeber Tijmen Jan Moser Michael A. Pelissier SEG Geophysics Reprint Series No. 29 Sergey Fomel, managing editor Evgeny Landa, volume editor Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3701993/frontmatter.pdf by guest on 29 September 2021 Society of Exploration Geophysicists 8801 S. Yale, Ste. 500 Tulsa, OK 74137-3575 U.S.A. # 2016 by Society of Exploration Geophysicists All rights reserved. This book or parts hereof may not be reproduced in any form without permission in writing from the publisher. Published 2016 Printed in the United States of America ISBN 978-1-931830-00-6 (Series) ISBN 978-1-56080-322-5 (Volume) Library of Congress Control Number: 2015951229 Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3701993/frontmatter.pdf by guest on 29 September 2021 Dedication We dedicate this volume to the memory Dr. Kamill Klem-Musatov. In reading this volume, you will find that the history of diffraction We worked with Kamill over a period of several years to compile theory was filled with many controversies and feuds as new theories this volume. This volume was virtually ready for publication when came to displace or revise previous ones. Kamill Klem-Musatov’s Kamill passed away. He is greatly missed. new theory also met opposition; he paid a great personal price in Kamill’s role in Classical and Modern Diffraction Theory goes putting forth his theory for the seismic diffraction forward problem. -
Intensity of Double-Slit Pattern • Three Or More Slits
• Intensity of double-slit pattern • Three or more slits Practice: Chapter 37, problems 13, 14, 17, 19, 21 Screen at ∞ θ d Path difference Δr = d sin θ zero intensity at d sinθ = (m ± ½ ) λ, m = 0, ± 1, ± 2, … max. intensity at d sinθ = m λ, m = 0, ± 1, ± 2, … 1 Questions: -what is the intensity of the double-slit pattern at an arbitrary position? -What about three slits? four? five? Find the intensity of the double-slit interference pattern as a function of position on the screen. Two waves: E1 = E0 sin(ωt) E2 = E0 sin(ωt + φ) where φ =2π (d sin θ )/λ , and θ gives the position. Steps: Find resultant amplitude, ER ; then intensities obey where I0 is the intensity of each individual wave. 2 Trigonometry: sin a + sin b = 2 cos [(a-b)/2] sin [(a+b)/2] E1 + E2 = E0 [sin(ωt) + sin(ωt + φ)] = 2 E0 cos (φ / 2) cos (ωt + φ / 2)] Resultant amplitude ER = 2 E0 cos (φ / 2) Resultant intensity, (a function of position θ on the screen) I position - fringes are wide, with fuzzy edges - equally spaced (in sin θ) - equal brightness (but we have ignored diffraction) 3 With only one slit open, the intensity in the centre of the screen is 100 W/m 2. With both (identical) slits open together, the intensity at locations of constructive interference will be a) zero b) 100 W/m2 c) 200 W/m2 d) 400 W/m2 How does this compare with shining two laser pointers on the same spot? θ θ sin d = d Δr1 d θ sin d = 2d θ Δr2 sin = 3d Δr3 Total, 4 The total field is where φ = 2π (d sinθ )/λ •When d sinθ = 0, λ , 2λ, .. -
Light and Matter Diffraction from the Unified Viewpoint of Feynman's
European J of Physics Education Volume 8 Issue 2 1309-7202 Arlego & Fanaro Light and Matter Diffraction from the Unified Viewpoint of Feynman’s Sum of All Paths Marcelo Arlego* Maria de los Angeles Fanaro** Universidad Nacional del Centro de la Provincia de Buenos Aires CONICET, Argentine *[email protected] **[email protected] (Received: 05.04.2018, Accepted: 22.05.2017) Abstract In this work, we present a pedagogical strategy to describe the diffraction phenomenon based on a didactic adaptation of the Feynman’s path integrals method, which uses only high school mathematics. The advantage of our approach is that it allows to describe the diffraction in a fully quantum context, where superposition and probabilistic aspects emerge naturally. Our method is based on a time-independent formulation, which allows modelling the phenomenon in geometric terms and trajectories in real space, which is an advantage from the didactic point of view. A distinctive aspect of our work is the description of the series of transformations and didactic transpositions of the fundamental equations that give rise to a common quantum framework for light and matter. This is something that is usually masked by the common use, and that to our knowledge has not been emphasized enough in a unified way. Finally, the role of the superposition of non-classical paths and their didactic potential are briefly mentioned. Keywords: quantum mechanics, light and matter diffraction, Feynman’s Sum of all Paths, high education INTRODUCTION This work promotes the teaching of quantum mechanics at the basic level of secondary school, where the students have not the necessary mathematics to deal with canonical models that uses Schrodinger equation. -
On an Experimentum Crucis for Optics
Research Report: On an Experimentum Crucis for Optics Ionel DINU, M.Sc. Teacher of Physics E-mail: [email protected] (Dated: November 17, 2010) Abstract Formulated almost 150 years ago, Thomas Young’s hypothesis that light might be a transverse wave has never been seriously questioned, much less subjected to experiment. In this article I report on an attempt to prove experimentally that Young’s hypothesis is untenable. Although it has certain limitations, the experiment seems to show that sound in air, a longitudinal wave, can be “polarized” by reflection just like light, and this can be used as evidence against Young’s hypothesis. Further refinements of the experimental setup may yield clearer results, making this report useful to those interested in the important issue of whether light is a transverse or a longitudinal wave. Introduction In the course of development of science, there has always been a close connection between sound and light [1]. Acoustics stimulated research and discovery in optics and vice-versa. Reflection and refraction being the first common points observed between the two, the analogy between optical and acoustical phenomena has been proven also in the case of interference, diffraction, Doppler effect, and even in their mode of production: sounds are produced by the vibration of macroscopic objects, while light is produced by vibrating molecules and atoms. The wave nature of sound implies that light is also a wave and, if matter must be present for sound to be propagated, then aether must exist in order to account for the propagation of light through spaces void of all the matter known to chemistry. -
Thomas Young the Man Who Knew Everything
Thomas Young The man Who Knew Everything Andrew Robinson marvels at the brain power and breadth of knowledge of the 18th-century polymath Thomas Young. He examines his relationship with his contemporaries, particularly with the French Egyptologist Champollion, and how he has been viewed subsequently by historians. ORTUNATE NEWTON, happy professor of natural philosophy at childhood of science!’ Albert the newly founded Royal Institution, F Einstein wrote in 1931 in his where in 1802-03 he delivered what foreword to the fourth edition of is generally regarded as the most far- Newton’s influential treatise Opticks, reaching series of lectures ever given originally published in 1704. by a scientist; a physician at a major London hospital, St George’s, for a Nature to him was an open book, quarter of a century; the secretary of whose letters he could read without the Admiralty’s Board of Longitude effort. ... Reflection, refraction, the and superintendent of its vital Nauti- formation of images by lenses, the cal Almanac from 1818 until his mode of operation of the eye, the death; and ‘inspector of calculations’ spectral decomposition and recom- for a leading life insurance company position of the different kinds of in the 1820s. In 1794, he was elected light, the invention of the reflecting a fellow of the Royal Society at the telescope, the first foundations of age of barely twenty-one, became its colour theory, the elementary theory foreign secretary at the age of thirty, of the rainbow pass by us in and turned down its presidency in procession, and finally come his his fifties. -
The Path Integral Approach to Quantum Mechanics Lecture Notes for Quantum Mechanics IV
The Path Integral approach to Quantum Mechanics Lecture Notes for Quantum Mechanics IV Riccardo Rattazzi May 25, 2009 2 Contents 1 The path integral formalism 5 1.1 Introducingthepathintegrals. 5 1.1.1 Thedoubleslitexperiment . 5 1.1.2 An intuitive approach to the path integral formalism . .. 6 1.1.3 Thepathintegralformulation. 8 1.1.4 From the Schr¨oedinger approach to the path integral . .. 12 1.2 Thepropertiesofthepathintegrals . 14 1.2.1 Pathintegralsandstateevolution . 14 1.2.2 The path integral computation for a free particle . 17 1.3 Pathintegralsasdeterminants . 19 1.3.1 Gaussianintegrals . 19 1.3.2 GaussianPathIntegrals . 20 1.3.3 O(ℏ) corrections to Gaussian approximation . 22 1.3.4 Quadratic lagrangians and the harmonic oscillator . .. 23 1.4 Operatormatrixelements . 27 1.4.1 Thetime-orderedproductofoperators . 27 2 Functional and Euclidean methods 31 2.1 Functionalmethod .......................... 31 2.2 EuclideanPathIntegral . 32 2.2.1 Statisticalmechanics. 34 2.3 Perturbationtheory . .. .. .. .. .. .. .. .. .. .. 35 2.3.1 Euclidean n-pointcorrelators . 35 2.3.2 Thermal n-pointcorrelators. 36 2.3.3 Euclidean correlators by functional derivatives . ... 38 0 0 2.3.4 Computing KE[J] and Z [J] ................ 39 2.3.5 Free n-pointcorrelators . 41 2.3.6 The anharmonic oscillator and Feynman diagrams . 43 3 The semiclassical approximation 49 3.1 Thesemiclassicalpropagator . 50 3.1.1 VanVleck-Pauli-Morette formula . 54 3.1.2 MathematicalAppendix1 . 56 3.1.3 MathematicalAppendix2 . 56 3.2 Thefixedenergypropagator. 57 3.2.1 General properties of the fixed energy propagator . 57 3.2.2 Semiclassical computation of K(E)............. 61 3.2.3 Two applications: reflection and tunneling through a barrier 64 3 4 CONTENTS 3.2.4 On the phase of the prefactor of K(xf ,tf ; xi,ti) .... -
Quantum Weirdness: a Beginner's Guide
Quantum Weirdness: A Beginner’s Guide Dr. Andrew Robinson Part 1 Introduction The Quantum Jump 9:38 AM About Me • From Bakewell • PhD in Physical Chemistry • Worked in Berlin, Liverpool, Birmingham • In Canada since 2000 • Worked at University of Saskatchewan • Moved to Ottawa in 2010 • Teach Physics at Carleton 9:38 AM In This Lecture Series • We will talk about • What does Quantum Mean? • Quantum Effects • What are the ramifications of Quantum Theory • How Quantum Theory impacts our everyday lives • I will show a few equations, but you don’t need to know any mathematics • Please ask questions at any time 9:38 AM Books “How to Teach Quantum Physics to your Dog” by Chad Orzel “30-Second Quantum Theory” By Brian Clegg (ed.) 9:38 AM Definition of “Quantum” Physics • A discrete quantity of energy proportional in magnitude to the frequency of the radiation it represents. Legal • A required or allowed amount, especially an amount of money legally payable in damages. 9:38 AM • Quantum Satis “as much as is sufficient“ – pharmacology and medicine • Quantum Salis “the amount which is enough” • Quantum comes from the Latin word quantus, meaning "how great". Used by the German Physicist Hermann von Helmholtz ( who was also a physician) in the context of the electron (quanta of electricity) Use by Einstein in 1905 "Lichtquanta” – particle of light 9:38 AM “Quantum Jump” & “Quantum Leap” • Colloquially “A sudden large increase or advance”. • In physics “A jump between two discrete energy levels in a quantum system” (Actually a rather small leap in terms of energy!) 9:38 AM Quantum Properties in Physics • Properties which can only take certain values When you are on the ladder, you must be on one of the steps: 4 1 3 2 Quantum 2 3 Numbers 1 4 9:38 AM • Not every quantity in physics is quantized • Your height from the ground when on the slide varies continuously Maximum height Minimum9:38 AM height • Whether you can treat the system as continuous, or quantum depends on the scale. -
Simple Cubic Lattice
Chem 253, UC, Berkeley What we will see in XRD of simple cubic, BCC, FCC? Position Intensity Chem 253, UC, Berkeley Structure Factor: adds up all scattered X-ray from each lattice points in crystal n iKd j Sk e j1 K ha kb lc d j x a y b z c 2 I(hkl) Sk 1 Chem 253, UC, Berkeley X-ray scattered from each primitive cell interfere constructively when: eiKR 1 2d sin n For n-atom basis: sum up the X-ray scattered from the whole basis Chem 253, UC, Berkeley ' k d k d di R j ' K k k Phase difference: K (di d j ) The amplitude of the two rays differ: eiK(di d j ) 2 Chem 253, UC, Berkeley The amplitude of the rays scattered at d1, d2, d3…. are in the ratios : eiKd j The net ray scattered by the entire cell: n iKd j Sk e j1 2 I(hkl) Sk Chem 253, UC, Berkeley For simple cubic: (0,0,0) iK0 Sk e 1 3 Chem 253, UC, Berkeley For BCC: (0,0,0), (1/2, ½, ½)…. Two point basis 1 2 iK ( x y z ) iKd j iK0 2 Sk e e e j1 1 ei (hk l) 1 (1)hkl S=2, when h+k+l even S=0, when h+k+l odd, systematical absence Chem 253, UC, Berkeley For BCC: (0,0,0), (1/2, ½, ½)…. Two point basis S=2, when h+k+l even S=0, when h+k+l odd, systematical absence (100): destructive (200): constructive 4 Chem 253, UC, Berkeley Observable diffraction peaks h2 k 2 l 2 Ratio SC: 1,2,3,4,5,6,8,9,10,11,12. -
Thomas Young's Research on Fluid Transients: 200 Years On
Thomas Young's research on fluid transients: 200 years on Arris S Tijsseling Alexander Anderson Department of Mathematics School of Mechanical and Computer Science and Systems Engineering Eindhoven University of Technology Newcastle University P.O. Box 513, 5600 MB Eindhoven Newcastle upon Tyne NE1 7RU The Netherlands United Kingdom ABSTRACT Thomas Young published in 1808 his famous paper (1) in which he derived the pressure wave speed in an incompressible liquid contained in an elastic tube. Unfortunately, Young's analysis was obscure and the wave speed was not explicitly formulated, so his achievement passed unnoticed until it was rediscovered nearly half a century later by the German brothers Weber. This paper briefly reviews Young's life and work, and concentrates on his achievements in the area of hydraulics and waterhammer. Young's 1808 paper is “translated” into modern terminology. Young's discoveries, though difficult for modern readers to identify, appear to include most if not all of the key elements which would subsequently be combined into the pressure rise equation of Joukowsky. Keywords waterhammer, fluid transients, solid transients, wave speed, history, Thomas Young NOTATION c sonic wave speed, m/s p fluid pressure, Pa D internal tube diameter, m R internal tube radius, m E Young’s modulus, Pa t time, s e tube wall thickness, m v velocity, m/s f elastic limit, Pa x length, m g gravitational acceleration, m/s 2 δ change, jump h height, pressure head, m ε longitudinal strain K fluid bulk modulus, Pa ρ mass density, kg/m -
Three Music-Theory Lessons
Three Music-Theory Lessons The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Rehding, Alexander. 2016. “Three Music-Theory Lessons.” Journal of the Royal Musical Association 141 (2) (July 2): 251–282. doi:10.1080/02690403.2016.1216025. Published Version doi:10.1080/02690403.2016.1216025 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:34220771 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#OAP Three Music Theory Lessons ALEXANDER REHDING To make the familiar strange and the strange familiar was as much a central feature of Novalis’ conception of Romanticism as it is a mainstay of semiotics.1 In this spirit, we will begin with what seems like a mundane description of music- theoretical practice. If we enter a music theory classroom in the western world, we would normally expect to find a number of objects. We see a blackboard, ideally with five-line staffs already printed on it. We also expect a piano in the room, and we would probably have some sheet music, perhaps with the quintessential music theory teaching material: Bach chorale harmonizations. Let’s further imagine it is this passage, the simple opening line of the Lutheran hymn “Wie schön leuchtet der Morgenstern,” from Bach’s Cantata BWV 1, shown in Fig. 1, that is marked on the board. -
Diffraction-Free Beams in Fractional Schrödinger Equation Yiqi Zhang1, Hua Zhong1, Milivoj R
www.nature.com/scientificreports OPEN Diffraction-free beams in fractional Schrödinger equation Yiqi Zhang1, Hua Zhong1, Milivoj R. Belić2, Noor Ahmed1, Yanpeng Zhang1 & Min Xiao3,4 We investigate the propagation of one-dimensional and two-dimensional (1D, 2D) Gaussian beams in Received: 14 December 2015 the fractional Schrödinger equation (FSE) without a potential, analytically and numerically. Without Accepted: 11 March 2016 chirp, a 1D Gaussian beam splits into two nondiffracting Gaussian beams during propagation, while a 2D Published: 21 April 2016 Gaussian beam undergoes conical diffraction. When a Gaussian beam carries linear chirp, the 1D beam deflects along the trajectoriesz = ±2(x − x0), which are independent of the chirp. In the case of 2D Gaussian beam, the propagation is also deflected, but the trajectories align along the diffraction cone z =+2 xy22 and the direction is determined by the chirp. Both 1D and 2D Gaussian beams are diffractionless and display uniform propagation. The nondiffracting property discovered in this model applies to other beams as well. Based on the nondiffracting and splitting properties, we introduce the Talbot effect of diffractionless beams in FSE. Fractional effects, such as fractional quantum Hall effect1, fractional Talbot effect2, fractional Josephson effect3 and other effects coming from the fractional Schrödinger equation (FSE)4, introduce seminal phenomena in and open new areas of physics. FSE, developed by Laskin5–7, is a generalization of the Schrödinger equation (SE) that contains fractional Laplacian instead of the usual one, which describes the behavior of particles with fractional spin8. This generalization produces nonlocal features in the wave function. In the last decade, research on FSE was very intensive9–17. -
The Doubling of the Degrees of Freedom in Quantum Dissipative Systems, and the Semantic Information Notion and Measure in Biosemiotics †
Proceedings The Doubling of the Degrees of Freedom in Quantum Dissipative Systems, and the Semantic † Information Notion and Measure in Biosemiotics Gianfranco Basti 1,*, Antonio Capolupo 2,3 and GiuseppeVitiello 2 1 Faculty of Philosophy, Pontifical Lateran University, 00120 Vatican City, Italy 2 Department of Physics “E:R. Caianiello”, University of Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy; [email protected] (A.C.); [email protected] (G.V.) 3 Istituto Nazionale di Fisica Nucleare (INFN), Gruppo Collegato di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy * Correspondence: [email protected]; Tel.: +39-339-576-0314 † Conference Theoretical Information Studies (TIS), Berkeley, CA, USA, 2–6 June 2019. Published: 11 May 2020 Keywords: semantic information; non-equilibrium thermodynamics; quantum field theory 1. The Semantic Information Issue and Non-Equilibrium Thermodynamics In the recent history of the effort for defining a suitable notion and measure of semantic information, starting points are: the “syntactic” notion and measure of information H of Claude Shannon in the mathematical theory of communications (MTC) [1]; and the “semantic” notion and measure of information content or informativeness of Rudolph Carnap and Yehoshua Bar-Hillel [2], based on Carnap’s theory of intensional modal logic [3]. The relationship between H and the notion and measure of entropy S in Gibbs’ statistical mechanics (SM) and then in Boltzmann’s thermodynamics is a classical one, because they share the same mathematical form. We emphasize that Shannon’s information measure is the information associated to an event in quantum mechanics (QM) [4]. Indeed, in the classical limit, i.e., whenever the classical notion of probability applies, S is equivalent to the QM definition of entropy given by John Von Neumann.