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Path Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics Dennis V. Perepelitsa MIT Department of Physics 70 Amherst Ave. Cambridge, MA 02142 Abstract We present the path integral formulation of quantum mechanics and demon- strate its equivalence to the Schr¨odinger picture. We apply the method to the free particle and quantum harmonic oscillator, investigate the Euclidean path integral, and discuss other applications. 1 Introduction A fundamental question in quantum mechanics is how does the state of a particle evolve with time? That is, the determination the time-evolution ψ(t) of some initial | i state ψ(t ) . Quantum mechanics is fully predictive [3] in the sense that initial | 0 i conditions and knowledge of the potential occupied by the particle is enough to fully specify the state of the particle for all future times.1 In the early twentieth century, Erwin Schr¨odinger derived an equation specifies how the instantaneous change in the wavefunction d ψ(t) depends on the system dt | i inhabited by the state in the form of the Hamiltonian. In this formulation, the eigenstates of the Hamiltonian play an important role, since their time-evolution is easy to calculate (i.e. they are stationary). A well-established method of solution, after the entire eigenspectrum of Hˆ is known, is to decompose the initial state into this eigenbasis, apply time evolution to each and then reassemble the eigenstates. That is, 1In the analysis below, we consider only the position of a particle, and not any other quantum property such as spin. 2 D.V. Perepelitsa n=∞ ψ(t) = exp [ iE t/~] n ψ(t ) n (1) | i − n h | 0 i| i n=0 X This (Hamiltonian) formulation works in many cases. -
An Introduction to Quantum Field Theory
AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ........................... -
Research Statement
D. A. Williams II June 2021 Research Statement I am an algebraist with expertise in the areas of representation theory of Lie superalgebras, associative superalgebras, and algebraic objects related to mathematical physics. As a post- doctoral fellow, I have extended my research to include topics in enumerative and algebraic combinatorics. My research is collaborative, and I welcome collaboration in familiar areas and in newer undertakings. The referenced open problems here, their subproblems, and other ideas in mind are suitable for undergraduate projects and theses, doctoral proposals, or scholarly contributions to academic journals. In what follows, I provide a technical description of the motivation and history of my work in Lie superalgebras and super representation theory. This is then followed by a description of the work I am currently supervising and mentoring, in combinatorics, as I serve as the Postdoctoral Research Advisor for the Mathematical Sciences Research Institute's Undergraduate Program 2021. 1 Super Representation Theory 1.1 History 1.1.1 Superalgebras In 1941, Whitehead dened a product on the graded homotopy groups of a pointed topological space, the rst non-trivial example of a Lie superalgebra. Whitehead's work in algebraic topol- ogy [Whi41] was known to mathematicians who dened new graded geo-algebraic structures, such as -graded algebras, or superalgebras to physicists, and their modules. A Lie superalge- Z2 bra would come to be a -graded vector space, even (respectively, odd) elements g = g0 ⊕ g1 Z2 found in (respectively, ), with a parity-respecting bilinear multiplication termed the Lie g0 g1 superbracket inducing1 a symmetric intertwining map of -modules. -
Mathematical Methods of Classical Mechanics-Arnold V.I..Djvu
V.I. Arnold Mathematical Methods of Classical Mechanics Second Edition Translated by K. Vogtmann and A. Weinstein With 269 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest V. I. Arnold K. Vogtmann A. Weinstein Department of Department of Department of Mathematics Mathematics Mathematics Steklov Mathematical Cornell University University of California Institute Ithaca, NY 14853 at Berkeley Russian Academy of U.S.A. Berkeley, CA 94720 Sciences U.S.A. Moscow 117966 GSP-1 Russia Editorial Board J. H. Ewing F. W. Gehring P.R. Halmos Department of Department of Department of Mathematics Mathematics Mathematics Indiana University University of Michigan Santa Clara University Bloomington, IN 47405 Ann Arbor, MI 48109 Santa Clara, CA 95053 U.S.A. U.S.A. U.S.A. Mathematics Subject Classifications (1991): 70HXX, 70005, 58-XX Library of Congress Cataloging-in-Publication Data Amol 'd, V.I. (Vladimir Igorevich), 1937- [Matematicheskie melody klassicheskoi mekhaniki. English] Mathematical methods of classical mechanics I V.I. Amol 'd; translated by K. Vogtmann and A. Weinstein.-2nd ed. p. cm.-(Graduate texts in mathematics ; 60) Translation of: Mathematicheskie metody klassicheskoi mekhaniki. Bibliography: p. Includes index. ISBN 0-387-96890-3 I. Mechanics, Analytic. I. Title. II. Series. QA805.A6813 1989 531 '.01 '515-dcl9 88-39823 Title of the Russian Original Edition: M atematicheskie metody klassicheskoi mekhaniki. Nauka, Moscow, 1974. Printed on acid-free paper © 1978, 1989 by Springer-Verlag New York Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis. -
Quantum Field Theory*
Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement. -
Path Probabilities for Consecutive Measurements, and Certain "Quantum Paradoxes"
Path probabilities for consecutive measurements, and certain "quantum paradoxes" D. Sokolovski1;2 1 Departmento de Química-Física, Universidad del País Vasco, UPV/EHU, Leioa, Spain and 2 IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013, Bilbao, Spain (Dated: June 20, 2018) Abstract ABSTRACT: We consider a finite-dimensional quantum system, making a transition between known initial and final states. The outcomes of several accurate measurements, which could be made in the interim, define virtual paths, each endowed with a probability amplitude. If the measurements are actually made, the paths, which may now be called "real", acquire also the probabilities, related to the frequencies, with which a path is seen to be travelled in a series of identical trials. Different sets of measurements, made on the same system, can produce different, or incompatible, statistical ensembles, whose conflicting attributes may, although by no means should, appear "paradoxical". We describe in detail the ensembles, resulting from intermediate measurements of mutually commuting, or non-commuting, operators, in terms of the real paths produced. In the same manner, we analyse the Hardy’s and the "three box" paradoxes, the photon’s past in an interferometer, the "quantum Cheshire cat" experiment, as well as the closely related subject of "interaction-free measurements". It is shown that, in all these cases, inaccurate "weak measurements" produce no real paths, and yield only limited information about the virtual paths’ probability amplitudes. arXiv:1803.02303v3 [quant-ph] 19 Jun 2018 PACS numbers: Keywords: Quantum measurements, Feynman paths, quantum "paradoxes" 1 I. INTRODUCTION Recently, there has been significant interest in the properties of a pre-and post-selected quan- tum systems, and, in particular, in the description of such systems during the time between the preparation, and the arrival in the pre-determined final state (see, for example [1] and the Refs. -
Mathematical Analysis of the Accordion Grating Illusion: a Differential Geometry Approach to Introduce the 3D Aperture Problem
Neural Networks 24 (2011) 1093–1101 Contents lists available at SciVerse ScienceDirect Neural Networks journal homepage: www.elsevier.com/locate/neunet Mathematical analysis of the Accordion Grating illusion: A differential geometry approach to introduce the 3D aperture problem Arash Yazdanbakhsh a,b,∗, Simone Gori c,d a Cognitive and Neural Systems Department, Boston University, MA, 02215, United States b Neurobiology Department, Harvard Medical School, Boston, MA, 02115, United States c Universita' degli studi di Padova, Dipartimento di Psicologia Generale, Via Venezia 8, 35131 Padova, Italy d Developmental Neuropsychology Unit, Scientific Institute ``E. Medea'', Bosisio Parini, Lecco, Italy article info a b s t r a c t Keywords: When an observer moves towards a square-wave grating display, a non-rigid distortion of the pattern Visual illusion occurs in which the stripes bulge and expand perpendicularly to their orientation; these effects reverse Motion when the observer moves away. Such distortions present a new problem beyond the classical aperture Projection line Line of sight problem faced by visual motion detectors, one we describe as a 3D aperture problem as it incorporates Accordion Grating depth signals. We applied differential geometry to obtain a closed form solution to characterize the fluid Aperture problem distortion of the stripes. Our solution replicates the perceptual distortions and enabled us to design a Differential geometry nulling experiment to distinguish our 3D aperture solution from other candidate mechanisms (see Gori et al. (in this issue)). We suggest that our approach may generalize to other motion illusions visible in 2D displays. ' 2011 Elsevier Ltd. All rights reserved. 1. Introduction need for line-ends or intersections along the lines to explain the illusion. -
Mathematical Foundations of Supersymmetry
MATHEMATICAL FOUNDATIONS OF SUPERSYMMETRY Lauren Caston and Rita Fioresi 1 Dipartimento di Matematica, Universit`adi Bologna Piazza di Porta S. Donato, 5 40126 Bologna, Italia e-mail: [email protected], fi[email protected] 1Investigation supported by the University of Bologna 1 Introduction Supersymmetry (SUSY) is the machinery mathematicians and physicists have developed to treat two types of elementary particles, bosons and fermions, on the same footing. Supergeometry is the geometric basis for supersymmetry; it was first discovered and studied by physicists Wess, Zumino [25], Salam and Strathde [20] (among others) in the early 1970’s. Today supergeometry plays an important role in high energy physics. The objects in super geometry generalize the concept of smooth manifolds and algebraic schemes to include anticommuting coordinates. As a result, we employ the techniques from algebraic geometry to study such objects, namely A. Grothendiek’s theory of schemes. Fermions include all of the material world; they are the building blocks of atoms. Fermions do not like each other. This is in essence the Pauli exclusion principle which states that two electrons cannot occupy the same quantum me- chanical state at the same time. Bosons, on the other hand, can occupy the same state at the same time. Instead of looking at equations which just describe either bosons or fermions separately, supersymmetry seeks out a description of both simultaneously. Tran- sitions between fermions and bosons require that we allow transformations be- tween the commuting and anticommuting coordinates. Such transitions are called supersymmetries. In classical Minkowski space, physicists classify elementary particles by their mass and spin. -
(Aka Second Quantization) 1 Quantum Field Theory
221B Lecture Notes Quantum Field Theory (a.k.a. Second Quantization) 1 Quantum Field Theory Why quantum field theory? We know quantum mechanics works perfectly well for many systems we had looked at already. Then why go to a new formalism? The following few sections describe motivation for the quantum field theory, which I introduce as a re-formulation of multi-body quantum mechanics with identical physics content. 1.1 Limitations of Multi-body Schr¨odinger Wave Func- tion We used totally anti-symmetrized Slater determinants for the study of atoms, molecules, nuclei. Already with the number of particles in these systems, say, about 100, the use of multi-body wave function is quite cumbersome. Mention a wave function of an Avogardro number of particles! Not only it is completely impractical to talk about a wave function with 6 × 1023 coordinates for each particle, we even do not know if it is supposed to have 6 × 1023 or 6 × 1023 + 1 coordinates, and the property of the system of our interest shouldn’t be concerned with such a tiny (?) difference. Another limitation of the multi-body wave functions is that it is incapable of describing processes where the number of particles changes. For instance, think about the emission of a photon from the excited state of an atom. The wave function would contain coordinates for the electrons in the atom and the nucleus in the initial state. The final state contains yet another particle, photon in this case. But the Schr¨odinger equation is a differential equation acting on the arguments of the Schr¨odingerwave function, and can never change the number of arguments. -
Physical Quantum States and the Meaning of Probability Michel Paty
Physical quantum states and the meaning of probability Michel Paty To cite this version: Michel Paty. Physical quantum states and the meaning of probability. Galavotti, Maria Carla, Suppes, Patrick and Costantini, Domenico. Stochastic Causality, CSLI Publications (Center for Studies on Language and Information), Stanford (Ca, USA), p. 235-255, 2001. halshs-00187887 HAL Id: halshs-00187887 https://halshs.archives-ouvertes.fr/halshs-00187887 Submitted on 15 Nov 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. as Chapter 14, in Galavotti, Maria Carla, Suppes, Patrick and Costantini, Domenico, (eds.), Stochastic Causality, CSLI Publications (Center for Studies on Language and Information), Stanford (Ca, USA), 2001, p. 235-255. Physical quantum states and the meaning of probability* Michel Paty Ëquipe REHSEIS (UMR 7596), CNRS & Université Paris 7-Denis Diderot, 37 rue Jacob, F-75006 Paris, France. E-mail : [email protected] Abstract. We investigate epistemologically the meaning of probability as implied in quantum physics in connection with a proposed direct interpretation of the state function and of the related quantum theoretical quantities in terms of physical systems having physical properties, through an extension of meaning of the notion of physical quantity to complex mathematical expressions not reductible to simple numerical values. -
Quantum Computing Joseph C
Quantum Computing Joseph C. Bardin, Daniel Sank, Ofer Naaman, and Evan Jeffrey ©ISTOCKPHOTO.COM/SOLARSEVEN uring the past decade, quantum com- underway at many companies, including IBM [2], Mi- puting has grown from a field known crosoft [3], Google [4], [5], Alibaba [6], and Intel [7], mostly for generating scientific papers to name a few. The European Union [8], Australia [9], to one that is poised to reshape comput- China [10], Japan [11], Canada [12], Russia [13], and the ing as we know it [1]. Major industrial United States [14] are each funding large national re- Dresearch efforts in quantum computing are currently search initiatives focused on the quantum information Joseph C. Bardin ([email protected]) is with the University of Massachusetts Amherst and Google, Goleta, California. Daniel Sank ([email protected]), Ofer Naaman ([email protected]), and Evan Jeffrey ([email protected]) are with Google, Goleta, California. Digital Object Identifier 10.1109/MMM.2020.2993475 Date of current version: 8 July 2020 24 1527-3342/20©2020IEEE August 2020 Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. sciences. And, recently, tens of start-up companies have Quantum computing has grown from emerged with goals ranging from the development of software for use on quantum computers [15] to the im- a field known mostly for generating plementation of full-fledged quantum computers (e.g., scientific papers to one that is Rigetti [16], ION-Q [17], Psi-Quantum [18], and so on). poised to reshape computing as However, despite this rapid growth, because quantum computing as a field brings together many different we know it. -
Renormalization and Effective Field Theory
Mathematical Surveys and Monographs Volume 170 Renormalization and Effective Field Theory Kevin Costello American Mathematical Society surv-170-costello-cov.indd 1 1/28/11 8:15 AM http://dx.doi.org/10.1090/surv/170 Renormalization and Effective Field Theory Mathematical Surveys and Monographs Volume 170 Renormalization and Effective Field Theory Kevin Costello American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Eric M. Friedlander Benjamin Sudakov MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 81T13, 81T15, 81T17, 81T18, 81T20, 81T70. The author was partially supported by NSF grant 0706954 and an Alfred P. Sloan Fellowship. For additional information and updates on this book, visit www.ams.org/bookpages/surv-170 Library of Congress Cataloging-in-Publication Data Costello, Kevin. Renormalization and effective fieldtheory/KevinCostello. p. cm. — (Mathematical surveys and monographs ; v. 170) Includes bibliographical references. ISBN 978-0-8218-5288-0 (alk. paper) 1. Renormalization (Physics) 2. Quantum field theory. I. Title. QC174.17.R46C67 2011 530.143—dc22 2010047463 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA.