DOCSLIB.ORG

Interpreting Quantum Mechanics in Terms of Random Discontinuous Motion of Particles

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Interpreting Quantum Mechanics in Terms of Random Discontinuous Motion of Particles Interpreting Quantum Mechanics in Terms of Random Discontinuous Motion of Particles by Shan Gao THESIS submitted for the degree of Doctor of Philosophy Unit for History and Philosophy of Science Centre for Time, SOPHI Faculty of Science Department of Philosophy Supervisors: Dr. Dean Rickles Prof. Huw Price University of Sydney Sydney, Australia February 2012 c Shan Gao 2011 All rights reserved This thesis is copyright. Subject to statutory exception and to provisions of relevant collective licensing agreements, no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, without the prior written permission of the author. This thesis is dedicated to Erwin Schr¨odinger,who introduced the wave function, discovered its equation named after him, and argued that quantum mechanics is in- complete by his famous cat paradox. Abstract This thesis is an attempt to reconstruct the conceptual foundations of quantum mechanics. First, we argue that the wave function in quantum mechanics is a description of random discontinuous motion of particles, and the modulus square of the wave function gives the probability density of the particles being in certain locations in space. Next, we show that the linear non-relativistic evolution of the wave function of an isolated system obeys the free Schr¨odingerequation due to the requirements of spacetime translation invariance and relativistic invariance. Thirdly, we argue that the random discontinuous motion of particles may lead to a stochastic, nonlinear collapse evolution of the wave function. A discrete model of energy-conserved wavefunction collapse is proposed and shown to be consistent with existing experiments and our macroscopic experience. In addition, we also give a critical analysis of the de Broglie-Bohm theory, the many-worlds interpretation and other dynamical collapse theories, and briefly discuss the issue of unifying quantum mechanics and special relativity. 4 Acknowledgement The idea of random discontinuous motion of particles came to my mind when I was a post- graduate at the Institute of Electronics, Chinese Academy of Sciences in 1993. I am happy that finally it has a more logical and satisfying formulation in this PhD thesis. During the writing of the thesis, I enjoyed financial support from the University of Sydney International Scholarship and Postgraduate Scholarship in Quantum Foundations provided by the Unit for HPS and Centre for Time (SOPHI), as well as from the Lucy Firth Scholarship in Philosophy provided by the Department of Philosophy. I would like to express my deepest gratitude to my supervisors, Dr. Dean and Prof. Huw, whose understanding, motivation and personal guidance was indispensable during my PhD studies. I am deeply indebted to them for their continuous encouragement, stimulating suggestions and healthy criticism which helped me greatly during my research and writing of this thesis. Their professional approach and accuracy was particularly beneficial to my study. I wish to express my warm and sincere thanks to the head of the Unit for HPS, Prof. Ofer, the postgraduate coordinator, Dr. Hans, and Mrs. Debby. I am grateful for their personal support and for their reassurance regarding technical conditions during my PhD work. I would like to take this opportunity to thank all the other members of the Unit for their help and encouragement throughout the years. I would like to gratefully thank Prof. David, Prof. Rod, Dr. Hans and Dr. Owen for helpful discussions and useful suggestions. I owe a debt of gratitude to all the PhD students here who I shared good times and had fun with. I am very grateful to the referees of my thesis too. Their comments and suggestions were indispensable to complete my dissertation. My deepest gratitude goes to my parents who spared no effort to provide the best possible environment for me to grow up in and their constant support during my extended studies. They have always allowed me to have the freedom of choice, and for that I am truly grateful. I am also thankful to my brother and to all my friends, as they have always been willing to share their thoughts with me and provide alternative viewpoints which have helped me think in a more holistic fashion. Finally, I am deeply indebted to my wife and my daughter for their unflagging love and support throughout my study; this dissertation would have been simply impossible without them. Moreover, they have never let me forget the true values of life. 5 Publications 1. On Di´osi-Penrose criterion of gravity-induced quantum collapse, International Journal of Theoretical Physics, 49, 849853 (2010). 2. The wave function and quantum reality, in Proceedings of the International Conference on Advances in Quantum Theory, A. Khrennikov, G. Jaeger, M. Schlosshauer, G. Weihs (eds), AIP Conference Proceedings 1327, 334-338 (2011). 3. Meaning of the wave function, International Journal of Quantum Chemistry, 111, 4124- 4138 (2011). 4. Is gravity an entropic force? Entropy special issue \Black Hole Thermodynamics", Jacob D. Bekenstein (eds), 13, 936-948 (2011). 5. A quantum physical argument for panpsychism, Forthcoming in Journal of Consciousness Studies, 2012. 6 Contents 1 Introduction 10 2 Meaning of the Wave Function 14 2.1 Standard quantum mechanics and impulse measurements........... 15 2.2 Weak measurements................................ 17 2.3 Protective measurements.............................. 18 2.3.1 Measurements with natural protection.................. 19 2.3.2 Measurements with artificial protection................. 21 2.3.3 Further discussions............................. 22 2.4 On the mass and charge density of a quantum system............. 23 2.4.1 A heuristic argument............................ 23 2.4.2 The answer of protective measurement.................. 24 2.4.3 A specific example............................. 24 2.5 The physical origin of mass and charge density................. 27 2.5.1 The mass and charge density is effective................. 27 2.5.2 The ergodic motion of a particle is discontinuous............ 28 2.5.3 An argument for random discontinuous motion............. 30 2.6 The wave function as a description of random discontinuous motion of particles 31 2.6.1 An analysis of random discontinuous motion of particles........ 32 2.6.2 Interpreting the wave function...................... 34 3 Schr¨odinger'sEquation and the Conservation Laws 36 3.1 Spacetime translation and its invariance..................... 36 3.2 Relativistic invariance............................... 38 3.3 Derivation of the free Schr¨odingerequation................... 40 3.4 Further discussions................................. 41 3.5 On the conservation of energy and momentum................. 42 4 A Suggested Solution to the Measurement Problem 47 4.1 The reality of wavefunction collapse....................... 48 4.1.1 Against the de Broglie-Bohm theory................... 49 4.1.2 Against the many-worlds interpretation................. 51 4.2 A conjecture on the origin of wavefunction collapse............... 53 4.2.1 The chooser in discrete time........................ 53 4.2.2 Energy conservation and the choices................... 56 4.2.3 In search of a deeper basis......................... 57 4.3 A discrete model of energy-conserved wavefunction collapse.......... 58 4.4 On the consistency of the model and experiments................ 65 4.4.1 Maintenance of coherence......................... 65 4.4.2 Rapid localization in measurement situations.............. 65 4.4.3 Emergence of the classical world..................... 67 4.5 Critical comments on other dynamical collapse models............. 69 4.5.1 Penrose's gravity-induced wavefunction collapse model......... 70 4.5.2 The CSL model............................... 73 7 CONTENTS 5 Random Discontinuous Motion and Relativity 76 5.1 Distorted picture of motion............................ 76 5.1.1 Single particle picture........................... 76 5.1.2 Picture of quantum entanglement..................... 78 5.1.3 Picture of wavefunction collapse..................... 79 5.2 On the absoluteness of simultaneity........................ 81 5.3 Collapse dynamics and preferred Lorentz frame................. 83 8 I think I can safely say that nobody understands quantum mechanics... Do not keep saying to yourself, if you can possible avoid it, \But how can it be like that?" because you will get `down the drain', into a blind alley from which nobody has escaped. Nobody knows how it can be like that. | Richard Feynman, 1964 1 Introduction Quantum mechanics, according to its Schr¨odingerpicture, is a non-relativistic theory about the wave function and its evolution. There are two main problems in the conceptual foun- dations of quantum mechanics. The first one concerns the physical meaning of the wave function in the theory. It has been widely argued that the probability interpretation is not wholly satisfactory because of resorting to the vague concept of measurement - though it is still the standard interpretation in textbooks nowadays. On the other hand, the meaning of the wave function is also in dispute in the alternative formulations of quantum mechanics such as the de Broglie-Bohm theory and the many-worlds interpretation (de Broglie 1928; Bohm 1952; Everett 1957; De Witt and Graham 1973). Exactly what does the wave function describe then? The second problem concerns the evolution of the wave function. It includes two parts. One part concerns the linear Schr¨odingerevolution. Why does the linear
Load more

Recommended publications
  • International Centre for Theoretical Physics
    XA<? 6 H 0 <! W IC/95/320 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS BOHM'S THEORY VERSUS DYNAMICAL REDUCTION G.C. Ghirardi and INTERNATIONAL ATOMIC ENERGY R. Grassi AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION MIRAMARE-TRIESTE VQL 'l 7 Ha ° 7 IC/95/320 ABSTRACT International Atomic Energy Agency and United Nations Educational scientific and Cultural Organization made that theories which exhibit parameter , INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS BOHM'S THEORY VERSUS DYNAMICAL REDUCTION1 1. Introduction. We share today's widespread opinion that Standard Quantum Mechanics (SQM), in spite of its enormous successes, has failed in giving a G.C. Ghirardi satisfactory picture of the world, as we perceive it. The difficulties about International Centre for Theoretical Physics, Trieste, Italy, the conceptual foundations of the theory arising, as is well known, from and the so-called objectification problem, have stimulated various attempts to Department of Theoretical Physics. University di Trieste, Trieste, Italy overcome them. Among these one should mention the search for a and deterministic completion of the theory, the many worlds and many minds intepretations, the so called environment induced superselection rules, the R. Grassi quantum histories approach and the dynamical reduction program. In this Department of Civil Engineering, University of Udine, Udine, Italy. paper we will focus our attention on the only available and precisely formulated examples of a deterministic completion and of a stochastic and nonlinear modification of SQM, i.e., Bohm's theory and the spontaneous reduction models, respectively. It is useful to stress that while the first theory is fully equivalent, from a predictive point of view, to SQM, the second one qualifies itself as a rival of SQM but with empirical divergence so small that it can claim all the same experimental support.
    [Show full text]
  • The Concept of Quantum State : New Views on Old Phenomena Michel Paty
    The concept of quantum state : new views on old phenomena Michel Paty To cite this version: Michel Paty. The concept of quantum state : new views on old phenomena. Ashtekar, Abhay, Cohen, Robert S., Howard, Don, Renn, Jürgen, Sarkar, Sahotra & Shimony, Abner. Revisiting the Founda- tions of Relativistic Physics : Festschrift in Honor of John Stachel, Boston Studies in the Philosophy and History of Science, Dordrecht: Kluwer Academic Publishers, p. 451-478, 2003. halshs-00189410 HAL Id: halshs-00189410 https://halshs.archives-ouvertes.fr/halshs-00189410 Submitted on 20 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. « The concept of quantum state: new views on old phenomena », in Ashtekar, Abhay, Cohen, Robert S., Howard, Don, Renn, Jürgen, Sarkar, Sahotra & Shimony, Abner (eds.), Revisiting the Foundations of Relativistic Physics : Festschrift in Honor of John Stachel, Boston Studies in the Philosophy and History of Science, Dordrecht: Kluwer Academic Publishers, 451-478. , 2003 The concept of quantum state : new views on old phenomena par Michel PATY* ABSTRACT. Recent developments in the area of the knowledge of quantum systems have led to consider as physical facts statements that appeared formerly to be more related to interpretation, with free options.
    [Show full text]
  • 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.
    [Show full text]
  • Lecture 9, P 1 Lecture 9: Introduction to QM: Review and Examples
    Lecture 9, p 1 Lecture 9: Introduction to QM: Review and Examples S1 S2 Lecture 9, p 2 Photoelectric Effect Binding KE=⋅ eV = hf −Φ energy max stop Φ The work function: 3.5 Φ is the minimum energy needed to strip 3 an electron from the metal. 2.5 2 (v) Φ is defined as positive . 1.5 stop 1 Not all electrons will leave with the maximum V f0 kinetic energy (due to losses). 0.5 0 0 5 10 15 Conclusions: f (x10 14 Hz) • Light arrives in “packets” of energy (photons ). • Ephoton = hf • Increasing the intensity increases # photons, not the photon energy. Each photon ejects (at most) one electron from the metal. Recall: For EM waves, frequency and wavelength are related by f = c/ λ. Therefore: Ephoton = hc/ λ = 1240 eV-nm/ λ Lecture 9, p 3 Photoelectric Effect Example 1. When light of wavelength λ = 400 nm shines on lithium, the stopping voltage of the electrons is Vstop = 0.21 V . What is the work function of lithium? Lecture 9, p 4 Photoelectric Effect: Solution 1. When light of wavelength λ = 400 nm shines on lithium, the stopping voltage of the electrons is Vstop = 0.21 V . What is the work function of lithium? Φ = hf - eV stop Instead of hf, use hc/ λ: 1240/400 = 3.1 eV = 3.1eV - 0.21eV For Vstop = 0.21 V, eV stop = 0.21 eV = 2.89 eV Lecture 9, p 5 Act 1 3 1. If the workfunction of the material increased, (v) 2 how would the graph change? stop a.
    [Show full text]
  • Origin of Probability in Quantum Mechanics and the Physical Interpretation of the Wave Function
    Origin of Probability in Quantum Mechanics and the Physical Interpretation of the Wave Function Shuming Wen ( [email protected] ) Faculty of Land and Resources Engineering, Kunming University of Science and Technology. Research Article Keywords: probability origin, wave-function collapse, uncertainty principle, quantum tunnelling, double-slit and single-slit experiments Posted Date: November 16th, 2020 DOI: https://doi.org/10.21203/rs.3.rs-95171/v2 License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License Origin of Probability in Quantum Mechanics and the Physical Interpretation of the Wave Function Shuming Wen Faculty of Land and Resources Engineering, Kunming University of Science and Technology, Kunming 650093 Abstract The theoretical calculation of quantum mechanics has been accurately verified by experiments, but Copenhagen interpretation with probability is still controversial. To find the source of the probability, we revised the definition of the energy quantum and reconstructed the wave function of the physical particle. Here, we found that the energy quantum ê is 6.62606896 ×10-34J instead of hν as proposed by Planck. Additionally, the value of the quality quantum ô is 7.372496 × 10-51 kg. This discontinuity of energy leads to a periodic non-uniform spatial distribution of the particles that transmit energy. A quantum objective system (QOS) consists of many physical particles whose wave function is the superposition of the wave functions of all physical particles. The probability of quantum mechanics originates from the distribution rate of the particles in the QOS per unit volume at time t and near position r. Based on the revision of the energy quantum assumption and the origin of the probability, we proposed new certainty and uncertainty relationships, explained the physical mechanism of wave-function collapse and the quantum tunnelling effect, derived the quantum theoretical expression of double-slit and single-slit experiments.
    [Show full text]
  • Statistics of a Free Single Quantum Particle at a Finite
    STATISTICS OF A FREE SINGLE QUANTUM PARTICLE AT A FINITE TEMPERATURE JIAN-PING PENG Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China Abstract We present a model to study the statistics of a single structureless quantum particle freely moving in a space at a finite temperature. It is shown that the quantum particle feels the temperature and can exchange energy with its environment in the form of heat transfer. The underlying mechanism is diffraction at the edge of the wave front of its matter wave. Expressions of energy and entropy of the particle are obtained for the irreversible process. Keywords: Quantum particle at a finite temperature, Thermodynamics of a single quantum particle PACS: 05.30.-d, 02.70.Rr 1 Quantum mechanics is the theoretical framework that describes phenomena on the microscopic level and is exact at zero temperature. The fundamental statistical character in quantum mechanics, due to the Heisenberg uncertainty relation, is unrelated to temperature. On the other hand, temperature is generally believed to have no microscopic meaning and can only be conceived at the macroscopic level. For instance, one can define the energy of a single quantum particle, but one can not ascribe a temperature to it. However, it is physically meaningful to place a single quantum particle in a box or let it move in a space where temperature is well-defined. This raises the well-known question: How a single quantum particle feels the temperature and what is the consequence? The question is particular important and interesting, since experimental techniques in recent years have improved to such an extent that direct measurement of electron dynamics is possible.1,2,3 It should also closely related to the question on the applicability of the thermodynamics to small systems on the nanometer scale.4 We present here a model to study the behavior of a structureless quantum particle moving freely in a space at a nonzero temperature.
    [Show full text]
  • The Uncertainty Principle (Stanford Encyclopedia of Philosophy) Page 1 of 14
    The Uncertainty Principle (Stanford Encyclopedia of Philosophy) Page 1 of 14 Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia Free The Uncertainty Principle First published Mon Oct 8, 2001; substantive revision Mon Jul 3, 2006 Quantum mechanics is generally regarded as the physical theory that is our best candidate for a fundamental and universal description of the physical world. The conceptual framework employed by this theory differs drastically from that of classical physics. Indeed, the transition from classical to quantum physics marks a genuine revolution in our understanding of the physical world. One striking aspect of the difference between classical and quantum physics is that whereas classical mechanics presupposes that exact simultaneous values can be assigned to all physical quantities, quantum mechanics denies this possibility, the prime example being the position and momentum of a particle. According to quantum mechanics, the more precisely the position (momentum) of a particle is given, the less precisely can one say what its momentum (position) is. This is (a simplistic and preliminary formulation of) the quantum mechanical uncertainty principle for position and momentum. The uncertainty principle played an important role in many discussions on the philosophical implications of quantum mechanics, in particular in discussions on the consistency of the so-called Copenhagen interpretation, the interpretation endorsed by the founding fathers Heisenberg and Bohr. This should not suggest that the uncertainty principle is the only aspect of the conceptual difference between classical and quantum physics: the implications of quantum mechanics for notions as (non)-locality, entanglement and identity play no less havoc with classical intuitions.
    [Show full text]
  • The Statistical Interpretation of Entangled States B
    The Statistical Interpretation of Entangled States B. C. Sanctuary Department of Chemistry, McGill University 801 Sherbrooke Street W Montreal, PQ, H3A 2K6, Canada Abstract Entangled EPR spin pairs can be treated using the statistical ensemble interpretation of quantum mechanics. As such the singlet state results from an ensemble of spin pairs each with an arbitrary axis of quantization. This axis acts as a quantum mechanical hidden variable. If the spins lose coherence they disentangle into a mixed state. Whether or not the EPR spin pairs retain entanglement or disentangle, however, the statistical ensemble interpretation resolves the EPR paradox and gives a mechanism for quantum “teleportation” without the need for instantaneous action-at-a-distance. Keywords: Statistical ensemble, entanglement, disentanglement, quantum correlations, EPR paradox, Bell’s inequalities, quantum non-locality and locality, coincidence detection 1. Introduction The fundamental questions of quantum mechanics (QM) are rooted in the philosophical interpretation of the wave function1. At the time these were first debated, covering the fifty or so years following the formulation of QM, the arguments were based primarily on gedanken experiments2. Today the situation has changed with numerous experiments now possible that can guide us in our search for the true nature of the microscopic world, and how The Infamous Boundary3 to the macroscopic world is breached. The current view is based upon pivotal experiments, performed by Aspect4 showing that quantum mechanics is correct and Bell’s inequalities5 are violated. From this the non-local nature of QM became firmly entrenched in physics leading to other experiments, notably those demonstrating that non-locally is fundamental to quantum “teleportation”.
    [Show full text]
  • Path Integral for the Hydrogen Atom
    Path Integral for the Hydrogen Atom Solutions in two and three dimensions Vägintegral för Väteatomen Lösningar i två och tre dimensioner Anders Svensson Faculty of Health, Science and Technology Physics, Bachelor Degree Project 15 ECTS Credits Supervisor: Jürgen Fuchs Examiner: Marcus Berg June 2016 Abstract The path integral formulation of quantum mechanics generalizes the action principle of classical mechanics. The Feynman path integral is, roughly speaking, a sum over all possible paths that a particle can take between fixed endpoints, where each path contributes to the sum by a phase factor involving the action for the path. The resulting sum gives the probability amplitude of propagation between the two endpoints, a quantity called the propagator. Solutions of the Feynman path integral formula exist, however, only for a small number of simple systems, and modifications need to be made when dealing with more complicated systems involving singular potentials, including the Coulomb potential. We derive a generalized path integral formula, that can be used in these cases, for a quantity called the pseudo-propagator from which we obtain the fixed-energy amplitude, related to the propagator by a Fourier transform. The new path integral formula is then successfully solved for the Hydrogen atom in two and three dimensions, and we obtain integral representations for the fixed-energy amplitude. Sammanfattning V¨agintegral-formuleringen av kvantmekanik generaliserar minsta-verkanprincipen fr˚anklassisk meka- nik. Feynmans v¨agintegral kan ses som en summa ¨over alla m¨ojligav¨agaren partikel kan ta mellan tv˚a givna ¨andpunkterA och B, d¨arvarje v¨agbidrar till summan med en fasfaktor inneh˚allandeden klas- siska verkan f¨orv¨agen.Den resulterande summan ger propagatorn, sannolikhetsamplituden att partikeln g˚arfr˚anA till B.
    [Show full text]
  • The Heisenberg Uncertainty Principle*
    OpenStax-CNX module: m58578 1 The Heisenberg Uncertainty Principle* OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 Abstract By the end of this section, you will be able to: • Describe the physical meaning of the position-momentum uncertainty relation • Explain the origins of the uncertainty principle in quantum theory • Describe the physical meaning of the energy-time uncertainty relation Heisenberg's uncertainty principle is a key principle in quantum mechanics. Very roughly, it states that if we know everything about where a particle is located (the uncertainty of position is small), we know nothing about its momentum (the uncertainty of momentum is large), and vice versa. Versions of the uncertainty principle also exist for other quantities as well, such as energy and time. We discuss the momentum-position and energy-time uncertainty principles separately. 1 Momentum and Position To illustrate the momentum-position uncertainty principle, consider a free particle that moves along the x- direction. The particle moves with a constant velocity u and momentum p = mu. According to de Broglie's relations, p = }k and E = }!. As discussed in the previous section, the wave function for this particle is given by −i(! t−k x) −i ! t i k x k (x; t) = A [cos (! t − k x) − i sin (! t − k x)] = Ae = Ae e (1) 2 2 and the probability density j k (x; t) j = A is uniform and independent of time. The particle is equally likely to be found anywhere along the x-axis but has denite values of wavelength and wave number, and therefore momentum.
    [Show full text]
  • Analysis of Nonlinear Dynamics in a Classical Transmon Circuit
    Analysis of Nonlinear Dynamics in a Classical Transmon Circuit Sasu Tuohino B. Sc. Thesis Department of Physical Sciences Theoretical Physics University of Oulu 2017 Contents 1 Introduction2 2 Classical network theory4 2.1 From electromagnetic fields to circuit elements.........4 2.2 Generalized flux and charge....................6 2.3 Node variables as degrees of freedom...............7 3 Hamiltonians for electric circuits8 3.1 LC Circuit and DC voltage source................8 3.2 Cooper-Pair Box.......................... 10 3.2.1 Josephson junction.................... 10 3.2.2 Dynamics of the Cooper-pair box............. 11 3.3 Transmon qubit.......................... 12 3.3.1 Cavity resonator...................... 12 3.3.2 Shunt capacitance CB .................. 12 3.3.3 Transmon Lagrangian................... 13 3.3.4 Matrix notation in the Legendre transformation..... 14 3.3.5 Hamiltonian of transmon................. 15 4 Classical dynamics of transmon qubit 16 4.1 Equations of motion for transmon................ 16 4.1.1 Relations with voltages.................. 17 4.1.2 Shunt resistances..................... 17 4.1.3 Linearized Josephson inductance............. 18 4.1.4 Relation with currents................... 18 4.2 Control and read-out signals................... 18 4.2.1 Transmission line model.................. 18 4.2.2 Equations of motion for coupled transmission line.... 20 4.3 Quantum notation......................... 22 5 Numerical solutions for equations of motion 23 5.1 Design parameters of the transmon................ 23 5.2 Resonance shift at nonlinear regime............... 24 6 Conclusions 27 1 Abstract The focus of this thesis is on classical dynamics of a transmon qubit. First we introduce the basic concepts of the classical circuit analysis and use this knowledge to derive the Lagrangians and Hamiltonians of an LC circuit, a Cooper-pair box, and ultimately we derive Hamiltonian for a transmon qubit.
    [Show full text]
  • The S-Matrix Formulation of Quantum Statistical Mechanics, with Application to Cold Quantum Gas
    THE S-MATRIX FORMULATION OF QUANTUM STATISTICAL MECHANICS, WITH APPLICATION TO COLD QUANTUM GAS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Pye Ton How August 2011 c 2011 Pye Ton How ALL RIGHTS RESERVED THE S-MATRIX FORMULATION OF QUANTUM STATISTICAL MECHANICS, WITH APPLICATION TO COLD QUANTUM GAS Pye Ton How, Ph.D. Cornell University 2011 A novel formalism of quantum statistical mechanics, based on the zero-temperature S-matrix of the quantum system, is presented in this thesis. In our new formalism, the lowest order approximation (“two-body approximation”) corresponds to the ex- act resummation of all binary collision terms, and can be expressed as an integral equation reminiscent of the thermodynamic Bethe Ansatz (TBA). Two applica- tions of this formalism are explored: the critical point of a weakly-interacting Bose gas in two dimensions, and the scaling behavior of quantum gases at the unitary limit in two and three spatial dimensions. We found that a weakly-interacting 2D Bose gas undergoes a superfluid transition at T 2πn/[m log(2π/mg)], where n c ≈ is the number density, m the mass of a particle, and g the coupling. In the unitary limit where the coupling g diverges, the two-body kernel of our integral equation has simple forms in both two and three spatial dimensions, and we were able to solve the integral equation numerically. Various scaling functions in the unitary limit are defined (as functions of µ/T ) and computed from the numerical solutions.
    [Show full text]