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Bouncing Oil Droplets, De Broglie's Quantum Thermostat And
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 28 August 2018 doi:10.20944/preprints201808.0475.v1 Peer-reviewed version available at Entropy 2018, 20, 780; doi:10.3390/e20100780 Article Bouncing oil droplets, de Broglie’s quantum thermostat and convergence to equilibrium Mohamed Hatifi 1, Ralph Willox 2, Samuel Colin 3 and Thomas Durt 4 1 Aix Marseille Université, CNRS, Centrale Marseille, Institut Fresnel UMR 7249,13013 Marseille, France; hatifi[email protected] 2 Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku, 153-8914 Tokyo, Japan; [email protected] 3 Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150,22290-180, Rio de Janeiro – RJ, Brasil; [email protected] 4 Aix Marseille Université, CNRS, Centrale Marseille, Institut Fresnel UMR 7249,13013 Marseille, France; [email protected] Abstract: Recently, the properties of bouncing oil droplets, also known as ‘walkers’, have attracted much attention because they are thought to offer a gateway to a better understanding of quantum behaviour. They indeed constitute a macroscopic realization of wave-particle duality, in the sense that their trajectories are guided by a self-generated surrounding wave. The aim of this paper is to try to describe walker phenomenology in terms of de Broglie-Bohm dynamics and of a stochastic version thereof. In particular, we first study how a stochastic modification of the de Broglie pilot-wave theory, à la Nelson, affects the process of relaxation to quantum equilibrium, and we prove an H-theorem for the relaxation to quantum equilibrium under Nelson-type dynamics. -
5.1 Two-Particle Systems
5.1 Two-Particle Systems We encountered a two-particle system in dealing with the addition of angular momentum. Let's treat such systems in a more formal way. The w.f. for a two-particle system must depend on the spatial coordinates of both particles as @Ψ well as t: Ψ(r1; r2; t), satisfying i~ @t = HΨ, ~2 2 ~2 2 where H = + V (r1; r2; t), −2m1r1 − 2m2r2 and d3r d3r Ψ(r ; r ; t) 2 = 1. 1 2 j 1 2 j R Iff V is independent of time, then we can separate the time and spatial variables, obtaining Ψ(r1; r2; t) = (r1; r2) exp( iEt=~), − where E is the total energy of the system. Let us now make a very fundamental assumption: that each particle occupies a one-particle e.s. [Note that this is often a poor approximation for the true many-body w.f.] The joint e.f. can then be written as the product of two one-particle e.f.'s: (r1; r2) = a(r1) b(r2). Suppose furthermore that the two particles are indistinguishable. Then, the above w.f. is not really adequate since you can't actually tell whether it's particle 1 in state a or particle 2. This indeterminacy is correctly reflected if we replace the above w.f. by (r ; r ) = a(r ) (r ) (r ) a(r ). 1 2 1 b 2 b 1 2 The `plus-or-minus' sign reflects that there are two distinct ways to accomplish this. Thus we are naturally led to consider two kinds of identical particles, which we have come to call `bosons' (+) and `fermions' ( ). -
8 the Variational Principle
8 The Variational Principle 8.1 Approximate solution of the Schroedinger equation If we can’t find an analytic solution to the Schroedinger equation, a trick known as the varia- tional principle allows us to estimate the energy of the ground state of a system. We choose an unnormalized trial function Φ(an) which depends on some variational parameters, an and minimise hΦ|Hˆ |Φi E[a ] = n hΦ|Φi with respect to those parameters. This gives an approximation to the wavefunction whose accuracy depends on the number of parameters and the clever choice of Φ(an). For more rigorous treatments, a set of basis functions with expansion coefficients an may be used. The proof is as follows, if we expand the normalised wavefunction 1/2 |φ(an)i = Φ(an)/hΦ(an)|Φ(an)i in terms of the true (unknown) eigenbasis |ii of the Hamiltonian, then its energy is X X X ˆ 2 2 E[an] = hφ|iihi|H|jihj|φi = |hφ|ii| Ei = E0 + |hφ|ii| (Ei − E0) ≥ E0 ij i i ˆ where the true (unknown) ground state of the system is defined by H|i0i = E0|i0i. The inequality 2 arises because both |hφ|ii| and (Ei − E0) must be positive. Thus the lower we can make the energy E[ai], the closer it will be to the actual ground state energy, and the closer |φi will be to |i0i. If the trial wavefunction consists of a complete basis set of orthonormal functions |χ i, each P i multiplied by ai: |φi = i ai|χii then the solution is exact and we just have the usual trick of expanding a wavefunction in a basis set. -
1 the Principle of Wave–Particle Duality: an Overview
3 1 The Principle of Wave–Particle Duality: An Overview 1.1 Introduction In the year 1900, physics entered a period of deep crisis as a number of peculiar phenomena, for which no classical explanation was possible, began to appear one after the other, starting with the famous problem of blackbody radiation. By 1923, when the “dust had settled,” it became apparent that these peculiarities had a common explanation. They revealed a novel fundamental principle of nature that wascompletelyatoddswiththeframeworkofclassicalphysics:thecelebrated principle of wave–particle duality, which can be phrased as follows. The principle of wave–particle duality: All physical entities have a dual character; they are waves and particles at the same time. Everything we used to regard as being exclusively a wave has, at the same time, a corpuscular character, while everything we thought of as strictly a particle behaves also as a wave. The relations between these two classically irreconcilable points of view—particle versus wave—are , h, E = hf p = (1.1) or, equivalently, E h f = ,= . (1.2) h p In expressions (1.1) we start off with what we traditionally considered to be solely a wave—an electromagnetic (EM) wave, for example—and we associate its wave characteristics f and (frequency and wavelength) with the corpuscular charac- teristics E and p (energy and momentum) of the corresponding particle. Conversely, in expressions (1.2), we begin with what we once regarded as purely a particle—say, an electron—and we associate its corpuscular characteristics E and p with the wave characteristics f and of the corresponding wave. -
Quantum Aspects of Life / Editors, Derek Abbott, Paul C.W
Quantum Aspectsof Life P581tp.indd 1 8/18/08 8:42:58 AM This page intentionally left blank foreword by SIR ROGER PENROSE editors Derek Abbott (University of Adelaide, Australia) Paul C. W. Davies (Arizona State University, USAU Arun K. Pati (Institute of Physics, Orissa, India) Imperial College Press ICP P581tp.indd 2 8/18/08 8:42:58 AM Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Quantum aspects of life / editors, Derek Abbott, Paul C.W. Davies, Arun K. Pati ; foreword by Sir Roger Penrose. p. ; cm. Includes bibliographical references and index. ISBN-13: 978-1-84816-253-2 (hardcover : alk. paper) ISBN-10: 1-84816-253-7 (hardcover : alk. paper) ISBN-13: 978-1-84816-267-9 (pbk. : alk. paper) ISBN-10: 1-84816-267-7 (pbk. : alk. paper) 1. Quantum biochemistry. I. Abbott, Derek, 1960– II. Davies, P. C. W. III. Pati, Arun K. [DNLM: 1. Biogenesis. 2. Quantum Theory. 3. Evolution, Molecular. QH 325 Q15 2008] QP517.Q34.Q36 2008 576.8'3--dc22 2008029345 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Photo credit: Abigail P. Abbott for the photo on cover and title page. Copyright © 2008 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. -
Qualification Exam: Quantum Mechanics
Qualification Exam: Quantum Mechanics Name: , QEID#43228029: July, 2019 Qualification Exam QEID#43228029 2 1 Undergraduate level Problem 1. 1983-Fall-QM-U-1 ID:QM-U-2 Consider two spin 1=2 particles interacting with one another and with an external uniform magnetic field B~ directed along the z-axis. The Hamiltonian is given by ~ ~ ~ ~ ~ H = −AS1 · S2 − µB(g1S1 + g2S2) · B where µB is the Bohr magneton, g1 and g2 are the g-factors, and A is a constant. 1. In the large field limit, what are the eigenvectors and eigenvalues of H in the "spin-space" { i.e. in the basis of eigenstates of S1z and S2z? 2. In the limit when jB~ j ! 0, what are the eigenvectors and eigenvalues of H in the same basis? 3. In the Intermediate regime, what are the eigenvectors and eigenvalues of H in the spin space? Show that you obtain the results of the previous two parts in the appropriate limits. Problem 2. 1983-Fall-QM-U-2 ID:QM-U-20 1. Show that, for an arbitrary normalized function j i, h jHj i > E0, where E0 is the lowest eigenvalue of H. 2. A particle of mass m moves in a potential 1 kx2; x ≤ 0 V (x) = 2 (1) +1; x < 0 Find the trial state of the lowest energy among those parameterized by σ 2 − x (x) = Axe 2σ2 : What does the first part tell you about E0? (Give your answers in terms of k, m, and ! = pk=m). Problem 3. 1983-Fall-QM-U-3 ID:QM-U-44 Consider two identical particles of spin zero, each having a mass m, that are con- strained to rotate in a plane with separation r. -
Molecular Energy Levels
MOLECULAR ENERGY LEVELS DR IMRANA ASHRAF OUTLINE q MOLECULE q MOLECULAR ORBITAL THEORY q MOLECULAR TRANSITIONS q INTERACTION OF RADIATION WITH MATTER q TYPES OF MOLECULAR ENERGY LEVELS q MOLECULE q In nature there exist 92 different elements that correspond to stable atoms. q These atoms can form larger entities- called molecules. q The number of atoms in a molecule vary from two - as in N2 - to many thousand as in DNA, protiens etc. q Molecules form when the total energy of the electrons is lower in the molecule than in individual atoms. q The reason comes from the Aufbau principle - to put electrons into the lowest energy configuration in atoms. q The same principle goes for molecules. q MOLECULE q Properties of molecules depend on: § The specific kind of atoms they are composed of. § The spatial structure of the molecules - the way in which the atoms are arranged within the molecule. § The binding energy of atoms or atomic groups in the molecule. TYPES OF MOLECULES q MONOATOMIC MOLECULES § The elements that do not have tendency to form molecules. § Elements which are stable single atom molecules are the noble gases : helium, neon, argon, krypton, xenon and radon. q DIATOMIC MOLECULES § Diatomic molecules are composed of only two atoms - of the same or different elements. § Examples: hydrogen (H2), oxygen (O2), carbon monoxide (CO), nitric oxide (NO) q POLYATOMIC MOLECULES § Polyatomic molecules consist of a stable system comprising three or more atoms. TYPES OF MOLECULES q Empirical, Molecular And Structural Formulas q Empirical formula: Indicates the simplest whole number ratio of all the atoms in a molecule. -
Atomic Excitation Potentials
ATOMIC EXCITATION POTENTIALS PURPOSE In this lab you will study the excitation of mercury atoms by colliding electrons with the atoms, and confirm that this excitation requires a specific quantity of energy. THEORY In general, atoms of an element can exist in a number of either excited or ionized states, or the ground state. This lab will focus on electron collisions in which a free electron gives up just the amount of kinetic energy required to excite a ground state mercury atom into its first excited state. However, it is important to consider all other processes which constantly change the energy states of the atoms. An atom in the ground state may absorb a photon of energy exactly equal to the energy difference between the ground state and some excited state, whereas another atom may collide with an electron and absorb some fraction of the electron's kinetic energy which is the amount needed to put that atom in some excited state (collisional excitation). Each atom in an excited state then spontaneously emits a photon and drops from a higher excited state to a lower one (or to the ground state). Another possibility is that an atom may collide with an electron which carries away kinetic energy equal to the atomic excitation energy so that the atom ends up in, say, the ground state (collisional deexcitation). Lastly, an atom can be placed into an ionized state (one or more of its electrons stripped away) if the collision transfers energy greater than the ionization potential of the atom. Likewise an ionized atom can capture a free electron. -
Theory and Experiment in the Quantum-Relativity Revolution
Theory and Experiment in the Quantum-Relativity Revolution expanded version of lecture presented at American Physical Society meeting, 2/14/10 (Abraham Pais History of Physics Prize for 2009) by Stephen G. Brush* Abstract Does new scientific knowledge come from theory (whose predictions are confirmed by experiment) or from experiment (whose results are explained by theory)? Either can happen, depending on whether theory is ahead of experiment or experiment is ahead of theory at a particular time. In the first case, new theoretical hypotheses are made and their predictions are tested by experiments. But even when the predictions are successful, we can’t be sure that some other hypothesis might not have produced the same prediction. In the second case, as in a detective story, there are already enough facts, but several theories have failed to explain them. When a new hypothesis plausibly explains all of the facts, it may be quickly accepted before any further experiments are done. In the quantum-relativity revolution there are examples of both situations. Because of the two-stage development of both relativity (“special,” then “general”) and quantum theory (“old,” then “quantum mechanics”) in the period 1905-1930, we can make a double comparison of acceptance by prediction and by explanation. A curious anti- symmetry is revealed and discussed. _____________ *Distinguished University Professor (Emeritus) of the History of Science, University of Maryland. Home address: 108 Meadowlark Terrace, Glen Mills, PA 19342. Comments welcome. 1 “Science walks forward on two feet, namely theory and experiment. ... Sometimes it is only one foot which is put forward first, sometimes the other, but continuous progress is only made by the use of both – by theorizing and then testing, or by finding new relations in the process of experimenting and then bringing the theoretical foot up and pushing it on beyond, and so on in unending alterations.” Robert A. -
15.1 Excited State Processes
15.1 Excited State Processes • both optical and dark processes are described in order to develop a kinetic picture of the excited state • the singlet-triplet split and Stoke's shift determine the wavelengths of emission • the fluorescence quantum yield and lifetime depend upon the relative rates of optical and dark processes • excited states can be quenched by other molecules in the solution 15.1 : 1/8 Excited State Processes Involving Light • absorption occurs over one cycle of light, i.e. 10-14 to 10-15 s • fluorescence is spin allowed and occurs over a time scale of 10-9 to 10-7 s • in fluid solution, fluorescence comes from the lowest energy singlet state S2 •the shortest wavelength in the T2 fluorescence spectrum is the longest S1 wavelength in the absorption spectrum T1 • triplet states lie at lower energy than their corresponding singlet states • phosphorescence is spin forbidden and occurs over a time scale of 10-3 to 1 s • you can estimate where spectral features will be located by assuming that S0 absorption, fluorescence and phosphorescence occur one color apart - thus a yellow solution absorbs in the violet, fluoresces in the blue and phosphoresces in the green 15.1 : 2/8 Excited State Dark Processes • excess vibrational energy can be internal conversion transferred to the solvent with very few S2 -13 -11 vibrations (10 to 10 s) - this T2 process is called vibrational relaxation S1 • a molecule in v = 0 of S2 can convert T1 iso-energetically to a higher vibrational vibrational relaxation intersystem level of S1 - this is called -
How the Many Worlds Interpretation Brings Common Sense to Paradoxical Quantum Experiments
How the Many Worlds Interpretation brings Common Sense to Paradoxical Quantum Experiments Kelvin J. McQueen and Lev Vaidman The many worlds interpretation of quantum mechanics (MWI) states that the world we live in is just one among many parallel worlds. It is widely believed that because of this commitment to parallel worlds, the MWI violates common sense. Some go so far as to reject the MWI on this basis. This is despite its myriad of advantages to physics (e.g. consistency with relativity theory, mathematical simplicity, realism, determinism, etc.). Here, we make the case that common sense in fact favors the MWI. We argue that causal explanations are commonsensical only when they are local causal explanations. We present several quantum mechanical experiments that seem to exhibit nonlocal “action at a distance”. Under the assumption that only one world exists, these experiments seem immune to local causal explanation. However, we show that the MWI, by taking all worlds together, can provide local causal explanations of the experiments. The MWI therefore restores common sense to physical explanation. 1. Introduction The purpose of explanation is to help us understand why things happen as they do. Typically, one explains an event by citing the causes of that event. For example, to explain why the window broke, one might cite the throwing of a stone. A particularly compelling and natural type of causal explanation is local causal explanation. In a local causal explanation there is spatiotemporal continuity between the cause and the explained effect. For example, the thrown stone explains the broken window in part because the stone traces out a continuous trajectory from the stone-thrower’s hand to the window. -
Lev VAIDMAN Date & Place of Birth
Name: Lev VAIDMAN Date & Place of Birth: 4 Sept 1955, Leningrad, Russia Date of arrival to Israel: 24 Sept 1973 Israeli Military Service: 10.12.77 { 10.03.80, officer of the IDF Marital Status: Married, 2 children A. Education 1972-1973 Leningrad University Physics 1973-1974 Ben-Gurion University Physics 1974-1976 Hebrew University Physics & Mathematics B.Sc. 1976 1980-1982 Weizmann Institute Physics M.Sc. 1982 1982-1987 Tel-Aviv University Physics Ph.D. 1987 Title of Master's thesis: Nonperturbative Sum Rules in Quantum Chromo-Dynamics Name of Supervisor: Prof. Hector Rubinstein Title of Doctoral Dissertation: The Problem of Measurement in Relativistic Quantum Theory Name of Supervisor: Prof. Yakir Aharonov B. Academic and professional experience 2005 - present U. of Tel-Aviv Full Professor 1998-2005 U. of Tel-Aviv Associate Professor 2000-2001 Centre for Quantum Comp., Oxford U-ty Visiting Professor 1995 - 1998 U. of Tel-Aviv Senior Lecturer 1998 April -June Institute of Advanced Studies, Jerusalem Visiting Professor 1996 July -August ITP, U. of California, Santa-Barbara Visiting Professor 1996 February-March Inst. of Exp. Physics, U. of Innsbruck Visiting Professor 1990 - 1995 U. of Tel-Aviv Senior Research Associate 1987 - 1990 U. of South Carolina Assistant Visiting Professor 1986 - 1987 U. of Tel-Aviv Senior Teaching Assistant 1 C. Active participation in scientific meetings. 1985 Microphysical Reality and Quantum Formalism, Urbino, Italy. 1986 New Techniques and Ideas in Quantum Measurements Theory, New-York. 1987 Frontiers of the Philosophy of Quantum Mechanics, Philadelphia. 1988 Bell's Theorem, Quantum Theory and Conceptions of the Universe, Fairfax, MD.