Theoretical Studies of Ground and Excited State Reactivity
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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Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 28 August 2018 doi:10.20944/preprints201808.0475.v1 Peer-reviewed version available at Entropy 2018, 20, 780; doi:10.3390/e20100780 Article Bouncing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. -
A Chemical Kinetics Network for Lightning and Life in Planetary Atmospheres P
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by St Andrews Research Repository The Astrophysical Journal Supplement Series, 224:9 (33pp), 2016 May doi:10.3847/0067-0049/224/1/9 © 2016. The American Astronomical Society. All rights reserved. A CHEMICAL KINETICS NETWORK FOR LIGHTNING AND LIFE IN PLANETARY ATMOSPHERES P. B. Rimmer and Ch Helling School of Physics and Astronomy, University of St Andrews, St Andrews, KY16 9SS, UK; [email protected] Received 2015 June 3; accepted 2015 October 22; published 2016 May 23 ABSTRACT There are many open questions about prebiotic chemistry in both planetary and exoplanetary environments. The increasing number of known exoplanets and other ultra-cool, substellar objects has propelled the desire to detect life andprebiotic chemistry outside the solar system. We present an ion–neutral chemical network constructed from scratch, STAND2015, that treats hydrogen, nitrogen, carbon, and oxygen chemistry accurately within a temperature range between 100 and 30,000 K. Formation pathways for glycine and other organic molecules are included. The network is complete up to H6C2N2O3. STAND2015 is successfully tested against atmospheric chemistry models for HD 209458b, Jupiter, and the present-day Earth using a simple one- dimensionalphotochemistry/diffusion code. Our results for the early Earth agree with those of Kasting for CO2,H2, CO, and O2, but do not agree for water and atomic oxygen. We use the network to simulate an experiment where varied chemical initial conditions are irradiated by UV light. The result from our simulation is that more glycine is produced when more ammonia and methane is present. -
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. -
An Improved Potential Energy Surface for the H2cl System and Its Use for Calculations of Rate Coefficients and Kinetic Isotope Effects Thomas C
Chemistry Publications Chemistry 8-1996 An Improved Potential Energy Surface for the H2Cl System and Its Use for Calculations of Rate Coefficients and Kinetic Isotope Effects Thomas C. Allison University of Minnesota - Twin Cities Gillian C. Lynch University of Minnesota - Twin Cities Donald G. Truhlar University of Minnesota - Twin Cities Mark S. Gordon Iowa State University, [email protected] Follow this and additional works at: http://lib.dr.iastate.edu/chem_pubs Part of the Chemistry Commons The ompc lete bibliographic information for this item can be found at http://lib.dr.iastate.edu/ chem_pubs/327. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. This Article is brought to you for free and open access by the Chemistry at Iowa State University Digital Repository. It has been accepted for inclusion in Chemistry Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. An Improved Potential Energy Surface for the H2Cl System and Its Use for Calculations of Rate Coefficients and Kinetic Isotope Effects Abstract We present a new potential energy surface (called G3) for the chemical reaction Cl + H2 → HCl + H. The new surface is based on a previous potential surface called GQQ, and it incorporates an improved bending potential that is fit ot the results of ab initio electronic structure calculations. Calculations based on variational transition state theory with semiclassical transmission coefficients corresponding to an optimized multidimensional tunneling treatment (VTST/OMT, in particular improved canonical variational theory with least-action ground-state transmission coefficients) are carried out for nine different isotopomeric versions of the abstraction reaction and six different isotopomeric versions of the exchange reaction involving the H, D, and T isotopes of hydrogen, and the new surface is tested by comparing these calculations to available experimental data. -
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. -
House Fly Attractants and Arrestante: Screening of Chemicals Possessing Cyanide, Thiocyanate, Or Isothiocyanate Radicals
House Fly Attractants and Arrestante: Screening of Chemicals Possessing Cyanide, Thiocyanate, or Isothiocyanate Radicals Agriculture Handbook No. 403 Agricultural Research Service UNITED STATES DEPARTMENT OF AGRICULTURE Contents Page Methods 1 Results and discussion 3 Thiocyanic acid esters 8 Straight-chain nitriles 10 Propionitrile derivatives 10 Conclusions 24 Summary 25 Literature cited 26 This publication reports research involving pesticides. It does not contain recommendations for their use, nor does it imply that the uses discussed here have been registered. All uses of pesticides must be registered by appropriate State and Federal agencies before they can be recommended. CAUTION: Pesticides can be injurious to humans, domestic animals, desirable plants, and fish or other wildlife—if they are not handled or applied properly. Use all pesticides selectively and carefully. Follow recommended practices for the disposal of surplus pesticides and pesticide containers. ¿/áepé4áaUÁí^a¡eé —' ■ -"" TMK LABIL Mention of a proprietary product in this publication does not constitute a guarantee or warranty by the U.S. Department of Agriculture over other products not mentioned. Washington, D.C. Issued July 1971 For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price 25 cents House Fly Attractants and Arrestants: Screening of Chemicals Possessing Cyanide, Thiocyanate, or Isothiocyanate Radicals BY M. S. MAYER, Entomology Research Division, Agricultural Research Service ^ Few chemicals possessing cyanide (-CN), thio- cyanate was slightly attractive to Musca domes- eyanate (-SCN), or isothiocyanate (~NCS) radi- tica, but it was considered to be one of the better cals have been tested as attractants for the house repellents for Phormia regina (Meigen). -
Molecular Excitation in the Interstellar Medium: Recent Advances In
Molecular excitation in the Interstellar Medium: recent advances in collisional, radiative and chemical processes Evelyne Roueff∗,† and François Lique∗,‡ Laboratoire Univers et Théories, Observatoire de Paris, 92190, Meudon, France, and LOMC - UMR 6294, CNRS-Université du Havre, 25 rue Philippe Lebon, BP 540, 76058, Le Havre, France E-mail: [email protected]; [email protected] Contents 1 Introduction 3 2 Collisional excitation 7 2.1 Methods . 9 2.1.1 Theory . 9 2.1.2 Potential energy surfaces . 10 2.1.3 Scattering Calculations . 13 arXiv:1310.8259v1 [physics.chem-ph] 30 Oct 2013 2.1.4 Experiments . 17 2.2 H2, CO and H2O molecules as benchmark systems . 19 2.2.1 H2 ....................................... 19 ∗To whom correspondence should be addressed †Laboratoire Univers et Théories, Observatoire de Paris, 92190, Meudon, France ‡LOMC - UMR 6294, CNRS-Université du Havre, 25 rue Philippe Lebon, BP 540, 76058, Le Havre, France 1 2.2.2 CO . 22 2.2.3 H2O...................................... 27 2.3 Other recent results . 30 2.3.1 CN / HCN / HNC . 31 2.3.2 CS / SiO / SiS / SO / SO2 .......................... 34 2.3.3 NH3 /NH................................... 38 2.3.4 O2 / OH / NO . 40 2.3.5 C2 /C2H/C3 /C4 / HC3N.......................... 41 2.3.6 Complex Organic Molecules : H2CO / HCOOCH3 / CH3OH . 45 + + + + + 2.3.7 Cations : CH / SiH / HCO /N2H / HOCO . 49 + 2.3.8 H3 ...................................... 52 − − 2.3.9 Anions : CN /C2H ............................ 52 2.4 Isotopologues . 55 3 Radiative and chemical excitation 57 3.1 Radiative effects . 57 3.2 Chemical effects . 59 3.3 Examples . -
8 Finding Transition States and Barrier Heights: First Order Saddle Points
8 Finding Transition States and Barrier Heights: First Order Saddle Points So far, you have already considered several systems that did not reside in a minimum of the potential energy surface. Both the bond breaking in the dissociation of H2 as well as the internal rotation of butane were examples of off-equilibrium processes, where the system moves from one equilibrium conformer, i.e. from one minimum, to another. It is intuitively clear that the likelihood of two equilibrium conformers to be transformed into each other by moving along some random path is negligible, and different methods have been established tofind reasonable paths for such transitions. A frequently used approach is transition state theory, which is concerned withfinding a realistic path be- tween potential energy minima. In this set of exercises, you will search for the transition state for a the cyclisation of a deprotonated chloropropanol to propylene oxide, and you willfind the minimum energy path that connects reactants to products via this transition state. 8.1 Synthesis of Propylene Oxide Corrosive propylene oxide (cf.figure 1) is the simplest chiral epoxide and of great indus- trial importance. The historically most important synthesis is based on hydrochlorination and ring closure of the resulting chloropropanols in a basic environment: The resulting epoxide is most often directly used as a racemate. a) What kind of reaction mechanism would you expect? Would you expect the stere- ochemistry at the chiral carbon to be preserved? Identify the chirality centre in the chloropropanoate and the product epoxide. b) Suggest possible transition state structues. Figure 1: Propylene oxide is a toxic, cancerogenic liquid that is produced on large indus- trial scale.