On the Logic Resolution of the Wave Particle Duality Paradox in Quantum Mechanics (Extended Abstract) M.A
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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. -
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Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 28 August 2018 doi:10.20944/preprints201808.0475.v1 Peer-reviewed version available at Entropy 2018, 20, 780; doi:10.3390/e20100780 Article Bouncing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 Mechanics
Quantum Mechanics Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Introduction 5 1.1 Intendedaudience................................ 5 1.2 MajorSources .................................. 5 1.3 AimofCourse .................................. 6 1.4 OutlineofCourse ................................ 6 2 Probability Theory 7 2.1 Introduction ................................... 7 2.2 WhatisProbability?.............................. 7 2.3 CombiningProbabilities. ... 7 2.4 Mean,Variance,andStandardDeviation . ..... 9 2.5 ContinuousProbabilityDistributions. ........ 11 3 Wave-Particle Duality 13 3.1 Introduction ................................... 13 3.2 Wavefunctions.................................. 13 3.3 PlaneWaves ................................... 14 3.4 RepresentationofWavesviaComplexFunctions . ....... 15 3.5 ClassicalLightWaves ............................. 18 3.6 PhotoelectricEffect ............................. 19 3.7 QuantumTheoryofLight. .. .. .. .. .. .. .. .. .. .. .. .. .. 21 3.8 ClassicalInterferenceofLightWaves . ...... 21 3.9 QuantumInterferenceofLight . 22 3.10 ClassicalParticles . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 25 3.11 QuantumParticles............................... 25 3.12 WavePackets .................................. 26 2 QUANTUM MECHANICS 3.13 EvolutionofWavePackets . 29 3.14 Heisenberg’sUncertaintyPrinciple . ........ 32 3.15 Schr¨odinger’sEquation . 35 3.16 CollapseoftheWaveFunction . 36 4 Fundamentals of Quantum Mechanics 39 4.1 Introduction .................................. -
Zero-Point Energy and Interstellar Travel by Josh Williams
;;;;;;;;;;;;;;;;;;;;;; itself comes from the conversion of electromagnetic Zero-Point Energy and radiation energy into electrical energy, or more speciÞcally, the conversion of an extremely high Interstellar Travel frequency bandwidth of the electromagnetic spectrum (beyond Gamma rays) now known as the zero-point by Josh Williams spectrum. ÒEre many generations pass, our machinery will be driven by power obtainable at any point in the universeÉ it is a mere question of time when men will succeed in attaching their machinery to the very wheel work of nature.Ó ÐNikola Tesla, 1892 Some call it the ultimate free lunch. Others call it absolutely useless. But as our world civilization is quickly coming upon a terrifying energy crisis with As you can see, the wavelengths from this part of little hope at the moment, radical new ideas of usable the spectrum are incredibly small, literally atomic and energy will be coming to the forefront and zero-point sub-atomic in size. And since the wavelength is so energy is one that is currently on its way. small, it follows that the frequency is quite high. The importance of this is that the intensity of the energy So, what the hell is it? derived for zero-point energy has been reported to be Zero-point energy is a type of energy that equal to the cube (the third power) of the frequency. exists in molecules and atoms even at near absolute So obviously, since weÕre already talking about zero temperatures (-273¡C or 0 K) in a vacuum. some pretty high frequencies with this portion of At even fractions of a Kelvin above absolute zero, the electromagnetic spectrum, weÕre talking about helium still remains a liquid, and this is said to be some really high energy intensities. -
Relativistic Quantum Mechanics 1
Relativistic Quantum Mechanics 1 The aim of this chapter is to introduce a relativistic formalism which can be used to describe particles and their interactions. The emphasis 1.1 SpecialRelativity 1 is given to those elements of the formalism which can be carried on 1.2 One-particle states 7 to Relativistic Quantum Fields (RQF), which underpins the theoretical 1.3 The Klein–Gordon equation 9 framework of high energy particle physics. We begin with a brief summary of special relativity, concentrating on 1.4 The Diracequation 14 4-vectors and spinors. One-particle states and their Lorentz transforma- 1.5 Gaugesymmetry 30 tions follow, leading to the Klein–Gordon and the Dirac equations for Chaptersummary 36 probability amplitudes; i.e. Relativistic Quantum Mechanics (RQM). Readers who want to get to RQM quickly, without studying its foun- dation in special relativity can skip the first sections and start reading from the section 1.3. Intrinsic problems of RQM are discussed and a region of applicability of RQM is defined. Free particle wave functions are constructed and particle interactions are described using their probability currents. A gauge symmetry is introduced to derive a particle interaction with a classical gauge field. 1.1 Special Relativity Einstein’s special relativity is a necessary and fundamental part of any Albert Einstein 1879 - 1955 formalism of particle physics. We begin with its brief summary. For a full account, refer to specialized books, for example (1) or (2). The- ory oriented students with good mathematical background might want to consult books on groups and their representations, for example (3), followed by introductory books on RQM/RQF, for example (4). -
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.