22.51 Course Notes, Chapter 9: Harmonic Oscillator
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1 Notation for States
The S-Matrix1 D. E. Soper2 University of Oregon Physics 634, Advanced Quantum Mechanics November 2000 1 Notation for states In these notes we discuss scattering nonrelativistic quantum mechanics. We will use states with the nonrelativistic normalization h~p |~ki = (2π)3δ(~p − ~k). (1) Recall that in a relativistic theory there is an extra factor of 2E on the right hand side of this relation, where E = [~k2 + m2]1/2. We will use states in the “Heisenberg picture,” in which states |ψ(t)i do not depend on time. Often in quantum mechanics one uses the Schr¨odinger picture, with time dependent states |ψ(t)iS. The relation between these is −iHt |ψ(t)iS = e |ψi. (2) Thus these are the same at time zero, and the Schrdinger states obey d i |ψ(t)i = H |ψ(t)i (3) dt S S In the Heisenberg picture, the states do not depend on time but the op- erators do depend on time. A Heisenberg operator O(t) is related to the corresponding Schr¨odinger operator OS by iHt −iHt O(t) = e OS e (4) Thus hψ|O(t)|ψi = Shψ(t)|OS|ψ(t)iS. (5) The Heisenberg picture is favored over the Schr¨odinger picture in the case of relativistic quantum mechanics: we don’t have to say which reference 1Copyright, 2000, D. E. Soper [email protected] 1 frame we use to define t in |ψ(t)iS. For operators, we can deal with local operators like, for instance, the electric field F µν(~x, t). -
An Introduction to Quantum Field Theory
AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ........................... -
Rotation and Spin and Position Operators in Relativistic Gravity and Quantum Electrodynamics
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 4 December 2019 doi:10.20944/preprints201912.0044.v1 Peer-reviewed version available at Universe 2020, 6, 24; doi:10.3390/universe6020024 1 Review 2 Rotation and Spin and Position Operators in 3 Relativistic Gravity and Quantum Electrodynamics 4 R.F. O’Connell 5 Department of Physics and Astronomy, Louisiana State University; Baton Rouge, LA 70803-4001, USA 6 Correspondence: [email protected]; 225-578-6848 7 Abstract: First, we examine how spin is treated in special relativity and the necessity of introducing 8 spin supplementary conditions (SSC) and how they are related to the choice of a center-of-mass of a 9 spinning particle. Next, we discuss quantum electrodynamics and the Foldy-Wouthuysen 10 transformation which we note is a position operator identical to the Pryce-Newton-Wigner position 11 operator. The classical version of the operators are shown to be essential for the treatment of 12 classical relativistic particles in general relativity, of special interest being the case of binary 13 systems (black holes/neutron stars) which emit gravitational radiation. 14 Keywords: rotation; spin; position operators 15 16 I.Introduction 17 Rotation effects in relativistic systems involve many new concepts not needed in 18 non-relativistic classical physics. Some of these are quantum mechanical (where the emphasis is on 19 “spin”). Thus, our emphasis will be on quantum electrodynamics (QED) and both special and 20 general relativity. Also, as in [1] , we often use “spin” in the generic sense of meaning “internal” spin 21 in the case of an elementary particle and “rotation” in the case of an elementary particle. -
Configuration Interaction Study of the Ground State of the Carbon Atom
Configuration Interaction Study of the Ground State of the Carbon Atom María Belén Ruiz* and Robert Tröger Department of Theoretical Chemistry Friedrich-Alexander-University Erlangen-Nürnberg Egerlandstraße 3, 91054 Erlangen, Germany In print in Advances Quantum Chemistry Vol. 76: Novel Electronic Structure Theory: General Innovations and Strongly Correlated Systems 30th July 2017 Abstract Configuration Interaction (CI) calculations on the ground state of the C atom are carried out using a small basis set of Slater orbitals [7s6p5d4f3g]. The configurations are selected according to their contribution to the total energy. One set of exponents is optimized for the whole expansion. Using some computational techniques to increase efficiency, our computer program is able to perform partially-parallelized runs of 1000 configuration term functions within a few minutes. With the optimized computer programme we were able to test a large number of configuration types and chose the most important ones. The energy of the 3P ground state of carbon atom with a wave function of angular momentum L=1 and ML=0 and spin eigenfunction with S=1 and MS=0 leads to -37.83526523 h, which is millihartree accurate. We discuss the state of the art in the determination of the ground state of the carbon atom and give an outlook about the complex spectra of this atom and its low-lying states. Keywords: Carbon atom; Configuration Interaction; Slater orbitals; Ground state *Corresponding author: e-mail address: [email protected] 1 1. Introduction The spectrum of the isolated carbon atom is the most complex one among the light atoms. The ground state of carbon atom is a triplet 3P state and its low-lying excited states are singlet 1D, 1S and 1P states, more stable than the corresponding triplet excited ones 3D and 3S, against the Hund’s rule of maximal multiplicity. -
Quantum Field Theory*
Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement. -
Discrete-Continuous and Classical-Quantum
Under consideration for publication in Math. Struct. in Comp. Science Discrete-continuous and classical-quantum T. Paul C.N.R.S., D.M.A., Ecole Normale Sup´erieure Received 21 November 2006 Abstract A discussion concerning the opposition between discretness and continuum in quantum mechanics is presented. In particular this duality is shown to be present not only in the early days of the theory, but remains actual, featuring different aspects of discretization. In particular discreteness of quantum mechanics is a key-stone for quantum information and quantum computation. A conclusion involving a concept of completeness linking dis- creteness and continuum is proposed. 1. Discrete-continuous in the old quantum theory Discretness is obviously a fundamental aspect of Quantum Mechanics. When you send white light, that is a continuous spectrum of colors, on a mono-atomic gas, you get back precise line spectra and only Quantum Mechanics can explain this phenomenon. In fact the birth of quantum physics involves more discretization than discretness : in the famous Max Planck’s paper of 1900, and even more explicitly in the 1905 paper by Einstein about the photo-electric effect, what is really done is what a computer does: an integral is replaced by a discrete sum. And the discretization length is given by the Planck’s constant. This idea will be later on applied by Niels Bohr during the years 1910’s to the atomic model. Already one has to notice that during that time another form of discretization had appeared : the atomic model. It is astonishing to notice how long it took to accept the atomic hypothesis (Perrin 1905). -
Forced Mechanical Oscillations
169 Carl von Ossietzky Universität Oldenburg – Faculty V - Institute of Physics Module Introductory laboratory course physics – Part I Forced mechanical oscillations Keywords: HOOKE's law, harmonic oscillation, harmonic oscillator, eigenfrequency, damped harmonic oscillator, resonance, amplitude resonance, energy resonance, resonance curves References: /1/ DEMTRÖDER, W.: „Experimentalphysik 1 – Mechanik und Wärme“, Springer-Verlag, Berlin among others. /2/ TIPLER, P.A.: „Physik“, Spektrum Akademischer Verlag, Heidelberg among others. /3/ ALONSO, M., FINN, E. J.: „Fundamental University Physics, Vol. 1: Mechanics“, Addison-Wesley Publishing Company, Reading (Mass.) among others. 1 Introduction It is the object of this experiment to study the properties of a „harmonic oscillator“ in a simple mechanical model. Such harmonic oscillators will be encountered in different fields of physics again and again, for example in electrodynamics (see experiment on electromagnetic resonant circuit) and atomic physics. Therefore it is very important to understand this experiment, especially the importance of the amplitude resonance and phase curves. 2 Theory 2.1 Undamped harmonic oscillator Let us observe a set-up according to Fig. 1, where a sphere of mass mK is vertically suspended (x-direc- tion) on a spring. Let us neglect the effects of friction for the moment. When the sphere is at rest, there is an equilibrium between the force of gravity, which points downwards, and the dragging resilience which points upwards; the centre of the sphere is then in the position x = 0. A deflection of the sphere from its equilibrium position by x causes a proportional dragging force FR opposite to x: (1) FxR ∝− The proportionality constant (elastic or spring constant or directional quantity) is denoted D, and Eq. -
Finally Making Sense of the Double-Slit Experiment
Finally making sense of the double-slit experiment Yakir Aharonova,b,c,1, Eliahu Cohend,1,2, Fabrizio Colomboe, Tomer Landsbergerc,2, Irene Sabadinie, Daniele C. Struppaa,b, and Jeff Tollaksena,b aInstitute for Quantum Studies, Chapman University, Orange, CA 92866; bSchmid College of Science and Technology, Chapman University, Orange, CA 92866; cSchool of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel; dH. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom; and eDipartimento di Matematica, Politecnico di Milano, 9 20133 Milan, Italy Contributed by Yakir Aharonov, March 20, 2017 (sent for review September 26, 2016; reviewed by Pawel Mazur and Neil Turok) Feynman stated that the double-slit experiment “. has in it the modular momentum operator will arise as particularly signifi- heart of quantum mechanics. In reality, it contains the only mys- cant in explaining interference phenomena. This approach along tery” and that “nobody can give you a deeper explanation of with the use of Heisenberg’s unitary equations of motion intro- this phenomenon than I have given; that is, a description of it” duce a notion of dynamical nonlocality. Dynamical nonlocality [Feynman R, Leighton R, Sands M (1965) The Feynman Lectures should be distinguished from the more familiar kinematical non- on Physics]. We rise to the challenge with an alternative to the locality [implicit in entangled states (10) and previously ana- wave function-centered interpretations: instead of a quantum lyzed in the Heisenberg picture by Deutsch and Hayden (11)], wave passing through both slits, we have a localized particle with because dynamical nonlocality has observable effects on prob- nonlocal interactions with the other slit. -
The Damped Harmonic Oscillator
THE DAMPED HARMONIC OSCILLATOR Reading: Main 3.1, 3.2, 3.3 Taylor 5.4 Giancoli 14.7, 14.8 Free, undamped oscillators – other examples k m L No friction I C k m q 1 x m!x! = !kx q!! = ! q LC ! ! r; r L = θ Common notation for all g !! 2 T ! " # ! !!! + " ! = 0 m L 0 mg k friction m 1 LI! + q + RI = 0 x C 1 Lq!!+ q + Rq! = 0 C m!x! = !kx ! bx! ! r L = cm θ Common notation for all g !! ! 2 T ! " # ! # b'! !!! + 2"!! +# ! = 0 m L 0 mg Natural motion of damped harmonic oscillator Force = mx˙˙ restoring force + resistive force = mx˙˙ ! !kx ! k Need a model for this. m Try restoring force proportional to velocity k m x !bx! How do we choose a model? Physically reasonable, mathematically tractable … Validation comes IF it describes the experimental system accurately Natural motion of damped harmonic oscillator Force = mx˙˙ restoring force + resistive force = mx˙˙ !kx ! bx! = m!x! ! Divide by coefficient of d2x/dt2 ! and rearrange: x 2 x 2 x 0 !!+ ! ! + " 0 = inverse time β and ω0 (rate or frequency) are generic to any oscillating system This is the notation of TM; Main uses γ = 2β. Natural motion of damped harmonic oscillator 2 x˙˙ + 2"x˙ +#0 x = 0 Try x(t) = Ce pt C, p are unknown constants ! x˙ (t) = px(t), x˙˙ (t) = p2 x(t) p2 2 p 2 x(t) 0 Substitute: ( + ! + " 0 ) = ! 2 2 Now p is known (and p = !" ± " ! # 0 there are 2 p values) p t p t x(t) = Ce + + C'e " Must be sure to make x real! ! Natural motion of damped HO Can identify 3 cases " < #0 underdamped ! " > #0 overdamped ! " = #0 critically damped time ---> ! underdamped " < #0 # 2 !1 = ! 0 1" 2 ! 0 ! time ---> 2 2 p = !" ± " ! # 0 = !" ± i#1 x(t) = Ce"#t+i$1t +C*e"#t"i$1t Keep x(t) real "#t x(t) = Ae [cos($1t +%)] complex <-> amp/phase System oscillates at "frequency" ω1 (very close to ω0) ! - but in fact there is not only one single frequency associated with the motion as we will see. -
1 the LOCALIZED QUANTUM VACUUM FIELD D. Dragoman
1 THE LOCALIZED QUANTUM VACUUM FIELD D. Dragoman – Univ. Bucharest, Physics Dept., P.O. Box MG-11, 077125 Bucharest, Romania, e-mail: [email protected] ABSTRACT A model for the localized quantum vacuum is proposed in which the zero-point energy of the quantum electromagnetic field originates in energy- and momentum-conserving transitions of material systems from their ground state to an unstable state with negative energy. These transitions are accompanied by emissions and re-absorptions of real photons, which generate a localized quantum vacuum in the neighborhood of material systems. The model could help resolve the cosmological paradox associated to the zero-point energy of electromagnetic fields, while reclaiming quantum effects associated with quantum vacuum such as the Casimir effect and the Lamb shift; it also offers a new insight into the Zitterbewegung of material particles. 2 INTRODUCTION The zero-point energy (ZPE) of the quantum electromagnetic field is at the same time an indispensable concept of quantum field theory and a controversial issue (see [1] for an excellent review of the subject). The need of the ZPE has been recognized from the beginning of quantum theory of radiation, since only the inclusion of this term assures no first-order temperature-independent correction to the average energy of an oscillator in thermal equilibrium with blackbody radiation in the classical limit of high temperatures. A more rigorous introduction of the ZPE stems from the treatment of the electromagnetic radiation as an ensemble of harmonic quantum oscillators. Then, the total energy of the quantum electromagnetic field is given by E = åk,s hwk (nks +1/ 2) , where nks is the number of quantum oscillators (photons) in the (k,s) mode that propagate with wavevector k and frequency wk =| k | c = kc , and are characterized by the polarization index s. -
VIBRATIONAL SPECTROSCOPY • the Vibrational Energy V(R) Can Be Calculated Using the (Classical) Model of the Harmonic Oscillator
VIBRATIONAL SPECTROSCOPY • The vibrational energy V(r) can be calculated using the (classical) model of the harmonic oscillator: • Using this potential energy function in the Schrödinger equation, the vibrational frequency can be calculated: The vibrational frequency is increasing with: • increasing force constant f = increasing bond strength • decreasing atomic mass • Example: f cc > f c=c > f c-c Vibrational spectra (I): Harmonic oscillator model • Infrared radiation in the range from 10,000 – 100 cm –1 is absorbed and converted by an organic molecule into energy of molecular vibration –> this absorption is quantized: A simple harmonic oscillator is a mechanical system consisting of a point mass connected to a massless spring. The mass is under action of a restoring force proportional to the displacement of particle from its equilibrium position and the force constant f (also k in followings) of the spring. MOLECULES I: Vibrational We model the vibrational motion as a harmonic oscillator, two masses attached by a spring. nu and vee! Solving the Schrödinger equation for the 1 v h(v 2 ) harmonic oscillator you find the following quantized energy levels: v 0,1,2,... The energy levels The level are non-degenerate, that is gv=1 for all values of v. The energy levels are equally spaced by hn. The energy of the lowest state is NOT zero. This is called the zero-point energy. 1 R h Re 0 2 Vibrational spectra (III): Rotation-vibration transitions The vibrational spectra appear as bands rather than lines. When vibrational spectra of gaseous diatomic molecules are observed under high-resolution conditions, each band can be found to contain a large number of closely spaced components— band spectra. -
Chapter 5 ANGULAR MOMENTUM and ROTATIONS
Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another.