8. Atoms in Electromagnetic Fields
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Quantum Theory of the Hydrogen Atom
Quantum Theory of the Hydrogen Atom Chemistry 35 Fall 2000 Balmer and the Hydrogen Spectrum n 1885: Johann Balmer, a Swiss schoolteacher, empirically deduced a formula which predicted the wavelengths of emission for Hydrogen: l (in Å) = 3645.6 x n2 for n = 3, 4, 5, 6 n2 -4 •Predicts the wavelengths of the 4 visible emission lines from Hydrogen (which are called the Balmer Series) •Implies that there is some underlying order in the atom that results in this deceptively simple equation. 2 1 The Bohr Atom n 1913: Niels Bohr uses quantum theory to explain the origin of the line spectrum of hydrogen 1. The electron in a hydrogen atom can exist only in discrete orbits 2. The orbits are circular paths about the nucleus at varying radii 3. Each orbit corresponds to a particular energy 4. Orbit energies increase with increasing radii 5. The lowest energy orbit is called the ground state 6. After absorbing energy, the e- jumps to a higher energy orbit (an excited state) 7. When the e- drops down to a lower energy orbit, the energy lost can be given off as a quantum of light 8. The energy of the photon emitted is equal to the difference in energies of the two orbits involved 3 Mohr Bohr n Mathematically, Bohr equated the two forces acting on the orbiting electron: coulombic attraction = centrifugal accelleration 2 2 2 -(Z/4peo)(e /r ) = m(v /r) n Rearranging and making the wild assumption: mvr = n(h/2p) n e- angular momentum can only have certain quantified values in whole multiples of h/2p 4 2 Hydrogen Energy Levels n Based on this model, Bohr arrived at a simple equation to calculate the electron energy levels in hydrogen: 2 En = -RH(1/n ) for n = 1, 2, 3, 4, . -
Electric Dipoles in Atoms and Molecules and the Stark Effect Publicação IF – 1663/2011
Electric Dipoles in Atoms and Molecules and the Stark Effect Instituto de Física, Universidade de São Paulo, CP 66.318 05315-970, São Paulo, SP, Brasil Publicação IF – 1663/2011 21/06/2011 Electric Dipoles of Atoms and Molecules and the Stark Effect M. Cattani Instituto de Fisica, Universidade de S. Paulo, C.P. 66318, CEP 05315−970 S. Paulo, S.P. Brazil . E−mail: [email protected] Abstract. This article was written for undergraduate and postgraduate students of physics. We analyze the electric dipole moments (EDM) of atoms and molecules when they are isolated and when are placed in a uniform static electric field. This was done this because usually in text books and articles this separation is not clearly displayed. Key words: Electric dipole moments of atoms and molecules; Stark effect . (I) Introduction . Our goal is to write an article for undergraduate and postgraduate students of physics to study the electric dipole moments (EDM) of atoms and molecules when they are isolated and when they are placed in a uniform static electric field . We have done this because many times in text books and articles this separation is not clearly displayed. The dipole of an isolated system will be named natural or permanent EDM and that one which is generated by an external field will be named induced EDM. So, we begin recalling the definition of EDM adopted in basic physics courses 1−4 for an isolated aggregate of charges. If in a given system positive charges + q and negative −q are concentrated at different points we say that it has an EDM. -
Principal, Azimuthal and Magnetic Quantum Numbers and the Magnitude of Their Values
268 A Textbook of Physical Chemistry – Volume I Principal, Azimuthal and Magnetic Quantum Numbers and the Magnitude of Their Values The Schrodinger wave equation for hydrogen and hydrogen-like species in the polar coordinates can be written as: 1 휕 휕휓 1 휕 휕휓 1 휕2휓 8휋2휇 푍푒2 (406) [ (푟2 ) + (푆푖푛휃 ) + ] + (퐸 + ) 휓 = 0 푟2 휕푟 휕푟 푆푖푛휃 휕휃 휕휃 푆푖푛2휃 휕휙2 ℎ2 푟 After separating the variables present in the equation given above, the solution of the differential equation was found to be 휓푛,푙,푚(푟, 휃, 휙) = 푅푛,푙. 훩푙,푚. 훷푚 (407) 2푍푟 푘 (408) 3 푙 푘=푛−푙−1 (−1)푘+1[(푛 + 푙)!]2 ( ) 2푍 (푛 − 푙 − 1)! 푍푟 2푍푟 푛푎 √ 0 = ( ) [ 3] . exp (− ) . ( ) . ∑ 푛푎0 2푛{(푛 + 푙)!} 푛푎0 푛푎0 (푛 − 푙 − 1 − 푘)! (2푙 + 1 + 푘)! 푘! 푘=0 (2푙 + 1)(푙 − 푚)! 1 × √ . 푃푚(퐶표푠 휃) × √ 푒푖푚휙 2(푙 + 푚)! 푙 2휋 It is obvious that the solution of equation (406) contains three discrete (n, l, m) and three continuous (r, θ, ϕ) variables. In order to be a well-behaved function, there are some conditions over the values of discrete variables that must be followed i.e. boundary conditions. Therefore, we can conclude that principal (n), azimuthal (l) and magnetic (m) quantum numbers are obtained as a solution of the Schrodinger wave equation for hydrogen atom; and these quantum numbers are used to define various quantum mechanical states. In this section, we will discuss the properties and significance of all these three quantum numbers one by one. Principal Quantum Number The principal quantum number is denoted by the symbol n; and can have value 1, 2, 3, 4, 5…..∞. -
Further Quantum Physics
Further Quantum Physics Concepts in quantum physics and the structure of hydrogen and helium atoms Prof Andrew Steane January 18, 2005 2 Contents 1 Introduction 7 1.1 Quantum physics and atoms . 7 1.1.1 The role of classical and quantum mechanics . 9 1.2 Atomic physics—some preliminaries . .... 9 1.2.1 Textbooks...................................... 10 2 The 1-dimensional projectile: an example for revision 11 2.1 Classicaltreatment................................. ..... 11 2.2 Quantum treatment . 13 2.2.1 Mainfeatures..................................... 13 2.2.2 Precise quantum analysis . 13 3 Hydrogen 17 3.1 Some semi-classical estimates . 17 3.2 2-body system: reduced mass . 18 3.2.1 Reduced mass in quantum 2-body problem . 19 3.3 Solution of Schr¨odinger equation for hydrogen . ..... 20 3.3.1 General features of the radial solution . 21 3.3.2 Precisesolution.................................. 21 3.3.3 Meanradius...................................... 25 3.3.4 How to remember hydrogen . 25 3.3.5 Mainpoints.................................... 25 3.3.6 Appendix on series solution of hydrogen equation, off syllabus . 26 3 4 CONTENTS 4 Hydrogen-like systems and spectra 27 4.1 Hydrogen-like systems . 27 4.2 Spectroscopy ........................................ 29 4.2.1 Main points for use of grating spectrograph . ...... 29 4.2.2 Resolution...................................... 30 4.2.3 Usefulness of both emission and absorption methods . 30 4.3 The spectrum for hydrogen . 31 5 Introduction to fine structure and spin 33 5.1 Experimental observation of fine structure . ..... 33 5.2 TheDiracresult ..................................... 34 5.3 Schr¨odinger method to account for fine structure . 35 5.4 Physical nature of orbital and spin angular momenta . -
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. -
22.51 Course Notes, Chapter 11: Perturbation Theory
11. Perturbation Theory 11.1 Time-independent perturbation theory 11.1.1 Non-degenerate case 11.1.2 Degenerate case 11.1.3 The Stark effect 11.2 Time-dependent perturbation theory 11.2.1 Review of interaction picture 11.2.2 Dyson series 11.2.3 Fermi’s Golden Rule 11.1 Time-independent perturbation theory Because of the complexity of many physical problems, very few can be solved exactly (unless they involve only small Hilbert spaces). In particular, to analyze the interaction of radiation with matter we will need to develop approximation methods36 . 11.1.1 Non-degenerate case We have an Hamiltonian = + ǫV H H0 where we know the eigenvalue of the unperturbed Hamiltonian and we want to solve for the perturbed case H0 = 0 + ǫV , in terms of an expansion in ǫ (with ǫ varying between 0 and 1). The solution for ǫ 1 is the desired Hsolution. H → We assume that we know exactly the energy eigenkets and eigenvalues of : H0 k = E(0) k H0 | ) k | ) As is hermitian, its eigenkets form a complete basis k k = 11. We assume at first that the energy spectrum H0 k | )( | is not degenerate (that is, all the E(0) are different, in the next section we will study the degenerate case). The k L eigensystem for the total hamiltonian is then ( + ǫV ) ϕ = E (ǫ) ϕ H0 | k)ǫ k | k)ǫ where ǫ = 1 is the case we are interested in, but we will solve for a general ǫ as a perturbation in this parameter: (0) (1) 2 (2) (0) (1) 2 (2) ϕ = ϕ + ǫ ϕ + ǫ ϕ + ..., E = E + ǫE + ǫ E + .. -
1.2 Formulation of the Schrödinger Equation for the Hydrogen Atom
2 1 Hydrogen orbitals 1.2 Formulation of the Schrodinger¨ equation for the hydrogen atom In this initial treatment, we will make some practical approximations and simplifi- cations. Since we are for the moment only trying to establish the general form of the hydrogenic wave functions, this will suffice. To start with, we will assume that the nucleus has zero extension. We place the origin at its position, and we ignore the centre-of-mass motion. This reduces the two-body problem to a single particle, the electron, moving in a central-field potential. To take the finite mass of the nu- cleus into account, we replace the electron mass with the reduced mass, m, of the two-body problem. Moreover, we will in this chapter ignore the effect on the wave function of relativistic effects, which automatically implies that we ignore the spins of the electron and of the nucleus. This makes us ready to formulate the Hamilto- nian. The potential is the classical Coulomb interaction between two particles of op- posite charges. With spherical coordinates, and with r as the radial distance of the electron from the origin, this is: Ze2 V(r) = − ; (1.1) 4pe0 r with Z being the charge state of the nucleus. The Schrodinger¨ equation is: h¯ 2 − ∇2y(r) +V(r)y = Ey(r) ; (1.2) 2m where the Laplacian in spherical coordinates is: 1 ¶ ¶ 1 ¶ ¶ 1 ¶ 2 ∇2 = r2 + sinq + : (1.3) r2 ¶r ¶r r2 sinq ¶q ¶q r2 sin2 q ¶j2 Since the potential is purely central, the solution to (1.2) can be factorised into a radial and an angular part, y(r;q;j) = R(r)Y(q;j). -
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. -
The Principal Quantum Number the Azimuthal Quantum Number The
To completely describe an electron in an atom, four quantum numbers are needed: energy (n), angular momentum (ℓ), magnetic moment (mℓ), and spin (ms). The Principal Quantum Number This quantum number describes the electron shell or energy level of an atom. The value of n ranges from 1 to the shell containing the outermost electron of that atom. For example, in caesium (Cs), the outermost valence electron is in the shell with energy level 6, so an electron incaesium can have an n value from 1 to 6. For particles in a time-independent potential, as per the Schrödinger equation, it also labels the nth eigen value of Hamiltonian (H). This number has a dependence only on the distance between the electron and the nucleus (i.e. the radial coordinate r). The average distance increases with n, thus quantum states with different principal quantum numbers are said to belong to different shells. The Azimuthal Quantum Number The angular or orbital quantum number, describes the sub-shell and gives the magnitude of the orbital angular momentum through the relation. ℓ = 0 is called an s orbital, ℓ = 1 a p orbital, ℓ = 2 a d orbital, and ℓ = 3 an f orbital. The value of ℓ ranges from 0 to n − 1 because the first p orbital (ℓ = 1) appears in the second electron shell (n = 2), the first d orbital (ℓ = 2) appears in the third shell (n = 3), and so on. This quantum number specifies the shape of an atomic orbital and strongly influences chemical bonds and bond angles. -
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.