Quantization of the E-M field
|
|
- Luke Sanders
- 5 years ago
- Views:
Transcription
1 April 6, 20 Lecture XXVI Quantization of the E-M field 2.0. Electric quadrupole transition If E transitions are forbidden by selection rules, then we consider the next term in the expansion of the spatial dependence of the field operator and the magnetic term in the interaction Hamiltonian. Recall that the e mcp A term in the interaction, that can connect initial and final states with a difference of a single photon f; n kλ H i; n kλ = e f e i(k x) p ɛ kλ i e iωt m 2ωV Expanding the exponent to first order, where k x is the small parameter (long wavelength and small atom) gives us f; n kλ H i; n kλ = e m 2ωV f ( + i(k x))p ɛ kλ i e iωt Typically the first order term contributes only if the zeroth order (dipole) is forbidden. Then f; n kλ H i; n kλ = i e m 2ωV f (k x)p ɛ kλ i e iωt Consider evaluation of the matrix element We can expand f (k x)p ɛ kλ i f (k x)(p ɛ kλ ) i = 2 f (k x)(p ɛ kλ) + (k p)(ɛ kλ x) i + 2 f (k x)(p ɛ kλ) (k p)(ɛ kλ x) i The first term 2 ((k x)(p ɛ kλ) + (k p)(ɛ kλ x)) = 2 k (xp + px) ɛ kλ As before, we can write the operator p = im [H 0, x] so that Finally xp + px = im [H 0, xx]. 2 k f xp + px i ɛ kλ = imω 2 k f xx i ɛ kλ.
2 This transition is the electric quadrupole (E2) transition. Since k ɛ = 0, where k i x i x j ɛ j = k i T ij ɛ j T ij = x i x j δ ij x 2. has zero trace and 5 independent components that can be written as a linear combination of Y2 m (spherical tensor operator). The the WE theorem tells use that the total angular momentum of initial and final states can change by at most 2. Spherical tensor operator Let s examine the tensor operator T ij in more detail. Recall that a cartesian tensor operator can be assembled from cartesian vector operators U and V according to W ij = U i V j As long as U and V transform as vectors under rotations then so will W ij. But the cartesian tensor does not transform irreducibly. We can write U i V j = U V δ ij + 2 (U iv j U j V i ) + 2 (U iv j + U j V i ) U V δ ij The first term transforms as a scalar, the second as a vector (namely U V) and the third term is a X symmetric traceless tensor. The 9 real parameters of the X cartesian tensor correspond to one parameter for the scalar, for the vector and 5 for the symmetric zero trace tensor. We can identify a correspondence between the spherical tensors with spherical harmonics. Y 0 = 4π cos θ = z 4π r T 0 = 4π V z Y ± = 8π sin x ± iy θe±iφ = T± V x ± iv = y 4π 2r 4π 2 ( ) 2 ( ) 2 x ± iy Y 2 ±2 = 2π sin2 θe ±2iφ = T±2 2 Vx ± iv = y 8π 2r 8π 2 Y 2 ± = 8π sin θ cos (x ± iy)z θe±iφ = T 2 (V x ± iv y ) 4π 2r 2 ± = V z 4π 2 Y2 0 = 6π ( (x + iy)/ 2(x iy)/ 2 cos2 θ ) = 6π 2z2 r 2 T0 2 = 6π 2 ( Vz 2 (V x + iv y )(V x iv y )/2 ) The above is obviously a special case, namely U = V but at least we see that we can write a cartesian tensor in spherical basis. More generally 2
3 T0 0 = U V = (U +V + U V + U 0 V 0 ) T q = (U V) q i 2 T 2 ±2 = U ± V ± T 2 ± = U ±V 0 + U 0 V ± 2 T 2 0 = U +V + 2U 0 V 0 + U V + 6 Indeed if X k q and Y k2 q 2 are irreducible spherical tensors under rotations then Z k q = q,q 2 kq k q q k 2 q 2 X k q Y k2 q 2 wnere kq k q k 2 q 2 is a Clebsch Gordon coefficient. We see that spherical tensors can be combined in the same way as angular momentum states as we can write analogously kq = q,q 2 kq k q q k 2 q 2 k q k 2 q 2 Back to the quadrupole operator Wigner Ekhart Theorem T ij T 2 ±2,±,0 = x i x j δ ij T k=2 q=±2,±,0 r 2 Y m=±2,±,0 l=2 We take advantage of the spherical tensor formalism to evaluate the matrix element for the electric quadrupole transition. Suppose the quantum numbers of initial and final state are α, j, m and α, j, m respectively. Then we need to evaluate α, j, m Tq 2 α, j, m The Wiger Ekhart theorem states that α, j, m T k q α, j, m = j, m k, q, j, m α j T k αj 2j + The Clebsch Gordon coefficient is zero unless j = m + q and k + j j k j We see that l 2. Parity is conserved in electromagnetic interactions. The quadrupole operator is even under the parity operation. Therefore, initial and final states must have the same parity so l = 2, 0. The wave functions of initial and final atomic states are R i (r)yl m (θ, φ) and R f ()Yl m (θ, φ) respectively. And we saw above that Tq k = r 2 Y q k (θ, φ). Therefore f T k q i = R f (r)r i (r)r 2 r 2 dr Yl m Y q dω k Y l m
4 We can use the triple integral for spherical harmonics dωyl m (2l (θ, φ)y m l (θ, φ)y m2 + )(2l 2 + ) l 2 (θ, φ) = l0 l 0l 2 0l m l 2 m 2 lm 4π(2l + ) The last Clebsch Gordon coefficient enforces the Wigner Ekhart selection rules. Note that for any given initial and final state, there will be a non-zero for only one of the five T k q. All that remains is the radial integrals Magnetic dipole transition The second term can be written 2 f (k x)(p ɛ kλ) (k p)(ɛ kλ x) i = 2 f (k ɛ kλ) (x p) i k ɛ kλ is the leading term in the plane-wave expansion of the magnetic field B and x p is the orbital angular momentum. This term contributes to magnetic dipole M transitions. The operator is constructed from Y m and corresponds to transitions between states with l. The M transition will be relevant when the E transition is zero, perhaps because initial and final states have δl even so no parity change. This might correspond to a spin flip. As stated above, the intrinsic parity of the photon is odd, so if initial and final states of the emitting atom have the same parity, it must be that the photon has some orbital angular momentum with respect to the atomic coordinate system. If the orbital angular momenum is kr = l =, then the photon would have had to be emitted at r = /k. But as we determined earlier, kr for typical energy differences between levels and so the rate for such a process is low. The next term in the interaction Hamiltonian is ( ) µ B = e σ (k ɛ) 2mc 2ωV to be compared to ( ) e L (k ɛ) 2mc 2ωV Clearly of the same order. 4
5 2.. PLANK RADIATION LAW 2. Plank radiation law Suppoe we have atoms that make transitions between state A and B as follows A γ + B The higher energy state A decays to state B with emission of a photon. Then state B absorbs a photon and transitions to A. If the system is in equilibrium then N(B)w abs = N(A)w emis where N(B), N(A) are the numbers of atoms in states A and B respectively. wabs is the probablity that an atom in the state B absorbs a photon and w emis is the probablity that an atom in state A emits a photon. If the atoms are in thermal equilibrium then N(B) N(A) = w emis = e EB/kT w abs e = E A/kT e ω/kt (2.) The transition probability from state A and n photons to the state B with n + photons w emis B, n γ + e ik x ɛ α pa k A, n γ 2 = (n γ + ) B e ik x ɛ α p A 2 The probability for transition from state B with n photons to state A with n is w abs A, n γ e +ik x ɛ α pa k A, n γ 2 = (n γ ) A e +ik x ɛ α p B 2 The ratio w emis = n + w abs n Together with Equation 2. we find that the number of photons in thermal equilibrium with wave number k and polarization ɛ is n = e ω/kt That is, we imagine that there is an oscillator with omega = c k and that it is in equilibrium at temperature T when it is in the n th energy level. Now imagine a box with walls that absorb and emit photons at all wavelengths and polarizations. It is filled with oscillators at every possible frequency. Each oscillator will be at energy E n = ω n e ω/kt The total number of oscillators per unit frequency is the density of states dn = 2 V 4πk2 dk (2π) = 2 V 4πω2 dω (2π) c The number of states per unit frequency per unit volume is ρ(ω) = 8πω2 (2π) c The total energy per unit frequency per unit volume is U(ω) = ρ(ω)e n = 8π ω (2π) c (e ω/kt ) 5
Tensor Operators and the Wigner Eckart Theorem
November 11, 2009 Tensor Operators and the Wigner Eckart Theorem Vector operator The ket α transforms under rotation to α D(R) α. The expectation value of a vector operator in the rotated system is related
More informationand for absorption: 2π c 3 m 2 ˆɛˆk,j a pe i k r b 2( nˆk,j +1 February 1, 2000
At the question period after a Dirac lecture at the University of Toronto, somebody in the audience remarked: Professor Dirac, I do not understand how you derived the formula on the top left side of the
More informationNotes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015)
Notes on x-ray scattering - M. Le Tacon, B. Keimer (06/2015) Interaction of x-ray with matter: - Photoelectric absorption - Elastic (coherent) scattering (Thomson Scattering) - Inelastic (incoherent) scattering
More informationSolutions to exam : 1FA352 Quantum Mechanics 10 hp 1
Solutions to exam 6--6: FA35 Quantum Mechanics hp Problem (4 p): (a) Define the concept of unitary operator and show that the operator e ipa/ is unitary (p is the momentum operator in one dimension) (b)
More information2. Electric Dipole Start from the classical formula for electric dipole radiation. de dt = 2. 3c 3 d 2 (2.1) qr (2.2) charges q
APAS 50. Internal Processes in Gases. Fall 999. Transition Probabilities and Selection Rules. Correspondence between Classical and Quantum Mechanical Transition Rates According to the correspondence principle
More informationFun With Carbon Monoxide. p. 1/2
Fun With Carbon Monoxide p. 1/2 p. 1/2 Fun With Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results p. 1/2 Fun With Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results C V (J/K-mole) 35 30 25
More information3. Quantum Mechanics in 3D
3. Quantum Mechanics in 3D 3.1 Introduction Last time, we derived the time dependent Schrödinger equation, starting from three basic postulates: 1) The time evolution of a state can be expressed as a unitary
More informationQuantum Mechanics: The Hydrogen Atom
Quantum Mechanics: The Hydrogen Atom 4th April 9 I. The Hydrogen Atom In this next section, we will tie together the elements of the last several sections to arrive at a complete description of the hydrogen
More informationLECTURES ON QUANTUM MECHANICS
LECTURES ON QUANTUM MECHANICS GORDON BAYM Unitsersity of Illinois A II I' Advanced Bock Progrant A Member of the Perseus Books Group CONTENTS Preface v Chapter 1 Photon Polarization 1 Transformation of
More informationWe consider the nonrelativistic regime so no pair production or annihilation.the hamiltonian for interaction of fields and sources is 1 (p
.. RADIATIVE TRANSITIONS Marh 3, 5 Leture XXIV Quantization of the E-M field. Radiative transitions We onsider the nonrelativisti regime so no pair prodution or annihilation.the hamiltonian for interation
More informationMore On Carbon Monoxide
More On Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results Jerry Gilfoyle The Configurations of CO 1 / 26 More On Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results Jerry Gilfoyle The Configurations
More informationRelativistic corrections of energy terms
Lectures 2-3 Hydrogen atom. Relativistic corrections of energy terms: relativistic mass correction, Darwin term, and spin-orbit term. Fine structure. Lamb shift. Hyperfine structure. Energy levels of the
More information( ) /, so that we can ignore all
Physics 531: Atomic Physics Problem Set #5 Due Wednesday, November 2, 2011 Problem 1: The ac-stark effect Suppose an atom is perturbed by a monochromatic electric field oscillating at frequency ω L E(t)
More informationPAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES AND TRANSITION DIPOLE MOMENT. OVERVIEW OF SELECTION RULES)
Subject Chemistry Paper No and Title Module No and Title Module Tag 8 and Physical Spectroscopy 5 and Transition probabilities and transition dipole moment, Overview of selection rules CHE_P8_M5 TABLE
More informationPhys 622 Problems Chapter 5
1 Phys 622 Problems Chapter 5 Problem 1 The correct basis set of perturbation theory Consider the relativistic correction to the electron-nucleus interaction H LS = α L S, also known as the spin-orbit
More informationPhysics 506 Winter 2008 Homework Assignment #4 Solutions. Textbook problems: Ch. 9: 9.6, 9.11, 9.16, 9.17
Physics 56 Winter 28 Homework Assignment #4 Solutions Textbook problems: Ch. 9: 9.6, 9., 9.6, 9.7 9.6 a) Starting from the general expression (9.2) for A and the corresponding expression for Φ, expand
More informationECEN 5005 Crystals, Nanocrystals and Device Applications Class 20 Group Theory For Crystals
ECEN 5005 Crystals, Nanocrystals and Device Applications Class 20 Group Theory For Crystals Laporte Selection Rule Polarization Dependence Spin Selection Rule 1 Laporte Selection Rule We first apply this
More informationTutorial Session - Exercises. Problem 1: Dipolar Interaction in Water Moleclues
Tutorial Session - Exercises Problem 1: Dipolar Interaction in Water Moleclues The goal of this exercise is to calculate the dipolar 1 H spectrum of the protons in an isolated water molecule which does
More informationClassical Field Theory: Electrostatics-Magnetostatics
Classical Field Theory: Electrostatics-Magnetostatics April 27, 2010 1 1 J.D.Jackson, Classical Electrodynamics, 2nd Edition, Section 1-5 Electrostatics The behavior of an electrostatic field can be described
More informationAngular Momentum. Classically the orbital angular momentum with respect to a fixed origin is. L = r p. = yp z. L x. zp y L y. = zp x. xpz L z.
Angular momentum is an important concept in quantum theory, necessary for analyzing motion in 3D as well as intrinsic properties such as spin Classically the orbital angular momentum with respect to a
More informationChemistry 795T. NC State University. Lecture 4. Vibrational and Rotational Spectroscopy
Chemistry 795T Lecture 4 Vibrational and Rotational Spectroscopy NC State University The Dipole Moment Expansion The permanent dipole moment of a molecule oscillates about an equilibrium value as the molecule
More information(a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron, but neglecting spin-orbit interactions.
1. Quantum Mechanics (Spring 2007) Consider a hydrogen atom in a weak uniform magnetic field B = Bê z. (a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron,
More informationNotes: Most of the material presented in this chapter is taken from Jackson, Chap. 2, 3, and 4, and Di Bartolo, Chap. 2. 2π nx i a. ( ) = G n.
Chapter. Electrostatic II Notes: Most of the material presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartolo, Chap... Mathematical Considerations.. The Fourier series and the Fourier
More informationModule I: Electromagnetic waves
Module I: Electromagnetic waves Lectures 10-11: Multipole radiation Amol Dighe TIFR, Mumbai Outline 1 Multipole expansion 2 Electric dipole radiation 3 Magnetic dipole and electric quadrupole radiation
More informationProblem 1: Spin 1 2. particles (10 points)
Problem 1: Spin 1 particles 1 points 1 Consider a system made up of spin 1/ particles. If one measures the spin of the particles, one can only measure spin up or spin down. The general spin state of a
More informationL z L L. Think of it as also affecting the angle
Quantum Mechanics and Atomic Physics Lecture 19: Quantized Angular Momentum and Electron Spin http://www.physics.rutgers.edu/ugrad/361 h / d/361 Prof. Sean Oh Last time Raising/Lowering angular momentum
More informationOptical Lattices. Chapter Polarization
Chapter Optical Lattices Abstract In this chapter we give details of the atomic physics that underlies the Bose- Hubbard model used to describe ultracold atoms in optical lattices. We show how the AC-Stark
More information129 Lecture Notes More on Dirac Equation
19 Lecture Notes More on Dirac Equation 1 Ultra-relativistic Limit We have solved the Diraction in the Lecture Notes on Relativistic Quantum Mechanics, and saw that the upper lower two components are large
More informationdoes not change the dynamics of the system, i.e. that it leaves the Schrödinger equation invariant,
FYST5 Quantum Mechanics II 9..212 1. intermediate eam (1. välikoe): 4 problems, 4 hours 1. As you remember, the Hamilton operator for a charged particle interacting with an electromagentic field can be
More informationOctober Entrance Examination: Condensed Matter Multiple choice quizzes
October 2013 - Entrance Examination: Condensed Matter Multiple choice quizzes 1 A cubic meter of H 2 and a cubic meter of O 2 are at the same pressure (p) and at the same temperature (T 1 ) in their gas
More informationClassical Field Theory
April 13, 2010 Field Theory : Introduction A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word classical is used in
More informationB2.III Revision notes: quantum physics
B.III Revision notes: quantum physics Dr D.M.Lucas, TT 0 These notes give a summary of most of the Quantum part of this course, to complement Prof. Ewart s notes on Atomic Structure, and Prof. Hooker s
More informationQuantization of the E-M field. 2.1 Lamb Shift revisited 2.1. LAMB SHIFT REVISITED. April 10, 2015 Lecture XXVIII
.. LAMB SHFT REVSTED April, 5 Lecture XXV Quantization of the E-M field. Lamb Shift revisited We discussed the shift in the energy levels of bound states due to the vacuum fluctuations of the E-M fields.
More informationPhysics 221A Fall 2005 Homework 11 Due Thursday, November 17, 2005
Physics 221A Fall 2005 Homework 11 Due Thursday, November 17, 2005 Reading Assignment: Sakurai pp. 234 242, 248 271, Notes 15. 1. Show that Eqs. (15.64) follow from the definition (15.61) of an irreducible
More informationSemi-Classical Theory of Radiative Transitions
Semi-Classical Theory of Radiative Transitions Massimo Ricotti ricotti@astro.umd.edu University of Maryland Semi-Classical Theory of Radiative Transitions p.1/13 Atomic Structure (recap) Time-dependent
More informationFORMULA SHEET FOR QUIZ 2 Exam Date: November 8, 2017
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II November 5, 207 Prof. Alan Guth FORMULA SHEET FOR QUIZ 2 Exam Date: November 8, 207 A few items below are marked
More informationCollisionally Excited Spectral Lines (Cont d) Diffuse Universe -- C. L. Martin
Collisionally Excited Spectral Lines (Cont d) Please Note: Contrast the collisionally excited lines with the H and He lines in the Orion Nebula spectrum. Preview: Pure Recombination Lines Recombination
More informationLecture 4 Quantum mechanics in more than one-dimension
Lecture 4 Quantum mechanics in more than one-dimension Background Previously, we have addressed quantum mechanics of 1d systems and explored bound and unbound (scattering) states. Although general concepts
More informationB. PHENOMENOLOGICAL NUCLEAR MODELS
B. PHENOMENOLOGICAL NUCLEAR MODELS B.0. Basic concepts of nuclear physics B.0. Binding energy B.03. Liquid drop model B.04. Spherical operators B.05. Bohr-Mottelson model B.06. Intrinsic system of coordinates
More informationDiscrete Transformations: Parity
Phy489 Lecture 8 0 Discrete Transformations: Parity Parity operation inverts the sign of all spatial coordinates: Position vector (x, y, z) goes to (-x, -y, -z) (eg P(r) = -r ) Clearly P 2 = I (so eigenvalues
More informationThe 3 dimensional Schrödinger Equation
Chapter 6 The 3 dimensional Schrödinger Equation 6.1 Angular Momentum To study how angular momentum is represented in quantum mechanics we start by reviewing the classical vector of orbital angular momentum
More informationCHAPTER 1. SPECIAL RELATIVITY AND QUANTUM MECHANICS
CHAPTER 1. SPECIAL RELATIVITY AND QUANTUM MECHANICS 1.1 PARTICLES AND FIELDS The two great structures of theoretical physics, the theory of special relativity and quantum mechanics, have been combined
More informationElectric and magnetic multipoles
Electric and magnetic multipoles Trond Saue Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech 2017 1 / 22 Multipole expansions In multipolar gauge the expectation value of the
More informationPhysics 221A Fall 1996 Notes 21 Hyperfine Structure in Hydrogen and Alkali Atoms
Physics 221A Fall 1996 Notes 21 Hyperfine Structure in Hydrogen and Alkali Atoms Hyperfine effects in atomic physics are due to the interaction of the atomic electrons with the electric and magnetic multipole
More informationATOMIC AND LASER SPECTROSCOPY
ALAN CORNEY ATOMIC AND LASER SPECTROSCOPY CLARENDON PRESS OXFORD 1977 Contents 1. INTRODUCTION 1.1. Planck's radiation law. 1 1.2. The photoelectric effect 4 1.3. Early atomic spectroscopy 5 1.4. The postulates
More informationLecture 7. both processes have characteristic associated time Consequence strong interactions conserve more quantum numbers then weak interactions
Lecture 7 Conserved quantities: energy, momentum, angular momentum Conserved quantum numbers: baryon number, strangeness, Particles can be produced by strong interactions eg. pair of K mesons with opposite
More informationA few principles of classical and quantum mechanics
A few principles of classical and quantum mechanics The classical approach: In classical mechanics, we usually (but not exclusively) solve Newton s nd law of motion relating the acceleration a of the system
More informationSection 5 Time Dependent Processes
Section 5 Time Dependent Processes Chapter 14 The interaction of a molecular species with electromagnetic fields can cause transitions to occur among the available molecular energy levels (electronic,
More information2m r2 (~r )+V (~r ) (~r )=E (~r )
Review of the Hydrogen Atom The Schrodinger equation (for 1D, 2D, or 3D) can be expressed as: ~ 2 2m r2 (~r, t )+V (~r ) (~r, t )=i~ @ @t The Laplacian is the divergence of the gradient: r 2 =r r The time-independent
More informationFinal Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall Duration: 2h 30m
Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall. ------------------- Duration: 2h 30m Chapter 39 Quantum Mechanics of Atoms Units of Chapter 39 39-1 Quantum-Mechanical View of Atoms 39-2
More information= ( Prove the nonexistence of electron in the nucleus on the basis of uncertainty principle.
Worked out examples (Quantum mechanics). A microscope, using photons, is employed to locate an electron in an atom within a distance of. Å. What is the uncertainty in the momentum of the electron located
More informationQuantum Mechanics II Lecture 11 (www.sp.phy.cam.ac.uk/~dar11/pdf) David Ritchie
Quantum Mechanics II Lecture (www.sp.phy.cam.ac.u/~dar/pdf) David Ritchie Michaelmas. So far we have found solutions to Section 4:Transitions Ĥ ψ Eψ Solutions stationary states time dependence with time
More informationChapter 9. Electromagnetic Radiation
Chapter 9. Electromagnetic Radiation 9.1 Photons and Electromagnetic Wave Electromagnetic radiation is composed of elementary particles called photons. The correspondence between the classical electric
More information8.05 Quantum Physics II, Fall 2011 FINAL EXAM Thursday December 22, 9:00 am -12:00 You have 3 hours.
8.05 Quantum Physics II, Fall 0 FINAL EXAM Thursday December, 9:00 am -:00 You have 3 hours. Answer all problems in the white books provided. Write YOUR NAME and YOUR SECTION on your white books. There
More informationElectrodynamics II: Lecture 9
Electrodynamics II: Lecture 9 Multipole radiation Amol Dighe Sep 14, 2011 Outline 1 Multipole expansion 2 Electric dipole radiation 3 Magnetic dipole and electric quadrupole radiation Outline 1 Multipole
More informationThe Northern California Physics GRE Bootcamp
The Northern California Physics GRE Bootcamp Held at UC Davis, Sep 8-9, 2012 Damien Martin Big tips and tricks * Multiple passes through the exam * Dimensional analysis (which answers make sense?) Other
More informationTotal Angular Momentum for Hydrogen
Physics 4 Lecture 7 Total Angular Momentum for Hydrogen Lecture 7 Physics 4 Quantum Mechanics I Friday, April th, 008 We have the Hydrogen Hamiltonian for central potential φ(r), we can write: H r = p
More informationLecture 4 Quantum mechanics in more than one-dimension
Lecture 4 Quantum mechanics in more than one-dimension Background Previously, we have addressed quantum mechanics of 1d systems and explored bound and unbound (scattering) states. Although general concepts
More informationEM radiation - Lecture 14
EM radiation - Lecture 14 1 Review Begin with a review of the potentials, fields, and Poynting vector for a point charge in accelerated motion. The retarded potential forms are given below. The source
More informationImplications of Time-Reversal Symmetry in Quantum Mechanics
Physics 215 Winter 2018 Implications of Time-Reversal Symmetry in Quantum Mechanics 1. The time reversal operator is antiunitary In quantum mechanics, the time reversal operator Θ acting on a state produces
More informationChapter 6: SYMMETRY IN QUANTUM MECHANICS
Chapter 6: SYMMETRY IN QUANTUM MECHANICS Since the beginning of physics, symmetry considerations have provided us with an extremely powerful and useful tool in o ur effort in understanding nature. Gradually
More informationPhysics 215 Quantum Mechanics I Assignment 8
Physics 15 Quantum Mechanics I Assignment 8 Logan A. Morrison March, 016 Problem 1 Let J be an angular momentum operator. Part (a) Using the usual angular momentum commutation relations, prove that J =
More informationAtoms 2012 update -- start with single electron: H-atom
Atoms 2012 update -- start with single electron: H-atom x z φ θ e -1 y 3-D problem - free move in x, y, z - easier if change coord. systems: Cartesian Spherical Coordinate (x, y, z) (r, θ, φ) Reason: V(r)
More informationQuantum Mechanics-I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras. Lecture - 21 Square-Integrable Functions
Quantum Mechanics-I Prof. Dr. S. Lakshmi Bala Department of Physics Indian Institute of Technology, Madras Lecture - 21 Square-Integrable Functions (Refer Slide Time: 00:06) (Refer Slide Time: 00:14) We
More informationAtomic Term Symbols and Energy Splitting. λ=5890 Å
Chemistry 362 Spring 2018 Dr. Jean M. Standard April 18, 2018 Atomic Term Symbols and Energy Splitting 1. Atomic Term Symbols and the Sodium D-Line The sodium D-line is responsible for the familiar orange
More information量子力学 Quantum mechanics. School of Physics and Information Technology
量子力学 Quantum mechanics School of Physics and Information Technology Shaanxi Normal University Chapter 9 Time-dependent perturation theory Chapter 9 Time-dependent perturation theory 9.1 Two-level systems
More informationElectronic Structure of Atoms. Chapter 6
Electronic Structure of Atoms Chapter 6 Electronic Structure of Atoms 1. The Wave Nature of Light All waves have: a) characteristic wavelength, λ b) amplitude, A Electronic Structure of Atoms 1. The Wave
More informationIntroduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world,
Introduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world, x p h π If you try to specify/measure the exact position of a particle you
More informationCold molecules: Theory. Viatcheslav Kokoouline Olivier Dulieu
Cold molecules: Theory Viatcheslav Kokoouline Olivier Dulieu Summary (1) Born-Oppenheimer approximation; diatomic electronic states, rovibrational wave functions of the diatomic cold molecule molecule
More informationQuantum Mechanics II
Quantum Mechanics II Prof. Boris Altshuler April, 20 Lecture 2. Scattering Theory Reviewed Remember what we have covered so far for scattering theory. We separated the Hamiltonian similar to the way in
More informationNuclear Shell Model. Experimental evidences for the existence of magic numbers;
Nuclear Shell Model It has been found that the nuclei with proton number or neutron number equal to certain numbers 2,8,20,28,50,82 and 126 behave in a different manner when compared to other nuclei having
More informationPhysics 221B Spring 2018 Notes 41 Emission and Absorption of Radiation
Copyright c 2018 by Robert G. Littlejohn Physics 221B Spring 2018 Notes 41 Emission and Absorption of Radiation 1. Introduction In these notes we will examine some simple examples of the interaction of
More informationTime dependent perturbation theory 1 D. E. Soper 2 University of Oregon 11 May 2012
Time dependent perturbation theory D. E. Soper University of Oregon May 0 offer here some background for Chapter 5 of J. J. Sakurai, Modern Quantum Mechanics. The problem Let the hamiltonian for a system
More information4. Spontaneous Emission october 2016
4. Spontaneous Emission october 016 References: Grynberg, Aspect, Fabre, "Introduction aux lasers et l'optique quantique". Lectures by S. Haroche dans "Fundamental systems in quantum optics", cours des
More informationNon-stationary States and Electric Dipole Transitions
Pre-Lab Lecture II Non-stationary States and Electric Dipole Transitions You will recall that the wavefunction for any system is calculated in general from the time-dependent Schrödinger equation ĤΨ(x,t)=i
More informationCoupling of Angular Momenta Isospin Nucleon-Nucleon Interaction
Lecture 5 Coupling of Angular Momenta Isospin Nucleon-Nucleon Interaction WS0/3: Introduction to Nuclear and Particle Physics,, Part I I. Angular Momentum Operator Rotation R(θ): in polar coordinates the
More informationAtomic spectra of one and two-electron systems
Atomic spectra of one and two-electron systems Key Words Term symbol, Selection rule, Fine structure, Atomic spectra, Sodium D-line, Hund s rules, Russell-Saunders coupling, j-j coupling, Spin-orbit coupling,
More informationExamination paper for FY3403 Particle physics
Department of physics Examination paper for FY3403 Particle physics Academic contact during examination: Jan Myrheim Phone: 900 75 7 Examination date: December 6, 07 Examination time: 9 3 Permitted support
More informationPhysics 221A Fall 1996 Notes 14 Coupling of Angular Momenta
Physics 1A Fall 1996 Notes 14 Coupling of Angular Momenta In these notes we will discuss the problem of the coupling or addition of angular momenta. It is assumed that you have all had experience with
More informationIntroduction to Quantum Physics and Models of Hydrogen Atom
Introduction to Quantum Physics and Models of Hydrogen Atom Tien-Tsan Shieh Department of Applied Math National Chiao-Tung University November 7, 2012 Physics and Models of Hydrogen November Atom 7, 2012
More informationLecture 11: Polarized Light. Fundamentals of Polarized Light. Descriptions of Polarized Light. Scattering Polarization. Zeeman Effect.
Lecture 11: Polarized Light Outline 1 Fundamentals of Polarized Light 2 Descriptions of Polarized Light 3 Scattering Polarization 4 Zeeman Effect 5 Hanle Effect Fundamentals of Polarized Light Electromagnetic
More informationPHYS 3327 PRELIM 1. Prof. Itai Cohen, Fall 2010 Friday, 10/15/10. Name: Read all of the following information before starting the exam:
PHYS 3327 PRELIM 1 Prof. Itai Cohen, Fall 2010 Friday, 10/15/10 Name: Read all of the following information before starting the exam: Put your name on the exam now. Show all work, clearly and in order,
More informationc E If photon Mass particle 8-1
Nuclear Force, Structure and Models Readings: Nuclear and Radiochemistry: Chapter 10 (Nuclear Models) Modern Nuclear Chemistry: Chapter 5 (Nuclear Forces) and Chapter 6 (Nuclear Structure) Characterization
More informationGian Gopal Particle Attributes Quantum Numbers 1
Particle Attributes Quantum Numbers Intro Lecture Quantum numbers (Quantised Attributes subject to conservation laws and hence related to Symmetries) listed NOT explained. Now we cover Electric Charge
More informationNotes on Quantum Mechanics
Notes on Quantum Mechanics K. Schulten Department of Physics and Beckman Institute University of Illinois at Urbana Champaign 405 N. Mathews Street, Urbana, IL 61801 USA (April 18, 2000) Preface i Preface
More informationIntroduction to Nuclear Physics Physics 124 Solution Set 4
Introduction to Nuclear Physics Physics 14 Solution Set 4 J.T. Burke January 3, 000 1 Problem 14 In making a back of the envelope calculation we must simplify the existing theory and make appropriate assumptions.
More informationLecture 5: Harmonic oscillator, Morse Oscillator, 1D Rigid Rotor
Lecture 5: Harmonic oscillator, Morse Oscillator, 1D Rigid Rotor It turns out that the boundary condition of the wavefunction going to zero at infinity is sufficient to quantize the value of energy that
More informationΨ t = ih Ψ t t. Time Dependent Wave Equation Quantum Mechanical Description. Hamiltonian Static/Time-dependent. Time-dependent Energy operator
Time Dependent Wave Equation Quantum Mechanical Description Hamiltonian Static/Time-dependent Time-dependent Energy operator H 0 + H t Ψ t = ih Ψ t t The Hamiltonian and wavefunction are time-dependent
More informationwhere n = (an integer) =
5.111 Lecture Summary #5 Readings for today: Section 1.3 (1.6 in 3 rd ed) Atomic Spectra, Section 1.7 up to equation 9b (1.5 up to eq. 8b in 3 rd ed) Wavefunctions and Energy Levels, Section 1.8 (1.7 in
More informationQuantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.
Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.
More informationSuperposition of electromagnetic waves
Superposition of electromagnetic waves February 9, So far we have looked at properties of monochromatic plane waves. A more complete picture is found by looking at superpositions of many frequencies. Many
More informationMOLECULAR SPECTROSCOPY
MOLECULAR SPECTROSCOPY First Edition Jeanne L. McHale University of Idaho PRENTICE HALL, Upper Saddle River, New Jersey 07458 CONTENTS PREFACE xiii 1 INTRODUCTION AND REVIEW 1 1.1 Historical Perspective
More informationJ10M.1 - Rod on a Rail (M93M.2)
Part I - Mechanics J10M.1 - Rod on a Rail (M93M.2) J10M.1 - Rod on a Rail (M93M.2) s α l θ g z x A uniform rod of length l and mass m moves in the x-z plane. One end of the rod is suspended from a straight
More informationElectromagnetic Theory I
Electromagnetic Theory I Final Examination 18 December 2009, 12:30-2:30 pm Instructions: Answer the following 10 questions, each of which is worth 10 points. Explain your reasoning in each case. Use SI
More informationChapter 3. Electromagnetic Theory, Photons. and Light. Lecture 7
Lecture 7 Chapter 3 Electromagnetic Theory, Photons. and Light Sources of light Emission of light by atoms The electromagnetic spectrum see supplementary material posted on the course website Electric
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 21 Quantum Mechanics in Three Dimensions Lecture 21 Physics 342 Quantum Mechanics I Monday, March 22nd, 21 We are used to the temporal separation that gives, for example, the timeindependent
More informationQuantum Theory of Angular Momentum and Atomic Structure
Quantum Theory of Angular Momentum and Atomic Structure VBS/MRC Angular Momentum 0 Motivation...the questions Whence the periodic table? Concepts in Materials Science I VBS/MRC Angular Momentum 1 Motivation...the
More informationAngular Momentum - set 1
Angular Momentum - set PH0 - QM II August 6, 07 First of all, let us practise evaluating commutators. Consider these as warm up problems. Problem : Show the following commutation relations ˆx, ˆL x = 0,
More informationAtoms 09 update-- start with single electron: H-atom
Atoms 09 update-- start with single electron: H-atom VII 33 x z φ θ e -1 y 3-D problem - free move in x, y, z - handy to change systems: Cartesian Spherical Coordinate (x, y, z) (r, θ, φ) Reason: V(r)
More information1 Electromagnetic Energy in Eyeball
µoβoλα Rewrite: 216/ First Version: Spring 214 Blackbody Radiation of the Eyeball We consider Purcell s back of the envelope problem [1] concerning the blackbody radiation emitted by the human eye, and
More information