Introduction To Special Relativity, Quantum Mechanics And .
Introduction toSpecial Relativity, Quantum Mechanics and Nuclear PhysicsforNuclear EngineersAlex F BielajewThe University of MichiganDepartment of Nuclear Engineering and Radiological Sciences2927 Cooley Building (North Campus)2355 Bonisteel BoulevardAnn Arbor, Michigan 48109-2104U. S. A.Tel: 734 764 6364Fax: 734 763 4540email: bielajew@umich.educ 2010–12 Alex F BielajewDecember 15, 2014
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PrefaceThis book arises from a series of hand-written notes I am continually revising in support oftwo courses I teach, each a three-credit (42 hour) junior-level course, NERS311 and NERS312: Elements of Nuclear Engineering and Radiological Sciences I and II at the Departmentof Nuclear Engineering and Radiological Sciences at the University of Michigan.More apt titles for these courses would be NERS311: Modern Physics and Quantum Mechanics and NERS312: Nuclear Physics because there is very little engineering in the coursecontent. Rather, we shall dwell on the sciences that underpin Nuclear Engineering and Radiological Sciences, for it is essential that we understand, in some detail, the nature of thestuff we are engineering. Nuclear materials and the resultant radiation are really some of themost dangerous (and interesting) things in the world, and, in my view at least, understandingthem, at least in some depth, is essential.These two courses assume, and make great use of, the earlier background courses in mathematics and physics. So, if you get stuck on some mathematics or physics concept, pleasedive into your old notes and texts, or ask questions. These things may have seemed tiredand dry when you learned them initially, but these courses will bring them to back to life,with some vigor, and a great deal of power. By their very nature, the consequences ofSpecial Relativity and Quantum Mechanics are counterintuitive. Our understanding of veryfast and/or very small objects is not reinforced by our everyday experiences. Consequently,the understanding of these phenomena falls to mathematical interpretation, that must berefined, to enable deeper understanding. Much like a blind person, whose other senses aresharpened to enable him or her to experience the world, so it is that mathematics becomesmore important. Equations really do speak, if you listen the right way.I would like to thank the students who took my first versions of there courses in the Fall of2005, and Winter of 2006. Your feedback convinced me that I should undertake this writing.Your support was deeply appreciated. To you taking NERS 311 or 312 now, this is verymuch a work-in-progress. There will be spelling and grammatical errors, sloppy English,missing figures, the occasional bad equation, and, once in a while, a logical argument thatdoes not make complete sense. If you find an error or can suggest an improvement, pleasebring it to me. I am most grateful for these. These notes may seem to be under constantrevision. They are, and it is a natural part of their development. Please bear with this. Oni
iithe plus side, they’re free!About Krane’s book, Modern Physics, Second editionThis is a decent text, good value for the money, but not perfect. If it were, these notes wouldnot exist! I hope these notes bring some added value to the material. In several cases, Idisagree with Krane’s approach, and will offer a little more rigor. In a few cases, I’ll disagreewith Krane’s interpretation of, in particular, Quantum Mechanics. This is not to say Krane’sapproach is wrong, because these topics are still being debated by theoretical physicists andnatural philosophers. However, I will try to justify my view of things.Two of the greatest things about Krane’s book are the collection of Questions and Problemsat the back of each Chapter. Good students should make use of these sections to the extentthat your interest, energy and time permit. I will provide more encouragement throughoutthe text.AFB, October 2, 20062008 updateIn 2008, the 311 course was reduced to 3 credits from 4, at the expense of eliminating mostof the Special Relativity material from the lectures. Most of that material is not essentialto obtaining a thorough understanding of Quantum Mechanics and Nuclear Physics. Whatis essential, and what will continue to be taught are the relativistic kinematical relations, aswe shall be learning about photons, that travel at the speed of light, and energetic electronsfrom β-decay, that have velocities close to the speed of light. I’ve elected to leave the ModernPhysics material intact. It’s good science culture, and perhaps an interested student or twowill be motivated to study this topic deeper.2009 updateExtensive revisions to most chapters have been undertaken. I’d like to thank Ms. LindaPark for her excellent and erudite proofreading of the technical and non-technical material.Linda, I am in awe of your attention-to-detail.A brief note on chapter headingsChapter and subchapter headings preceded by a dagger† do not exist in Krane’s book, andare here for supplementary, prerequisite, or co-requisite study. They will be covered on aneed-to basis. Chapter and subchapter headings preceded by an asterisk are generally notcovered in teaching the NERS 311/312 editions of the class.AFB, July 16, 2009
Contents1 Introduction11.1About this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.2Modern Physics, Quantum Mechanics and Nuclear Physics . . . . . . . . . .21.3Some History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101.4Review of Classical Physics. . . . . . . . . . . . . . . . . . . . . . . . . . .13Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131.5Units and Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .221.6Significant Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .221.7Theory, Experiment, Law. . . . . . . . . . . . . . . . . . . . . . . . . . . .231.8†Basic Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .231.4.11.8.1Accounting for Estimated Error for Independent Quantities . . . . . .23Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .241.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .241.11 Supplementary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .241.92 The Special Theory of Relativity272.1 Classical Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .272.2 The Michelson-Morley Experiment . . . . . . . . . . . . . . . . . . . . . . .292.3 Einstein’s Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .312.4 The Lorentz Transformation . . . . . . . . . . . . . . . . . . . . . . . . . .312.5Relativistic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .372.6Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45iii
ivCONTENTS2.7Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 The Particlelike Properties of Electromagnetic Radiation45473.1Review of Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . .473.2The Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . .473.3Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .483.4The Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .483.5Other Photon Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .483.6What is a Photon? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .484 The Wavelike Properties of Particles494.1De Broglie’s Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .494.2Uncertainty Relationships for Classical Waves . . . . . . . . . . . . . . . . .504.3Heisenberg Uncertainty Relationships . . . . . . . . . . . . . . . . . . . . . .504.4Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .504.5Probability and Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . .524.6The Probability Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . .525 The Schrödinger Equation in 1D535.1Justifying the Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . .545.2The Schrödinger Recipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . .555.3Probability Densities and Normalization . . . . . . . . . . . . . . . . . . . .555.4Applications of Scattering in 1D . . . . . . . . . . . . . . . . . . . . . . . . .565.4.1Time-Independent Scattering Applications . . . . . . . . . . . . . . .565.5Applications of Time-Independent Bound States . . . . . . . . . . . . . . . .585.6Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .595.7†59Time-Dependent Perturbations . . . . . . . . . . . . . . . . . . . . . . . . .5.7.1Fermi’s Golden Rule #2 . . . . . . . . . . . . . . . . . . . . . . . . .595.7.2The Lorentz Distribution . . . . . . . . . . . . . . . . . . . . . . . . .636 The Rutherford-Bohr Model of the Atom65
CONTENTSv6.1Basic Properties of Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . .656.2The Thomson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .656.3The Rutherford Nuclear Atom . . . . . . . . . . . . . . . . . . . . . . . . . .656.4Line Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .656.5The Bohr Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .656.6The Franck-Hertz Experiment . . . . . . . . . . . . . . . . . . . . . . . . . .656.7The Correspondence Principle . . . . . . . . . . . . . . . . . . . . . . . . . .656.8Deficiencies of the Bohr Model . . . . . . . . . . . . . . . . . . . . . . . . . .657 The Hydrogen Atom677.0.1Central force, two-body systems in Classical Mechanics . . . . . . . .677.0.2Central force, two-body systems in Quantum Mechanics. . . . . . .687.1The Schrödinger Equation in 3D . . . . . . . . . . . . . . . . . . . . . . . . .687.2The Hydrogenic Atom Wave Functions . . . . . . . . . . . . . . . . . . . . .707.3Radial Probability Densities . . . . . . . . . . . . . . . . . . . . . . . . . . .707.4Angular Momentum and Probability Densities . . . . . . . . . . . . . . . . .707.5Intrinsic Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .707.6Energy Level and Spectroscopic Notation . . . . . . . . . . . . . . . . . . . .707.7The Zeeman Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .707.8Fine Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .708 Many-Electron Atoms718.1The Pauli Exclusion Principle . . . . . . . . . . . . . . . . . . . . . . . . . .718.2Electronic States in Many-Electron Atoms . . . . . . . . . . . . . . . . . . .718.3The Periodic Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .718.4Properties of the Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . .718.5X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .718.6Optical Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .718.7Addition of Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . . . .718.8Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71
viCONTENTS9 Review of Classical Physics Relevant to Nuclear Physics7310 Nuclear Properties7510.1 The Nuclear Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7810.1.1 Application to spherical charge distributions . . . . . . . . . . . . . .8010.1.2 Nuclear shape data from electron scattering experiments . . . . . . .8810.1.3 Nuclear size from spectroscopy measurements . . . . . . . . . . . . .8910.2 Mass and Abundance of Nuclei . . . . . . . . . . . . . . . . . . . . . . . . .9410.3 Nuclear Binding Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9510.4 Angular Momentum and Parity . . . . . . . . . . . . . . . . . . . . . . . . .9910.5 Nuclear Magnetic and Electric Moments . . . . . . . . . . . . . . . . . . . . 10010.5.1 Magnetic Dipole Moments of Nucleons . . . . . . . . . . . . . . . . . 10010.5.2 Quadrupole Moments of Nuclei . . . . . . . . . . . . . . . . . . . . . 10211 The Force Between Nucleons11911.1 The Deuteron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12311.2 Nucleon-nucleon scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12711.3 Proton-proton and neutron-neutron interactions . . . . . . . . . . . . . . . . 12711.4 Properties of the nuclear force . . . . . . . . . . . . . . . . . . . . . . . . . . 12811.5 The exchange force model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12812 Nuclear Models13112.1 The Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13412.2 Even-Z, even-N Nuclei and Collective Structure . . . . . . . . . . . . . . . . 14912.2.1 The Liquid Drop Model of the Nucleus . . . . . . . . . . . . . . . . . 15013 Radioactive Decay15913.1 The Radioactive Decay Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 15913.2 Quantum Theory of Radioactive Decay . . . . . . . . . . . . . . . . . . . . . 16313.3 Production and Decay of Radioactivity . . . . . . . . . . . . . . . . . . . . . 17413.4 Growth of Daughter Activities . . . . . . . . . . . . . . . . . . . . . . . . . . 178
CONTENTSvii13.5 Types of Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18113.6 Natural Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18413.7 Radioactive Dating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18413.8 Units for Measuring Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 18514 α Decay18914.1 Why α Decay Occurs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18914.2 Basic α Decay Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18914.3 α Decay Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19114.4 Theory of α Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19214.4.1 Comparison with Measurements . . . . . . . . . . . . . . . . . . . . . 19714.5 Angular momentum and parity in α decay . . . . . . . . . . . . . . . . . . . 19814.6 α-decay spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20015 β Decay20115.1 Energy release in β decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20315.2 Fermi’s theory of β decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20715.3 Experimental tests of Fermi’s theory . . . . . . . . . . . . . . . . . . . . . . 21315.4 Angular momentum and parity selection rules . . . . . . . . . . . . . . . . . 21415.4.1 Matrix elements for certain special cases . . . . . . . . . . . . . . . . 21615.5 Comparative half-lives and forbidden decays . . . . . . . . . . . . . . . . . . 21715.6 Neutrino physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21715.7 Double-β Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21815.8 β-delayed electron emission. . . . . . . . . . . . . . . . . . . . . . . . . . . 21815.9 Non-conservation of parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21815.10β spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21816 γ Decay21916.1 Energetics of γ decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22216.2 Classical Electromagnetic Radiation . . . . . . . . . . . . . . . . . . . . . . . 224
viiiCONTENTS16.2.1 A general and more sophisticated treatment of classical multipole fields 22816.3 Transition to Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 23016.4 Angular momentum and parity selection rules . . . . . . . . . . . . . . . . . 23316.5 Angular Distribution and Polarization Measurements . . . . . . . . . . . . . 23516.6 Internal Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23517 Nuclear Reactions24317.1 Types of Reactions and Conservation Laws . . . . . . . . . . . . . . . . . . . 24317.1.1 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24517.1.2 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24617.2 Energetics of Nuclear Reactions . . . . . . . . . . . . . . . . . . . . . . . . . 24717.3 Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25417.4 Reaction Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25517.5 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25517.6 Coulomb Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25517.7 Nuclear Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25517.8 Scattering and Reaction Cross Sections . . . . . . . . . . . . . . . . . . . . . 25517.8.1 Partial wave analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 25717.9 The Optical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26217.10Compound-Nucleus Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . 26317.11Direct Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26517.12Resonance Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26618 † Mathematical Techniques and Notation Used in this Book27518.1 Vectors and Operators in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . 27518.1.1 Some common coordinate system representations . . . . . . . . . . . 27618.2 Common Trigonometric Relations . . . . . . . . . . . . . . . . . . . . . . . . 28018.3 Common Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 28018.4 Complex Numbers or Functions . . . . . . . . . . . . . . . . . . . . . . . . . 28118.5 3D Differential Operators in a Cartesian Coordinate System . . . . . . . . . 282
CONTENTS18.6 3D Differential Operators in a Cylindrical Coordinate System18.7 3D Differential Operators in a Spherical Coordinate Systemix. . . . . . . . 282. . . . . . . . . 28318.8 Dirac, Kronecker Deltas, Heaviside Step-Function . . . . . . . . . . . . . . . 28318.9 Taylor/MacLaurin Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
xCONTENTS
Chapter 1Introduction1.1About this bookThis book arises from a series of hand-written notes I am continually revising in support oftwo courses I teach, each a three-credit (42 hour) junior-level course, NERS311 and NERS312: Elements of Nuclear Engineering and Radiological Sciences I and II at the Departmentof Nuclear Engineering and Radiological Sciences at the University of Michigan.More apt titles for these courses would be NERS311: Modern Physics and Quantum Mechanics and NERS312: Nuclear Physics because there is very little engineering in the coursecontent. Rather, we shall dwell on the sciences that underpin Nuclear Engineering and Radiological Sciences, for it is essential that we understand, in some detail, the nature of thestuff we are engineering. Nuclear materials and the resultant radiation are really some of themost dangerous (and interesting) things in the world, and, in my view at least, understandingthem, at least in some depth, is essential.These two courses assume, and make great use of, the earlier background courses in mathematics and physics. So, if you get stuck on some mathematics or physics concept, pleasedive into your old notes and texts, or ask questions. These things may have seemed tiredand dry when you learned them initially, but these courses will bring them to back to life,with some vigor, and a great deal of power. By their very nature, the consequences ofSpecial Relativity and Quantum Mechanics are counterintuitive. Our understanding of veryfast and/or very small objects is not reinforced by our everyday experiences. Consequently,the understanding of these phenomen
Special Relativity, Quantum Mechanics and Nuclear Physics for Nuclear Engineers Alex F Bielajew The University of Michigan Department of Nuclear Engineering and Radiological Sciences 2927 Cooley Building (North Campus) 2355 Bonisteel Boulevard Ann Arbor, Michigan 48109-2104 U. S. A. Tel:
1 RELATIVITY I 1 1.1 Special Relativity 2 1.2 The Principle of Relativity 3 The Speed of Light 6 1.3 The Michelson–Morley Experiment 7 Details of the Michelson–Morley Experiment 8 1.4 Postulates of Special Relativity 10 1.5 Consequences of Special Relativity 13 Simultaneity and the Relativity
The theory of relativity is split into two parts: special and general. Albert Einstein came up with the spe-cial theory of relativity in 1905. It deals with objects mov-ing relative to one another, and with the way an observer's experience of space and time depends on how she is mov-ing. The central ideas of special relativity can be formu-
According to the quantum model, an electron can be given a name with the use of quantum numbers. Four types of quantum numbers are used in this; Principle quantum number, n Angular momentum quantum number, I Magnetic quantum number, m l Spin quantum number, m s The principle quantum
1. Quantum bits In quantum computing, a qubit or quantum bit is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics.
6. Quantum Theory and Relativity 6.1. Introduction 6.2. Einstein's special theory of relativity 6.3. Minkowski diagrams 6.4. The Klein-Gordon equation 6.5. The Dirac equation 6.6. Relativistic quantum eld theory 6.6.1. Introduction 6.6.2. Quantum eld theory as a many particle theory 6.6.3. Fock space and its operators 6.6.4. The scalar .
Introduction Special Relativity General Relativity Curriculum Books The Geometry of Special Relativity Tevi
Theory of Relativity. Einstein's General Theory of Relativity by Asghar Qadir. Einstein'sGeneralTheoryofRelativity ByAsgharQadir . Relativity: An Introduction to the Special Theory (World Scientific 1989) or equivalent, but do not have a sound background of Geometry. It can be used
The Quantum Nanoscience Laboratory (QNL) bridges the gap between fundamental quantum physics and the engineering approaches needed to scale quantum devices into quantum machines. The team focuses on the quantum-classical interface and the scale-up of quantum technology. The QNL also applies quantum technology in biomedicine by pioneering new
