Foldy-Wouthuysen Transformation with Dirac Matrices in Chiral Representation. V.P.Neznamov RFNC-VNIIEF, , Sarov, Nizhniy Novgorod region
|
|
- Hugh Steven Moody
- 5 years ago
- Views:
Transcription
1 Foldy-Wouthuysen Transormaton wth Drac Matrces n Chral Representaton V.P.Neznamov RFNC-VNIIEF, 679, Sarov, Nzhny Novgorod regon Abstract The paper oers an expresson o the general Foldy-Wouthuysen transormaton n the chral representaton o Drac matrces and n the presence o boson elds B ( xt, ) nteractng wth ermon eld ( x, t) ψ.
2 The papers [], [] dscuss the theory o nteractng quantum elds n the Foldy-Wouthuysen representaton [3]. These papers oer, n partcular, the relatvstc nonlocal Hamltonan H n the orm o a seres n terms o powers o charge e. Quantum electrodynamcs n the Foldy-Wouthuysen ( ) representaton has been ormulated usng Hamltonan H and some quantum electrodynamcs processes have been calculated wthn the lowest-order perturbaton theory. As a result, the concluson has been made that the representaton descrbes some quas-classc states n the quantum eld theores. Both partcles and antpartcles are avalable n these states. Partcles, as well as antpartcles, nteract wth each other. However, there s no nteracton o real partcles wth antpartcles such nteracton s possble only n ntermedate (vrtual) states. The representaton modcaton s requred to take nto account real partcle/antpartcle nteractons. In the papers [], [] such modcaton has been made usng the symmetry dentcal to the sotopc spn symmetry owng to nvarance o nal physcal results under change o sgns n the mass terms o Drac Hamltonan H D and Hamltonan H. In the moded Foldy- Wouthuysen representaton, real ermons and antermons can be n two states 3 characterzed by the values o the thrd component o the sotopc spn T = ± ; real ermons and antermons nteractng wth each other must have opposte sgns o 3 T. Quantum electrodynamcs n the moded representaton s nvarant under P -, C -, T -transormatons. Volatons o the ntroduced symmetry o the sotopc spn lead to the correspondng volaton o CP -nvarance. The Standard Model n the moded representaton was ormulated n the papers [], [4]. It has been shown that ormulaton o the theory n the moded representaton doesn t requre that Hggs bosons should oblgatorly nteract wth ermons to preserve the SU()- nvarance, whereas all the rest theoretcal and expermental mplcatons o the Standard model obtaned n the Drac representaton are preserved. In such a case, Hggs bosons are responsble only or the gauge nvarance o the boson sector o the ± theory and nteract only wth gauge bosons W,Z, gluons and photons.
3 3 In the papers mentoned above, the energy representaton o Drac matrces derved by Drac hmsel s used: σ I I α =, β γ, γ, γ γ α σ = = = =. () I I The queston arses: what changes o the Foldy-Wouthuysen transormaton orm wll result rom usng the chral representaton o Drac matrces? σ I I α =, β γ, γ, γ γ α σ = = = = () I I The chral representaton () s commonly used n the modern gauge eld theores and n the Standard Model, n partcular. To answer the queston above, rst consder the structure o equatons descrbng the components o the wave unctons ψ D ( x ) or the two representatons o Drac matrces consdered n the paper. In relatons (), () and below the system o unts wth = c = s used; x, p, A k k are 4-vectors; the nner product s taken as x y = x y x y, =,,,3, k =,,3;, k = p = ; σ are Paul matrces; α = ; ψ D ( x) s the ourcomponent wave uncton, ϕ( x), χ( x), ψ ( x), ψ ( x) are the two-component x α, = k, k =,,3 wave unctons. R The ollowng operator relatons are vald or the ree Drac equaton wth representaton (): ϕ( x) pψd( x) = ( αp+ βm) ψd( x); ψd( x) = ; χ( x) pϕ( x) = σ pχ( x) + mϕ( x) ; pχ( x) = σ pϕ( x) mχ( x) (3) χ = p + m σ pϕ; ϕ = p m σ pχ ( ) ( ) Wth representaton (), relaton (3) looks lke ψ R ( x) pψd( x) = ( αp+ βm) ψd( x); ψd( x) = ; ψ L( x) L
4 pψr( x) = σ pψr( x) + mψl( x) ; pψl( x) = σ pψl( x) + mψr( x) p σ p ψl( x) = ψr( x) = ( p + σ p) mψr( x) m p + σ p ψr( x) = ψl( x) = ( p σ p) mψl( x) m Relatons (4) use the operator equalty: p = E = p + m. 4 (4) Comparson between relatons (3) and (4) shows that wth the substtuton below, m σ p, β γ () these relatons transorm nto each other. The Foldy-Wouthuysen transormatons or the energy and chral representatons o Drac matrces also transorm nto each other, substtuton () s made. The Foldy-Wouthuysen transormatons or ree moton o ermons (wthout boson elds B ( x) ) look lke ( U ) m βγ σ p = + E m en ( U ) σ p γβm = + (7) E σ p chr Expresson (6) s wrtten or the energy representaton o Drac matrces; expresson (7) s the desred one descrbng the Foldy-Wouthuysen transormaton wth Drac matrces n the chral representaton; n expressons (6), (7) E = p + m. Matrces (6), (7) are untary and U α p+ βm U = βe; en en ( ) ( )( ) chr U α p+ βm U = γ E chr ( ) ( )( ) ; I boson elds B ( x) nteractng wth ermon eld ψ ( x ) are present, the desred Foldy-Wouthuysen transormaton wth Drac matrces n the chral representaton chr chr 3 ( U ) and the ermon Hamltonan ( H = γ qk + q K + q K +...; qs the couplng 3 (6) (8) (9)
5 constant) o the correspondng orm can be obtaned usng the algorthm descrbed n the Res.[], [] along wth substtuton (). For example, expressons or operators C and N ormng the bass or Hamltonan o nteracton n the Foldy-Wouthuysen representaton (see [], []) can be wrtten n the ollowng orm or Drac matrces n the chral representaton: chr chr even C = ( U ) qα B ( U ) = qr( B LB L) R qr( αb LαBL) R; γβ m σ p () L=, R= σ p E chr chr ( ) ( ) odd N = U qα B U = qr( LB B L) R qr( LαB αbl) R () even, odd n expressons (), () denote the even and odd parts o the correspondng operator (see [], []). Thus, the general Foldy-Wouthuysen transormaton wth Drac matrces n the chr o chr сhr сhr сhr chral representaton U = ( U ) ( + δ + δ + δ3 +...), as well as the ermon Hamltonan n the Foldy-Wouthuysen representaton chr chr chr 3 chr = γ H E qk q K q K en en can be obtaned rom the correspondng expressons or U, H wth Drac matrces n the energy representaton (see [], []) wth substtuton m σ p, β γ. Certanly, physcal results do not depend on the chosen representaton o Drac matrces. A researcher chooses a certan representaton or the sake o convenence. The Appendx below gves examples o calculatons or two quantum eletrodynamcs processes n Foldy-Wouthusyen representaton usng Drac matrces n the chral representaton.
6 6 Reerences. V.P.Neznamov. Physcs o Partcles and Nucle,Vol.37, N, pp.86-3, (6).. V.P.Neznamov. Voprosy Atomno Nauk I Tekhnk. Ser.: Teoretcheskaya Prkladnaya Fzka. 4. Issues -. P.4; hep-th/4. 3. L.L.Foldy and S.A.Wouthuysen, Phys.Rev 78, 9 (9). 4. V.P.Neznamov. The Standard Model n the moded Foldy-Wouthuysen representaton, hep-th/447, ().
7 7 Appendx Calculaton o some quantum electrodynamcs processes n representaton wth Drac matrces n the chral representaton [], []. The matrx elements and calculatons below are gven usng the notatons rom Ze. Electron scatterng n Coulomb eld s A(x) =. 4π х Feynman dagram o the process s shown n Fg.. Fgure. Electron scatterng n Coulomb eld S = d x( Ψ (x,p,s )) K A Ψ (x,p,s ) = 4 ( + ) ( + ) δ (E E ) o U s p C A p U s = < > = ( π ) Ze δ(e E ) E σ p γ βm γ βm + E + σ p = U s U s,q = p p. q ( π ) E E + σ p E + σ p E K The entry K A made or the sake o convenence actually means that wth A ( x) =. In other words, A ( ) occupes the postons determned by A K expresson (6) rom the paper []. The same s vald or the entry C A. The transton rom K A to C A has been made accordng to expresson (39) rom the paper []. x
8 8 Then, havng the matrx element S and usng the common methods one can obtan the derental Mott scatterng cross-secton that takes the orm o the Rutherord cross-secton n non-relatvstc case. dσ 4Z α E E σ p m + σ p m = Sp dω q E ( E σ p )( E σ p) E ( E σ p)( E σ p ) σ p ; E p e E = E = E,p= p = p,p p = p cos θ, β =, α = ; E 4π Further transormatons gve us the derental Mott scatterng cross-secton: dσ Z α θ = β sn d 4θ Ω. 4p β sn. The sel-energy o electron n the second-order perturbaton theory. Feynman dagrams o the processes are shown n Fg.. Fgure. The electron sel-energy
9 4 ( ) dk ( p) = K 4 ( p; p k; ν ) K ( p k; p; ) = ν = k p k βe p k ( π) β E( p) k E( p k) 9 ( ) ( + β ) K ( p; p k; ) ν = K ( p k; p; ν = ) + ( + β ) ( ; ) + C p p k C ( p k; p) + β E( p) k E( p k) ( β ) + N ( p; p k) ( ; ) ( ) ( ) N p E p k E p k k p β + ( + ) ( + ) In vew o βψ ψ,) or a ree electron (p =m ) the rst two tems n the = ntegrand are mutually annhlatng and expressons C ( p;p )C ( p ;p), N ( p;p )N ( p ;p) have the orm C ( p;p )C ( p ;p) = E σ p m E σ p m + + σ p = E + ( E σ p)( E σ p) E ( E+ σ p)( σ p) E σ p m m E + σ p m m E α α α α ( E σ p) ( E σ p) E + + ( E+ σ p) ( σ p) σ p = ( EE+ pp+ m ), E EE p = p k, E = m + ( p k ) ; N ( p;p )N ( p ;p) = E σ p γβm γβm E σ p γβm γβm + + σ p = E ( E σ p) ( E σ p) E ( E σ p) ( E σ p) E σ p γβm γβm E+ σ p γβm γβm α α α α E E + σ p E+ σ p E E+ σ p E + σ p σ p = ( EE+ pp+ m ). E EE
10 Hence, 4 () e d k pk+ m ( р) = 4 E( p) ( π ) k [( p k) m ] and, wth regard to spnor normalzaton, ths expresson s smlar to the expresson descrbng the mass operator n Drac representaton.
ECEN 5005 Crystals, Nanocrystals and Device Applications Class 19 Group Theory For Crystals
ECEN 5005 Crystals, Nanocrystals and Devce Applcatons Class 9 Group Theory For Crystals Dee Dagram Radatve Transton Probablty Wgner-Ecart Theorem Selecton Rule Dee Dagram Expermentally determned energy
More informationColored and electrically charged gauge bosons and their related quarks
Colored and electrcally charged gauge bosons and ther related quarks Eu Heung Jeong We propose a model of baryon and lepton number conservng nteractons n whch the two states of a quark, a colored and electrcally
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechancs Rajdeep Sensarma! sensarma@theory.tfr.res.n ecture #9 QM of Relatvstc Partcles Recap of ast Class Scalar Felds and orentz nvarant actons Complex Scalar Feld and Charge conjugaton
More informationwhere the sums are over the partcle labels. In general H = p2 2m + V s(r ) V j = V nt (jr, r j j) (5) where V s s the sngle-partcle potental and V nt
Physcs 543 Quantum Mechancs II Fall 998 Hartree-Fock and the Self-consstent Feld Varatonal Methods In the dscusson of statonary perturbaton theory, I mentoned brey the dea of varatonal approxmaton schemes.
More informationIngeniare. Revista Chilena de Ingeniería ISSN: Universidad de Tarapacá Chile
Ingenare. Revsta Chlena de Ingenería ISSN: 78-9 facng@uta.cl Unversdad de Tarapacá Chle Torres-Slva, éctor IRAC MATRICES IN CIRAL REPRESENTATION AN TE CONNECTION WIT TE ELECTRIC FIEL PARALLEL TO TE MAGNETIC
More informationA how to guide to second quantization method.
Phys. 67 (Graduate Quantum Mechancs Sprng 2009 Prof. Pu K. Lam. Verson 3 (4/3/2009 A how to gude to second quantzaton method. -> Second quantzaton s a mathematcal notaton desgned to handle dentcal partcle
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationNote on the Electron EDM
Note on the Electron EDM W R Johnson October 25, 2002 Abstract Ths s a note on the setup of an electron EDM calculaton and Schff s Theorem 1 Basc Relatons The well-known relatvstc nteracton of the electron
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationThis model contains two bonds per unit cell (one along the x-direction and the other along y). So we can rewrite the Hamiltonian as:
1 Problem set #1 1.1. A one-band model on a square lattce Fg. 1 Consder a square lattce wth only nearest-neghbor hoppngs (as shown n the fgure above): H t, j a a j (1.1) where,j stands for nearest neghbors
More informationRobert Eisberg Second edition CH 09 Multielectron atoms ground states and x-ray excitations
Quantum Physcs 量 理 Robert Esberg Second edton CH 09 Multelectron atoms ground states and x-ray exctatons 9-01 By gong through the procedure ndcated n the text, develop the tme-ndependent Schroednger equaton
More informationQuantum Mechanics I - Session 4
Quantum Mechancs I - Sesson 4 Aprl 3, 05 Contents Operators Change of Bass 4 3 Egenvectors and Egenvalues 5 3. Denton....................................... 5 3. Rotaton n D....................................
More informationLecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The non-relativistic Schrödinger equation was obtained by noting that the Hamiltonian 2
P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The non-relatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons
More informationSupplemental document
Electronc Supplementary Materal (ESI) for Physcal Chemstry Chemcal Physcs. Ths journal s the Owner Socetes 01 Supplemental document Behnam Nkoobakht School of Chemstry, The Unversty of Sydney, Sydney,
More informationUniversity of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2014
Lecture 16 8/4/14 Unversty o Washngton Department o Chemstry Chemstry 452/456 Summer Quarter 214. Real Vapors and Fugacty Henry s Law accounts or the propertes o extremely dlute soluton. s shown n Fgure
More information5.03, Inorganic Chemistry Prof. Daniel G. Nocera Lecture 2 May 11: Ligand Field Theory
5.03, Inorganc Chemstry Prof. Danel G. Nocera Lecture May : Lgand Feld Theory The lgand feld problem s defned by the followng Hamltonan, h p Η = wth E n = KE = where = m m x y z h m Ze r hydrogen atom
More informationPHYS 215C: Quantum Mechanics (Spring 2017) Problem Set 3 Solutions
PHYS 5C: Quantum Mechancs Sprng 07 Problem Set 3 Solutons Prof. Matthew Fsher Solutons prepared by: Chatanya Murthy and James Sully June 4, 07 Please let me know f you encounter any typos n the solutons.
More informationQuantum Field Theory III
Quantum Feld Theory III Prof. Erck Wenberg February 16, 011 1 Lecture 9 Last tme we showed that f we just look at weak nteractons and currents, strong nteracton has very good SU() SU() chral symmetry,
More informationEPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski
EPR Paradox and the Physcal Meanng of an Experment n Quantum Mechancs Vesseln C Nonnsk vesselnnonnsk@verzonnet Abstract It s shown that there s one purely determnstc outcome when measurement s made on
More informationRFNC-VNIIEF, , Sarov, N.Novgorod region
The Standard Model Without Higgs Bosons in the Fermion Sector V.P. Neznamov FNC-VNIIEF, 679, Sarov, N.Novgorod region e-mail: Neznamov@vniie.ru Abstract The paper addresses the construction o the Standard
More informationx = , so that calculated
Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to
More informationSummary with Examples for Root finding Methods -Bisection -Newton Raphson -Secant
Summary wth Eamples or Root ndng Methods -Bsecton -Newton Raphson -Secant Nonlnear Equaton Solvers Bracketng Graphcal Open Methods Bsecton False Poston (Regula-Fals) Newton Raphson Secant All Iteratve
More informationChange. Flamenco Chuck Keyser. Updates 11/26/2017, 11/28/2017, 11/29/2017, 12/05/2017. Most Recent Update 12/22/2017
Change Flamenco Chuck Keyser Updates /6/7, /8/7, /9/7, /5/7 Most Recent Update //7 The Relatvstc Unt Crcle (ncludng proof of Fermat s Theorem) Relatvty Page (n progress, much more to be sad, and revsons
More informationDynamics of a Superconducting Qubit Coupled to an LC Resonator
Dynamcs of a Superconductng Qubt Coupled to an LC Resonator Y Yang Abstract: We nvestgate the dynamcs of a current-based Josephson juncton quantum bt or qubt coupled to an LC resonator. The Hamltonan of
More informationOne can coose te bass n te 'bg' space V n te form of symmetrzed products of sngle partcle wavefunctons ' p(x) drawn from an ortonormal complete set of
8.54: Many-body penomena n condensed matter and atomc pyscs Last moded: September 4, 3 Lecture 3. Second Quantzaton, Bosons In ts lecture we dscuss second quantzaton, a formalsm tat s commonly used to
More informationCHARACTERISTICS OF COMPLEX SEPARATION SCHEMES AND AN ERROR OF SEPARATION PRODUCTS OUTPUT DETERMINATION
Górnctwo Geonżynera Rok 0 Zeszyt / 006 Igor Konstantnovch Mladetskj * Petr Ivanovch Plov * Ekaterna Nkolaevna Kobets * Tasya Igorevna Markova * CHARACTERISTICS OF COMPLEX SEPARATION SCHEMES AND AN ERROR
More informationHW #6, due Oct Toy Dirac Model, Wick s theorem, LSZ reduction formula. Consider the following quantum mechanics Lagrangian,
HW #6, due Oct 5. Toy Drac Model, Wck s theorem, LSZ reducton formula. Consder the followng quantum mechancs Lagrangan, L ψ(σ 3 t m)ψ, () where σ 3 s a Paul matrx, and ψ s defned by ψ ψ σ 3. ψ s a twocomponent
More informationLinear Momentum. Equation 1
Lnear Momentum OBJECTIVE Obsere collsons between two carts, testng or the conseraton o momentum. Measure energy changes durng derent types o collsons. Classy collsons as elastc, nelastc, or completely
More information19 Quantum electrodynamics
Modern Quantum Feld Theory 77 9 Quantum electrodynamcs 9. Gaugng the theory Consderng the Drac Lagrangan L D = @/ m we observed the presence of a U( symmetry! e, assocated wth the Noether current j µ =
More informationHomework & Solution. Contributors. Prof. Lee, Hyun Min. Particle Physics Winter School. Park, Ye
Homework & Soluton Prof. Lee, Hyun Mn Contrbutors Park, Ye J(yej.park@yonse.ac.kr) Lee, Sung Mook(smlngsm0919@gmal.com) Cheong, Dhong Yeon(dhongyeoncheong@gmal.com) Ban, Ka Young(ban94gy@yonse.ac.kr) Ro,
More information763622S ADVANCED QUANTUM MECHANICS Solution Set 1 Spring c n a n. c n 2 = 1.
7636S ADVANCED QUANTUM MECHANICS Soluton Set 1 Sprng 013 1 Warm-up Show that the egenvalues of a Hermtan operator  are real and that the egenkets correspondng to dfferent egenvalues are orthogonal (b)
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationSolution 1 for USTC class Physics of Quantum Information
Soluton 1 for 018 019 USTC class Physcs of Quantum Informaton Shua Zhao, Xn-Yu Xu and Ka Chen Natonal Laboratory for Physcal Scences at Mcroscale and Department of Modern Physcs, Unversty of Scence and
More informationNon-interacting Spin-1/2 Particles in Non-commuting External Magnetic Fields
EJTP 6, No. 0 009) 43 56 Electronc Journal of Theoretcal Physcs Non-nteractng Spn-1/ Partcles n Non-commutng External Magnetc Felds Kunle Adegoke Physcs Department, Obafem Awolowo Unversty, Ile-Ife, Ngera
More informationRepresentation theory and quantum mechanics tutorial Representation theory and quantum conservation laws
Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac
More informationYukawa Potential and the Propagator Term
PHY304 Partcle Physcs 4 Dr C N Booth Yukawa Potental an the Propagator Term Conser the electrostatc potental about a charge pont partcle Ths s gven by φ = 0, e whch has the soluton φ = Ths escrbes the
More informationThe optimal delay of the second test is therefore approximately 210 hours earlier than =2.
THE IEC 61508 FORMULAS 223 The optmal delay of the second test s therefore approxmately 210 hours earler than =2. 8.4 The IEC 61508 Formulas IEC 61508-6 provdes approxmaton formulas for the PF for smple
More informationPhys304 Quantum Physics II (2005) Quantum Mechanics Summary. 2. This kind of behaviour can be described in the mathematical language of vectors:
MACQUARIE UNIVERSITY Department of Physcs Dvson of ICS Phys304 Quantum Physcs II (2005) Quantum Mechancs Summary The followng defntons and concepts set up the basc mathematcal language used n quantum mechancs,
More informationConstraints on multiparticle production in scalar field theory from classical simulations
EPJ Web o Conerences 191, 221 (218) https://do.org/1.151/epjcon/218191221 Constrants on multpartcle producton n scalar eld theory rom classcal smulatons S.V. Demdov 1,2, and B.R. Farkhtdnov 1,2, 1 Insttute
More informationApplied Nuclear Physics (Fall 2004) Lecture 23 (12/3/04) Nuclear Reactions: Energetics and Compound Nucleus
.101 Appled Nuclear Physcs (Fall 004) Lecture 3 (1/3/04) Nuclear Reactons: Energetcs and Compound Nucleus References: W. E. Meyerhof, Elements of Nuclear Physcs (McGraw-Hll, New York, 1967), Chap 5. Among
More informationMathematical Preparations
1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the
More informationarxiv:cond-mat/ v2 [cond-mat.mes-hall] 3 Jan 2006
arxv:cond-mat/0210519v2 [cond-mat.mes-hall] 3 Jan 2006 Non Equlbrum Green s Functons for Dummes: Introducton to the One Partcle NEGF equatons Magnus Paulsson Dept. of mcro- and nano-technology, NanoDTU,
More informationThe Dirac Monopole and Induced Representations *
The Drac Monopole and Induced Representatons * In ths note a mathematcally transparent treatment of the Drac monopole s gven from the pont of vew of nduced representatons Among other thngs the queston
More informationFermi-Dirac statistics
UCC/Physcs/MK/EM/October 8, 205 Fer-Drac statstcs Fer-Drac dstrbuton Matter partcles that are eleentary ostly have a type of angular oentu called spn. hese partcles are known to have a agnetc oent whch
More information5 The Rational Canonical Form
5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces
More informationFormulas for the Determinant
page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationLagrangian Field Theory
Lagrangan Feld Theory Adam Lott PHY 391 Aprl 6, 017 1 Introducton Ths paper s a summary of Chapter of Mandl and Shaw s Quantum Feld Theory [1]. The frst thng to do s to fx the notaton. For the most part,
More informationA particle in a state of uniform motion remain in that state of motion unless acted upon by external force.
The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,
More informationTranslational Equations of Motion for A Body Translational equations of motion (centroidal) for a body are m r = f.
Lesson 12: Equatons o Moton Newton s Laws Frst Law: A artcle remans at rest or contnues to move n a straght lne wth constant seed there s no orce actng on t Second Law: The acceleraton o a artcle s roortonal
More informationRate of Absorption and Stimulated Emission
MIT Department of Chemstry 5.74, Sprng 005: Introductory Quantum Mechancs II Instructor: Professor Andre Tokmakoff p. 81 Rate of Absorpton and Stmulated Emsson The rate of absorpton nduced by the feld
More informationChapter 5 rd Law of Thermodynamics
Entropy and the nd and 3 rd Chapter 5 rd Law o hermodynamcs homas Engel, hlp Red Objectves Introduce entropy. Derve the condtons or spontanety. Show how S vares wth the macroscopc varables,, and. Chapter
More informationA Theory of Weak Interaction Dynamics
Open Access Lbrary Journal 2016, Volume 3, e3264 ISSN Onlne: 2333-9721 ISSN Prnt: 2333-970 A Theory of Weak Interacton Dynamcs Elahu Comay Charactell Ltd., Tel-Avv, Israel How to cte ths paper: Comay,
More informationPhysics 30 Lesson 31 The Bohr Model of the Atom
Physcs 30 Lesson 31 The Bohr Model o the Atom I. Planetary models o the atom Ater Rutherord s gold ol scatterng experment, all models o the atom eatured a nuclear model wth electrons movng around a tny,
More informationPhysics 2A Chapter 3 HW Solutions
Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C
More informationDensity matrix. c α (t)φ α (q)
Densty matrx Note: ths s supplementary materal. I strongly recommend that you read t for your own nterest. I beleve t wll help wth understandng the quantum ensembles, but t s not necessary to know t n
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationarxiv: v2 [cond-mat.str-el] 5 Sep 2010
Classfcaton of Gapped Symmetrc Phases n 1D Spn Systems Xe Chen, 1 Zheng-Cheng Gu, 2 and Xao-Gang Wen 1, 3 1 Department of Physcs, Massachusetts Insttute of Technology, Cambrdge, Massachusetts 02139, USA
More informationComputational Biology Lecture 8: Substitution matrices Saad Mneimneh
Computatonal Bology Lecture 8: Substtuton matrces Saad Mnemneh As we have ntroduced last tme, smple scorng schemes lke + or a match, - or a msmatch and -2 or a gap are not justable bologcally, especally
More informationThe Dirac Equation for a One-electron atom. In this section we will derive the Dirac equation for a one-electron atom.
The Drac Equaton for a One-electron atom In ths secton we wll derve the Drac equaton for a one-electron atom. Accordng to Ensten the energy of a artcle wth rest mass m movng wth a velocty V s gven by E
More informationCHAPTER 4. Vector Spaces
man 2007/2/16 page 234 CHAPTER 4 Vector Spaces To crtcze mathematcs for ts abstracton s to mss the pont entrel. Abstracton s what makes mathematcs work. Ian Stewart The man am of ths tet s to stud lnear
More informationAn Algorithm to Solve the Inverse Kinematics Problem of a Robotic Manipulator Based on Rotation Vectors
An Algorthm to Solve the Inverse Knematcs Problem of a Robotc Manpulator Based on Rotaton Vectors Mohamad Z. Al-az*, Mazn Z. Othman**, and Baker B. Al-Bahr* *AL-Nahran Unversty, Computer Eng. Dep., Baghdad,
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationSlepton NLG (Non-linear gauge) in GRACE/SUSY-loop
Slepton NLG (Non-lnear gauge n GRACE/SUSY-loop M. Jbo (Speaker, T. Ishkawa and M. Kuroda 3 Chba Unversty o Coerce KEK 3 Mej GakunUnversty CPP, 3rd Coputatonal Partcle Physcs orkshop at KEK (Tsukuba, Japan,
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More information) is the unite step-function, which signifies that the second term of the right-hand side of the
Casmr nteracton of excted meda n electromagnetc felds Yury Sherkunov Introducton The long-range electrc dpole nteracton between an excted atom and a ground-state atom s consdered n ref. [1,] wth the help
More informationThe non-negativity of probabilities and the collapse of state
The non-negatvty of probabltes and the collapse of state Slobodan Prvanovć Insttute of Physcs, P.O. Box 57, 11080 Belgrade, Serba Abstract The dynamcal equaton, beng the combnaton of Schrödnger and Louvlle
More informationQuantum Particle Motion in Physical Space
Adv. Studes Theor. Phys., Vol. 8, 014, no. 1, 7-34 HIKARI Ltd, www.-hkar.co http://dx.do.org/10.1988/astp.014.311136 Quantu Partcle Moton n Physcal Space A. Yu. Saarn Dept. of Physcs, Saara State Techncal
More informationSome Notes on Field Theory
Some Notes on Feld Theory Eef van Beveren Centro de Físca Teórca Departamento de Físca da Faculdade de Cêncas e Tecnologa Unversdade de Combra Portugal http://cft.fs.uc.pt/eef May 20, 2014 Contents 1
More informationLecture 14: Forces and Stresses
The Nuts and Bolts of Frst-Prncples Smulaton Lecture 14: Forces and Stresses Durham, 6th-13th December 2001 CASTEP Developers Group wth support from the ESF ψ k Network Overvew of Lecture Why bother? Theoretcal
More informationQED: Quantum Electrodynamics
Chapter 2 QED: Quantum Electrodynamcs 2. Negatve-Energy States: Antpartcles 2.. Settng the Stage: Non-Relatvstc Quantum Mechancs In non-relatvstc Quantum Mechancs, t was seen that (n the standard poston
More informationFinite Difference Method
7/0/07 Instructor r. Ramond Rump (9) 747 698 rcrump@utep.edu EE 337 Computatonal Electromagnetcs (CEM) Lecture #0 Fnte erence Method Lecture 0 These notes ma contan coprghted materal obtaned under ar use
More informationInvestigating a Possible Dynamical Origin of the Electroweak Scale
Investgatng a Possble Dynamcal Orgn of the Electroweak Scale Unversty of Southern Denmark Master's Thess n Partcle Physcs Martn Rosenlyst Jørgensen CP 3 -Orgns Supervsed by Ass. Prof. Mads Toudal Frandsen,
More informationAppendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis
A Appendx for Causal Interacton n Factoral Experments: Applcaton to Conjont Analyss Mathematcal Appendx: Proofs of Theorems A. Lemmas Below, we descrbe all the lemmas, whch are used to prove the man theorems
More information1 Interactions and Green functions
Interactng Quantum Feld Theory 1 D. E. Soper 2 Unversty of Oregon Physcs 665, Quantum Feld Theory February 2001 1 Interactons and Green functons In these sectons, we dscuss perturbaton theory for the nteractng
More informationSymmetries and nonequilibrium thermodynamics
PHYSICAL REVIEW E 95, 06208 (207 Symmetres and nonequlbrum thermodynamcs Sergo Bordel * Laboratory o Cell Culture, Insttute o Cardology, Lthuanan Unversty o Health Scences, Sulėlų ave. 7, Kaunas, Lthuana
More informationElectrical double layer: revisit based on boundary conditions
Electrcal double layer: revst based on boundary condtons Jong U. Km Department of Electrcal and Computer Engneerng, Texas A&M Unversty College Staton, TX 77843-318, USA Abstract The electrcal double layer
More information9 Characteristic classes
THEODORE VORONOV DIFFERENTIAL GEOMETRY. Sprng 2009 [under constructon] 9 Characterstc classes 9.1 The frst Chern class of a lne bundle Consder a complex vector bundle E B of rank p. We shall construct
More informationBinding energy of a Cooper pairs with non-zero center of mass momentum in d-wave superconductors
Bndng energ of a Cooper pars wth non-zero center of mass momentum n d-wave superconductors M.V. remn and I.. Lubn Kazan State Unverst Kremlevsaa 8 Kazan 420008 Russan Federaton -mal: gor606@rambler.ru
More informationSolution 1 for USTC class Physics of Quantum Information
Soluton 1 for 017 018 USTC class Physcs of Quantum Informaton Shua Zhao, Xn-Yu Xu and Ka Chen Natonal Laboratory for Physcal Scences at Mcroscale and Department of Modern Physcs, Unversty of Scence and
More information1 Rabi oscillations. Physical Chemistry V Solution II 8 March 2013
Physcal Chemstry V Soluton II 8 March 013 1 Rab oscllatons a The key to ths part of the exercse s correctly substtutng c = b e ωt. You wll need the followng equatons: b = c e ωt 1 db dc = dt dt ωc e ωt.
More informationElectron scattering form factors with core-polarization effects on 6 Li nucleus. Khalid S. Jassim and Fouad A. MAJEED
Kua s Frst Conerence or physcs, specal Issue (ISSN 077-5830), 6-7 October 00 Electron scatterng orm actors wth core-polarzaton eects on 6 L nucleus. Khald S. Jassm and Fouad A. MAJEED,Department o Physcs,
More informationSIO 224. m(r) =(ρ(r),k s (r),µ(r))
SIO 224 1. A bref look at resoluton analyss Here s some background for the Masters and Gubbns resoluton paper. Global Earth models are usually found teratvely by assumng a startng model and fndng small
More informationPh 219a/CS 219a. Exercises Due: Wednesday 23 October 2013
1 Ph 219a/CS 219a Exercses Due: Wednesday 23 October 2013 1.1 How far apart are two quantum states? Consder two quantum states descrbed by densty operators ρ and ρ n an N-dmensonal Hlbert space, and consder
More informationChapter 3 Differentiation and Integration
MEE07 Computer Modelng Technques n Engneerng Chapter Derentaton and Integraton Reerence: An Introducton to Numercal Computatons, nd edton, S. yakowtz and F. zdarovsky, Mawell/Macmllan, 990. Derentaton
More informationEEE 241: Linear Systems
EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationThis chapter illustrates the idea that all properties of the homogeneous electron gas (HEG) can be calculated from electron density.
1 Unform Electron Gas Ths chapter llustrates the dea that all propertes of the homogeneous electron gas (HEG) can be calculated from electron densty. Intutve Representaton of Densty Electron densty n s
More informationV.C The Niemeijer van Leeuwen Cumulant Approximation
V.C The Nemejer van Leeuwen Cumulant Approxmaton Unfortunately, the decmaton procedure cannot be performed exactly n hgher dmensons. For example, the square lattce can be dvded nto two sublattces. For
More informationHigh-Field Laser Physics (2,3) ETH Zürich Spring Semester 2010 H. R. Reiss
Hgh-Feld Laser Physcs (,3) ETH Zürch Sprng Semester 1 H. R. Ress DIPOLE APPROXIMATION (LONG-WAVELENGTH APPROXIMATION) The dpole approxmaton [or electrc-dpole approxmaton (EDA)] s used almost unversally
More informationPOINCARE ALGEBRA AND SPACE-TIME CRITICAL DIMENSIONS FOR PARASPINNING STRINGS
POINCARE ALGEBRA AND SPACE-TIME CRITICAL DIMENSIONS FOR PARASPINNING STRINGS arxv:hep-th/0055v 7 Oct 00 N.BELALOUI, AND H.BENNACER LPMPS, Département de Physque, Faculté des Scences, Unversté Mentour Constantne,
More informationIndeterminate pin-jointed frames (trusses)
Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all
More informationThe Lorentz group. Generate the group SO(3,1) To construct representa;ons a more convenient (non- Hermi;an) basis is. i j ijk k. j ijk k.
Fermons S 1 2 The Lorentz group Rotatons J Boosts K [ J, J ] ε J [ J, K ] ε K [ K, K ] ε J j jk k j jk k j jk k } Generate the group SO(3,1) ( M ( x x ) J M K M ) 1 ρσ ρ σ x σ ρ x 2 ε jk jk 0 To construct
More informationSnce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t
8.5: Many-body phenomena n condensed matter and atomc physcs Last moded: September, 003 Lecture. Squeezed States In ths lecture we shall contnue the dscusson of coherent states, focusng on ther propertes
More informationSpring Force and Power
Lecture 13 Chapter 9 Sprng Force and Power Yeah, energy s better than orces. What s net? Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi IN THIS CHAPTER, you wll learn how to solve problems
More information1 Vectors over the complex numbers
Vectors for quantum mechancs 1 D. E. Soper 2 Unversty of Oregon 5 October 2011 I offer here some background for Chapter 1 of J. J. Sakura, Modern Quantum Mechancs. 1 Vectors over the complex numbers What
More informationTREATMENT OF THE TURNING POINT IN ADK-THEORY INCLUDING NON-ZERO INITIAL MOMENTA
41 Kragujevac J. Sc. 5 (00) 41-46. TREATMENT OF THE TURNING POINT IN ADK-THEORY INCLUDING NON-ZERO INITIAL MOMENTA Vladmr M. Rstć a and Tjana Premovć b a Faculty o Scence, Department o Physcs, Kragujevac
More informationExperiment 5 Elastic and Inelastic Collisions
PHY191 Experment 5: Elastc and Inelastc Collsons 7/1/011 Page 1 Experment 5 Elastc and Inelastc Collsons Readng: Bauer&Westall: Chapter 7 (and 8, or center o mass deas) as needed Homework 5: turn n the
More informationEinstein-Podolsky-Rosen Paradox
H 45 Quantum Measurement and Spn Wnter 003 Ensten-odolsky-Rosen aradox The Ensten-odolsky-Rosen aradox s a gedanken experment desgned to show that quantum mechancs s an ncomplete descrpton of realty. The
More information