Previous knowlegde required. Spherical harmonics and some of their properties. Angular momentum. References. Angular momentum operators
|
|
- Georgiana Warren
- 6 years ago
- Views:
Transcription
1 // vious owg ui phica haoics a so o thi poptis Goup thoy Quatu chaics pctoscopy H. Haga 8 phica haoics Rcs Bia. iv «Iucib Tso thos A Itouctio o chists» Acaic ss D.A. c Quai.D. io «hii hysiu Appoch oécuai» Duo R.cWy «Quatu chaics: thos a basic appicatios» gao ss Agua otu Rotatioa spctoscopy Hyog ato pi NR ER tc pusio pi obit coupig ysta i phica haoics phica haoics Agua otu opatos ( ) z x i y x i y i i [ x y ] iz [ y z ] ix [ z x ] iy ( ) ( ) ( ) ( ) is abitay ovtioa phas choic (oo a hoty) :. osuc : phica haoics Agua otu opatos x y z si cot cosϕ i z y i ϕ y z x cos cot siϕ i x z i ϕ x x y i y x i ϕ x y z si si si ϕ si si si phica haoics z
2 // phica haoics phica haoics ( ) Θ( ) Φ( ) i ( ) A Φ Φ i ( ) Φ is oaiz Φ ( ) Φ( ) A A A phica haoics Th uctio o ca b xpss as a g poyoia Θ ( ) ( x) x cos x ( x) phica haoics 8 ( x) x ( x) ( x ) ( x) ( x x) phica haoics phica haoics Obita agua otu : a a itgs Noaisatio a othogoaity si opx cougat ( ) i ( cos ) ( ) phica haoics phica haoics phica haoics yty poptis Th stats > o ix spa a iucib pstatio D o th iiit otatio goup R. This ipis that i o appis a abitay otatio D(βγ) to th stat > o obtais a ia cobiatio o th copt st o stats > with th sa. phica haoics phica haoics [ ] R x y R z i [ ] yz R [ ] xz R i [ ] xy R phica haoics
3 // oupig o two agua ota Exap : ato with ctos o i a p obita o i a p obita Not : o spi-obit coupig p obita : p obita : - possib pouct stats o th o : tc phica syty (R ) wavuctios a iguctios o ( ( ) ) a z z z phica haoics oupig o two agua ota yty popty: p p taso as D () pouct stat tasos as D () D (). Ga u: D () D () D () D (-) D ( - ) D () D () D () D () D () () othooa uctios which a ia cobiatios phica haoics ob: oupig o two agua ota cacuat th coicits (vcto coupig o bsch-goa coicits) ( ) ( ) ( ) ( ) ( ) ( ) phica haoics oupig o two agua ota I th cas o uivat p ctos w ot that th putatio o th abs a av th stats o a uchag whi thy chag sig o. o a w a sigt spi uctio (atisytic) a o a tipt spi uctio (sytic): ts A D o th coiguatio p. phica haoics oupig o two agua ota phica haoics oupig o two agua ota Vcto coupig coicits phica haoics 8 Th agua otu iguctios o a othooa st. utipyig both sis o th uatio abov by > a itgatig o obtais : scaa pouct (a vau)
4 // phica haoics oupig o two agua ota optis: uss a - (tiag u) Ra scaa pouct ipis that <a b> <b a> phica haoics oupig o two agua ota Th othogoaity o th vcto-coupig atix ipis phica haoics - sybo Th - sybos ca b cacuat xacty. I th past tabs o ths sybos hav b pubish toay o is - sybo cacuatos o th wb g o i athatica. Not th chag o sig o btw th vcto coupig coicit a th sybo. phica haoics - sybo phica haoics - sybo phica haoics - sybo acuatio o so o th vcto coupig ts i th atix abov
5 // phica haoics - sybo o syty poptis o th - sybos: Ev putatio sa sig O putatio: i i ust b v i phica haoics - sybo si Itga o sphica haoics Not: o copx cougat i this xpssio phica haoics Th Wig-Ecat Tho I spctoscopy w us th oowig syty popty: Th atix t b i a O ψ ψ uss i Γ Γ Γ Howv this tho os ot xpoit a syty poptis. Th Wig-Ecat Tho ats atix ts to coupig coicits K T K is a costat ipt o i I th vcto coupig coicit is pac by a - sybo o obtais T T phica haoics 8 Th Wig-Ecat Tho T T Th ast t is ca a uc atix t. psts ay aitioa uatu ub cssay to spciy th stat. phica haoics Th Wig-Ecat Tho I th opato is a sphica haoic opatig o stats > th uc atix ts ca b cacuat as oows: Usig th ga itgatio o sphica haoics Usig th Wig Ecat tho phica haoics Th Wig-Ecat Tho I o uss Racah s oais sphica haoics:.8..
6 // phica haoics atix ts si si A tab o ths vaus ca b ou i:.ugao.taab a H.Kaiua utipts o Tasitio-ta Ios i ystas Acaic ss Nw o a oo p.. phica haoics - sybos Itouc to scib th coupig o agua ota. It ca b xpss i ts o - sybos: ) ( a Th Wig -sybos a tu by th athatica uctio ixybo[ ]. aso: phica haoics Ecto-cto pusio atix ts o itctoic pusio: Expasio o / usig g poyoias: cos cos ϕ ϕ ω ω > < phica haoics Ecto-cto pusio t us i th opatos: / cos ω This cospos to a scaa pouct o th vctos (i). > < This tatt aows to spaat th aia cotibutio o th agua pat xpss by th. phica haoics Ecto-cto pusio : xap p s s R R p p > < Th itgas a ca at itgas. phica haoics ouob spittig I this cas t. W ca cacuat th ativ gy o th ts () () t D () D E E E Th gou stat t is giv by Hu s u: -axiu spi utipicity (a N spis a paa i N< i p N ) -ach poctio o th obita otu is th agst aow by th xcusio u.n (p ) : (-) ½ ½ Etat oata
The Real Hydrogen Atom
T Ra Hydog Ato ov ad i fist od gt iddt of :.6V a us tubatio toy to dti: agti ffts si-obit ad yfi -A ativisti otios Aso av ab sift du to to sfitatio. Nd QD Dia q. ad dds o H wavfutio at sou of ti fid. Vy
More informationToday s topic 2 = Setting up the Hydrogen Atom problem. Schematic of Hydrogen Atom
Today s topic Sttig up th Hydog Ato pobl Hydog ato pobl & Agula Motu Objctiv: to solv Schödig quatio. st Stp: to dfi th pottial fuctio Schatic of Hydog Ato Coulob s aw - Z 4ε 4ε fo H ato Nuclus Z What
More informationThe Hydrogen Atom. Chapter 7
Th Hyog Ato Chapt 7 Hyog ato Th vy fist pobl that Schöig hislf tackl with his w wav quatio Poucig th oh s gy lvls a o! lctic pottial gy still plays a ol i a subatoic lvl btw poto a lcto V 4 Schöig q. fo
More informationECEN 5005 Crystals, Nanocrystals and Device Applications Class 14 Group Theory For Crystals
ECEN 5005 Cryta Naocryta ad Dvic Appicatio Ca 14 Group Thory For Cryta Spi Aguar Motu Quatu Stat of Hydrog-ik Ato Sig Ectro Cryta Fid Thory Fu Rotatio Group 1 Spi Aguar Motu Spi itriic aguar otu of ctro
More informationChapter 11 Solutions ( ) 1. The wavelength of the peak is. 2. The temperature is found with. 3. The power is. 4. a) The power is
Chapt Solutios. Th wavlgth of th pak is pic 3.898 K T 3.898 K 373K 885 This cospods to ifad adiatio.. Th tpatu is foud with 3.898 K pic T 3 9.898 K 50 T T 5773K 3. Th pow is 4 4 ( 0 ) P σ A T T ( ) ( )
More informationand integrated over all, the result is f ( 0) ] //Fourier transform ] //inverse Fourier transform
NANO 70-Nots Chapt -Diactd bams Dlta uctio W d som mathmatical tools to dvlop a physical thoy o lcto diactio. Idal cystals a iiit this, so th will b som iiitis lii about. Usually, th iiit quatity oly ists
More informationRelation between wavefunctions and vectors: In the previous lecture we noted that:
Rlatio tw wavuctios a vctos: I th pvious lctu w ot that: * Ψm ( x) Ψ ( x) x Ψ m Ψ m which claly mas that th commo ovlap itgal o th lt must a i pouct o two vctos. I what ss is ca w thi o th itgal as th
More informationHomework 1: Solutions
Howo : Solutos No-a Fals supposto tst but passs scal tst lthouh -f th ta as slowss [S /V] vs t th appaac of laty alty th path alo whch slowss s to b tat to obta tavl ts ps o th ol paat S o V as a cosquc
More informationInstrumentation for Characterization of Nanomaterials (v11) 11. Crystal Potential
Istumtatio o Chaactizatio o Naomatials (v). Cystal Pottial Dlta uctio W d som mathmatical tools to dvlop a physical thoy o lcto diactio om cystal. Idal cystals a iiit this, so th will b som iiitis lii
More informationChemistry 342 Spring, The Hydrogen Atom.
Th Hyrogn Ato. Th quation. Th first quation w want to sov is φ This quation is of faiiar for; rca that for th fr partic, w ha ψ x for which th soution is Sinc k ψ ψ(x) a cos kx a / k sin kx ± ix cos x
More informationQuantum Mechanics Lecture Notes 10 April 2007 Meg Noah
The -Patice syste: ˆ H V This is difficut to sove. y V 1 ˆ H V 1 1 1 1 ˆ = ad with 1 1 Hˆ Cete of Mass ˆ fo Patice i fee space He Reative Haitoia eative coodiate of the tota oetu Pˆ the tota oetu tota
More informationFI 3103 Quantum Physics
7//7 FI 33 Quantum Physics Axan A. Iskana Physics of Magntism an Photonics sach oup Institut Tknoogi Banung Schoing Equation in 3D Th Cnta Potntia Hyognic Atom 7//7 Schöing quation in 3D Fo a 3D pobm,
More informationPLS-CADD DRAWING N IC TR EC EL L RA IVE ) R U AT H R ER 0. IDT FO P 9-1 W T OO -1 0 D EN C 0 E M ER C 3 FIN SE W SE DE EA PO /4 O 1 AY D E ) (N W AN N
A IV ) H 0. IT FO P 9-1 W O -1 0 C 0 M C FI S W S A PO /4 O 1 AY ) ( W A 7 F 4 H T A GH 1 27 IGO OU (B. G TI IS 1/4 X V -S TO G S /2 Y O O 1 A A T H W T 2 09 UT IV O M C S S TH T ) A PATO C A AY S S T
More informationIntro to QM due: February 8, 2019 Problem Set 12
Intro to QM du: Fbruary 8, 9 Prob St Prob : Us [ x i, p j ] i δ ij to vrify that th anguar ontu oprators L i jk ɛ ijk x j p k satisfy th coutation rations [ L i, L j ] i k ɛ ijk Lk, [ L i, x j ] i k ɛ
More informationShape parameterization
Shap paatization λ ( θ, φ) α ( θ ) λµ λµ, φ λ µ λ axially sytic quaupol axially sytic octupol λ α, α ± α ± λ α, α ±,, α, α ±, Inian Institut of Tchnology opa Hans-Jügn Wollshi - 7 Octupol collctivity coupling
More informationPower Spectrum Estimation of Stochastic Stationary Signals
ag of 6 or Spctru stato of Stochastc Statoary Sgas Lt s cosr a obsrvato of a stochastc procss (). Ay obsrvato s a ft rcor of th ra procss. Thrfor, ca say:
More informationOutlines. Part 1 Fundamentals. Atomic Structure---An Overview. The Bohr Model 2/2/2011. PHY5937:ST-Nanofabrication using FIB. v r
//0 PHY5937:ST-Naofabicaio usig FIB Chow Dp. of Physics Uivsiy of Ca Foia Ouis Pa Fuaas of Io-Soi Iacios Pa Physics of FIB Isu Pa 3 Appicaio of FIB Isu Pa 4 Naofabicaio wih cos a phoos Pa 5 Oh io ba achiqus
More informationP a g e 5 1 of R e p o r t P B 4 / 0 9
P a g e 5 1 of R e p o r t P B 4 / 0 9 J A R T a l s o c o n c l u d e d t h a t a l t h o u g h t h e i n t e n t o f N e l s o n s r e h a b i l i t a t i o n p l a n i s t o e n h a n c e c o n n e
More informationWindowing in FIR Filter Design. Design Summary and Examples
Lctur 3 Outi: iowig i FIR Fitr Dsig. Dsig Summary a Exams Aoucmts: Mitrm May i cass. i covr through FIR Fitr Dsig. 4 ost, 5% ogr tha usua, 4 xtra ays to comt (u May 8) Mor tais o say Thr wi b o aitioa
More informationClassical Theory of Fourier Series : Demystified and Generalised VIVEK V. RANE. The Institute of Science, 15, Madam Cama Road, Mumbai
Clssil Thoy o Foi Sis : Dmystii Glis VIVEK V RANE Th Istitt o Si 5 Mm Cm Ro Mmbi-4 3 -mil ss : v_v_@yhoooi Abstt : Fo Rim itgbl tio o itvl o poit thi w i Foi Sis t th poit o th itvl big ot how wh th tio
More informationCreative Office / R&D Space
Ga 7th A t St t S V Nss A issi St Gui Stt Doos St Noiga St Noiga St a Csa Chaz St Tava St Tava St 887 itt R Buig, CA 9400 a to Po St B i Co t Au St B Juipo Sa B sb A Siv Si A 3 i St t ssi St othhoo Wa
More information8(4 m0) ( θ ) ( ) Solutions for HW 8. Chapter 25. Conceptual Questions
Solutios for HW 8 Captr 5 Cocptual Qustios 5.. θ dcrass. As t crystal is coprssd, t spacig d btw t plas of atos dcrass. For t first ordr diffractio =. T Bragg coditio is = d so as d dcrass, ust icras for
More informationSchool of Aerospace Engineering Origins of Quantum Theory. Measurements of emission of light (EM radiation) from (H) atoms found discrete lines
Ogs of Quatu Thoy Masuts of sso of lght (EM adato) fo (H) atos foud dsct ls 5 4 Abl to ft to followg ss psso ν R λ c λwavlgth, νfqucy, cspd lght RRydbg Costat (~09,7677.58c - ),,, +, +,..g.,,.6, 0.6, (Lya
More information2 tel
Us. Timeless, sophisticated wall decor that is classic yet modern. Our style has no limitations; from traditional to contemporar y, with global design inspiration. The attention to detail and hand- craf
More informationPHY 309: QUANTUM MECHANICS I (3 UNITS) COURSE GUIDE
PHY 9: QUANTUM MECHANICS I UNITS COURSE GUIDE I Qutu Mchics I PHY 9, you t bout th iqucis of Cssic Mchics th ffots by Physicists to ss th shotcoigs o th ptfo of Qutu Mchics. You hv t oo t th thtic foutio
More informationCh. 6 Free Electron Fermi Gas
Ch. 6 lcto i Gas Coductio lctos i a tal ov fl without scattig b io cos so it ca b cosidd as if walitactig o f paticls followig idiac statistics. hfo th coductio lctos a fqutl calld as f lcto i gas. Coductio
More informationNotes 12 Asymptotic Series
ECE 6382 Fall 207 David R. Jackso otes 2 Asymptotic Series Asymptotic Series A asymptotic series (as ) is of the form a ( ) f as = 0 or f a + a a + + ( ) 2 0 2 ote the asymptotically equal to sig. The
More information* Meysam Mohammadnia Department of Nuclear Engineering, East Tehran Branch, Islamic Azad University, Tehran, Iran *Author for Correspondence
Indian Jouna o Fundanta and ppid Li Scincs ISSN: 65 Onin n Opn ccss, Onin Intnationa Jouna vaiab at www.cibtch.og/sp.d/js///js.ht Vo. S, pp. 7-/Mysa Rsach tic CQUISITION N NLYSIS OF FLUX N CURRENT COEFFICIENTS
More informationA L A BA M A L A W R E V IE W
A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N
More informationMIL-DTL-5015 Style Circular Connectors
--01 y i oo /- i ooi --01 /- i i oi i --01 oi iio oo. io oiio o, oi i o i o o - -o-. /- i i oi i 12 i o iz o 10 o, i o o iz o #1 o #0 7 i o i o oo i iy o iio. o, i oo, i, oiio i i i o y oi --01, - o: i
More informationMultidimensional Laplace Transforms over Quaternions, Octonions and Cayley-Dickson Algebras, Their Applications to PDE
dac i Pu Mathatic 63-3 http://dxdoiog/436/ap3 Pubihd Oi Mach (http://scipog/oua/ap) Mutidiioa Lapac Tafo o uatio Octoio ad Cayy-Dico gba Thi ppicatio to PDE Sgy Victo Ludoy Dpatt of ppid Mathatic Moco
More informationENGG 1203 Tutorial. Difference Equations. Find the Pole(s) Finding Equations and Poles
ENGG 03 Tutoial Systms ad Cotol 9 Apil Laig Obctivs Z tasfom Complx pols Fdbac cotol systms Ac: MIT OCW 60, 6003 Diffc Equatios Cosid th systm pstd by th followig diffc quatio y[ ] x[ ] (5y[ ] 3y[ ]) wh
More informationF(F \m 1,m 2, ), which is suitable for large, small, r N and 0<q<1, where. is the incomplete Gamma function ratio and
A Asyptotic Expasio fo th No-tal -Distibtio y Jia azah ahoo Dpatt of Mathatics ollg of Ecatio 6 Abstact A asyptotic xpasio is iv fo th o-ctal -istibtio (\ ) hich is sitabl fo lag sall N a
More informationMagnetic effects and the peculiarity of the electron spin in Atoms
Magtic ffcts ad t pculiaity of t lcto spi i Atos Pit Za Hdik otz Nobl Piz 90 Otto t Nobl 9 Wolfgag Pauli Nobl 95 ctu Nots tuctu of Matt: Atos ad Molculs; W. Ubacs T obital agula otu of a lcto i obit iclassical
More informationIn the name of Allah Proton Electromagnetic Form Factors
I th a of Allah Poto Elctoagtc o actos By : Maj Hazav Pof A.A.Rajab Shahoo Uvsty of Tchology Atoc o acto: W cos th tactos of lcto bas wth atos assu to b th gou stats. Th ct lcto ay gt scatt lastcally wth
More information19TH Ave. Monterey Blvd. 87th St DALY CITY. St Fran
t St Mak 19TH A a D i Oa k a Oc a Mt B A Siv Si A St t 3 v aa M ui i B Da Gu up C a Pkw v aa si at hg ut 87th St v B ho s Ba Gu a au p Hi Missi St aa So Highights Siv Bothhoo Wa Joh A Mi ssi St v Juipo
More informationT h e C S E T I P r o j e c t
T h e P r o j e c t T H E P R O J E C T T A B L E O F C O N T E N T S A r t i c l e P a g e C o m p r e h e n s i v e A s s es s m e n t o f t h e U F O / E T I P h e n o m e n o n M a y 1 9 9 1 1 E T
More informationDISCRETE-TIME RANDOM PROCESSES
DISCRT-TIM RNDOM PROCSSS Rado Pocsss Dfiitio; Ma ad vaiac; autocoatio ad autocovaiac; Ratiosip btw ado vaiabs i a sig ado pocss; Coss-covaiac ad coss-coatio of two ado pocsss; Statioa Rado Pocsss Statioait;
More informationEE243 Advanced Electromagnetic Theory Lec # 22 Scattering and Diffraction. Reading: Jackson Chapter 10.1, 10.3, lite on both 10.2 and 10.
Appid M Fa 6, Nuuth Lctu # V //6 43 Advancd ctomagntic Thoy Lc # Scatting and Diffaction Scatting Fom Sma Obcts Scatting by Sma Dictic and Mtaic Sphs Coction of Scatts Sphica Wav xpansions Scaa Vcto Rading:
More informationShor s Algorithm. Motivation. Why build a classical computer? Why build a quantum computer? Quantum Algorithms. Overview. Shor s factoring algorithm
Motivation Sho s Algoith It appas that th univs in which w liv is govnd by quantu chanics Quantu infoation thoy givs us a nw avnu to study & tst quantu chanics Why do w want to build a quantu coput? Pt
More informationMath 7409 Homework 2 Fall from which we can calculate the cycle index of the action of S 5 on pairs of vertices as
Math 7409 Hoewok 2 Fall 2010 1. Eueate the equivalece classes of siple gaphs o 5 vetices by usig the patte ivetoy as a guide. The cycle idex of S 5 actig o 5 vetices is 1 x 5 120 1 10 x 3 1 x 2 15 x 1
More informationBellman-F o r d s A lg o r i t h m The id ea: There is a shortest p ath f rom s to any other verte that d oes not contain a non-negative cy cle ( can
W Bellman Ford Algorithm This is an algorithm that solves the single source shortest p ath p rob lem ( sssp ( f ind s the d istances and shortest p aths f rom a source to all other nod es f or the case
More informationPriority Search Trees - Part I
.S. 252 Pro. Rorto Taassa oputatoal otry S., 1992 1993 Ltur 9 at: ar 8, 1993 Sr: a Q ol aro Prorty Sar Trs - Part 1 trouto t last ltur, w loo at trval trs. or trval pot losur prols, ty us lar spa a optal
More informationToday s topics. How did we solve the H atom problem? CMF Office Hours
CMF Offc ous Wd. Nov. 4 oo-p Mo. Nov. 9 oo-p Mo. Nov. 6-3p Wd. Nov. 8 :30-3:30 p Wd. Dc. 5 oo-p F. Dc. 7 4:30-5:30 Mo. Dc. 0 oo-p Wd. Dc. 4:30-5:30 p ouly xa o Th. Dc. 3 Today s topcs Bf vw of slctd sults
More informationDe Moivre s Theorem - ALL
De Moivre s Theorem - ALL. Let x ad y be real umbers, ad be oe of the complex solutios of the equatio =. Evaluate: (a) + + ; (b) ( x + y)( x + y). [6]. (a) Sice is a complex umber which satisfies = 0,.
More informationTowards Healthy Environments for Children Frequently asked questions (FAQ) about breastfeeding in a contaminated environment
Ta a i f i Fu a ui (FQ) abu bafi i a aia i Su b i abu i ia i i? Y; u b i. ia aia a aui a u i; ia aii, bafi u a a aa i a ai f iiai f i ia i i. If ifa b a, a i, u fi i a b bu f iuia i iui ii, PB, u, aa,
More informationWeek 10 Spring Lecture 19. Estimation of Large Covariance Matrices: Upper bound Observe. is contained in the following parameter space,
Week 0 Sprig 009 Lecture 9. stiatio of Large Covariace Matrices: Upper boud Observe ; ; : : : ; i.i.d. fro a p-variate Gaussia distributio, N (; pp ). We assue that the covariace atrix pp = ( ij ) i;jp
More informationValley Forge Middle School Fencing Project Facilities Committee Meeting February 2016
Valley Forge iddle chool Fencing roject Facilities ommittee eeting February 2016 ummer of 2014 Installation of Fencing at all five istrict lementary chools October 2014 Facilities ommittee and
More information(Reference: sections in Silberberg 5 th ed.)
ALE. Atomic Structur Nam HEM K. Marr Tam No. Sctio What is a atom? What is th structur of a atom? Th Modl th structur of a atom (Rfrc: sctios.4 -. i Silbrbrg 5 th d.) Th subatomic articls that chmists
More informationAnouncements. Conjugate Gradients. Steepest Descent. Outline. Steepest Descent. Steepest Descent
oucms Couga Gas Mchal Kazha (6.657) Ifomao abou h Sma (6.757) hav b pos ol: hp://www.cs.hu.u/~msha Tch Spcs: o M o Tusay afoo. o Two paps scuss ach w. o Vos fo w s caa paps u by Thusay vg. Oul Rvw of Sps
More informationUse precise language and domain-specific vocabulary to inform about or explain the topic. CCSS.ELA-LITERACY.WHST D
Lesson eight What are characteristics of chemical reactions? Science Constructing Explanations, Engaging in Argument and Obtaining, Evaluating, and Communicating Information ENGLISH LANGUAGE ARTS Reading
More informationStability of Quadratic and Cubic Functional Equations in Paranormed Spaces
IOSR Joua o Matheatics IOSR-JM e-issn 8-578, p-issn 9-765. Voue, Issue Ve. IV Ju - Aug. 05, - www.iosouas.og Stabiit o uadatic ad ubic Fuctioa Equatios i aaoed Spaces Muiappa, Raa S Depatet o Matheatics,
More informationOrthogonal transformations
Orthogoal trasformatios October 12, 2014 1 Defiig property The squared legth of a vector is give by takig the dot product of a vector with itself, v 2 v v g ij v i v j A orthogoal trasformatio is a liear
More information8.1 8.2 9.1 9.2 9.3 10.1 10.2 11.1 12.1 12.2 13.1 14.2 15.1 15.2 16.1 17.1 17.2 8.1 8.2 9.1 9.2 10.1 10.2 10.3 11.1 12.1 12.2 13.1 13.2 14.1 14.1 14.2 15.1 16.1 16.2 16.3 16.4 16.5 16.6 W A W C 12 U A
More informationSoftware Process Models there are many process model s in th e li t e ra t u re, s om e a r e prescriptions and some are descriptions you need to mode
Unit 2 : Software Process O b j ec t i ve This unit introduces software systems engineering through a discussion of software processes and their principal characteristics. In order to achieve the desireable
More informationOn Jackson's Theorem
It. J. Cotm. Math. Scics, Vol. 7, 0, o. 4, 49 54 O Jackso's Thom Ema Sami Bhaya Datmt o Mathmatics, Collg o Educatio Babylo Uivsity, Babil, Iaq mabhaya@yahoo.com Abstact W ov that o a uctio W [, ], 0
More informationI N A C O M P L E X W O R L D
IS L A M I C E C O N O M I C S I N A C O M P L E X W O R L D E x p l o r a t i o n s i n A g-b eanste d S i m u l a t i o n S a m i A l-s u w a i l e m 1 4 2 9 H 2 0 0 8 I s l a m i c D e v e l o p m e
More informationheliozoan Zoo flagellated holotrichs peritrichs hypotrichs Euplots, Aspidisca Amoeba Thecamoeba Pleuromonas Bodo, Monosiga
Figures 7 to 16 : brief phenetic classification of microfauna in activated sludge The considered taxonomic hierarchy is : Kingdom: animal Sub kingdom Branch Class Sub class Order Family Genus Sub kingdom
More informationTable of C on t en t s Global Campus 21 in N umbe r s R e g ional Capac it y D e v e lopme nt in E-L e ar ning Structure a n d C o m p o n en ts R ea
G Blended L ea r ni ng P r o g r a m R eg i o na l C a p a c i t y D ev elo p m ent i n E -L ea r ni ng H R K C r o s s o r d e r u c a t i o n a n d v e l o p m e n t C o p e r a t i o n 3 0 6 0 7 0 5
More informationExecutive Committee and Officers ( )
Gifted and Talented International V o l u m e 2 4, N u m b e r 2, D e c e m b e r, 2 0 0 9. G i f t e d a n d T a l e n t e d I n t e r n a t i o n a2 l 4 ( 2), D e c e m b e r, 2 0 0 9. 1 T h e W o r
More informationGRADED QUESTIONS ON COMPLEX NUMBER
E /Math-I/ GQ/Comple umer GRADED QUESTINS N CMPEX NUMBER. The umer of the form + i y where ad y are real umers ad i = - i. e.( i ) is called a comple umer ad it is deoted y z i.e. z = + i y.the comple
More informationCOMPLEX NUMBERS AND DE MOIVRE'S THEOREM SYNOPSIS. Ay umber of the form x+iy where x, y R ad i = - is called a complex umber.. I the complex umber x+iy, x is called the real part ad y is called the imagiary
More informationFuzzy Reasoning and Optimization Based on a Generalized Bayesian Network
Fuy R O B G By Nw H-Y K D M Du M Hu Cu Uvy 48 Hu Cu R Hu 300 Tw. @w.u.u.w A By w v wy u w w uy. Hwv u uy u By w y u v w uu By w w w u vu vv y. T uy v By w w uy v v uy. B By w uy. T uy v uy. T w w w- uy.
More informationThe z Transform. The Discrete LTI System Response to a Complex Exponential
The Trasform The trasform geeralies the Discrete-time Forier Trasform for the etire complex plae. For the complex variable is sed the otatio: jω x+ j y r e ; x, y Ω arg r x + y {} The Discrete LTI System
More informationSOLUTION FOR HOMEWORK 7, STAT np(1 p) (α + β + n) + ( np + α
SOLUTION FOR HOMEWORK 7, STAT 6331 1 Exerc733 Here we just recall that MSE(ˆp B ) = p(1 p) (α + β + ) + ( p + α 2 α + β + p) 2 The you plug i α = β = (/4) 1/2 After simplificatios MSE(ˆp B ) = 4( 1/2 +
More informationThe local orthonormal basis set (r,θ,φ) is related to the Cartesian system by:
TIS in Sica Cooinats As not in t ast ct, an of t otntias tat w wi a wit a cnta otntias, aning tat t a jst fnctions of t istanc btwn a atic an so oint of oigin. In tis cas tn, (,, z as a t Coob otntia an
More informationWorksheet: Taylor Series, Lagrange Error Bound ilearnmath.net
Taylor s Thorm & Lagrag Error Bouds Actual Error This is th ral amout o rror, ot th rror boud (worst cas scario). It is th dirc btw th actual () ad th polyomial. Stps:. Plug -valu ito () to gt a valu.
More informationIYGB. Special Extension Paper E. Time: 3 hours 30 minutes. Created by T. Madas. Created by T. Madas
YGB Special Extesio Paper E Time: 3 hours 30 miutes Cadidates may NOT use ay calculator. formatio for Cadidates This practice paper follows the Advaced Level Mathematics Core ad the Advaced Level Further
More informationDiscrete Fourier Series and Transforms
Lctur 4 Outi: Discrt Fourir Sris ad Trasforms Aoucmts: H 4 postd, du Tus May 8 at 4:3pm. o at Hs as soutios wi b avaiab immdiaty. Midtrm dtais o t pag H 5 wi b postd Fri May, du foowig Fri (as usua) Rviw
More informationVector Spherical Harmonics
Vector Spherica Haronics Lecture Introduction Previousy we have seen that the Lapacian operator is different when operating on a vector or a scaar function. We avoided this probe by etting the Lapacian
More informationHelping you learn to save. Pigby s tips and tricks
Hlpg yu lan t av Pigby tip and tick Hlpg vy littl av Pigby ha bn tachg hi find all abut ny and hw t av f what ty want. Tuffl i avg f a nw tappy bubbl d and Pi can t wait t b abl t buy nw il pat. Pigby
More informationCS229 Lecture notes. Andrew Ng
CS229 Lecture notes Andrew Ng Part IX The EM agorithm In the previous set of notes, we taked about the EM agorithm as appied to fitting a mixture of Gaussians. In this set of notes, we give a broader view
More informationA Review of Complex Arithmetic
/0/005 Rviw of omplx Arithmti.do /9 A Rviw of omplx Arithmti A omplx valu has both a ral ad imagiary ompot: { } ad Im{ } a R b so that w a xprss this omplx valu as: whr. a + b Just as a ral valu a b xprssd
More informationI M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o
I M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o u l d a l w a y s b e t a k e n, i n c l u d f o l
More informationBeechwood Music Department Staff
Beechwood Music Department Staff MRS SARAH KERSHAW - HEAD OF MUSIC S a ra h K e rs h a w t r a i n e d a t t h e R oy a l We ls h C o l le g e of M u s i c a n d D ra m a w h e re s h e ob t a i n e d
More informationThe tight-binding method
Th tight-idig thod Wa ottial aoach: tat lcto a a ga of aly f coductio lcto. ow aout iulato? ow aout d-lcto? d Tight-idig thod: gad a olid a a collctio of wa itactig utal ato. Ovla of atoic wav fuctio i
More informationc. What is the average rate of change of f on the interval [, ]? Answer: d. What is a local minimum value of f? Answer: 5 e. On what interval(s) is f
Essential Skills Chapter f ( x + h) f ( x ). Simplifying the difference quotient Section. h f ( x + h) f ( x ) Example: For f ( x) = 4x 4 x, find and simplify completely. h Answer: 4 8x 4 h. Finding the
More information[ m] x = 0.25cos 20 t sin 20 t m
. x.si ( 5 s [ ] CHAPER OSCILLAIONS x ax (.( ( 5 6. s s ( ( ( xax. 5.7 s s. x.si [] x. cos s Whe, x a x.5. s 5s.6 s x. x( x cos + si a f ( ( [ ] x.5cos +.59si. ( ( cos α β cosαcos β + siαsi β x Acos φ
More informationECE507 - Plasma Physics and Applications
ECE57 - Pasma Physics ad Appicatios Lecture 9 Prof. Jorge Rocca ad r. Ferado Tomase epartmet of Eectrica ad Computer Egieerig Low temperature pasmas Low temperature pasmas are a very iterestig subject
More informationChapter 4 Postulates & General Principles of Quantum Mechanics
Chapter 4 Postulates & Geeral Priciples of Quatu Mechaics Backgroud: We have bee usig quite a few of these postulates already without realizig it. Now it is tie to forally itroduce the. State of a Syste
More informationSupplemental Material: Proofs
Proof to Theorem Supplemetal Material: Proofs Proof. Let be the miimal umber of traiig items to esure a uique solutio θ. First cosider the case. It happes if ad oly if θ ad Rak(A) d, which is a special
More informationVISUALIZATION OF TRIVARIATE NURBS VOLUMES
ISUALIZATIO OF TRIARIATE URS OLUMES SAMUELČÍK Mat SK Abstact. I ths pap fcs patca st f f-f bcts a ts sazat. W xt appach f g cs a sfacs a ppa taat s bas z a -sp xpsss. O a ga s t saz g paatc s. Th sazat
More informationVII. Central Potentials
VII. Cnta Potntias Bfo going any futh with angua ontu, it is bst to bgin using th ations w aady hav so that w can gt so ida what thy a good fo. Phaps th bst appication of th angua ontu ignfunctions w dat
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 20
ECE 6341 Sprig 016 Prof. David R. Jackso ECE Dept. Notes 0 1 Spherical Wave Fuctios Cosider solvig ψ + k ψ = 0 i spherical coordiates z φ θ r y x Spherical Wave Fuctios (cot.) I spherical coordiates we
More informationCHAPTER-11 The SCHRODINGER EQUATION in 3D
Lt Nots PH 4/5 C 598 A. La osa INTODUCTION TO QUANTU CHANICS CHAPT- Th SCHODING QUATION in 3D Dsiption of th otion of two intatin patis. Gna as of an abita intation potntia. Cas whn th potntia pns on on
More informationBBU Codes Overview. Outline Introduction Beam transport Equation How to solve (BBU-R, TDBBUU, bi, MATBBU etc.) Comparison of BBU codes
BBU Codes Oveview asau Sawamua ad Ryoichi Hajima JAERI Outie Itoductio Beam tasot Equatio How to sove BBU-R, DBBUU, bi, ABBU etc. Comaiso of BBU codes Itoductio ERL cuet imited by Beam Beaku asvese defect
More information5.61 Physical Chemistry Exam III 11/29/12. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Chemistry Chemistry Physical Chemistry.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Chemistry Chemistry - 5.61 Physical Chemistry Exam III (1) PRINT your name on the cover page. (2) It is suggested that you READ THE ENTIRE EXAM before
More informationLECTURE 13 Filling the bands. Occupancy of Available Energy Levels
LUR 3 illig th bads Occupacy o Availabl rgy Lvls W hav dtrmid ad a dsity o stats. W also d a way o dtrmiig i a stat is illd or ot at a giv tmpratur. h distributio o th rgis o a larg umbr o particls ad
More information3/31/ Product of Inertia. Sample Problem Sample Problem 10.6 (continue)
/1/01 10.6 Product of Inertia Product of Inertia: I xy = xy da When the x axis, the y axis, or both are an axis of symmetry, the product of inertia is zero. Parallel axis theorem for products of inertia:
More information/ / MET Day 000 NC1^ INRTL MNVR I E E PRE SLEEP K PRE SLEEP R E
05//0 5:26:04 09/6/0 (259) 6 7 8 9 20 2 22 2 09/7 0 02 0 000/00 0 02 0 04 05 06 07 08 09 0 2 ay 000 ^ 0 X Y / / / / ( %/ ) 2 /0 2 ( ) ^ 4 / Y/ 2 4 5 6 7 8 9 2 X ^ X % 2 // 09/7/0 (260) ay 000 02 05//0
More informationNoise in electronic components.
No lto opot5098, JDS No lto opot Th PN juto Th ut thouh a PN juto ha fou opot t: two ffuo ut (hol fo th paa to th aa a lto th oppot to) a thal at oty ha a (hol fo th aa to th paa a lto th oppot to, laka
More informationObjectives. We will also get to know about the wavefunction and its use in developing the concept of the structure of atoms.
Modue "Atomic physics and atomic stuctue" Lectue 7 Quantum Mechanica teatment of One-eecton atoms Page 1 Objectives In this ectue, we wi appy the Schodinge Equation to the simpe system Hydogen and compae
More informationYamaha Virago V-twin. Instruction manual with visual guide for Yamaha XV
Yamaha Virago V-twin Instruction manual with visual guide for Yamaha XV700-1100 PHOTO HOWN FOR ILLU TRATION PURPO E ONLY We o use a o e pie e housi g a d s all si gle to e oils fo i p o ed ope aio. If
More information( r) 3. For constant speed, 8. Mutual inductance M = Þ f M
LL ND PEN ES/JEE (Mai)/55 ENHUSS & LEDER CURSE RGE : JEE (MN) 5 LL ND PEN ES # DE : 5 5 st y : Majo st Patt : JEE (Mai) NSWER KEY u. 4 5 6 7 8 9 4 5 6 7 8 9 s. 4 u. 4 5 6 7 8 9 4 5 6 7 8 9 4 s. 4 4 4 u.
More informationWhat are S M U s? SMU = Software Maintenance Upgrade Software patch del iv ery u nit wh ich once ins tal l ed and activ ated prov ides a point-fix for
SMU 101 2 0 0 7 C i s c o S y s t e m s, I n c. A l l r i g h t s r e s e r v e d. 1 What are S M U s? SMU = Software Maintenance Upgrade Software patch del iv ery u nit wh ich once ins tal l ed and activ
More informationLWC 434 East First Street 4440 Garwood Place
//0 :00: 0 0 U S S W I. V S U W W W W S S W W X U W S S W V V S S S V W W W I S V W W. S UI UI IU I I W U I W I W U I W SUY I W S W S W U I IS SI W I II S W SIS VV W SWI W U S (U) VIZ S U S IU V I VIZ
More informationChapter Taylor Theorem Revisited
Captr 0.07 Taylor Torm Rvisitd Atr radig tis captr, you sould b abl to. udrstad t basics o Taylor s torm,. writ trascdtal ad trigoomtric uctios as Taylor s polyomial,. us Taylor s torm to id t valus o
More informationE F. and H v. or A r and F r are dual of each other.
A Duality Thom: Consid th following quations as an xampl = A = F μ ε H A E A = jωa j ωμε A + β A = μ J μ A x y, z = J, y, z 4π E F ( A = jω F j ( F j β H F ωμε F + β F = ε M jβ ε F x, y, z = M, y, z 4π
More informationS U E K E AY S S H A R O N T IM B E R W IN D M A R T Z -PA U L L IN. Carlisle Franklin Springboro. Clearcreek TWP. Middletown. Turtlecreek TWP.
F R A N K L IN M A D IS O N S U E R O B E R T LE IC H T Y A LY C E C H A M B E R L A IN T W IN C R E E K M A R T Z -PA U L L IN C O R A O W E N M E A D O W L A R K W R E N N LA N T IS R E D R O B IN F
More informationFourier Transforms. Convolutions. Capturing what s important. Last Time. Linear Image Transformation. Invertible Transforms.
orr Trasors Rq rad: Captr 7 92 &P Adso Soc ad ra (adot o) Opt rad: Hor 7 & 8 P 8 Last T Cooto trs: a/box tr Gassa tr t drc tr Lapaca o Gassa tr Ed Dtcto Cootos Cooto s coptatoay costy ad a copx oprato
More information