components along the X and Y axes. Based on time-dependent perturbation theory, it can be shown that if the populations
|
|
- Chester Mitchell
- 5 years ago
- Views:
Transcription
1 Block quations to th vctor paradigm Assum that thr ar N N α +N β idntical, isolatd spins alignd along an trnal magntic fild in th Z dirction with th population ratio of N α and N β givn b a Boltmann distribution (Eqn 1). N α g β ΔE / kt N β N N (1) Whr g N is th magntogric ratio of nuclus N, β Ν is th nuclar magnton for nuclus N, ΔE is th nrg diffrnc btwn th α and β stats, k is Boltmann s constant, and T is th absolut tmpratur. At thrmal quilibrium N α > N β, and thr will b a nt (classical) macroscopic magntiation alignd along th Z ais with no magntiation componnts along th X and Y as. Basd on tim-dpndnt prturbation thor, it can b shown that if th populations of N α and N β ar prturbd awa from thir thrmal quilibrium valus, th sstm will dca ponntiall back to th quilibrium valus. Eqn. whr Z N α -N β Z Z T1 Th paramtr T 1 is th longitudinal or spin-lattic rlaation tim and is a charactristic of th spin sstm. Th () macroscopic magntiation can b rotatd to li in an dirction through th application of an oscillating magntic fild. Th componnts of th magntiation along th transvrs as dca to ro (Eqns. 3). X T X (3) Y Y T Th paramtr T is th transvrs or spin-spin rlaation tim. n gnral, this rlaation tim is diffrnt than T 1. To complt th dscription of th motion of th macroscopic magntiation, th spin angular momntum, a strictl quantum mchanical quantit, must b includd. Although this is a quantum ffct, it can b thought of as th ffct of a groscopic motion of a bar magnt placd in an trnal magntic fild. Th angular momntum du to th groscopic motion causs th magntiation to prcss around th trnal magntic fild, just as a groscop prcsss in a gravitational fild. Th quation of motion, tmporaril in th absnc of rlaation, is givn as th cross product of th magntiation and th magntic fild (Eqn 4).
2 (4) γ N ( H ) Hr rprsnts th macroscopic magntiation containing angular momntum du to th groscopic motion and H is a uniform magntic fild; γ N is th magntogric ratio of th nuclus. Th cross product of Eqn. 4 is givn in dtrminant form in Eqn 5. i j k H H H N X Y Z X Y Z For an trnal magntic fild along th Z ais, on obtains Eqn. 6. γ (5) i j k γ N X Y Z H Z Th Larmor prcssion frqunc of th rotation is givn b γ N H; for protons (γ Η 4.55 H/T) in a 14.1 Tsla (6) magntic fild this frqunc is 6 H. Th chmical shifts of common nucli ar tpicall in th rang of th audio frquncis (- kh). Th larg diffrnc btwn th Larmor prcssion frqunc and th chmical shift frquncis suggsts that th motion can b simplifid b th introduction of a rotating coordinat sstm. Essntiall, this is a trick to rmov th larg not-vr-intrsting Larmor frqunc and lav onl th important frquncis du to th chmical shifts. t can b thought of as appling a fictitious magntic fild opposing that of th 14.1 T trnal magntic fild. Th frqunc for a nuclus prcssing actl at th Larmor frqunc (on-rsonanc) undr ths conditions bcoms ro. For rsonancs that ar not on-rsonanc, th frqunc is non-ro and will prcss in th rotating fram. athmaticall, th H in Eqn. 4-6 is rplacd with an ffctiv fild H H-ω L /γ. This is quivalnt to th as rotating at an angular vlocit of th Larmor frqunc. Upon solving th dtrminant in Eqn. 6, Eqns. 7 ar obtaind. X Y Ω Ω Z Ω is th chmical shift frqunc in th rotating fram, -γ N H Z. Th particular algbraic sign for Ω ariss from th X Y (7) convntion that a positiv chmical shift frqunc is assumd to b gratr than th rotating fram frqunc. Th dirction of motion is givn b a right handd coordinat sstm, ->->-->-. Equations 7 indicat that th magntiation vctor will prcss (rotat) around th Z ais indfinitl. Combining Eqns. 7 with th rlaation quations (Eqns and 3) ilds th Block quations for prcssion of th magntiation in a static magntic fild (Eqns. 8).
3 X Y Ω Y T ΩX T X Y ( ) Z Z T1 Ths quations dscrib th volution of a magntiation vctor of an isolatd spin during a fr prcssion priod. Whn an RF fild is applid th Block quations ar transformd to thos in Eqns 9. Whr Ω is th chmical shift offst; ω is th rotating fram frqunc; B 1, ω, and φ ar th magnitud, frqunc, and phas of th applid RF fild, rspctivl. (8) *1/ T * Ω+ * ω * Ω *1/ T * ω * ω+ * ω ( )* ω γb1 cos( φ) ω γb1 sin( φ) Ω ω ω RF 1/ T (9) 1 n pulsd NR spctroscop, th applid radio frqunc is usuall much gratr than th offst frqunc du to chmical shifts and th duration of th RF puls is much shortr than 1/T 1 and 1/T. Undr ths conditions th trms arising from th applid RF fild dominat and th Block quations ar givn b Eqn 1. * ω * ω * ω+ * ω (1) From th solution of th st of diffrntial quations in Eqns. 8, th linshap for th NR rsonanc can b dtrmind. Thr ar two componnts of th linshap: on is in-phas with th citing RF fild and on is 9 out-of- phas. Th Lorntian absorption linshap (in-phas) is givn and th disprsiv (out-of-phas) linshap ar givn in Eqn. 11. A plot of ths linshaps ar plottd in Figur 1. On usful paramtr is th linwih of th absorption at half-hight (Δν 1/ ). This is givn as 1/πT H.
4 T A k* 1 + T Δ ω (11) ΔωT D k* 1 + T Δ Eqns. 9 can b also writtn in matri form (Eqn. 1). ω 1/ T Ω ω d T Ω 1/ ω + ω ω 1/ T 1 *1/ T 1 (1) Ths quations form a st of inhomognous diffrntial quations. This ariss sinc th rats of th X and Y componnts dpnd onl th magnitud of th Y and X magntiation, rspctivl, whras th Z magntiation dpnds on th diffrnc from th quilibrium stat. Th inhomognous charactr of ths quations maks th calculation of th propagation of th magntiation in a puls squnc difficult if th Hamiltonian is tim dpndnt, as it is in all usful puls squncs. olutions to ths quations provid a dtaild mthod to follow spin dnamics in th prsnc of rlaation and RF filds. A mthod to convrt th st of inhomognous quations to homognous is dscribd blow. Ω ω d ω Ω ω ω (13) Thr ar two simplifications that ar commonl applid to liminat th inhomognous charactr: olution 1: f rlaation is ignord, a homognous st of diffrntial quations is obtaind (Eqn. 13). This simplification givs ris to th simpl vctor dscription of th volution of th magntiation vctor. Th magnitud of th vctor rmains constant in this simplification. olution : rlaation is introducd during fr prcssion priods, but sinc th Hamiltonian during a puls dos not commut with th dnsit oprator, rlaation is ignord during all Figur 1. Lorntian absorption and disprsion linshaps. Not that th disprsion lin has finit amplitud ovr a much widr rang than th absorption lin. This is important to rmmbr in stting spctral wihs.
5 pulss. This is not a bad approimation whn th puls tim is much shortr than th rlaation rats. n this simplification th lngth of th vctor dos not rmain constant. Whil th introduction of rlaation during th fr prcssion priods maks a computation a bit mor ralistic, th simplst approach of totall ignoring rlaation lads to a vr intuitiv pictur of th motion of th magntiation. Th formal solution to Eqn. 13 givn in Eqn. 14. Ω ω * t Ω ω ω ω t (14) Or, b simplifing, on obtains Eqn. 15 with R bing th rat matri arising from th groscopic motion of th nuclar spin. A compact vrsion of this quation is givn in Eqn. 15. t R Th ponntial matri can b simplifid b various mthods (Appndi 1) to ild th simpl matri quation in Eqn 16, whr A is a 3 X 3 rotation matri in thr dimnsional spac. * t t A n th spcial cas whr onl chmical shift is prsnt (fr prcssion ilding rotation around th Z ais), matri A bcoms simpl a Z-ais rotation matri (Eqn. 17). Th transvrs componnts and volv. (15) (16) Ω Ω * t cos( Ωt) sin( Ωt) sin( t) cos( t) Ω Ω 1 (17) f th frqunc du to th applid RF fild B 1 (along th X ais) is much gratr than th chmical shift Ω, th lattr can b ignord and a simpl X-ais rotation matri is obtaind for an X ais rotation (Eqn. 18). ω * t ω 1 cos( wt) sin( wt) sin( wt ) cos( wt ) (18) Changing th phas of th applid RF fild to gnrat a Y ais rotation ilds Eqn. 19. ω * t ω cos( wt ) sin( wt) 1 sin( wt ) cos( wt ) (19)
6 Ths rotation matrics ar usd to calculat th (approimat) trajctor of th magntiation in th absnc of rlaation. f thr is mor than on non-commuting (Appndi ) oprator in th ponntial matri, thn th rsulting rotation matrics ar no longr along th X, Y, or Z ais. n othr words if th RF fild strngth along th X ais is comparabl to th chmical shift offst, thn th rotation occurs simultanousl around th Z and Y ais. This dos not lad to a simpl rotation matri around on of th X, Y, or Z ais, but a rotation about an ais that is tiltd awa from all of th as. As an ampl of two simultanous, non-commuting rotations, considr a RF puls causing a rotation about th X ais at a frqunc ω 1 for a tim t with a simultanous Z rotation du to a chmical shift offst Ω. Th ponntial rat matri and th corrsponding rotation matri is givn in Eqn. Not that th frqunc ω is th vctor sum of ω 1 and Ω, a consqunc of th introduction of a rotating coordinat sstm. This lads to a rotation ais that is tiltd out of th XY plan having a magnitud gratr than ithr ω 1 or Ω individuall. ω Ω Ω ω Ω + cos sin 1 cos ( ) ( ω t) ( ω t) ( ω t) 1 1 Ω ω ω ω ω Ω ω1 * t ω1 Ω ω1 sin ( ω ) cos( ω ) sin( ωt ) ω ω t t ω1ω ω 1 Ω ω 1 + ω ω ω ω ( 1 cos( ωt) ) sin( ωt) cos( ωt) () ω1 ω Ω + Th unwild natur of ths rotation matrics can b bpassd through th us of similarit transforms to mov th sstm into a tiltd fram of rfrnc, prform a rotation, and thn rturn th sstm back from th tiltd fram. Eqn. 1 givs th appropriat similarit transform for this situation. Acting on a column vctor at th right, th rightmost rotation R (tan -1 Ω/ω 1 ) rotats th tiltd rotation ais to along th X ais. Th nt R rotation rotats th magntiation vctor with considration that th frqunc of th rotation is gratr than ω 1. 1 Ω Ω + ω 1 1 Ω R tan R * ω1t R tan ω1 ω1 ω 1 (1) For situations in which th chmical shift frqunc is similar to th RF, th motion of th magntiation is no longr a simpl rotation about th X, Y or Z as. n th analsis of puls squncs, it is commonl assumd that th applid RF is infinitl gratr than th chmical shift frqunc lading to th simplification in Eqn. 1. n th prsnc of rlaation, th us of similarit transforms is not possibl (s blow).
7 Th rotation matrics obtaind from th ponntiation of th rat matri of nuclar prcssion lad to th wll known volution of th product oprators for a singl spin. Th magntiation for a singl isolatd spin can b rprsntd b th column matri in Eqn 3. (3) At quilibrium, th magntiation vctor lis along th Z ais and is rprsntd b Eqn (4) Appling a θ rotation of th magntiation vctor around th X ais ilds Eqn cos( θ ) sin( θ) sin( θ) sin( θ ) cos( θ) 1 cos( θ) (5) Or in standard product oprator trms on obtains Eqn 6. θ cos( θ ) sin( θ ) Th tnsion to mor than on isolatd spin involvs th Bloch quations for ach spin individuall. For two spins in th absnc of rlaation, this lads to th 6X6 matri in Eqn. 7. Not that sinc th spins ar isolatd no matri lmnts connct th and spins. (6) ω Ω Ω ω d ω ω ω Ω ω Ω ω ω (7) Th solution is th sam as abov. Th sam simplifications for chmical shift and RF rotations appl as for th singl spin cas. That is, sinc diffrnt spins alwas commut simultanous and rotations.g. + ild simpl spin spcific rotation matrics around X, Y, or Z, but simultanous non-commuting rotations such as + or + do not
8 giv simpl rotation matrics. imultanous non-commuting rotations can b dalt with b similarit transforms (tiltd frams) or b solving th ponntial matri numricall. Rlaation can b rintroducd to this dscription b svral mthods. A vr clvr mthod is to transform th inhomognous quation into a homognous on b adding a row and column to th matri to introduc th diffrnc from quilibrium (Jnr,Lvitt, Allard, Ernst##). An ampl of this is shown in Eqn. 8. E/ E/ d 1/ T Ω ω (8) Ω 1/ T ω 1/ ( T1* ) ω ω 1/ T1 Th lmnt (1,4) of th matri introducs th quilibrium magntiation. This lads to Eqn. 9 for th tim dpndnc of in th prsnc of a RF fild, which is idntical to that givn in Eqn 14. Equation 8 can b solvd using th matri ponntial. Th rsulting matri is not a simpl rotation matri sinc th rlaation trms do not prsrv th lngth of th vctor. Howvr whn rlaation occurs during pulss, this mthod ilds a complt dscription. d d ( 1/ 1* ω ω 1/ 1 ) T + + T ( ω ω 1/ 1( )) + T (9)
9 For a htronuclar spin sstm a st of diffrntial quations can b st up similar to th following (no rlaation). d Ω + ω πj From ths diffrntial quations, th rat matri is obtaind Ω ω πj Ω ω πj ω ω J Ω ω π Ω ω π J ω ω πj Ω ω ω ω d π J Ω ω ω ω πj Ω ω ω ω πj Ω ω ω ω ω ω Ω Ω ω ω Ω Ω ω ω Ω Ω ω ω Ω Ω ω ω ω ω This is th matri rprsntation for th volution of th dnsit oprator in th Cartsian product oprator basis in th absnc of rlaation. t has a solution of th form t R * t Again if onl on non-commuting rotation is usd thn th rsulting matri is a simpl rotation matri, hr as 3 dimnsional subspacs of 15 dimnsional spac. n mab, mor familiar nomnclatur d σ ihσ i[ H, σ ]
10 Th solution to this quation in rgular oprators is iht σ ( t ) * σ () * iht
11 Appndi 1 Connction of ponntial rat matrics with rotation oprators. (dimnsional cas) Rat matri R, rat a, tim t 1 R 1 Rat Rat + ( Rat) + ( Rat) +! 3! ( Rat ) + ( Rat) + + Rat + ( Rat) + ( Rat) +! 4! 3! 5! at at ! 1 4! 1 at + at + at 1 3! 1 5! ( at) ( at) at ( at) ( at)! 1 + 4! ! 1 5! 1 cos( at)* + sin( at)* R cos( at) sin( at) sin( at) cos( at) D rotation matri Rat E 1 cos( at) sin( at) 1 cos( at) sin( at) cos( at) sin( at) +
12 Commuting oprators A* B B* A * * Non-commuting oprators A*B -B*A * * 1 1 1
13 Appndi 3. Eponntial commuting oprators. A + B + ( A+ B) + 1 ( A+ B) +! 1 A + A+ A +! B + B+ 1 B +! A B 1 + ( A+ B) + ( A+ AB+ B) +! comparing th scond ordr trms 1 ( A+ B ) A+ AB+ BA+ B! and 1 ( A+ AB+ B )! A+ B A B onl if ABBA
11: Echo formation and spatial encoding
11: Echo formation and spatial ncoding 1. What maks th magntic rsonanc signal spatiall dpndnt? 2. How is th position of an R signal idntifid? Slic slction 3. What is cho formation and how is it achivd?
More informationBackground: We have discussed the PIB, HO, and the energy of the RR model. In this chapter, the H-atom, and atomic orbitals.
Chaptr 7 Th Hydrogn Atom Background: W hav discussd th PIB HO and th nrgy of th RR modl. In this chaptr th H-atom and atomic orbitals. * A singl particl moving undr a cntral forc adoptd from Scott Kirby
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationText: WMM, Chapter 5. Sections , ,
Lcturs 6 - Continuous Probabilit Distributions Tt: WMM, Chaptr 5. Sctions 6.-6.4, 6.6-6.8, 7.-7. In th prvious sction, w introduc som of th common probabilit distribution functions (PDFs) for discrt sampl
More informationAs the matrix of operator B is Hermitian so its eigenvalues must be real. It only remains to diagonalize the minor M 11 of matrix B.
7636S ADVANCED QUANTUM MECHANICS Solutions Spring. Considr a thr dimnsional kt spac. If a crtain st of orthonormal kts, say, and 3 ar usd as th bas kts, thn th oprators A and B ar rprsntd by a b A a and
More informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
More informationHydrogen Atom and One Electron Ions
Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial
More informationDerivation of Electron-Electron Interaction Terms in the Multi-Electron Hamiltonian
Drivation of Elctron-Elctron Intraction Trms in th Multi-Elctron Hamiltonian Erica Smith Octobr 1, 010 1 Introduction Th Hamiltonian for a multi-lctron atom with n lctrons is drivd by Itoh (1965) by accounting
More informationSection 11.6: Directional Derivatives and the Gradient Vector
Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationThe Relativistic Stern-Gerlach Force C. Tschalär 1. Introduction
Th Rlativistic Strn-Grlach Forc C. Tschalär. Introduction For ovr a dcad, various formulations of th Strn-Grlach (SG) forc acting on a particl with spin moving at a rlativistic vlocity in an lctromagntic
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationThe van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012
Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor
More informationUniversity of Illinois at Chicago Department of Physics. Thermodynamics & Statistical Mechanics Qualifying Examination
Univrsity of Illinois at Chicago Dpartmnt of hysics hrmodynamics & tatistical Mchanics Qualifying Eamination January 9, 009 9.00 am 1:00 pm Full crdit can b achivd from compltly corrct answrs to 4 qustions.
More informationClassical Magnetic Dipole
Lctur 18 1 Classical Magntic Dipol In gnral, a particl of mass m and charg q (not ncssarily a point charg), w hav q g L m whr g is calld th gyromagntic ratio, which accounts for th ffcts of non-point charg
More informationDIFFERENTIAL EQUATION
MD DIFFERENTIAL EQUATION Sllabus : Ordinar diffrntial quations, thir ordr and dgr. Formation of diffrntial quations. Solution of diffrntial quations b th mthod of sparation of variabls, solution of homognous
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationSearch sequence databases 3 10/25/2016
Sarch squnc databass 3 10/25/2016 Etrm valu distribution Ø Suppos X is a random variabl with probability dnsity function p(, w sampl a larg numbr S of indpndnt valus of X from this distribution for an
More informationThere is an arbitrary overall complex phase that could be added to A, but since this makes no difference we set it to zero and choose A real.
Midtrm #, Physics 37A, Spring 07. Writ your rsponss blow or on xtra pags. Show your work, and tak car to xplain what you ar doing; partial crdit will b givn for incomplt answrs that dmonstrat som concptual
More informationsurface of a dielectric-metal interface. It is commonly used today for discovering the ways in
Surfac plasmon rsonanc is snsitiv mchanism for obsrving slight changs nar th surfac of a dilctric-mtal intrfac. It is commonl usd toda for discovring th was in which protins intract with thir nvironmnt,
More information1.2 Faraday s law A changing magnetic field induces an electric field. Their relation is given by:
Elctromagntic Induction. Lorntz forc on moving charg Point charg moving at vlocity v, F qv B () For a sction of lctric currnt I in a thin wir dl is Idl, th forc is df Idl B () Elctromotiv forc f s any
More informationForces. Quantum ElectroDynamics. α = = We have now:
W hav now: Forcs Considrd th gnral proprtis of forcs mdiatd by xchang (Yukawa potntial); Examind consrvation laws which ar obyd by (som) forcs. W will nxt look at thr forcs in mor dtail: Elctromagntic
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More informationDIELECTRIC AND MAGNETIC PROPERTIES OF MATERIALS
DILCTRIC AD MAGTIC PROPRTIS OF MATRIALS Dilctric Proprtis: Dilctric matrial Dilctric constant Polarization of dilctric matrials, Typs of Polarization (Polarizability). quation of intrnal filds in liquid
More information22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.
Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M
More informationQuasi-Classical States of the Simple Harmonic Oscillator
Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats
More informationSelf-Adjointness and Its Relationship to Quantum Mechanics. Ronald I. Frank 2016
Ronald I. Frank 06 Adjoint https://n.wikipdia.org/wiki/adjoint In gnral thr is an oprator and a procss that dfin its adjoint *. It is thn slf-adjoint if *. Innr product spac https://n.wikipdia.org/wiki/innr_product_spac
More informationOn the Hamiltonian of a Multi-Electron Atom
On th Hamiltonian of a Multi-Elctron Atom Austn Gronr Drxl Univrsity Philadlphia, PA Octobr 29, 2010 1 Introduction In this papr, w will xhibit th procss of achiving th Hamiltonian for an lctron gas. Making
More informationCHAPTER 1. Introductory Concepts Elements of Vector Analysis Newton s Laws Units The basis of Newtonian Mechanics D Alembert s Principle
CHPTER 1 Introductory Concpts Elmnts of Vctor nalysis Nwton s Laws Units Th basis of Nwtonian Mchanics D lmbrt s Principl 1 Scinc of Mchanics: It is concrnd with th motion of matrial bodis. odis hav diffrnt
More information2. Laser physics - basics
. Lasr physics - basics Spontanous and stimulatd procsss Einstin A and B cofficints Rat quation analysis Gain saturation What is a lasr? LASER: Light Amplification by Stimulatd Emission of Radiation "light"
More informationMA 262, Spring 2018, Final exam Version 01 (Green)
MA 262, Spring 218, Final xam Vrsion 1 (Grn) INSTRUCTIONS 1. Switch off your phon upon ntring th xam room. 2. Do not opn th xam booklt until you ar instructd to do so. 3. Bfor you opn th booklt, fill in
More informationB. water content (WC) = 0.4 WC CSF WC GM WC WM, or WC = = mol/l.
9.. A. Rlvant quation: M (TR, TE) = M ( TR /T ) TE / T M (NAA) = 38. M (cratin) = 9.6 M (cholin) = 98.4 M (watr) = 7,4 B. watr contnt (WC) =.4 WC CSF +.5 WC GM +. WC WM, or WC =.4. 55.6 +.5.87 55.6 +..83
More information1 Isoparametric Concept
UNIVERSITY OF CALIFORNIA BERKELEY Dpartmnt of Civil Enginring Spring 06 Structural Enginring, Mchanics and Matrials Profssor: S. Govindj Nots on D isoparamtric lmnts Isoparamtric Concpt Th isoparamtric
More information2.3 Matrix Formulation
23 Matrix Formulation 43 A mor complicatd xampl ariss for a nonlinar systm of diffrntial quations Considr th following xampl Exampl 23 x y + x( x 2 y 2 y x + y( x 2 y 2 (233 Transforming to polar coordinats,
More informationIntroduction to the quantum theory of matter and Schrödinger s equation
Introduction to th quantum thory of mattr and Schrödingr s quation Th quantum thory of mattr assums that mattr has two naturs: a particl natur and a wa natur. Th particl natur is dscribd by classical physics
More information6. The Interaction of Light and Matter
6. Th Intraction of Light and Mattr - Th intraction of light and mattr is what maks lif intrsting. - Light causs mattr to vibrat. Mattr in turn mits light, which intrfrs with th original light. - Excitd
More informationu 3 = u 3 (x 1, x 2, x 3 )
Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats.
More informationSundials and Linear Algebra
Sundials and Linar Algbra M. Scot Swan July 2, 25 Most txts on crating sundials ar dirctd towards thos who ar solly intrstd in making and using sundials and usually assums minimal mathmatical background.
More information1 General boundary conditions in diffusion
Gnral boundary conditions in diffusion Πρόβλημα 4.8 : Δίνεται μονοδιάτατη πλάκα πάχους, που το ένα άκρο της κρατιέται ε θερμοκραία T t και το άλλο ε θερμοκραία T 2 t. Αν η αρχική θερμοκραία της πλάκας
More informationdx equation it is called a second order differential equation.
TOPI Diffrntial quations Mthods of thir intgration oncption of diffrntial quations An quation which spcifis a rlationship btwn a function, its argumnt and its drivativs of th first, scond, tc ordr is calld
More informationContemporary, atomic, nuclear, and particle physics
Contmporary, atomic, nuclar, and particl physics 1 Blackbody radiation as a thrmal quilibrium condition (in vacuum this is th only hat loss) Exampl-1 black plan surfac at a constant high tmpratur T h is
More informationElectromagnetics Research Group A THEORETICAL MODEL OF A LOSSY DIELECTRIC SLAB FOR THE CHARACTERIZATION OF RADAR SYSTEM PERFORMANCE SPECIFICATIONS
Elctromagntics Rsarch Group THEORETICL MODEL OF LOSSY DIELECTRIC SLB FOR THE CHRCTERIZTION OF RDR SYSTEM PERFORMNCE SPECIFICTIONS G.L. Charvat, Prof. Edward J. Rothwll Michigan Stat Univrsit 1 Ovrviw of
More informationDirect Approach for Discrete Systems One-Dimensional Elements
CONTINUUM & FINITE ELEMENT METHOD Dirct Approach or Discrt Systms On-Dimnsional Elmnts Pro. Song Jin Par Mchanical Enginring, POSTECH Dirct Approach or Discrt Systms Dirct approach has th ollowing aturs:
More informationIntro to Nuclear and Particle Physics (5110)
Intro to Nuclar and Particl Physics (5110) March 09, 009 Frmi s Thory of Bta Dcay (continud) Parity Violation, Nutrino Mass 3/9/009 1 Final Stat Phas Spac (Rviw) Th Final Stat lctron and nutrino wav functions
More informationDifferential Equations
UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs
More informationSAFE HANDS & IIT-ian's PACE EDT-15 (JEE) SOLUTIONS
It is not possibl to find flu through biggr loop dirctly So w will find cofficint of mutual inductanc btwn two loops and thn find th flu through biggr loop Also rmmbr M = M ( ) ( ) EDT- (JEE) SOLUTIONS
More informationMassachusetts Institute of Technology Department of Mechanical Engineering
Massachustts Institut of Tchnolog Dpartmnt of Mchanical Enginring. Introduction to Robotics Mid-Trm Eamination Novmbr, 005 :0 pm 4:0 pm Clos-Book. Two shts of nots ar allowd. Show how ou arrivd at our
More informationCollisions between electrons and ions
DRAFT 1 Collisions btwn lctrons and ions Flix I. Parra Rudolf Pirls Cntr for Thortical Physics, Unirsity of Oxford, Oxford OX1 NP, UK This rsion is of 8 May 217 1. Introduction Th Fokkr-Planck collision
More informationTitle: Vibrational structure of electronic transition
Titl: Vibrational structur of lctronic transition Pag- Th band spctrum sn in th Ultra-Violt (UV) and visibl (VIS) rgions of th lctromagntic spctrum can not intrprtd as vibrational and rotational spctrum
More informationME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002
3.4 Forc Analysis of Linkas An undrstandin of forc analysis of linkas is rquird to: Dtrmin th raction forcs on pins, tc. as a consqunc of a spcifid motion (don t undrstimat th sinificanc of dynamic or
More informationPartial Derivatives: Suppose that z = f(x, y) is a function of two variables.
Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv
More informationChapter 7b Electron Spin and Spin- Orbit Coupling
Wintr 3 Chm 356: Introductory Quantum Mchanics Chaptr 7b Elctron Spin and Spin- Orbit Coupling... 96 H- atom in a Magntic Fild: Elctron Spin... 96 Total Angular Momntum... 3 Chaptr 7b Elctron Spin and
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More informationMCE503: Modeling and Simulation of Mechatronic Systems Discussion on Bond Graph Sign Conventions for Electrical Systems
MCE503: Modling and Simulation o Mchatronic Systms Discussion on Bond Graph Sign Convntions or Elctrical Systms Hanz ichtr, PhD Clvland Stat Univrsity, Dpt o Mchanical Enginring 1 Basic Assumption In a
More informationph People Grade Level: basic Duration: minutes Setting: classroom or field site
ph Popl Adaptd from: Whr Ar th Frogs? in Projct WET: Curriculum & Activity Guid. Bozman: Th Watrcours and th Council for Environmntal Education, 1995. ph Grad Lvl: basic Duration: 10 15 minuts Stting:
More informationIntroduction to Condensed Matter Physics
Introduction to Condnsd Mattr Physics pcific hat M.P. Vaughan Ovrviw Ovrviw of spcific hat Hat capacity Dulong-Ptit Law Einstin modl Dby modl h Hat Capacity Hat capacity h hat capacity of a systm hld at
More information5.80 Small-Molecule Spectroscopy and Dynamics
MIT OpnCoursWar http://ocw.mit.du 5.80 Small-Molcul Spctroscopy and Dynamics Fall 008 For information about citing ths matrials or our Trms of Us, visit: http://ocw.mit.du/trms. Lctur # 3 Supplmnt Contnts
More informationVII. Quantum Entanglement
VII. Quantum Entanglmnt Quantum ntanglmnt is a uniqu stat of quantum suprposition. It has bn studid mainly from a scintific intrst as an vidnc of quantum mchanics. Rcntly, it is also bing studid as a basic
More informationElements of Statistical Thermodynamics
24 Elmnts of Statistical Thrmodynamics Statistical thrmodynamics is a branch of knowldg that has its own postulats and tchniqus. W do not attmpt to giv hr vn an introduction to th fild. In this chaptr,
More informationOutline. Thanks to Ian Blockland and Randy Sobie for these slides Lifetimes of Decaying Particles Scattering Cross Sections Fermi s Golden Rule
Outlin Thanks to Ian Blockland and andy obi for ths slids Liftims of Dcaying Particls cattring Cross ctions Frmi s Goldn ul Physics 424 Lctur 12 Pag 1 Obsrvabls want to rlat xprimntal masurmnts to thortical
More informationPHYSICS 489/1489 LECTURE 7: QUANTUM ELECTRODYNAMICS
PHYSICS 489/489 LECTURE 7: QUANTUM ELECTRODYNAMICS REMINDER Problm st du today 700 in Box F TODAY: W invstigatd th Dirac quation it dscribs a rlativistic spin /2 particl implis th xistnc of antiparticl
More informationSec 2.3 Modeling with First Order Equations
Sc.3 Modling with First Ordr Equations Mathmatical modls charactriz physical systms, oftn using diffrntial quations. Modl Construction: Translating physical situation into mathmatical trms. Clarly stat
More information10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationVSMN30 FINITA ELEMENTMETODEN - DUGGA
VSMN3 FINITA ELEMENTMETODEN - DUGGA 1-11-6 kl. 8.-1. Maximum points: 4, Rquird points to pass: Assistanc: CALFEM manual and calculator Problm 1 ( 8p ) 8 7 6 5 y 4 1. m x 1 3 1. m Th isotropic two-dimnsional
More information5. B To determine all the holes and asymptotes of the equation: y = bdc dced f gbd
1. First you chck th domain of g x. For this function, x cannot qual zro. Thn w find th D domain of f g x D 3; D 3 0; x Q x x 1 3, x 0 2. Any cosin graph is going to b symmtric with th y-axis as long as
More informationCoupled Pendulums. Two normal modes.
Tim Dpndnt Two Stat Problm Coupld Pndulums Wak spring Two normal mods. No friction. No air rsistanc. Prfct Spring Start Swinging Som tim latr - swings with full amplitud. stationary M +n L M +m Elctron
More informationLorentz force rotor formulation.
Lorntz forc rotor formulation. Ptr Joot ptr.joot@gmail.com March 18, 2009. Last Rvision: Dat : 2009/03/2321 : 19 : 46 Contnts 1 Motivation. 1 2 In trms of GA. 1 2.1 Omga bivctor............................
More informationNote If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.
. (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold
More information4. Money cannot be neutral in the short-run the neutrality of money is exclusively a medium run phenomenon.
PART I TRUE/FALSE/UNCERTAIN (5 points ach) 1. Lik xpansionary montary policy, xpansionary fiscal policy rturns output in th mdium run to its natural lvl, and incrass prics. Thrfor, fiscal policy is also
More information2. Background Material
S. Blair Sptmbr 3, 003 4. Background Matrial Th rst of this cours dals with th gnration, modulation, propagation, and ction of optical radiation. As such, bic background in lctromagntics and optics nds
More informationAS 5850 Finite Element Analysis
AS 5850 Finit Elmnt Analysis Two-Dimnsional Linar Elasticity Instructor Prof. IIT Madras Equations of Plan Elasticity - 1 displacmnt fild strain- displacmnt rlations (infinitsimal strain) in matrix form
More informationBroadband All-Angle Negative Refraction by Phononic Crystals
Supplmntar Information Broadband All-Angl Ngativ Rfraction b Phononic Crstals Yang Fan Li, Fi Mng, Shiwi Zhou, Ming-Hui Lu and Xiaodong Huang 1 Optimization algorithm and procss Bfor th optimization procss,
More informationWhat are those βs anyway? Understanding Design Matrix & Odds ratios
Ral paramtr stimat WILD 750 - Wildlif Population Analysis of 6 What ar thos βs anyway? Undrsting Dsign Matrix & Odds ratios Rfrncs Hosmr D.W.. Lmshow. 000. Applid logistic rgrssion. John Wily & ons Inc.
More informationSchrodinger Equation in 3-d
Schrodingr Equation in 3-d ψ( xyz,, ) ψ( xyz,, ) ψ( xyz,, ) + + + Vxyz (,, ) ψ( xyz,, ) = Eψ( xyz,, ) m x y z p p p x y + + z m m m + V = E p m + V = E E + k V = E Infinit Wll in 3-d V = x > L, y > L,
More informationPhys 344 Lect 17 March 16 th,
Phs 344 Lct 7 March 6 th, 7 Fri. 3/ 6 C.8.-.8.7, S. B. Gaussian, 6.3-.5 quipartion, Mawll HW7 S. 3,36, 4 B.,,3 Mon. 3/9 Wd. 3/ Fri. 3/3 S 6.5-7 Partition Function S A.5 Q.M. Background: Bos and Frmi S
More informationGEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES. Eduard N. Klenov* Rostov-on-Don, Russia
GEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES Eduard N. Klnov* Rostov-on-Don, Russia Th articl considrs phnomnal gomtry figurs bing th carrirs of valu spctra for th pairs of th rmaining additiv
More informationThe pn junction: 2 Current vs Voltage (IV) characteristics
Th pn junction: Currnt vs Voltag (V) charactristics Considr a pn junction in quilibrium with no applid xtrnal voltag: o th V E F E F V p-typ Dpltion rgion n-typ Elctron movmnt across th junction: 1. n
More informationELECTRON-MUON SCATTERING
ELECTRON-MUON SCATTERING ABSTRACT Th lctron charg is considrd to b distributd or xtndd in spac. Th diffrntial of th lctron charg is st qual to a function of lctron charg coordinats multiplid by a four-dimnsional
More informationCO-ORDINATION OF FAST NUMERICAL RELAYS AND CURRENT TRANSFORMERS OVERDIMENSIONING FACTORS AND INFLUENCING PARAMETERS
CO-ORDINATION OF FAST NUMERICAL RELAYS AND CURRENT TRANSFORMERS OVERDIMENSIONING FACTORS AND INFLUENCING PARAMETERS Stig Holst ABB Automation Products Swdn Bapuji S Palki ABB Utilitis India This papr rports
More information3 Finite Element Parametric Geometry
3 Finit Elmnt Paramtric Gomtry 3. Introduction Th intgral of a matrix is th matrix containing th intgral of ach and vry on of its original componnts. Practical finit lmnt analysis rquirs intgrating matrics,
More informationSolution: APPM 1360 Final (150 pts) Spring (60 pts total) The following parts are not related, justify your answers:
APPM 6 Final 5 pts) Spring 4. 6 pts total) Th following parts ar not rlatd, justify your answrs: a) Considr th curv rprsntd by th paramtric quations, t and y t + for t. i) 6 pts) Writ down th corrsponding
More informationIntroduction to the Fourier transform. Computer Vision & Digital Image Processing. The Fourier transform (continued) The Fourier transform (continued)
Introduction to th Fourir transform Computr Vision & Digital Imag Procssing Fourir Transform Lt f(x) b a continuous function of a ral variabl x Th Fourir transform of f(x), dnotd by I {f(x)} is givn by:
More informationProblem Set #2 Due: Friday April 20, 2018 at 5 PM.
1 EE102B Spring 2018 Signal Procssing and Linar Systms II Goldsmith Problm St #2 Du: Friday April 20, 2018 at 5 PM. 1. Non-idal sampling and rcovry of idal sampls by discrt-tim filtring 30 pts) Considr
More informationNEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA
NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals
More information1 1 1 p q p q. 2ln x x. in simplest form. in simplest form in terms of x and h.
NAME SUMMER ASSIGNMENT DUE SEPTEMBER 5 (FIRST DAY OF SCHOOL) AP CALC AB Dirctions: Answr all of th following qustions on a sparat sht of papr. All work must b shown. You will b tstd on this matrial somtim
More informationPARTICLE MOTION IN UNIFORM GRAVITATIONAL and ELECTRIC FIELDS
VISUAL PHYSICS ONLINE MODULE 6 ELECTROMAGNETISM PARTICLE MOTION IN UNIFORM GRAVITATIONAL and ELECTRIC FIELDS A fram of rfrnc Obsrvr Origin O(,, ) Cartsian coordinat as (X, Y, Z) Unit vctors iˆˆj k ˆ Scif
More informationME311 Machine Design
ME311 Machin Dsign Lctur 4: Strss Concntrations; Static Failur W Dornfld 8Sp017 Fairfild Univrsit School of Enginring Strss Concntration W saw that in a curvd bam, th strss was distortd from th uniform
More informationDifferentiation of Exponential Functions
Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of
More information2F1120 Spektrala transformer för Media Solutions to Steiglitz, Chapter 1
F110 Spktrala transformr för Mdia Solutions to Stiglitz, Chaptr 1 Prfac This documnt contains solutions to slctd problms from Kn Stiglitz s book: A Digital Signal Procssing Primr publishd by Addison-Wsly.
More informationMaxwell s Equations on a Yee Grid
Instructor Dr. Ramond Rumpf (95) 77 6958 rcrumpf@utp.du 57 Computational lctromagntics (CM) Lctur # Mawll s quations on a Y Grid Lctur Ths nots ma contain coprightd matrial obtaind undr fair us ruls. Distribution
More informationDynamic Modelling of Hoisting Steel Wire Rope. Da-zhi CAO, Wen-zheng DU, Bao-zhu MA *
17 nd Intrnational Confrnc on Mchanical Control and Automation (ICMCA 17) ISBN: 978-1-6595-46-8 Dynamic Modlling of Hoisting Stl Wir Rop Da-zhi CAO, Wn-zhng DU, Bao-zhu MA * and Su-bing LIU Xi an High
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More informationHomotopy perturbation technique
Comput. Mthods Appl. Mch. Engrg. 178 (1999) 257±262 www.lsvir.com/locat/cma Homotopy prturbation tchniqu Ji-Huan H 1 Shanghai Univrsity, Shanghai Institut of Applid Mathmatics and Mchanics, Shanghai 272,
More informationde/dx Effectively all charged particles except electrons
de/dx Lt s nxt turn our attntion to how chargd particls los nrgy in mattr To start with w ll considr only havy chargd particls lik muons, pions, protons, alphas, havy ions, Effctivly all chargd particls
More informationDesign Guidelines for Quartz Crystal Oscillators. R 1 Motional Resistance L 1 Motional Inductance C 1 Motional Capacitance C 0 Shunt Capacitance
TECHNICAL NTE 30 Dsign Guidlins for Quartz Crystal scillators Introduction A CMS Pirc oscillator circuit is wll known and is widly usd for its xcllnt frquncy stability and th wid rang of frquncis ovr which
More informationlinarly acclratd []. Motivatd by ths complications w latr xamind mor carfully th cts of quantum uctuations for an lctron moving in a circular orbit [2
OSLO-TP 3-98 April-998 Acclratd Elctrons and th Unruh Ect Jon Magn Linaas Dpartmnt of Physics P.O. Box 048 Blindrn N-036 Oslo Norway Abstract Quantum cts for lctrons in a storag ring ar studid in a co-moving,
More informationDetermination of Vibrational and Electronic Parameters From an Electronic Spectrum of I 2 and a Birge-Sponer Plot
5 J. Phys. Chm G Dtrmination of Vibrational and Elctronic Paramtrs From an Elctronic Spctrum of I 2 and a Birg-Sponr Plot 1 15 2 25 3 35 4 45 Dpartmnt of Chmistry, Gustavus Adolphus Collg. 8 Wst Collg
More information