CHAPTER 5: Lie Differentiation and Angular Momentum
|
|
- Kenneth Rogers
- 5 years ago
- Views:
Transcription
1 CHAPTER 5: Le Dfferentaton and Angular Momentum Jose G. Vargas 1 Le dfferentaton Kähler s theory of angular momentum s a specalzaton of hs approach to Le dfferentaton. We could deal wth the former drectly, but we do not want to mss ths opportunty to show you both, as they are jewels. As an exercse, readers can at each step specalze the Le theory to rotatons. 1.1 Of Le dfferentaton and angular momentum For rotatons around the z axs, we have φ = x y y x. (1.1) On the left of (1.1), we have a partal dervatve. On the rght, we have an example of what Kähler defnes as a Le operator,.e. X = α (x 1, x 2,...x n ), (1.2) wthout explctly resortng to vector felds and ther flows. See secton 16 of hs 1962 paper. Incdentally, / does not respond to the concept of vector feld of those authors. For more on these concepts n Cartan and Kähler, see secton 8.1 of my book Dfferental Geometry for Physcsts and Mathematcans. Contrary to what one may read n the lterature, not all concepts of vector feld are equvalent. One would lke to make (1.2) nto a partal dervatve. When I had already wrtten most of ths secton, I realzed that t was not good enough to refer readers to Kähler s 1960 paper n order how to do that; untl one gets hold of that paper (n German, by the way), many readers would not be able to understand ths secton. So, we have added the last subsecton of ths secton to effect such a change nto a partal dervatve. 66
2 Followng Kähler, we wrte the operator (1.1) as χ 3 snce we may extend the concept to any plane. We shall later use χ k = x x j xj x, (1.3) where (, j, k) consttutes any of the three cyclc permutatons of (1, 2, 3), ncludng the unty. Here, the coordnates are Cartesan. Startng wth chapter 2 posted n ths web ste (the frst one to be taught n the Kähler calculus phases (II and III) of the summer school), we have not used tangent-valued dfferental forms, not even tangent vector felds. Let us be more precse. We wll encounter expressons that can be vewed as components of vector-valued dfferental 1 forms because of the way they transform when changng bases. But those components are extractons from formulas arsng n manpulatons, wthout the need to ntroduce nvarant objects of whch those expressons may be vewed as components. The not resortng to tangent-valued quanttes wll reman the case n ths chapter, even when dealng wth total angular momentum; the three components wll be brought together nto just one element of the algebra of scalar-valued dfferental forms. 1.2 Le operators as partal dervatves Cartan and Kähler defned Le operators by (1.2) (n arbtrary coordnate systems!) and appled them to dfferental forms. A subrepttous dffculty wth ths operator s that the partal dervatves take place under dfferent condtons as to what s mantaned constant for each of them. Ths has consequences when appled to dfferental forms. In subsecton 1.8, we reproduce Kähler s dervaton of the Le dervatve as a sngle partal dervatve wth respect to a coordnate y n from other coordnate systems, X = α (x) = y n. (1.4) Hs proof of (1.4) makes t obvous why he chose the notaton y n Let u be a dfferental form of grade p, u = 1 p! a 1... p dx 1... dx p, (1.5) n arbtrary coordnate systems. Exceptonally, summaton does not take place over a bass of dfferental p forms, but over all values of the ndces. Ths notaton s momentarly used to help readers connect wth formulas n n Kähler s 1960 paper. 67
3 Our startng pont wll be Xa 1... p = α (x) a 1... p = a 1... p y n. (1.6) 1.3 Non-nvarant form of Le dfferentaton In subsecton 1.8, we derve Xu = 1 p! α a 1... p dx 1... dx p + dα e u, (1.7) wth the operator e as n prevous chapters. Assume that the α s were constants. The last term would drop out. Hence, for X gven by / and for u gven by a 1... p dx 1... dx p, we have X (a 1... p dx 1... dx p ) = (a 1... p dx 1... dx p ) = a 1... p dx 1... p, (1.8) where dx 1... p stands for dx 1... dx p. Ths allows us to rewrte (1.7) as Xu = 1 [ (a1 p! α... p dx 1... p ] ) + dα e u, (1.9) It s then clear that Xu = α u + dα e u, (1.10) In 1962, Kähler used (1.10) as startng pont for a comprehensve treatment of le dfferentaton. The frst term on the rght of (1.10) may look as suffcent to represent the acton of X on u, and then be overlooked n actual computatons. In subsecton 1.8, we show that ths s not so. We now focus on the frst term snce t s the one wth whch one can become confused n actual practce wth Le dervatves. Notce agan that, f the α s are constants and the constants (0, 0,..1, 0,...0) n partcular the last term n all these equatons vanshes. So, we have X(cu) = c u, (1.11) for a equal to a constant c. But X [a(x)u] = a(x) u (Wrong!) 68
4 s wrong. When n doubt wth specal cases of Le dfferentatons, resort to (1.10). The terms on the rght of equatons (1.7) to (1.10) are not nvarant under changes of bases. So, f u were the state dfferental form for a partcle, none of these terms could be consdered as propertes of the partcle, say ts orbtal and spn angular momenta. 1.4 Invarant form of Le dfferentaton Kähler subtracted α ω k e k u from the frst term n (1.10) and smultaneously added t to the second term. Thus he obtaned Xu = α d u + (dα) e u, (1.12) snce α u x α ω k e k u = α d u, (1.13) and where we have defned (dα) as (dα) dα + α ω k. (1.14) One may vew dα + α k ωk as the contravarant components of what Cartan and Kähler call the exteror dervatve of a vector feld of components α. By components as vector, we mean those quanttes whch contracted wth the elements of a feld of vector bases yeld the sad exteror dervatve. Both dfferental-form-valued vector feld and vectorfeld-valued dfferental 1 form are legtmate terms for a quantty of that type. The correspondng covarant components are (dα) = dα α h ω h. (1.15) If you do not fnd (1.15) n the sources from whch you learn dfferental geometry, and much more so f your knowledge of ths subject s confned to the tensor calculus, please refer agan to my book Dfferental Geometry for Physcsts and Mathematcans. Of course, f you do not need to know thngs n such a depth, just beleve the step from (1.14) to (1.15). We are usng Kaehler s notaton, or stayng very close to t. Nevertheless, there s a more Cartanan way of dealng wth the contents of ths and the next subsectons. See subsecton 1.7. In vew of the consderatons made n the prevous sectons, we further have Xu = α d u + (dα) e u (1.16) All three terms n (1.12) and (1.16) are nvarant under coordnate transformatons. The two terms on the rght do not mx when performng a change of bass. Ths was not the case wth the two terms on the rght of (1.7) and (1.10), even though ther form mght nduce one to beleve otherwse. 69
5 1.5 Acton of a Le operator on the metrc s coeffcents Followng Kähler we ntroduce the dfferental 1 form α wth components α,.e. α = α k dx k = g k α dx k. (1.17) If the α were components of a vector feld, the α k would be ts covarant components. But both of them are here components of the dfferental form α. We defne d α k by Hence, on account of (1.15), Therefore, (dα) = (d α k )dx k. (1.18) d α k α,k α h Γ h k. (1.19) d α k + d k α = α,k + α k, α h Γ h k α h Γ h k (1.20) In a coordnate system where α = 0 ( < n) and α n = 1, we have α, k = (g p α p ), k = g p, k α p = g n, k, (1.21) and, therefore, On the other hand, α, k +α k, = g n, k +g nk,. (1.22) α l Γ lk + α l Γ kl = 2Γ nk = g n,k + g nk, g k,n, (1.23) From (1.20), (1.22) and (1.23), we obtan d k α + d α k = g k x n. (1.24) 1.6 Kllng symmetry and the Le dervatve When the metrc does not depend on x n, (1.24) yelds d k α + d α k = 0. (1.25) We then have that Indeed, e dα = 2(dα). (1.26) e dα = e d(α k dx k )] = e [(α k,m α m,k )(dx m dx k )] = (α k, α,k )dx k, (1.27) 70
6 where the parenthess around dx m dx k s meant to sgnfy that we sum over a bass of dfferental 2 forms, rather than for all values of and k. By vrtue of (1.18), (1.19) and (1.25), we have 2(dα) = (d α k d k α )dx k = [(α,k α h Γ h We now use that Γ h k = Γ k h k) (α k, α h Γk h )]dx k. (1.28) n coordnate bases, and, therefore, 2(dα) = (α,k α k, )dx k = e dα. (1.29) Hence (1.26) follows, and (1.16) becomes Xu = α d u 1 2 e dα e u. (1.30) Notce that we have just got Xu n pure terms of dfferental forms, unlke (1.16), where (dα) makes mplct reference to the dfferentaton of a tensor feld. An easy calculaton (See Kähler 1962) yelds Hence, 2e dα = dα u u dα. (1.31) Xu = α d u dα u 1 u dα, (1.32) 4 whch s our fnal expresson for the Le dervatve of a dfferental form f that dervatve s assocated wth a Kllng symmetry. 1.7 Remarks for mprovng the Kähler calculus The Kähler calculus s a superb calculus, and yet Cartan would have wrtten t f alve. The man concern here s the use of coordnate bases. We saw n chapter one the dsadvantage they have when compared wth the orthonormal ω s; these are dfferental nvarants that defne a dfferentable manfold endowed wth a metrc. In ths secton, the dsadvantage les n that one needs to have extreme care when rasng and lowerng ndces, whch s not a problem wth orthonormal bases snce one smply multples by one or mnus one. Add to that the fact that dx does not make sense snce there are not covarant curvlnear coordnates. On the other hand, ω s well defned. Consder next the Kllng symmetry, (1.25). The d k α are assocated wth the covarant dervatve of a vector feld. But they could also be assocated wth the covarant dervatves of a dfferental 1-form. Indeed, we defne (d α) k by d α = (α k, α l Γ l k )dx k (d α) k dx k. (1.33) 71
7 But Thus d k α α k, α h Γ h k. (1.34) (d α) k = d k α (1.35) and the argument of the prevous two sectons could have been carred out wth covarant dervatves of dfferental forms wthout nvokng components of vector felds. 1.8 Dervaton of Le dfferentaton as partal dfferentaton Because the treatment of vector felds and Le dervatves n the modern lterature s what t s, we now proceed to show how a Le operator as defned by Kähler (and by Cartan, except that he dd not use ths termnology but nfntesmal operator) can be reduced to a partal dervatve. Consder the dfferental system λ = α (x 1,... x n ), (1.36) the α not dependng on λ. One of n ndependent constant of the moton (.e. lne ntegrals) s then addtve to λ. It can then be consdered to be λ tself. Denote as y ( = 1, n 1) a set of n 1 such ntegrals, ndependent among themselves and ndependent of λ, to whch we shall refer as y n. The y s ( = 1, n) consttute a new coordnate system and we have x = x (y 1,, y n ). (1.37) In the new coordnate system, the Le operator reads X = β / y. Its acton on a scalar functon s β f f = αl = xl f = f y x l λ x l y. (1.38) n We rewrte u (gven by (1.5)), as and then u = 1 p! a 1... p 1 u y = 1 a 1... p 1 n p! y n y (p 1)! a 1... p y n y 1 p p y p dyk 1... dy kp, (1.39) y dyk 1... dy kp + ( p x 1 ) x 2 y k 1 dyk 1 y k 2 p y kp dyk 2... dy kp. (1.40) 72
8 We now use that and that ( x 1 ) y n y k dyk 1 1 Hence a 1... p y n = y k 1 = a 1... p ( x 1 y n y n = α a 1... p (1.41) ) ( dy k 1 x 1 = d y n ) = dα 1. (1.42) Xu = u y n = 1 p! α a 1... p dx 1... p + 1 p! a 1... p dα dx 2... dx p. (1.43) and fnally 2 Angular momentum Xu = 1 u α p! x + dα e u, (1.44) The components of the angular momentum operators actng on scalar functons are gven by (1.3), and therefore and α k = x j dx + x dx j, (2.1) dα k = dx j dx + dx dx j = 2dx dx j 2w k. (2.2) Hence χ k u = x u u xj xj x w k u 1 2 u w k. (2.3) The last two terms consttute the component k of the spn operator. It s worth gong back to (1.7) and (1.10), where we have the entangled germs of the orbtal and spn operator, f we replace χwth χ k. It does not make sense to speak of spn as ntrnsc angular momentum untl u represents a partcle, whch would not be the case at ths pont. Kähler denotes the total angular momentum as K + 1, whch he defnes as 3 (K + 1)u = χ u w. (2.4) =1 He then shows by straghtforward algebra that K(K + 1) = χ χ χ 2 3. (2.5) He also develops the expresson for (K + 1) untl t becomes (K + 1)u = u dx rdr + 73 x u x + 3 (u ηu) + gηu (2.6) 2
9 and also (K + 1)u = ζ ζu rdr + x u x + 3 (u ηu) + gηu, (2.7) 2 where η s as n prevous chapters, where ζ reverses the order of all the dfferental 1 form factors n u and where g dx e. Ths expresson for (K + 1)u s used n the next secton. 3 Harmonc dfferentals n E 3 {0} As could be expected, the equaton u = 0 (whch n E 3 {0} concdes wth u = 0) has an nfnte varety of solutons. One seeks solutons that are proper functons of the total angular momentum operator. One becomes more specfc and specalzes to those whose coeffcents are harmonc functons of the Cartesan coordnates. One need only focus on those that belong to the even subalgebra, snce we can obtan the other ones by product of the even ones wth the unt dfferental form of grade three. One recovers generalty by formng lnear combnatons. Those from the even subalgebra can be wrtten as u = a + v, where a and v are dfferental 0 form and 2 form respectvely. Kähler shows that the acton of the dfferental operator K on these dfferental forms f homogeneous of degree h s gven by Ku = (h + 1)a + (h + 1)v 2da rdr. (3.1) Kähler shows that for these u to be proper dfferentals of K wth proper value k t s necessary and suffcent that the followng equatons be satsfed (h + 1)a = ka, (h + 1)v 2da rdr = kv. (3.2) We stop the argument at ths pont, havng shown a role that the operator K plays n fndng the sought solutons. We may retake the argument at some pont n the future. 4 The fne structure of the hydrogen atom 5 Entry pont for research n analyss wth the Kähler calculus 74
Difference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationAffine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
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 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 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 informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
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 informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
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 informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More informationIntegrals and Invariants of Euler-Lagrange Equations
Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationSpin-rotation coupling of the angularly accelerated rigid body
Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationLecture 20: Noether s Theorem
Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external
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 informationFrom Biot-Savart Law to Divergence of B (1)
From Bot-Savart Law to Dvergence of B (1) Let s prove that Bot-Savart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of Bot-Savart. The dervatve s wth respect to
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 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 informationOne Dimension Again. Chapter Fourteen
hapter Fourteen One Dmenson Agan 4 Scalar Lne Integrals Now we agan consder the dea of the ntegral n one dmenson When we were ntroduced to the ntegral back n elementary school, we consdered only functons
More informationMA 323 Geometric Modelling Course Notes: Day 13 Bezier Curves & Bernstein Polynomials
MA 323 Geometrc Modellng Course Notes: Day 13 Bezer Curves & Bernsten Polynomals Davd L. Fnn Over the past few days, we have looked at de Casteljau s algorthm for generatng a polynomal curve, and we have
More information12. The Hamilton-Jacobi Equation Michael Fowler
1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
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 informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationProjective change between two Special (α, β)- Finsler Metrics
Internatonal Journal of Trend n Research and Development, Volume 2(6), ISSN 2394-9333 www.jtrd.com Projectve change between two Specal (, β)- Fnsler Metrcs Gayathr.K 1 and Narasmhamurthy.S.K 2 1 Assstant
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 informationProfessor Terje Haukaas University of British Columbia, Vancouver The Q4 Element
Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca The Q Element Ths document consders fnte elements that carry load only n ther plane. These elements are sometmes referred to
More informationQuantum Mechanics I Problem set No.1
Quantum Mechancs I Problem set No.1 Septembe0, 2017 1 The Least Acton Prncple The acton reads S = d t L(q, q) (1) accordng to the least (extremal) acton prncple, the varaton of acton s zero 0 = δs = t
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 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 informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationTensor Analysis. For orthogonal curvilinear coordinates, ˆ ˆ (98) Expanding the derivative, we have, ˆ. h q. . h q h q
For orthogonal curvlnear coordnates, eˆ grad a a= ( aˆ ˆ e). h q (98) Expandng the dervatve, we have, eˆ aˆ ˆ e a= ˆ ˆ a h e + q q 1 aˆ ˆ ˆ a e = ee ˆˆ ˆ + e. h q h q Now expandng eˆ / q (some of the detals
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
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 informationMechanics Physics 151
Mechancs Physcs 5 Lecture 0 Canoncal Transformatons (Chapter 9) What We Dd Last Tme Hamlton s Prncple n the Hamltonan formalsm Dervaton was smple δi δ p H(, p, t) = 0 Adonal end-pont constrants δ t ( )
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
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 informationLinear Feature Engineering 11
Lnear Feature Engneerng 11 2 Least-Squares 2.1 Smple least-squares Consder the followng dataset. We have a bunch of nputs x and correspondng outputs y. The partcular values n ths dataset are x y 0.23 0.19
More informationSpecial Relativity and Riemannian Geometry. Department of Mathematical Sciences
Tutoral Letter 06//018 Specal Relatvty and Reannan Geoetry APM3713 Seester Departent of Matheatcal Scences IMPORTANT INFORMATION: Ths tutoral letter contans the solutons to Assgnent 06. BAR CODE Learn
More informationCausal Diamonds. M. Aghili, L. Bombelli, B. Pilgrim
Causal Damonds M. Aghl, L. Bombell, B. Plgrm Introducton The correcton to volume of a causal nterval due to curvature of spacetme has been done by Myrhem [] and recently by Gbbons & Solodukhn [] and later
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationFor now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results.
Neural Networks : Dervaton compled by Alvn Wan from Professor Jtendra Malk s lecture Ths type of computaton s called deep learnng and s the most popular method for many problems, such as computer vson
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 informationPHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationLorentz Group. Ling Fong Li. 1 Lorentz group Generators Simple representations... 3
Lorentz Group Lng Fong L ontents Lorentz group. Generators............................................. Smple representatons..................................... 3 Lorentz group In the dervaton of Drac
More informationLinear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.
Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationCALCULUS CLASSROOM CAPSULES
CALCULUS CLASSROOM CAPSULES SESSION S86 Dr. Sham Alfred Rartan Valley Communty College salfred@rartanval.edu 38th AMATYC Annual Conference Jacksonvlle, Florda November 8-, 202 2 Calculus Classroom Capsules
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationLagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013
Lagrange Multplers Monday, 5 September 013 Sometmes t s convenent to use redundant coordnates, and to effect the varaton of the acton consstent wth the constrants va the method of Lagrange undetermned
More informationare called the contravariant components of the vector a and the a i are called the covariant components of the vector a.
Non-Cartesan Coordnates The poston of an arbtrary pont P n space may be expressed n terms of the three curvlnear coordnates u 1,u,u 3. If r(u 1,u,u 3 ) s the poston vector of the pont P, at every such
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 informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationCS 468 Lecture 16: Isometry Invariance and Spectral Techniques
CS 468 Lecture 16: Isometry Invarance and Spectral Technques Justn Solomon Scrbe: Evan Gawlk Introducton. In geometry processng, t s often desrable to characterze the shape of an object n a manner that
More informationThree views of mechanics
Three vews of mechancs John Hubbard, n L. Gross s course February 1, 211 1 Introducton A mechancal system s manfold wth a Remannan metrc K : T M R called knetc energy and a functon V : M R called potental
More informationMechanics Physics 151
Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed mult-partcle systems! Internal and external forces! Laws of acton and
More informationMATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples
More information8.6 The Complex Number System
8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want
More informationTHE WEIGHTED WEAK TYPE INEQUALITY FOR THE STRONG MAXIMAL FUNCTION
THE WEIGHTED WEAK TYPE INEQUALITY FO THE STONG MAXIMAL FUNCTION THEMIS MITSIS Abstract. We prove the natural Fefferman-Sten weak type nequalty for the strong maxmal functon n the plane, under the assumpton
More informationAssortment Optimization under MNL
Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenue-maxmzng assortment of products to offer when the prces of products are fxed.
More informationTHE SUMMATION NOTATION Ʃ
Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
More informationBezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0
Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the
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 informationMAE140 - Linear Circuits - Fall 13 Midterm, October 31
Instructons ME140 - Lnear Crcuts - Fall 13 Mdterm, October 31 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationCSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography
CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve
More informationWeek 9 Chapter 10 Section 1-5
Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,
More informationMAE140 - Linear Circuits - Winter 16 Midterm, February 5
Instructons ME140 - Lnear Crcuts - Wnter 16 Mdterm, February 5 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationTHE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions
THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George
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 informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
More informationPhysics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationwhere a is any ideal of R. Lemma 5.4. Let R be a ring. Then X = Spec R is a topological space Moreover the open sets
5. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationPHYS 705: Classical Mechanics. Hamilton-Jacobi Equation
1 PHYS 705: Classcal Mechancs Hamlton-Jacob Equaton Hamlton-Jacob Equaton There s also a very elegant relaton between the Hamltonan Formulaton of Mechancs and Quantum Mechancs. To do that, we need to derve
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 informationAdvanced Circuits Topics - Part 1 by Dr. Colton (Fall 2017)
Advanced rcuts Topcs - Part by Dr. olton (Fall 07) Part : Some thngs you should already know from Physcs 0 and 45 These are all thngs that you should have learned n Physcs 0 and/or 45. Ths secton s organzed
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 information10. Canonical Transformations Michael Fowler
10. Canoncal Transformatons Mchael Fowler Pont Transformatons It s clear that Lagrange s equatons are correct for any reasonable choce of parameters labelng the system confguraton. Let s call our frst
More informationWeek 5: Neural Networks
Week 5: Neural Networks Instructor: Sergey Levne Neural Networks Summary In the prevous lecture, we saw how we can construct neural networks by extendng logstc regresson. Neural networks consst of multple
More informationCONJUGACY IN THOMPSON S GROUP F. 1. Introduction
CONJUGACY IN THOMPSON S GROUP F NICK GILL AND IAN SHORT Abstract. We complete the program begun by Brn and Squer of charactersng conjugacy n Thompson s group F usng the standard acton of F as a group of
More informationLecture Notes 7: The Unruh Effect
Quantum Feld Theory for Leg Spnners 17/1/11 Lecture Notes 7: The Unruh Effect Lecturer: Prakash Panangaden Scrbe: Shane Mansfeld 1 Defnng the Vacuum Recall from the last lecture that choosng a complex
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More information(δr i ) 2. V i. r i 2,
Cartesan coordnates r, = 1, 2,... D for Eucldean space. Dstance by Pythagoras: (δs 2 = (δr 2. Unt vectors ê, dsplacement r = r ê Felds are functons of poston, or of r or of {r }. Scalar felds Φ( r, Vector
More informationChapter 4 The Wave Equation
Chapter 4 The Wave Equaton Another classcal example of a hyperbolc PDE s a wave equaton. The wave equaton s a second-order lnear hyperbolc PDE that descrbes the propagaton of a varety of waves, such as
More informationSolutions Homework 4 March 5, 2018
1 Solutons Homework 4 March 5, 018 Soluton to Exercse 5.1.8: Let a IR be a translaton and c > 0 be a re-scalng. ˆb1 (cx + a) cx n + a (cx 1 + a) c x n x 1 cˆb 1 (x), whch shows ˆb 1 s locaton nvarant and
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 informationSolution Thermodynamics
Soluton hermodynamcs usng Wagner Notaton by Stanley. Howard Department of aterals and etallurgcal Engneerng South Dakota School of nes and echnology Rapd Cty, SD 57701 January 7, 001 Soluton hermodynamcs
More informationSo far: simple (planar) geometries
Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationLecture Note 3. Eshelby s Inclusion II
ME340B Elastcty of Mcroscopc Structures Stanford Unversty Wnter 004 Lecture Note 3. Eshelby s Incluson II Chrs Wenberger and We Ca c All rghts reserved January 6, 004 Contents 1 Incluson energy n an nfnte
More information), it produces a response (output function g (x)
Lnear Systems Revew Notes adapted from notes by Mchael Braun Typcally n electrcal engneerng, one s concerned wth functons of tme, such as a voltage waveform System descrpton s therefore defned n the domans
More information