CHAPTER 5: Lie Differentiation and Angular Momentum

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