Adding Angular Momenta

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1 Adding Angular Monta Michal Fowlr, UVa /8/07 Introduction Considr a syst having two angular onta, for xapl an lctron in a hydrogn ato having both orbital angular ontu and spin Th kt spac for a singl angular ontu has an orthonoral basis, so for two angular onta an obvious orthonoral basis is th st of dirct product kts,, What dos this an, xactly? Suppos th first angular ontu has agnitud J = +, and is in th stat α,, and J ( ) siilarly th scond angular ontu J is in th stat = β, = Evidntly th probability aplitud for finding th first spin in stat and at th sa ti th scond in is α β, and w dnot that stat by,, How to handl ths dirct product spacs will bco clar on xaining spcific xapls, as w do blow, bginning with two spins on-half Now th su of two angular onta J = J + J is itslf an angular ontu, oprating in a spac with a coplt basis, This is asy to prov: th coponnts of J satisfy Ji, J = i ε ikjk, and siilarly for th coponnts of J Th coponnts of J cout with th coponnts of J, of cours, fro which it follows idiatly that th vctor coponnts of J = J+ J do indd oby th angular ontu coutation rlations: and rcall that th coutation rlations wr sufficint to dtrin th allowd sts of ignvalus W shall prov latr that th ignstats, of J, Jz ar a coplt basis for th product spac of th ignkts of J, J, Jz, Jz to stablish this, w ust first find th possibl allowd valus of th total angular ontu quantu nubr Hr w hav, thn, two diffrnt orthonoral bass for what is vidntly th sa vctor spac In practical applications, it oftn turns out that w hav to translat fro on of ths bass to th othr Our prsnt task is to construct th appropriat transforation: w accoplish this by finding th cofficints of any, in th,, basis (Ths ar calld th Clbsch-Gordan cofficints)

2 W shall build gradually, bginning with adding two spins on-half, thn a spin on-half with an orbital angular ontu, finally two gnral angular onta This is a vry iportant part of quantu chanics: w giv vry dtail Radrs alrady sowhat failiar with th subct ay find this a bit tdious, thy can glanc ovr th introductory xapls and go to th gnral cas Adding Two Spins: th Basis Stats and Spin Oprators Th ost lntary xapl of a syst having two angular onta is th hydrogn ato in its ground stat Th orbital angular ontu is zro, th lctron has spin angular ontu, and th proton has spin Th spac of possibl stats of th lctron spin has th two basis kts and, (also 0 variously writtn as +, ;, ; χ+, χ!) th basis proton spin kts ar and 0 p p, so th possibl stats of th cobind syst ar kts in th dirct product spac which has a basis of four kts: p, p, p, p using p as shorthand for p Not hr that w v writtn th kts in alphabtical ordr with as th first lttr, as th scond That is to say, w v first writtn all th kts having as th first lttr, tc For th or gnral cas of adding to, to b considrd shortly, w ll ordr th kts in th sa alphabtical way, writing first all th kts having =, and so on down to =, so th possibl sts ar: ( ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ),,,,,,,,, Th dinsionality of this spac is thn ( +) ( + ) Now th first block of + lnts all hav th sa -coponnt of, that is, =, th nxt block has =, and so on Think about what this ans for constructing a rotation oprator acting on th kts in this spac: if it oprats only on th angular ontu, it will chang th factors i ultiplying th blocks, if th oprator rotats only, it will oprat within ach block, all th blocks bing changd in th sa way To gt a fling for how this works in practic, w go back to th siplst cas, two spins onhalf Th spac is four-dinsional, having basis

3 3 p, p, p, p Any oprator acting on th spins will b rprsntd by a 4 4 atrix, bst thought of as a atrix ad up of blocks: an oprator acting on th proton spin acts within th blocks, on oprating on th lctron spin acts on th blocks thslvs, rgardd as singl ntitis Lt s look at a fw xapls Rcall that th raising oprator for a singl spin is th atrix 0 S + = So what is th raising oprator for th lctron spin? 0 0 S I Ip = = W us bold to dnot atrics Th pattrn is clar: th big structur (in bold abov), that of th four blocks, rflct th 0 structur of th lctron spin oprator S + =, within thos blocks (of which only on survivs) th idntity oprator I = acts on th proton spin 0 Th oprator that raiss th proton spin is: σ I S + p = = + 0 σ What about th oprator that raiss both lctron and proton spin? In this cas, th pattrn of 0 blocks, and th pattrn within ach block, ust both b, so 0 0 S σ Sp = =

4 4 Thr is only on nonzro atrix lnt bcaus only on br of th bas survivs this opration If two spins intract (via thir agntic onts, for xapl) in a way that prsrvs total angular ontu, a possibl tr in th Hailtonian would b S S +, rprsntd by: SSp = = S Sp p Rprsnting th Rotation Oprator for Two Spins Rcall fro th lctur on spin that th rotation oprator on a singl spin on-half is ( ) ( ) ( θ ) iθ nˆ J i ( /)( nˆ ) cos ( σ) / θ σ ˆ θ D R n I i nˆ θ = = = sin in th spinor spac As w stablishd, this atrix oprator has th for ( θ ) inz ( θ ) ( inx + ny) ( θ ) ( x + y) sin ( θ / ) cos ( θ / ) + zsin ( θ / ) a b cos / sin / sin / * * = b a in n in with a + b = This st of unitary atrics for a rprsntation of th rotation group in th sns that th total rsulting fro two succssiv rotations is givn by th atrix which is th atrix product of thos corrsponding to th two rotations Fro th discussion in th prvious sction, it should b clar that in th product spac of th two spins, th rprsntation of th rotation oprator both spins of cours undrgoing th sa rotation is: a b a b a ab ab b a * * b * * * * * * b a b a ab aa bb a b = * * * * ab bb aa a b * a b * a b b a * * * * * * * * * * b a b a b a b a b a

5 5 This st of 4 4 atrics, again with a + b =, ust also for a rprsntation of th rotation group ovr th four-dinsional spac W shall shortly discovr that this rprsntation can b siplifid, but to achiv that w nd to analyz th stats in trs of total angular ontu Rprsnting Stats of Two Spins in Trs of Total Angular Montu W r now rady to look at total spin stats for th ground-stat (zro orbital angular ontu) hydrogn ato Considr first th stat with both lctron and proton spin pointing upwards, Th z- coponnt of th total spin is Sz = Sz + S p z, so S z = Labling th total spin stat s,, w hav a stat with =, so s = (To confir that this stat indd has s = w can apply th total-spin raising oprator S+ = S+ + S p + Sinc both coponnt spins hav axiu valu, p S+ s, = ( S S ) 0 s =, but S + only givs zro whn acting on th = s br of a ultiplt ) W find, thn, that =, s whr w v addd th suffix s to ak clar that th nubrs in th last kt signify s, for th total spin Th total spin s =, bing a total angular ontu ignstat, has a triplt of valus, =, 0,,, s bing th top br Th = 0 br is found by applying th lowring oprator to : which togthr with p ( )( p) S = S + S = + p p givs S = S, =, 0, s s ( ), 0 = + s Obviously, th third br of th triplt,, s = But this triplt only accounts for thr basis stats in th s, total angular ontu rprsntation A fourth stat, orthogonal to ths thr and noralizd, is ( ) This has = 0, and also has s = 0, asily chckd by noting that th total spin raising oprator

6 6 S+ = S+ + S p + valu acting on this stat givs zro, so th stat has th axiu allowd for its s To suariz: in th total angular ontu s, rprsntation for two spins on-half, th four basis stats ar,,, 0,,, 0, 0 This orthonoral basis spans th sa s s s s spac as th othr orthonoral st,,, Our construction of th s, stats abov aounts to finding on st of basis kts in trs of th othrs Not that sinc both sts of basis kts ar orthonoral, apping a vctor fro on st to th othr is a unitary transforation But thr s or: th cofficints w found xprssing on basis kt in th othr basis ar all ral This ans that if any kt has ral cofficints in on basis, it dos in th othr For this spcial cas of all ral cofficints, a unitary transforation is trd orthogonal Th orthogonal transforation xprssing on bas in trs of th othr is asy to construct:, s, s = 0,0 0 0 s, s Th atrix is orthogonal and sytric, so is its own invrs Gotrically, s = ans th coponnt spins ar paralll, for s = 0 thy ar antiparalll This 3 3 can b statd or prcisly: S S = S S S, so for s =, S S = ( 4 4) = / 4, 3 and for s = 0 S S = 4 This aks it asy to construct proction oprators into th s = 0 3 P = s= S S / + and s = subspacs: ( ) 4 Rprsnting th Rotation Oprator in th Total Angular Montu Basis W v alrady stablishd that th rotation oprator, acting on th two spin syst, can b rprsntd by a 4 4 atrix, and that th nw (total angular ontu) basis can b rachd fro th original (two sparat spin) basis by th orthogonal transforation givn xplicitly abov Thrfor, pr-and post-ultiplying th two-spin rotation oprator will in fact giv a 4 4 atrix rprsntation of th rotation oprator in th nw total angular ontu basis Howvr, that approach isss th point: first, th singlt stat ( ) angular ontu, and so is not changd by rotation has zro

7 7 Scond, th triplt stat has angular ontu on, so rotation oprators ust act on it ust as w found arlir for an angular ontu on: ˆ iθ n J ˆ ˆ () n J n J D ( R( θ )) = = I + ( cosθ ) isin θ This ans that, as far as rotations ar concrnd, th spac spannd by th four kts 0,0,,,,0,, s s s s is actually a su of two sparat subspacs, th on-dinsional spac 0,0 s, and th thr-dinsional spac having basis,,, 0,, Undr s s s rotation, a vctor in on of ths subspacs stays thr: thr ar no cross trs in th atrix ixing th spacs I 0 This ans that th rotation atrix has th for whr R 3 is th 3 3 atrix for spin 0 R3 on, I is ust th trivial atrix in th singlt subspac, in othr words, and th O s ar 3 and 3 sts of zros A stat of th spins can of cours b a su of coponnts in th two subspacs, for xapl ( ) ( ) = + + Rducibl and Irrducibl Group Rprsntations W bgan our discussion of two spins on-half by xaining proprtis of spin oprators in th four-dinsional product spac of th two two-dinsional spin spacs, and wnt on to construct a four-dinsional rprsntation of th gnral rotation oprator in that spac: a atrix rprsntation of th rotation group But whn th two-spin syst is labld in trs of total angular ontu, w find that in fact this four-dinsional rotation oprator is a su of a thr-dinsional rotation, and a trivial idntity rotation for an angular ontu zro stat Th four-dinsional oprator can b diagonalizd : th spac split into a thr dinsional spac and a on-dinsional spac that don t ix undr rotation, and any stat of th syst is a su of kts fro th two spacs This is oftn xprssd by saying th product spac of two spins on-half is th su of a spin on spac and a spin zro spac, and writtn = 0 Putting in th dinsionalitis of th spacs in this quation, = 3+

8 8 This sipl chck on total dinsionality sts th pattrn for or coplicatd product spacs xaind blow Th 4 4 rprsntation of th rotation oprator is said to b a rducibl rprsntation: it can b rducd to a su of sallr dinsional rprsntations An irrducibl rprsntation is on in which thr ar no subspacs invariant undr all rotations Rcall that w constructd th rducibl 4 4 rprsntation by taking a dirct product of th spin on-half rprsntations of th rotation group Th quation = 0 w usd abov to dscrib th kt spacs quivalntly dscribs th rotation group rprsntations within thos subspacs On ight wondr why w would bothr to build two diffrnt bass for th sa vctor spac Th rason is that diffrnt probls nd diffrnt bass For a syst of two spins in an xtrnal agntic fild, not intracting with ach othr, th indpndnt spins basis, tc, is natural On th othr hand, for a hydrogn ato in no xtrnal fild, but including an intraction btwn th spins (which ar alignd with th agntic dipol onts of th particls) th, basis is th right on: th intraction Hailtonian is proportional to S S p, which can b ( )( ) ( )( ) p p p p p writtn Sx + isy Sx isy + Sx isy Sx + isy + SzSz, whr w rcogniz th raising and lowring oprators for th individual spins This ans that th stat, for xapl, cannot b an ignstat if th spin tr in th Hailtonian is S S p, but th stats, ar ignstats bcaus S S p couts with th total angular ontu and its coponnts But what would b a good basis for a hydrogn ato, including th S xtrnal agntic fild? That is a nic xrcis for th radr S p tr, and in an Adding a Spin to an Orbital Angular Montu In this sction, w considr a hydrogn ato in a stat with nonzro orbital angular ontu, L 0 Such orbital otion is quivalnt to an lctric currnt loop and gnrats a agntic fild Th agntic dipol ont associatd with th lctron spin intracts with this fild, th appropriat Hailtonian having a tr proportional to L S, and is trd th spin-orbit intraction Th proton also has a agntic ont, but that is thr ordrs of agnitud sallr than th lctron s, so w ll nglct it for now Th spin-orbit intraction L S is ost naturally analyzd in th basis stats of total angular ontu,,, whr J = L + S (s th analogous discussion of th spin-spin intraction abov) Writ th orbital angular ontu ignstats l, l and th spin stats s, s whr, = and, = Th product spac l, l s, s is ( l + ) dinsional: a singl kt in this product spac would b fully dscribd by l, ; s,, but sinc both l, s ar l s

9 9 constant throughout th probl, th only actual variabls ar or copact for,, for xapl, l s l s l s, l s so w ll writ th kt in th Th axiu possibl angular ontu coponnt in th z-dirction is clarly( l + ), for th stat l, In th total angular ontu rprsntation, this ust b th stat, = l+, l+ So th two diffrnt bass hav a coon br: l+ l+ = l,, l s In th total angular ontu, rprsntation, l+, l+ is th top stat of a ultiplt having ( ) l+ + = l + brs Th proof is th sa as that for adding two spins on-half: th total spin raising oprator givs zro acting on this stat And, ust as for th spin-spin cas, th nxt br down of th ultiplt is gnratd by applying th lowring oprator: Thrfor J l+, l+ = l+ l+, l ( ) = L + S l, l s = l l, + l, l s l s l l +, l = l, l l s l s l +, + l+ This stat l+, l lis in th = l subspac, which is two-dinsional, having basis vctors l, and l, l s l s in th l s rprsntation So it ust hav two basis vctors in th rprsntation as wll Th othr kt ust b orthogonal to l+, l and noralizd: it can only b l l, l = l, l l s l s l+ l+, W v rprsntd this nw kt in as th top stat of a = l ultiplt It s asy to chck that this is indd th cas: it has = l, and J + acting on it givs zro, so it has to b th top br of its ultiplt

10 0 Th only abiguity is an ovrall phas: th Condon-Shortly convntion is that th highst - stat of th largr coponnt angular ontu is assignd a positiv cofficint is th top stat of a nw ultiplt having ( l ) = l + and = l takn togthr hav ( l + ) So l, l + = l brs Th two ultiplts brs, and thrfor span th whol ( l + ) dinsional spac Th rst of th basis vctors ar gnratd by rpatd application of th lowring oprator in th two ultiplts Th rason thr ar only two ultiplts in this probl is that thr ar only two ways th spin on-half can point rlativ to th orbital angular ontu Rcalling that for th two spins w xprssd th product spac a su of a spin spac and a spin 0 spac, = 0, th analogous quation hr is ( ) ( ) l = l+ l For th gnral cas of adding angular onta, with, + ultiplts ar gnratd, corrsponding to th nubr of possibl rlativ orintations of th two angular onta Adding Two Angular Monta: th Gnral Cas Th spac of kts dscribing two angular onta, is th dirct product of two spacs ach for a singl angular ontu, but th dirct product natur of th kts is usually not ad xplicit, for xapl,, is usually writtn as a singl kt, ;, Just as in th xapls abov, sinc, ar fixd throughout, thy don t nd to b writtn into vry kt, w ll ust dnot th kt by,, or, whn daling with nurical valus, appnd as a suffix, for xapl,3 Th kts, + + dinsional product spac of th two angular onta: thy ar th ignstats of th coplt st of couting variabls J, J, J, J for a coplt orthonoral basis of th ( )( ) z z Total Angular Montu Basis Stats Thr is of cours an altrnativ coplt orthogonal basis of th spac of th two angular onta: for total angular ontu J = J+ J, a diffrnt st of coplt couting variabls is: J, J, J, Jz (This is not th sa st of stats as in th prvious paragraph: for xapl, J dos not cout with Jz Chck it out!) W shall stablish latr in th lctur that th allowd valus of total angular ontu rang fro = + to =, ust as on would naïvly xpct

11 This altrnativ st is a bttr basis st for two angular onta intracting with ach othr an intraction tr lik J J can chang, but not = +, or J As always, w r taking J, J to b constants throughout, so th significant variabls hr ar J and Jz, and w writ th stats siply as, or whn w hav nurical valus, 3,, following th notation introducd abov Of cours, J, = ( + ),, and J, =, z Going fro On Basis to th Othr: th Clbsch-Gordan Cofficints How do w writ a stat, in trs of th stats,? Furthror, how do w prov th nw st of stats, is a coplt basis for th spac? W know that th st of stats,,, is a coplt basis, sinc th whol spac is a product spac of th and spacs, which ar spannd by th sts, rspctivly Thrfor, th idntity oprator can b writtn = I,, = = It follows that any total angular ontu ignkt, can b xprssd as a su ovr th basis vctors, :, =,,, = = Th cofficints,, ar calld th Clbsch Gordan cofficints, oftn writtn CG cofficints On idiat proprty of th CG cofficints is that,, = 0 unlss = + This follows fro th oprator idntity Jz = Jz + J z takn btwn a bra and a kt fro diffrnt bass,, J, =, J + J, z z z and J,, z =,, ( J J ), ( ) + = +, z z

12 so ( ),, = 0 W alrady know that th axiu valu of is, and of is, so th axiu valu of is Thrfor, th axiu valu of = +, bcaus if it could go any highr, + thr would b a highr sowhr in th spac, contradicting = + For th st,, thr is on kt having this axial valu of :, Equally, in th st of stats, thr is only on with th axial : + +, Thrfor, ths two kts ust b idntical (stting th arbitrary phas factor qual to on):, = +, is th top kt in a ultiplt having ( ) Now, + = + + brs Th nxt-to-top br of th ultiplt is gnratd as bfor by applying th lowring oprator to both rprsntations: giving ( ) J +, + = J + J, so ( + ) + + = +,,, +, + =, +, + + and by xact analogy with th spin orbit cas, th othr subspac is basis stat in th = + +, + =, +, + + with th appropriat (Condon Shortly) sign convntion for > This is th top br of a ultiplt having = +, and so ( + ) + = ( + ) brs (chckd as usual by applying and gtting zro) J +

13 3 To procd furthr, th lowring oprator is applid onc or, to ntr th = + subspac In th rprsntation, this has thr indpndnt basis vctors (providd > ):,,,,, But only two kts hav bn lowrd in th rprsntation th issing third kt in th = + subspac ust b th top br of anothr nw ultiplt having = +, and so brs Not that th cofficints gnratd by th lowring oprators ar all ral, so all thr in th = + subspac can b writtn in trs of th ( ) + 3 kts kts with ral cofficints This procss can b rpatd until th dinsionality of th spac, fro th in add to a total dinsionality ultiplts gnratd span th spac Rcall that th rprsntation, is ( )( ) + + Th ultiplts ( ) ( ) ( ) but whr do w stop? Coon sns suggsts that for >, th iniu total angular ontu ust b = Coon sns is not ncssarily to b trustd, but it is clar that all th brs of th ultiplts in gnratd by using th lowring oprator, followd by introducing a nw orthogonal ultiplt top br ach ti, as dscribd abov, ar indpndnt orthonoral kts, and if w stop at =, th total nubr gnratd is (Us ( n+ ) = ( + ) n= 0 + ( n+ ) = ( + )( + ) n= ) This stablishs that including all total angular onta btwn and + dos in fact giv a coplt basis spanning th spac, so ( ) ( ) ( ) = + + Calculating Clbsch-Gordan Cofficints Using Rcursion Rlations Th sch prsntd abov, constructing a succssion of ultiplts bginning fro th highst stat and using th Condon-Shortly convntion to sttl signs, will gnrat all th CG cofficints Howvr, anothr approach provs usful in latr work Rcall that by finding

14 4 atrix lnts of Jz = Jz + J z btwn a, bra and a, kt, w stablishd that th Clbsch-Gordan cofficints ar zro unlss = + A paralll valuation of atrix lnts of J± = J ± + J ± yilds a rlationship btwn thr CG cofficints: yilds, J, =, J, +, J, ( + ) ( + ),, + = ( ) ( ) ( ) ( ) +,, + +,, whr J + acting to th lft rducs by on (Hr, obviously, w ust choos = + to hav nonzro cofficints) To visualiz what s going on with all ths cofficints, rbr can tak + valus and can tak + valus, so for givn, vry possibl stat of th two spins can b rprsntd by a dot on a ( + ) ( + ) grid: hr s = 3, = : = = = 3, = Each dot rprsnts an, stat for Th four blu dots hav = + = For givn total, th cofficints for th thr rd dots ar connctd by a J + rcursion rlation How do ths dots rlat to th CG cofficints? th top right-hand dot (3, ) uniquly rprsnts th = 5, = 5 stat of total angular ontu Th nxt dots down, (, ) and (3, ), corrspond to two CG cofficints for = 5 and two diffrnt CG cofficints for = 4 If w now pick on valu of lss than +, ach dot in th grid will corrspond to on cofficint

15 5 Not that having fixd, th grid will b curtaild: lt s tak = 3, so = + = 3 at ost Thn th grid loss its far cornrs: = = Nonzro grid brs for = 3 Lt us xain for this fixd which CG cofficints ar whr in this curtaild grid Thr ar a total of + = 7 stats for = 3,,, 3 Th top stat, = 3, = 3, or 3,3, is givn by thr cofficints on th top diagonal lin (it s in a thr-dinsional subspac, and orthogonal to = 5 and = 4 ultiplt brs 5,3, 4,3 which ar also in th = 3 subspac) W r not at this point calculating ths cofficints, w r ust trying to find th a ho Applying th lowring oprator to 3,3 givs a vctor in th four-dinsional = subspac, th cofficints would blong to th nxt diagonal down, which has four lnts (This subspac also includs th top br of th = ultiplt) Using th lowring oprator on or ti w ntr th fiv-dinsional = subspac but that is th axiu nubr of dinsions in this probl, sinc angular onta 3 and cannot b addd to giv a = 0 scalar Having now, for this particular ad fro +, found whr all th CG cofficints for all th + ultiplt brs ar locatd, w shall s how thy can all b systatically calculatd using th rcursion rlations gnratd by J = J + J ± ± ± W v appd th rcursion rlations on th diagra: givn,, th thr rd dots at (, ), (, ), (, ) (with =, = in this xapl) locat th thr CG cofficints satisfying th linar quation abov fro, J, =, J, +, J, + + +

16 6 so if two of th ar known th third is givn Siilarly, th paralll quation gnratd by J J J links th thr grn dots, at,, +,,, + = + ( ) ( ) ( ) W bgin th coputation of th CG cofficints with th blu dot ( = 3, = 0), th point on th lading arrow dgs Lt us arbitrarily assign a valu to this point If w ak it th top br of a grn triangl, that will link it to th dot blow and to a dot to th right which is off th array Th dot off th array aks zro contribution, so w hav an quation giving th valu of th cofficint at th dot blow th blu dot as a ultipl of th valu on th blu dot W can thn continu down to th nxt dot W could instad hav gon up fro th blu dot using incoplt rd triangls in fact w can continu around th dg of th whol array Thn, onc th valus along th dgs ar fixd, th rcursion triangls can b usd to ov inward and find th rst Th point of this sction is to stablish that, apart fro an ovrall ultiplicativ constant that ust b fixd by noralization, all th CG cofficints for this valu of can b found fro th rcursion rlations alon Th rason this is iportant is bcaus th sa algbraic structur, and thrfor th sa rcursion rlations, ar usd to dfin sphrical tnsors, so thy can also b cobind using th sa CG cofficints (W still nd a sign convntion hr to prsnt a coplt tabl: so far, th diffrnt valus of total hav arbitrary rlativ phass)

Addition of angular momentum

Addition 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