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 ffct of a potntial. W now dvlop gnral tchniqus for this. Stats Suppos w hav a stat which has svral parts. For xampl, w might dscrib a particl by th product of a spinor and a spatial wav function, ψ x, t χ θ, ϕ whr χ is a two componnt spinor. Or, w might hav a composit particl such as a proton, mad up of thr quarks, p uud This lattr involvs both th spatial wav function and th spinor for ach quark, p U x, t χ u θ, ϕ U x, t χ u θ, ϕ D x, t χ d θ, ϕ Each of th wav functions may hav orbital angular momntum dscribd by l, m l stats, whil ach spinor is a, m s, so th total angular momntum is built from th -fold product, l, m l, m s l, m l, m s l d, m dl, m s Evn if all thr quarks ar in th l 0 ground stat, thr ar 8 possibl combinations of th spins. W nd tchniqus to find all possibl total angular momntum stats, j, m, for such systms. Whn w hav products lik this, w think of thm as gnral outr products and writ th combind spin stat with a gnric product,. For xampl, a systm comprisd of an lctron and a positron will hav a combind spin stat,, m, m This is no diffrnt than th products w writ for sparation of variabls problms. It simply mans that ach spinor can tak ach of its stats indpndntly of th othr. If our lctron-positron systm has th lctron in a spin up stat, and th positron with its spin in th plus dirction along th x-axis, χ, χ,,
thn th total stat is χ χ,,,,,,, For mor stats, w just continu th product. For xampl, ight possibl spin stats for th ground stat of th positron will hav th form, m, m, m Oprators u Angular momntum is additiv, so th oprators rprsnting dynamical variabl of angular momntum, Ĵ, will add whn w hav multipl particls. Thus, for th lctron-positron systm, masuring th total z-componnt of spin amounts to masuring th z-componnt of spin of ach particl and adding thm, u Ĵ J J whr ach of th oprators on th right only acts on its corrsponding spinor. Thus, for th stat dscribd abov, th action of Ĵ is Ĵ χ χ J J,,,, J,,,, J,,,, J,, J,,, J,, J,,,,,,,,,,, Sinc th positron is not in an ignstat of J, th combind systm is not in an ignstat of Ĵ. Similar considrations apply to th othr componnt spin oprators, Ĵ, Ĵ, Ĵ±. Also, th angular momntum oprators for componnts of angular momntum for diffrnt particls commut, [Ĵ i, Ĵ j 0 [Ĵ i, Ĵ j [Ĵ i, Ĵ j i ε ijk Ĵ k i ε ijk Ĵ k d
As an altrnativ to writing suprscript labls, Ĵ, Ĵ, w could writ th total Ĵ vctor as Ĵ ˆ ˆ Ĵ This way, vrything is writtn in th product spac. Howvr, th suprscript notation is simplr for most purposs. Notic that th total angular momntum vctor, Ĵ i Ĵ i Ĵ i satisfis th sam commutation rlations as that of ach individual particl: [Ĵi Ĵj, [Ĵ i Ĵ i, Ĵ j Ĵ j [Ĵ i, Ĵ j [Ĵ i, Ĵ j [Ĵ i, Ĵ j [Ĵ i, Ĵ j i ε ijk Ĵ k 0 0 i ε ijk Ĵ k i ε ijk Ĵ k This mans that th total vctor will also b dscribd by j, m stats. If w hav an quivalnc btwn som j, m stat and any product or sum of product stats, of th form j, m j, m j, m w will always hav so that Ĵ j, m Ĵ Ĵ j, m j, m m j, m Ĵ j, m j, m j, m Ĵ j, m m j, m m j, m j, m m j, m j, m m j, m m m j, m j, m m m m that is, th total z-componnt of spin is always th sum of th individual z-componnts. For th total spin, w must comput Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ If w writ th dot product as Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ iĵ Ĵ Ĵ Ĵ iĵ Ĵ iĵ Ĵ iĵ Ĵ Ĵ Ĵ Ĵ iĵ Ĵ iĵ Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ iĵ Ĵ iĵ Ĵ
thn w can valuat Ĵ for various product stats. For th gnral cas, considr th highst valu of m for ach particl, Ĵ j, j j, j Ĵ j, j j, j Ĵ j, j j, j Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ j, j j, j j j j j j j j, j j, j j j j j j, j j, j so that this stat is an ignstat of Ĵ with ignvalu j j, and w writ j, j j, j, j, j j, j j, j W can find j j of stats j, j, j, j by acting with th lowring oprator. Howvr, w hav j j possibl product stats, but only j j stats of th form j, j, m. To find mor of th possibl combinations, considr th j, j, j, j stat, givn by Ĵ j, j, j, j Ĵ Ĵ j, j j, j j, j j, j j, j, j, j Ĵ j, j j, j j, j j, j j, j Ĵ j, j j, j j, j, j, j j j j j j, j j, j j j j j j, j j, j j j, j j, j j j, j j Whil this stat has total j j j, thr is a scond combination of j, j j, j and j, j j, j which is orthogonal to this on, found by intrchanging th cofficints on th right and changing th rlativ sign, α j j, j j, j j j, j j, j Acting with Ĵ on this stat will tll us what rprsntation it blongs to. Act first on ach part. Applying Ĵ th first trm givs Ĵ j, j j, j Ĵ j, j j, j Ĵ j, j j, j Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ j, j j, j whil th scond trm givs j j j j j j j, j j, j j j j j j j j j j, j j, j j j j j j j j j, j j, j j j j, j j, j Ĵ j, j j, j Ĵ j, j j, j Ĵ j, j j, j Ĵ Ĵ Ĵ Ĵ Ĵ Ĵ j, j j, j j j j j j j j, j j, j j j j j j j j j j, j j, j j j j j j j j, j j, j j j j, j j, j 4
Putting this all togthr, w find th action of Ĵ on α Ĵ α Ĵ j j, j j, j j j, j j, j j j j j j j j j j, j j, j j j j j, j j, j j j j j j j j j, j j, j j j j j, j j, j j j j j j j j j, j j, j j j j j j j j j, j j, j j j j j α j j j j α so w hav j j j. This may or may not xhaust all possibilitis. If not, w can lowr twic from th top stat to gt linar combinations of th thr m j stats, j, j j, j, j, j j, j, j, j j, j Th two sts of stats w hav found, j j, m and j j, m account for two linarly indpndnt combinations of ths, so if thr rmain mor dgrs of frdom w can find a third combination orthogonal to ths. It will hav j lowr by, giving a j j, m rprsntation. W continu in this mannr until w hav j j stats. Sinc th j, m account for j of th dgrs of frdom, this occurs whn Solving for K, j j K j j k k0 K K j j k k0 k0 K j j K K j j K 0 K K j j j j K K j j 4j j K j j ± 4 j j j j j j ± j j j, j K K Thrfor, th rprsntations includ all j from j j to th first of j j j j or j j j j to occur. If j > j, thn j j j > 0 will b th first to occur; if j > j thn j j > 0 will occur first, so in ithr cas, th lowst valu is j j and w will hav rprsntations j j, m
j j, m. j j, m and will hav xactly accountd for all stats of th systm. Exampl : Two spin / particls Th simplst nontrivial addition coms whn w combin two spin- particls to gt four stats of th form, m, m With th highstj j j, and stpping down to j j 0, w xpct j, j 0 stats. Bgin with th highst stat,,,, and apply Ĵ Ĵ Ĵ, Ĵ, Ĵ Ĵ,,, 0 Ĵ,,, Ĵ,, 0,,,, 0,,,, Lowring again, w complt th j triplt, Ĵ, 0 Ĵ Ĵ,,,,,, Ĵ, Ĵ,, whr w us th fact that Ĵ, 0. Thn,,,,,,,, so th full triplt is,,,, 0,,,,,,, 4 4,
Thr is on rmaining stat, and it must b th singl j 0 stat. Sinc w must also hav m 0, it must b constructd from th m m 0 combinations,,, and,,. Also, it must b orthogonal to th othr thr stats. This is immdiat for th,, and,,, whil for th, 0 stat orthogonality forcs us to writ 0, 0,,,, This complts th idntification of stats. 4 Exampl : Ral x matrix: Add two j stats W hav notd bfor that a ral, matrix can b dcomposd into a -dim trac trm, a -dim spac of antisymmtric matrics, and a -dim spac of traclss, symmtric matrics. W considr this dcomposition in trms of irrducibl rprsntations. Sinc th, m stats form a -dim vctor spac, w can think of a matrix as an outr product of two of thm,, m, m W hav j j, so th rang of total j should b from j j j to j j j 0. Thus, w hav thr irrducibl rprsntations,, m, m 0, 0 of dimnsions j, that is,, and as xpctd. To comput th stats in dtail, w start with th j stats,,,,, 0,,, 0,,,, 0, 0,,, 0,, 0, 0,,,,, Th j stats start with th uniqu stat orthogonal to,, namly, 0,,, 0. Thrfor,, 0,,, 0,,,,,, 0,, 0, 0,, Th final stat is th uniqu normalizd stat orthogonal to both:, 0,,, 0, 0,,, 0,,,, 7
and built as a linar combination 0, 0 α,, β, 0, 0 γ,, sinc this is th most gnral combination with m 0. Orthogonality btwn 0, 0 and, 0 shows that α γ, thn orthogonality btwn 0, 0 and, 0 givs α β α 0 so that β α. This mans that 0, 0 α,,, 0, 0,, and w choos α to normaliz, giving th final on of th nin stats, 0, 0,,, 0, 0,, Exampl : Add j and j / stats Suppos w want to add, m and, m angular momnta. Thn thr ar j j stats of th form, m, m and w xpct th total angular momntum to run from j down to j, that is, to gt th rquisit stats. stats, m 4 stats, m stats, m. Th j / stats W start with th highst stat, and apply th lowring oprator: Ĵ, 7,,,,, Ĵ Ĵ,,,, 0,, 0,,,, 0 This givs th scond stat,,,,,, 0 8
Notic that th stat is normalizd and that ach trm has a total m, i.., for th first trm on th right, and for th scond trm, 0. Continu, lowring four mor tims: Ĵ, Ĵ Ĵ,,,, 0 7, Ĵ,, Ĵ,, Ĵ,, 0 Ĵ,, 0,,,,, 0,, 0 0,,,,,,, 0,, 0,, so w hav, 0,, 0, Onc again, th stat is normalizd and m m in ach trm. Continu Ĵ, Ĵ 0 Ĵ,, Ĵ 0 Ĵ,, 0,, 0, 0,, 0, and thrfor,, 0,, 0, 0 0,,, 0 Ĵ Ĵ 0, 0 0,, 0, 0 0,,,,,, 0 Notic th symmtry in th cofficints btwn this stat and th prvious on. Th symmtry continus, with th nxt lowring lading us to,,,,, 0 and finally, th lowst stat is uniqu just lik th highst stat. Just to chck th consistncy, w work it out: Ĵ, Ĵ Ĵ,, Ĵ Ĵ,, 0, Ĵ,, Ĵ,, 9
Ĵ,,, 0 Ĵ, 0 0,, 0,, 0 so that, combining th trms,,,, as xpctd. Collcting th full multiplt,,,,,,, 0,, 0,,,,,. Th j / stats Th scond stat,,,,,,,, involvd a linar combination of two product stats, stat built from ths, j, m,,, 0 0, 0,,,,,, 0 0, 0, 0 0,,,,,, 0, and,, 0, so thr is a uniqu scond,, 0 As shown abov, this stat will hav a total angular momntum of j. It is th highst stat, as can b sn by noting that m m in ach trm. Thrfor,,,,,, 0 and w may lowr to find th rmaining stats,,,,,, 0,, 0,, so, 8,,,, 0,, 0
Lowring again,, and finally,, 8,,,,, 8,,,,,,, 0, 0, 8,, 0,,, 0 8,,,,,,, 0 which is what w xpct for th lowst stat. Just as a chck, try to lowr this last stat again: 0,, 0 W thrfor hav all of th, m stats,,,, 8,,,,,. Th j / stats,,,,,, 0, 0,,, 0, 0, 8,,,,,,,, 0 W hav xhaustd all stats with m m and m m, but thr rmain two linar combinations unaccountd for. W hav two stats built from,,,,, 0,,, and two stats built out of,,,,, 0,,, Considr th two stats built from th first tripl, with m m,, 0,, 0,, 8,,,, 0 0, 0,,,,
Ths ar orthogonal to on anothr, sinc,, 8 0 0 0 0 and thr is xactly on mor normalizd stat orthogonal to both of ths. This will b th j, m stat. Start with an arbitrary linar combination,, α,, β,, 0 γ,, and dmand th vanishing of th innr products 0,, 0,, Canclling th dnominators, so that β 8α γ from th scond, laving Thrfor, so our stat is, and normalization rquirs 0 α 0 β 0 γ 8 α β γ 0 α β γ 0 8α β γ 0 α 8α γ γ α 4 α γ γ α γ γ α β 8α γ 8α 8 α α α,, α,, 0 α,,, α α,, Finally,,,,,, 0,,
Now lowr to find th final stat,,,,, and w hav th complt doublt:,,,,,,,,,,,,, 0,,, 0,, 0, 0,,,,,,, 0,