Addition of angular momentum
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1 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 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 (spin- and pion (spin-, w may writ m for m, and M for M, 0, w s that th outr produc will hav a combind spin stat m M, m, M This is just lik an outr product of vctors, u i v j, forming a matrix, xcpt ths two vctors liv in spacs of diffrnt dimnsion so that m M is a matrix with two rows and thr columns. Th lctron and pion stats bhav indpndntly of on anothr. If our lctron-pion systm has th lctron in a spin up stat in th plus dirction along th x-axis, and th pion has its angular momntum in th positiv z dirction, χ (, ψ,
2 thn th total stat is χ ψ (,,, (,,,, ( 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 u W would lik to r-xprss such compound stats in trms of stats, j, m of total angular momntum j and z-componnt, m. Oprators 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-pion systm, masuring th total z- componnt of spin amounts to masuring th z-componnt of spin of ach particl and adding thm, u d Ĵ Ĵ Ĵ whr ach of th oprators on th right only acts on its corrsponding spinor. In a matrix rprsntation, this sms lik an odd combination, sinc th matrics ar of diffrnt sizs: ( [Ĵ mm [Ĵ 0 MM To writ this with formal prcision acting on th product stat w writ ach as a pair, [Ĵ [Ĵ [ MM AB mm [ mm [Ĵ MM Ĵ ˆ ˆ Ĵ whr A, B tak six valus ovr th pairs (m, M. Th action of any product oprator, Â ˆB on a product stat is givn by ( (Â ˆB ( ψ χ (Â ψ ˆB χ Notic that if w us this ordrd pair notation, w don t nd th suprscripts. Eithr notation is clar with th undrstanding that Ĵ( ss only th lctron stat and Ĵ( only th positron. For xampl, for th stat dscribd abov, th action of Ĵ is ( Ĵ (χ ψ (Ĵ Ĵ,,,, (Ĵ, Ĵ,, (,, Ĵ,
3 ((, (,,,,, (,,,,, (,,, Sinc th lctron is not in an ignstat of J (, th combind systm is not in an ignstat of Ĵ. Howvr, if both stats ar ignstats, so is th combind stat. If w lt χ ψ,,, thn Ĵ (χ ψ (Ĵ Ĵ,, Ĵ,,, Ĵ,,, Similar considrations apply to any othr spin oprators, Ĵ, Ĵ, Ĵ±. and so on. Notic that th angular momntum oprators for componnts of angular momntum for diffrnt particls commut, [Ĵ i, Ĵ j 0 [Ĵ i, Ĵ j i ε ijk Ĵk [Ĵ i, Ĵ j i ε ijk Ĵk Significantly, th spin vctor for th total angular momntum, Ĵ i Ĵ ( i Ĵ ( i satisfis th fundamntal commutation rlations: [Ĵi Ĵj, ĴiĴj ĴjĴi (Ĵ i Ĵ i (Ĵ j Ĵ j (Ĵ j Ĵ j (Ĵ i Ĵ i (Ĵ i Ĵj Ĵ i Ĵj Ĵ i Ĵ j Ĵ i Ĵj (Ĵ j Ĵi Ĵ j Ĵ i Ĵ j Ĵ i Ĵ j Ĵ i (Ĵ i Ĵj Ĵ i Ĵj Ĵ i Ĵ j Ĵ i Ĵj (Ĵ j Ĵi Ĵ j Ĵ i Ĵ j Ĵ i Ĵ j Ĵ i [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. This mans that w ar abl to st up a -, onto quality btwn stats j, m of total angular momntum, and linar combinations of products stats of constitunt particls. Equivalnc of simpl and composit stats Bcaus th sam algbra holds for th composit stat as for th individual particl stats, ignstats of th two ar th sam. This lts us stablish uniqu rlationships btwn th individual particl angular
4 momntum and th total angular momntum of th systm. W stablish a systmatic way of driving this corrspondnc. If w hav an quivalnc btwn som j, m stat and any product of stats, of th form j, m j, m j, m w will always hav 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 so that 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( ( Using Ĵ± Ĵ ± iĵ for ach particl, w notic that Ĵ ( Ĵ ( ( (Ĵ iĵ ( ( (Ĵ iĵ ( so that adding ths togthr, dot product as allowing us to writ Ĵ as Ĵ ( Ĵ ( iĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ ( iĵ ( ( (Ĵ Ĵ ( Ĵ ( Ĵ ( iĵ ( Ĵ ( Ĵ ( iĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ( Ĵ ( Ĵ ( Ĵ ( Ĵ Ĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ (. Thrfor, w may writ th Ĵ ( ( Ĵ Ĵ ( Ĵ ( Ĵ ( Ĵ ( Ĵ ( for th product stats. For th gnral cas, apply q.( to 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 4
5 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. Rcalling that Ĵ ± j, m j (j m (m ± j, m ± ( th ffct of th lowring oprator is Ĵ j, j, j, j ( (Ĵ Ĵ ( j, j j, j (j, j (j, j (j, j (j, j j, j, j, j Ĵ ( 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 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
6 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 K (j (j ( (j j k Solving for 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.
7 4 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, ( (, 0 Ĵ (, 0 ( (Ĵ Ĵ (,, (, 0 (,,,, Ĵ (, (,,,,,, 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. 7
8 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: and built as a linar combination, 0,,, 0, 0,,, 0 (,,,, 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 8
9 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 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:,,, 0 Ĵ, 7, ( (Ĵ ( Ĵ ( ( Ĵ ( Ĵ,,,,, 0 ( Ĵ ( Ĵ,,,, 0 9
10 ,, (,,,,,, 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,, 0 0,,, 0 ( (Ĵ Ĵ ( 0, 0 0, 0,,,, 0 and thrfor,, 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,, ( Ĵ,, 0 ( Ĵ,, 0,, 0 0,, 0 so that, combining th trms,,,, 0
11 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 Lowring again,,,, 8, 8,,,, 8,,,,,, 0,,, 0,, 0, 8,,,,, 0
12 and finally,,,,, 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,,,,, 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,,, Ths ar orthogonal to on anothr, sinc,,, 0 0, ,, 0,,
13 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 0 α 0 β 0 γ 8 α β γ 0 α β γ 0 8α β γ 0 α ( 8α γ γ, Thrfor, α 4 α γ γ α γ γ α β 8α γ 8α 8 α α so our stat is, and normalization rquirs Finally,, α,, α,, 0 α,, Now lowr to find th final stat,,,,,,, α ( α,,,,,,,,, 0, 0,, 0,,,,,,, 0
14 and w hav th complt doublt:,,,,,,,,, 0, 0,,,, Exrcis: Comput all ight total angular momntum stats of th combination of thr spin- particls,, m, m, m by first driving th total angular momntum of th combination of th first two, m, m as a triplt, m and a singlt 0, 0, thn combining ach of ths with th rmaining spin- stat:, m, m 0, 0, m Chck that you hav all ight stats. 4
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
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