![Sclufion : 98 j = 0 the h on=0 and the statement Correct. obvioeusty 9f j = ½, let j. is w hore Consider paulis maticas . K](//img.homeworklib.com/questions/003ef320-78a3-11ec-916a-a104c8e4271b.png?x-oss-process=image/resize,w_560)
![and, m> <t m/ul% m) =(h m/exr(-inig6) mw, As the makin efements op Îg in the Se cond te rm are indepenhnt op /. Ka Rinear hom](//img.homeworklib.com/questions/0100f860-78a3-11ec-97c1-fbbdaf4687d0.png?x-oss-process=image/resize,w_560)
![we Can expand j, m) = | j+½m) in the Coupling representation using te rms op the uacoupling epmcentaio i+% m) = G |j, m + Y=](//img.homeworklib.com/questions/01ab4b10-78a3-11ec-bf9b-25614a7752f6.png?x-oss-process=image/resize,w_560)
![For example, <두- 주 [y/L.ryi-그Jn기주- 주>= m+ 2j 3(a,CosP +b, Sin 1%) SAn(lo%) (Sin Piy 2j-n 2li+%). 2(j+%)-A This the 8tafe men](//img.homeworklib.com/questions/026ef000-78a3-11ec-932c-cfb338ba4f6c.png?x-oss-process=image/resize,w_560)
Sclufion : 98 j = 0 the h on=0 and the statement Correct. obvioeusty 9f j = ½, let j. is w hore Consider pauli's maticas . K = x,y or 2. Since, oe have for -lhe 2init mafrix, u x = Constant (tian) 3! 31 21 41 Sin x. Thus ue ) = 2 xp(-iRiyA) = exp(-i) Sin P/2 Sin P/2
and, m> <t m/ul% m) =(h m/exr(-inig6) mw, As the makin efements op Îg in the Se cond te rm are indepenhnt op /. Ka Rinear homugenemus form p Cos (o) and Sin/P/2) and the Statement is Corereeet also. P the STAF ment is Com ree ct for j. that is, <j,mluj; m) =(;,m/expl-ini) 2j-n (an)" @here A, depends on j, m, m' i,e; An = Aplįm, m'). %3D that the stafement we shall is aco Carreck fox jt ½. Lef ĵ =Î + 3, Where the quarnfum numbers and Ĵ, are jand 2 respectively. prove
we Can expand j, m) = | j+½'m) in the Coupling representation using te rms op the uacoupling epmcentaio i+% m) = G |j, m + Y=>\ Ya." Ve> +C_| j, m - %½>l '½), and G (independent of B) are Clebsch- Gordan Coefficients. n. whore A ppleying the esxpansion to we reeduce the procedure to calcufating mafrix eleme nts the =%).
For example, <두- 주 [y/L.ryi-그Jn기주- '주>= m+ 2j 3(a,CosP +b, Sin 1%) SAn(lo%) (Sin Piy 2j-n 2li+%). 2(j+%)-A This the 8tafe ment is also walid the for j+Yg: Thet is to Say mafrix elementr polynamiade og degree 2j with res pect to the varciables cot are Coo (MA) s (2) nod Sinf A/2).