Question

If U(,) refers to a rotation through an angle ß about the y-axis, show that the matrix elements (j, m|U(B, Ý)|j, m'), -ism, m' Sj, are polynomials of degree 2j with respect to the variables sin (6/2) and cos (B/2). Here [j, m) refers to an eigenstate of the square and z-component of the angular momentum: j2|j, m) = jlj +1) ħaj, m), İzlj, m) = mħ|j, m).

Answer 1

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).