Question 5 Spin and Angular Momentum Addition: 10 Points 5a)…

Questiоn 5 Spin аnd Angulаr Mоmentum Additiоn: 10 Points 5а) Identify all the operators from J2, L2, S2, Lz, Sz that commute with Jz. (We are using standard notation e.g. J = L + S.) [5 points] Hints: Use                1.) J2=Jx2+Jy2+Jz2  (and similarly for L2 and S2),                                   2.) [A2,B]= A[A,B]+[A,B]A                                   3.)    [Jx​,Jy​]=iℏJz​,     [Jy​,Jz​]=iℏJx​,    [Jz​,Jx​]=iℏJy​    (and similarly for L and S)   5b) Write |j = 5/2, mj= 3/2〉as a sum over |l = 2, ml〉|s = 1/2, ms〉states using Clebsch-Gordon coefficients. [5 points] Hints: ml can take values of -2,-1,0,1,2 and ms can take values of -1/2, +1/2, with the restriction that mj=3/2=ml+ms The only two combinations that satisfy mj=3/2=ml+ms are ml=2, ms=-1/2 and mj=1 and ms=+1/2. Use the j=5/2 and mj=3/2 vertical line in the CG table to get the appropriate CG coefficients.