Question

5. Let S-S, + S2 + S, be the total angular momentum of three spin 1/2 particles (whose orbital variables will be ignored). Let | ε1, ε2, ε,' be the eigenstates common to Sla, S2t, ș3s, of respective eigenvalues 1 h/2, e2 h/2, E3ћ/2. Give a basis of eigenvectors common to S2 and S., in terms of the kets Ιει, ε2,E3 Do these two operators form a C.S.C.O.? (Begin by adding two of the spins, then add the partial angular momentum so obtained to the third one.)

Answer 1

There are three spin - 1/2 partial the Hilbert space of three spins, 1/2 (x) 1/2 (x) 1/2 = (1(x) 0) (x) 1/2 1/2 (x) 1/2 (x) 1/2 = 3/2m(+) 1/2 (+) 1/2 for s = 3/2, the components are plusminus 3/2, plusminus (m) s = 1/2, the component are plusminus are plusminus 1/2 (n) the eigenstate of the total spin and component is, 1s_12, s, m > to 1s_1, s_2, s_3) s = 1/2, 11, 3/2, 3/2 > = Vertical-bar 1/2, 1/2, 1/2 > - (1) and Vertical-bar 1, 3/2, - 3/2 > = Vertical-bar - 1/2, - 1/2, - 1/2 > (2) s: Vertical-bar , n > = squareroot s(s + 1) - n(n - 1) Vertical-bar s, n - 1 > squareroot 1/2 (1/2 + 1) - 1/2(1/2 - 1) squareroot 3/4 + 1/4 = 1 squareroot 3/2 times 5/2 - 3/2 times 1/2 = squareroot 3 Acting s, on (1) rightdoublearrow h squareroot 3/2 (3/2 + 1) - 3/2 (3/2 - 1) Vertical-bar 1, 3/2,