Question

The Cartesian components Sx, Sy, and Sz, of spin satisfy the commutation relation [Sx, Sy] = ihsz Measurements of Sy are performed on an ensemble of systems prepared in the state given by 14 ) = (3+i) I+) + (1 + 5i) |-) where =) are the Sy eigenkets v. Give the possible outcomes of an Sx measurement and the probability for each outcome. vi. Determine the exact uncertainty Osz in Sy and compare with the smallest value it can have that you obtained using the Uncertainty Relation. vii. Give the matrix representation of the normalized state vector [).

Answer 1

42 and we have, H7= 1-7) S2 and I / 8x 1t7x = h 147x and § 1-7x = -t 17 ² (ity at te (1+7x-1-%) Nor malized N17 = 34 (ta+172) + let (1931-1-74) => 147 = (at 6i) 177x + GV2 are (2-4 ;) 172 => 147 = atsi 1 +7 + 1-2i 1 -> 3 r2 v Possible outcome of Sx measurment is to with probability. 2²+3² and the with probability. I + 2? <s> <sxY <4 1 Saly) = e n-Ja <S>=<415²147 ) = 352 5 - 18 18 18 2 (vi) Tsx <Sx) = 18 t² 13 는 t 3 こ 1 2 2 ay ti 4 8 182 q 8 =7 sx ti 65 & . 81 2 7865 Chech with the previous questions eign (vii) in Sa basis matrix representation of 147is 3+ 1751 • Normalized