Question

(b) in a direct way Problem #5-20 PTS A spin % system is in the state l) in the usual S2 eigenstate basis IT) - What is the probability that a measurement of Sx yields a value? basis |T) - (a) and | )-(1 2

Answer 1

The state of the system is Vertical-bar psi > = 1/squareroot 5 (1 2). Now let Vertical-bar phi > = (alpha beta) be the Eigen basis for the operator S_x, The operator S_x is given by When in acts on its Eigen state we get S_x Vertical-bar phi > = S_x (alpha beta) = h/2 (0 1 1 0) (alpha beta) = lambda (alpha beta) The Eigen values of this operator is given by Vertical-bar S_x - lambda I Vertical-bar = Vertical-bar 0 - lambda h/2 h/2 0 - lambda Vertical-bar = > lambda = plusminus h/2 This given us h/2 (0 1 1 0) (alpha beta) = plusminus h/2 (alpha beta) From this h/2 (0 1 1 0) (alpha beta) = plusminus h/2 (alpha beta) = > (beta alpha) = plusminus (alpha beta) = > beta = plusminus alpha Normalization gives us Vertical-bar alpha Vertical-bar ^2 + Vertical-bar beta Vertical-bar ^2 = 1 = > beta = plusminus 1/squareroot 2 = > alpha = 1/squareroot 2 \r\n The Eigen states are x^+_x = (1/squareroot 2 1/squareroot 2), x^-_x = (1/squareroot 2 -1/squareroot 2)