Question

Consider a ph<sica stem whose state spuce, which is three -dinensional , is s panned by an orthonormal basis 7 l t > , 1 2 > ,'3> ỉ . In this basis- Wo obser vab les A and B re represenfed by th matrice E a -Vt where a, ans b are positive real constants. The syste s in t itia stte nsanlS normali 2atinn Cnnst ant Al lhe observable f is measuredon and tn mazimum Possible Value was obtained What is the s tra system in I l> tate vecfor o th Sy5Tem mmedlate ? ter the measurment ? what is the Pro ba bilt to obtain 2b whe observable B is measured on ystem in I t.> ?

Answer 1

a) Given Vertical-bar Psi_0 > = N( Vertical-bar 1 > + (1 - 2i) Vertical-bar 2 > + (1 + squareroot 2i) Vertical-bar 3 >) < Psi_0 Vertical-bar Psi_0 > = 1 Vertical-bar N Vertical-bar ^2 = (1 + (1 + 2i) (1 - 2i) + (1 + squareroot 2 i) (1 - squareroot 2 i)) = 1 Vertical-bar N Vertical-bar ^2 (1 + 1 + 4 + 1 + 2) = 1. Vertical-bar N Vertical-bar ^2 9 = 1 N = 1/3 < Psi_0 Vertical-bar A Vertical-bar Psi_0 > / < Psi_0 Vertical-bar Psi_0 > = a_0/9 (1 1 + 2i 1 - squareroot 2 i) (1 -squareroot 2i 1 squareroot 2i 0 -squareroot 2i 1 squareroot 2i 1) (1 1 - 2i 1 + squareroot 2i) = a_0/9 (1 1 + 2i 1 - squareroot 2i) (1 + squareroot 2i (1 - 2i) + (1 + squareroot 2i) -squareroot 2i + squareroot 2i (1 + squareroot 2i) 1 - squareroot 2i (1 - 2i) + (1 + squareroot 2i)) = a_0/9 (1 1 + 2i 1 - squareroot 2i) (1 + squareroot 2i