Answer Happy

Consider a two-state system and two different observables A and B, represented by the operators 3 2i 0 A = (-₁2) and 8 = (18) B 3 (a) Do the operators  and Ê commute? What does your answer imply about the corresponding observables? (b) Find the eigenvectors and eigenvalues of  and Â. 3 Now let the initial state of the system be given by the vector = √ ( ³ ). (c) If I measure  on this state, what is the probability of finding the value 1? If I B measure on this state, what is the probability of finding 1? (d) Now I first measure  on the state, followed immediately afterwards by B. What is the probability that I will find the value 1 for A and also 1 for B? (e) Now I first measure B on the state, followed immediately afterwards by A. What is the probability that I will find 1 for B and 1 for A? (f) Do your answers for the probability agree? If not, explain why not. [Note for (d) and (e): Don't forget that after measuring an operator on the state, the state will change, that is it will collapse to an eigenfunction of the operator you measured.]