1. The problem statement, all variables and given/known data A quantum system has Hamiltonian H with normalised eigenstates ψn and corresponding energies En (n = 1,2,3...). A linear operator Q is defined by its action on these states: Qψ1 = ψ2 Qψ2 = ψ1 Qψn = 0, n>2 Show that Q has eigenvalues 1 and -1 and find the corresponding normalised eigenstates ζ1 and ζ2, in terms of energy eigenstates. Calculate <H> in each of the states ζ1 and ζ2. A measurement of Q is made at time=0, and the result 1 is obtained. The system is then left undisturbed for a time t, at which instant another measurement of Q is made. What is the probability that the result will again be 1? Show that the probability is 0 if the measurement is made after a time T = [itex]\pi[/itex]ħ/(E2 - E1), assuming E2 - E1> 0. 2. Relevant equations 3. The attempt at a solution I found ζ1 = (ψ1 + ψ2)/√2 ζ2 = (ψ1 - ψ2)/√2 and <H> = (E1 + E2)/2 for both. I have trouble doing the second part. Doesnt the system collapse into ζ1 given we know this is the state at time =0? so probability will be 1?