quantum system time evolution

1. The problem statement, all variables and given/known data



A quantum system has Hamiltonian H with normalised eigenstates ψ_n and corresponding energies E_n (n = 1,2,3...). A linear operator Q is defined by its action on these states:



Qψ₁ = ψ₂

Qψ₂ = ψ₁

Qψ_n = 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?