Hamilton operator affecting observable
'm working on this problem "Consider an experiment on a system that can be described using two basis functions. In this experiment, you begin in the ground state of Hamiltonian H0 at time t1. You have an apparatus that can change the Hamiltonian suddenly from H0 to H1. You turn this apparatus on at time t1. Then, at time tD > t1, you perform a measurement of an observable,D. In matrix notation, the Hamiltonians and the operator D are given below:
H0 = [0 4; 4 6], H1 = [1 0; 0 3]"
b) If you perform many, many measurements, what will be the average observed value of D as a function of t1 and tD?
d)You perform the experiment, but suspect your apparatus is malfunctioning and
turning off at some systematic time t2 between t1 and tD. In other words, at some time t2, you suspect the Hamiltonian is reverting to H0. What qualitative effect would this have on your results from part b)? [Note: this only requires a qualitative description, not a full calculation.]"
So I found that <D> = (2/5)(exp(2i[tDt1])+exp(2i[tDt1])+1) which would be the answer in b). I can't seem to figure out d). I was thinking that because H0 is the ground state, the energy would be lower than the one in b). Therefor D would be lower as well. But than again, the two eigenvalues of H0 is 2 an 8. Now 2 is lower than the eigenvalues of H1, but 8 is higher than both the eigenvalues of H1. How can I know which state we are in?
