Given the initial state of a system at time t_0:
∣\psi ⟩=(4−i)∣1⟩+(−2+5i)∣2⟩+(3+2i)∣3⟩,
where {∣n⟩,n=1,2,3}is an orthonormal basis over the Hilbert space. The Hamiltonian of the system is expressed in this basis as:
H=E2(−i∣1⟩⟨2∣+i∣2⟩⟨1∣+3∣2⟩⟨2∣+3∣2⟩⟨3∣+3∣3⟩⟨2∣),
where E is a real constant with dimensions of energy.
(a) What are the possible values to obtain from a measurement of energy? What is the associated probability for each?
(b) Determine the state evolved at an arbitrary time ∣\psi ,t0;t⟩.
(c) Find the expected value of the energy based on the initial condition and subsequently at an arbitrary time t. Compare your results.