Answer:
(a) Find first-order and second-order correction to energy eigenvalues are solved in the attached picture below. same as (b) and (c)
(b) Compare new energy eigenvalues with the exact eigenvalues for the perturbed system
(c) Find the eigenstates of the perturbed system up to first order
The above was solved considering a 2-level quantum system.
Step-by-step explanation: