Q. Consider the wavefunction of an electron defined by the following ID Hamiltonian: H(x) = _ d 2maz tax + Be " Cos[xx] Outline how you could solve for the energy of the ground state wavefunction for this Hamiltonian using the (A) Finite Difference Approximation, (B) 1st order Perturbation Theory, and (C) Variational Theory. (A) In the case of Finite Difference, finish when you have written the Hamiltonian matrix from which you must find the eigenvalues and eigenvectors, but there is no need to diagonalize it. Use the positions x= 0, x=-2, and x= 2 to evaluate the Hamitlonian. State how you would use the eigenvalues of the Hamiltonian to compute the ground state energy. (B) For Ist Order Perturbation theory, identify your baseline solution that you could leverage, and write out the integral that you would need to solve to evaluate the energy of the ground-state using perturbation theory. Describe if there are any relationships between the parameters in the Hamiltonian above that must be true for your application of Ist order Perturbation Theory to be accurate. (C) For Variational theory, write down your trial wavefunction you will use and identify the variational parameter in your trial function that you would use. Write down the integrals and procedure that would be needed to identify the optimal variational solution for the trial wavefunction. Go up until the point that you need to evaluate actual integrals and then stop before evaluating the integrals. Suggest a way that you could improve your trial wavefunction solution beyond your current guess