Just C, D and E please!

Consider a particle in a box Hamiltonian with a slightly slanted bottom, attributed to the presence of a uniform electric field across the length of the box. The Hamiltonian of this system is: h2 H= V2 + (x) 2m where (x) = \(x g) is defined on (0,a] and is otherwise (1 is positive). Note this is the potential drawn in class given as an example of pure second order perturbation theory. Hint: The following integral will be useful: 2X 16 V12 (01W62) 972 = [ * ( ) sin (79) sin (299) da = a Also note that V21 = V12. (a) [5pts] Use the first two lowest lying levels of the particle in a box potential to construct the Hamiltonian matrix that describes the system with the slanted bottom. Use the integral given to write the matrix elements in terms of the given constants and E and Ey, the unperturbed energy levels. (b) [4pts) What are the energies, up to first order (using perturbation theory), of the two lowest lying states? Are these any different than the unperturbed energies? (c) [5pts] Use the formulae for non-degenerate perturbation theory to obtain the new eigenvalues (i.e., energies) and eigenstates (e.g. wavefunctions) of the particle in a box with a slanted bottom. You should use the matrix elements found in a) and only consider the two lowest lying levels. (d) [5pts] We could have included higher lying levels of the unperturbed system to make a better approximation of the new ground state wave function. If we allowed for all possible states to get mixed in, and calculated the correction to the wave function to 2nd order in non-degenerate perturbation theory, there would still only be a certain set of states that would adjust the ground state wave functions. Find the general rule for which unperturbed states would contribute. (Hint: Think about the symmetry of the quantum mechanical integrals.] (e) [2pts] Compare the ground state energy of the unperturbed particle in a box to this example. Does the net effect of the slanted potential raise or lower the ground state energy? (The unperturbed particle in a box formulae can be found on the first midterm equation sheet on Canvas.]