4. (5 points) Using your knowledge of perturbation theory, predict the changes in the wave function and energies for the following modified particle-in-a-box systems with the given wave function y; in the figures. Compute a number for the energies and compare it to the original energy, but just sketch the perturbed wave functions and show the changes. P, i1sthen=1 particle-in-a-box wave function. (2 P, isthen=2 particle-in-a-box wave function. (h) Consider a harmonic oscillator that is split into a double well potential. Consider only the v = 2 wave function at the energy shown. v=2 T v=1 v=0 \\.'\\ ,, S~ ? i. Using principles from perturbation theory, suggest what happens to the wave function and its energy due to the change in the potential, thinking physically about the perturbation to the original potential. ii. Using principles from variational theory, what other harmonic oscillator wave function would you add to the usual ,_, wave function to accomplish the required shift in energy? Why? Try to preserve the symmetry of the v = 2 state