Variational Principle

4 The variational principle (10 points) a) Normalize the trial wave-function U(I) = JAr(a - I) if0

4 The variational principle (10 points) a) Normalize the trial wave-function A.T(a x) if 0 < x < a 0 otherwise and use it to get an upper bound on the ground-state energy of the infinite square well. How is it compared to the exact result? [5 points] b) Considering now a generalized function of the form A [x(a if 0 < c < a otherwise where p is a real number, the expectation value of the Hamiltonian becomes 112 2p('lp+ l) 2ma2 2p 1 Determine the value of p that minimizes the ground-state energy [3 points] c) What is the value of the ground-state energy obtained by using the optimal value of p and how does it compare to the result obtained at point a) and to the exact result? [2 points]