5. Variational analysis of the potential V(x) = or* [20 points] We are considering the SE 2m dr2 tax y = Ey. (a) Perform a change of coordinates, setting r = Bu and determine the constant B so that the differential equation takes the form 2 duz + (u' - e ) 1 = 0. How is E given in terms of the unit-free constant e? Physics 8.05, Quantum Physics II, Fall 2008 (b) Use an algebraic manipulator that can handle differential equations to determine the value of the constant e for the ground state energy to six digits accuracy (e ~ 0.67). For this try integrating the equation numerically starting at u = 0 setting (0) = 1 and '(0) =0 (why is this derivative zero?). Only for discrete values of e the solution does not blow up as u becomes large. The lowest such value of e is the one you are looking for. (c) Write a candidate wavefunction for the variational principle and determine an upper bound for the first excited energy. (d) Use the algebraic manipulator to determine the next-to-lowest value of e (to three digits accurary) and compare with your variational estimate