Question

Consider a particle of mass ?? in a box or length ?? in a potential ??(??) = { 0, ?? ? [0, ??] +?, ????????????????? . Now, consider a sawtooth perturbation: ?? ? (??) = { ??(?? ? ?? 4 ), ?? ? [ ?? 4 ; ?? 2 ] ??( 3?? 4 ? ??), ?? ? [ ?? 2 ; 3?? 4 ] 0, ????????????????? . Consider the basis of the first two states of the particle-in-a-box problem, {??1 (??), ??2(??)} a) Assuming the wavefunction is sought for in the form ??(??) = ??1??1 (??) + ??2??2(??), where ??1 and ??2 are the linear variational parameters. Compute the matrix of the perturbed Hamiltonian in the basis {??1 (??), ??2(??)}. b) Solve the secular equation and determine the variational energy of the lowest energy level (within the given basis). c) Find the variational normalized wavefunction corresponding to the lowest energy level (within the given basis) d) Compute the 1-st order correction to energy of the ground state, ??0 (1) , using perturbation theory. How does the energy corrected to the 1-st order, ??0 (0) + ??0 (1) , compares to the variational energy of the lowest energy state? e) Compute the 2-nd order correction to energy of the ground state, ??0 (2) , using perturbation theory. How does the energy corrected to the 2-nd order, ??0 (0) + ??0 (1) + ??0 (2) , compares to the variational energy of the lowest energy state? f) Compute a normalized ground-state wavefunction corrected to the first order of perturbation theory. How does it compare to the variational solution?