Question

4. (20 points). 5-function perturbation. Consider a particle in a one-dimensional infinite square well with boundaries at x--a and x-a. We introduce the following δ-function perturbation at V'(x) 00(z). a. Compute the first-order corrections to the energies of the particle induced by the perturbation b. Recall that you solved this problem exactly in problem set 4 (Griffiths 2.43). Compare your perturbation theory result to the exact solution

Answer 1

Psi^0_n (x) = squareroot 2/2a cos (n pi x/2a) = squareroot 1/a cos (n pi x/2a) Now, the first order correction to energy E_1 = < psi^0_n Vertical-bar v' Vertical-bar psi^0_n > = alpha/a integral^+ a_-a cos^2 n pi x/2a middot delta (x) dx = 2 alpha/a integral^a_0 cos^2 n pi x/2a delta (x) dx \r\n = 2 alpha/a cos^2 n pi a/2a E'_m = 2 alpha/a cos^2 n pi/2 = 2 alpha/a for n = even = 0 for n = odd b) psi (x) = A sin k x + B cos k x 0 lessthanorequalto x lessthanorequalto a -(1) with k = squareroot 2mE/h^2 even solution, psi(x) = psi(-x) = -A sin kx + B cos kx (-a lessthanorequalto x lessthanorequalto 0) -(ii) Now, the boundary condition, psi continuous at 0 B = B psi' continuous at 0, so, from eq^n (i) and (ii) AK + AK = 2 m alpha/h^2 B therefore B = h^2 k/m alpha A.