Question

11. Show that the following pairs of wave functions are orthogonal over the indicated range. (a) e- ar' and (2ax2-1)e-2ar',-ος x o where α is a constant that is greater than zero. (b) (2ř)e-rrza° and-e-riza, cos θ over the interval OS r so,oses, os

Answer 1

psi_1 = e^- d x^2/2 psi_2 (x) = 2 (alpha x^2 - 1) e^- d x^2/2 orthogonality condition is = integral^infinity_- infinity psi_1 (x) psi_2 (x) d x = 0 Now, integral^infinity_- infinity e^- alpha x^2/2 (2 alpha x^2 - 1) e^- alpha x^2/2 d x = 2 alpha integral^infinity_- infinity x^2 e^alpha x^2 d x - integral^infinity_- infinity e^- alpha x^2 d x using integral^infinity_- infinity x^n e^- a x^2 d x = squareroot n + 1/2/a^(n + 1)/2 if n is even = 2 alpha squareroot 3/2/alpha^3/2 - squareroot 1/2/alpha^1/2 = 2 alpha 1/2 times squareroot pi/alpha^3/2 - pi/alpha^1/2 = squareroot pi/alpha^1/2 - squareroot pi/alpha^1/2 = 0 So, integral^infinity_- infinity psi_1 (x) psi_2 (x) d x = 0 psi (x) and psi_2 (x) are orthogonal to each other (b) psi_1 (r) =