A particle in an infinite square well V ( x ) = 0 for O S x Sa oo for x a has a normalized wave function that is an even mixture of its two first stationary-state eigenfunctions: P ( x, 1 ) = A (wi(x)e -int + us (x)e-it ) where A is a normalization constant and the W, are the orthonormal eigenfunctions for the infinite well: 1 if n =m a. Show that the integral * (x, 1) (x, 1) dx does not depend on time. -00 b. Using the fact that * (x, t) P (x, 1) dx = 1, find the normalization constant A. -Do Answer: A = V 2 c. Find |P (x, 1) -= 4 (x, () 4 (x, 1). Using the fact that the eigenfunctions for the infinite well are real, express | P (x, t) - in terms of y1, 42 and a sinusoidal function of the form cos (kot) where k is an integer and @ is given by @o = 2ma2 h Answer: |4 (x, t) | = = [47+ 42] + 4142 cos (3Wot)