A particle in an infinite square well V(x) = So for 0 5 x Sa co for x a has a normalized wave function that is an even mixture of its two first stationary-state eigenfunctions: P(x, 1 ) = A y(x)e in it + us ( x )e in where A is a normalization constant and the y, are the orthonormal eigenfunctions for the infinite well: 1 if n = m y,(x) ym(x)dx = onm = 10 if n # m -DO a. Show that the integral "P*(x, t) P (x, t) dx does not depend on time. -00 b. Using the fact that " (x, 1) 4 (x, 1) dx = 1, find the normalization constant A. -00 Answer: A = V Zc. Find P (x, t) -= * (x, t) P (x, 1). Using the fact that the eigenfunctions for the infinite well are real, express | P (x, t) in terms of y1, y, and a sinusoidal function E1 of the form cos (koot ) where k is an integer and @, is given by do = = 2ma2 er: ( " (x,D) |? = = ly?+431+4,42 cos( d. Compute (x). Note that it oscillates in time. 3w0 i. What is the frequency of the oscillations in (x)? Answer: f = 2 n 16a ii. What is the amplitude of the oscillations in (x)? Answer: 9TT 2 e. Compute ( H) Answer: ( H ) = 2 ( E, + E2)