Consider the infinite square well potential V(x) = 0 for 0 < x < a and V(x) = infinity otherwise, wh ...

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Consider the infinite square well potential V(x) = 0 for 0 < x < a and V(x) = infinity otherwise, where a > 0. The initial wave function is Psi(x, 0) = A sin^2(pi x / a). (a) Determine A. (b) Express Psi(x, t) as the infinite sum of the eigenfunctions psi_n(x) = sqrt(2/a) sin(n pi x / a), that is, Psi(x, t) = sum over n of c_n psi_n(x) e^(-i E_n t) (taking hbar = 1). You may find the following identity useful: the integral from 0 to 1 of sin^2(pi x) sin(n pi x) dx equals 4/((4n - n^3) pi) if n is odd, and 0 if n is even. (c) Evaluate the sum over all positive odd integers n of 1/(n^2 (n - 2)^2 (n + 2)^2).