A particle in an infinite square well V(x) = 0 for O S x Sa co for x a has, at time t = 0, the wave function P (x,0) = y(x) = A (3 sin (xx/a) - sin (4xx/a)). a. Write P (x,0) = y (x) as a linear combination yr(x) = > CY, of the eigenfunctions 4,(x) of the time independent Schrodinger equation for this infinite square well. Answer: P (x,0) = y (x) = A (3451 - 4/4) Hint: This is easily done by inspection - notice that sin (nax/a) = - 4x, (x). b. Normalize y (x) Hint: You will find helpful to use the orthogonality relations of the W,(x): 1 if n = m Wn(x) Wm(x) dx = onm = 0 if n #m Answer: A = c. Write the normalized time-dependent wave function P(x, () in terms of the eigenfunctions w,,(x) and the energies E,. Notice that if P(x,0) = cy, (x) + CAYA(x) then P(x, 1) = CIP,(x, t) + CAP' 4(x, 1) Answer: P(x, t) = /10 10