particle in a one dimensional box

. Consider a particle of mass m which can move freely along the x-axis anywhere from x+-a/2 to x= a/2, but which is strictly prohibited from being found outside this region.  The particle bounces back and forth between the wall at x=a/2 of box.  The walls are assumed to be completely impenetrable, no matter how energetic is the particle.  Using the following wave function and time dependent Schroedinger  equation:


y(x,t)= Acos (px/a)e^-I(E/h)t,  for -a/2,x,a/2

                                0                   for x£-a/2  or x³a/2


Determine the lowest energy state E for the system


Determine the expectation values of x, p, x², and p².  Explain where these values come from please.