Consider a particle of mass m that can move in a one-dimensionalbox of size L with the edges of the box at x=0 and x = L. Thepotential is zero inside the box and infinite outside.
You may need the following integrals:
∫ 0 1 d y sin ⡠( n π y ) 2 = 1 / 2 ,  for allinteger  n
∫ 0 1 d y sin ⡠( n π y ) 2 y = 1 / 4 ,  for allinteger  n
∫ 0 1 d y sin ⡠( n π y ) 2 y 2 = 1 6 − 1 16 π 2,  for all integer  n
(a) Write down the time-independent Schrodingerequation describing the system, and determine whether ψ = sin ⡠( 2π x / L )is an energy eigenstate by checking whether it satisfiesthe Schrodinger equation. If it is an energy eigenstate, what isthe energy? [10 pts]
(b) Suppose we measure the position of theparticle in the state in part (a) repeatedly and average theresults. Determine the result. [10 pts]
(c) What are the most likely places to find theparticle in a single measurement if it is in the state in part (a)?[10 pts]
