If we only consider motion of a particle in the x-direction, the eigenfunctions of hamiltonian ),where φ1m sin 2 tltx for the particle in a box are {φ,u n = 1,2 3 ifo xslhere l 0, otherwise. is the length of the box in the x-direction. 1. confirm the eigenfunctions are normalized. 2. calculate the expectation values of position and momentum operators, and p, for du. how do they change with n? ) are orthogonal to each other when m n. 3. show the eigenfunctions {pn, n = 1,2,3 some equation you may need: sin α × sin β 2 [cos(α-β)-cos(α + β)]. cos α × cos β = 2 [cos (α-β) + cos (α + β), sin 2α = 2 sin α × cos α. fx cos(x) dx = x sin(x) + cos(x) + c.