Consider the hamiltonian for the two-dimensional motion of a particle of mass j. j- in a hru. n1onic oscillator potential: (a) show that the energy eigenvalues are given by en = (n + 1)/iw. where the i n t e g e r n = n 1 + n 2 , w i t h n 1, n 2 = 0 , i , 2 , . . (b) expresstheoperatorlz=xpy- /¥,; (~ i ~ ) al= -. x+-px 2 h j. j-w ~ a*~ i a) and a2= - y+-py 2 / i j. j-w ~ and the corresponding raising operators a)' and al .give a symmetry argument showingthat[h, iz]= 0. it indeed vanishes. (c) determine the correct linear combination of the energ) eigenstates with en- ergy £ 1 = 2nw that are eigenstates of lz by diagonalizing the matrix repre- sentation of lz restricted to this subspace of states.