Problem 1: Consider this problem from the class notes. Solve part (a) of the problem using the density matrix formalism. (a) The density operator at time t=0 is r(0)=|ne><ne|. Write down the density matrix r(0) in the eigenbasis of the Hamiltonian {|n1>, |n2>}. (b) Write down the matrix of H in this basis and find the matrix of [H,r(0)]. (c) Find at t=0 and solve for rnm(t). (d) Find the probability that a measurement at time t will yield an electron neutrino.
	Problem 2: A quantum system can exist in two states |a0> and |a1>, which are normalized eigenstates of the observable A with eigenvalues 0 and 1. A second observable B is defined by B|a0>=7|a0>-24i|a1>, B|a1>=24i|a0>-7|a1>. (a) Find the eigenstates of B. (b) The system is in the state |a0> when B is measured. Immediately afterwards A is measured. Find the probability that a measurement of A gives the result 0.
	Problem 3: Quantum mechanics is often conveniently formulated in terms of matrix operators.  Let {|i>} be an orthonormal basis for the state space E. (a) Prove that for any operator A , where Tr denotes the trace. (b) Derive the condition that must be satisfied for the product of two Hermitian operators to be itself a Hermitian operator. (c) Prove that the trace of a matrix operator is invariant under a change of representation, i.e. a change of basis.