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 ρ(0) = |νe><νe|.  Write down the density matrix ρ(0) in the eigenbasis of the Hamiltonian {|ν1>, |ν2>}.
(b) Write down the matrix of H in this basis and find the matrix of [H,ρ(0)].
(c) Find dρnm/dt at t = 0 and solve for ρnm(t).
(d) Find the probability that a measurement at time t will yield an electron neutrino.

#### (c)  dρ/dt = (iħ)-1[H,ρ]. dρ11/dt = dρ21/dt = 0.  The diagonal elements of the matrix do not change. dρ12/dt = -(iħ)-1(E1 -E2)ρ12,  dρ21/dt = -(iħ)-1(E2 -E21)ρ21. Therefore for n ≠ m, ρnm(t) = ρnm(0)exp(-i(En - Em)t/ħ).

(d)  The projector into the subspace spanned by |νe> is |νe><νe|.
The probability that a measurement at time t will yield an electron neutrino is
Pνe(t) = Tr{ρ(t)|νe><νe|}.
(ρ(t)|νe><νe|)nm = ∑jρnj(t)ρjm(0).
(ρ(t)|νe><νe|)11 = ρ11(t)ρ11(0) + ρ12(t)ρ21(0) = sin4θ + sin2θ cos2θ exp(-i(E1 - E2 )t/ħ).
(ρ(t)|νe><νe|)122 = ρ21(t)ρ12(0) + ρ22(t)ρ22(0) = cos4θ + sin2θ cos2θ exp(i(E1 - E2 )t/ħ).
Tr{ρ(t)|νe><νe|} = sin4θ + cos4θ + 2sin2θ cos2θ cos((E1 - E2 )t/ħ).
The off-diagonal elements of ρ are responsible for the interference effects.