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) = |ν_e><ν_e|.  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 dρ_nm/dt at t = 0 and solve for ρ_nm(t).
(d) Find the probability that a measurement at time t will yield an electron neutrino.

Solution:
(a)  The density operator at time t = 0 is ρ(0) = |ν_e><ν_e|.   The density matrix has elements
ρ_nm = <ν_n|ν_e><ν_e|ν_m>,  (n, m = 1,2).
ρ₁₁ = sin²θ,  ρ₁₂ = -sinθ cosθ, 
ρ₂₁ = -sinθ cosθ,  ρ₂₂ = cos²θ.

(b)   The matrix of H as has elements
H₁₁ = E₁,  H₁₂ = 0, 
H₂₁ = 0,  H₂₂ = E₂.

[H,ρ] = Hρ - ρH.  The matrix if [H,ρ] has elements
[H,ρ]₁₁ = 0,   [H,ρ]₁₂ = (E₂ - E₁)sinθ cosθ, 
[H,ρ]₂₁ = (E₁ - E₂)sinθ cosθ,   [H,ρ]₂₂ = 0.

(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, ρ_nm(t) = ρ_nm(0)exp(-i(E_n - E_m)t/ħ).

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