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## Homework Statement

Considering just the electron and the muon neutrino, The Free hamiltonian is written in this basis as:

H = 1/2 (E + Δ Cos[θ]) |Ve><Ve| + 1/2 Δ Sin[θ] |Vu><Ve| + 1/2 Δ Sin[θ]|Ve><Vu| + 1/2 (E - Δ Cos[θ]) |Vu><Vu|

Since H is not diagonal in this basis, they will exhibit oscillations between them. Find the eigenstates |V1> and |V2> along with the eigenvalues E1 and E2.

## Homework Equations

Book gives us:

|Ve> = Cos(θ) |V1> - Sin(θ)

|V2> |Vu> = Sin(θ)|V1> +Cos(θ) |V2>

## The Attempt at a Solution

I tried to do:

solve for the Eigenvalues using the eq: H-λI = 0

I arrived at two eigenvalues E1=1/2 (Δ +E) and E2=1/2 (E-Δ).

I put these two eigenvalues back in, and solve for the EigenVectors using H|v1> = E2 |v1>.

Then H|v2> = E2 |v2>...

I arrived at:

|v1> = {{(1+Cos(θ))/Sin(θ)},{1}}

|v2> = {{(1+Cos(θ))/Sin(θ)},{1}}

Now I Don't think I made an error in the math per say, but I do feel as though i should have had to do something to the non diagonal H before i attempted to find the eigenvalues?