and because H is a hermitian operator, 〈a|b〉 = 〈b|a〉, so
〈a|Ĥ|a〉 = δ(1 + |〈a|b〉|²);
〈a|Ĥ|b〉 = 〈a|a〉 δ 〈a|b〉 + 〈a|b〉 δ 〈b|b〉 =
〈a|Ĥ|b〉 = δ (〈a|b〉 + 〈a|b〉) = δ 2〈a|b〉;
because of the hermition property,
〈b|Ĥ|a〉 = 〈a|Ĥ|b〉 = δ 2〈a|b〉;
finally,
〈b|Ĥ|b〉 = 〈b|(|a〉 δ 〈a| + |b〉 δ 〈b|)|b〉 =
〈b|Ĥ|b〉 = 〈b|a〉 δ 〈a|b〉 + 〈b|b〉 δ 〈b|b〉 =
〈b|Ĥ|b〉 = δ(〈b|a〉〈a|b〉 + 1),
〈b|Ĥ|b〉 = δ(1 + |〈a|b〉|²).
So, the Hamiltonian operator Ĥ ≐
δ ⎛ 1 + |〈a|b〉|² 2〈a|b〉 ⎞
⎝ 2〈a|b〉 1 + |〈a|b〉|² ⎠.
The eigenstates, which I will call |1〉 and |2〉 can be obtained by diagonalizing the Hamiltonian matrix. The first eigenvalue equations are
Ĥ|1〉 = E₁|1〉 and Ĥ|2〉 = E₂|2〉, with the eigenstates represented by the vector matrices, respectively,
⎛α₁⎞ ⎛α₂⎞
⎝β₁⎠ and ⎝β₂⎠.
δ ⎛ 1 + |〈a|b〉|² 2〈a|b〉 ⎞ ⎛α₁⎞ = E₁ ⎛α₁⎞
⎝ 2〈a|b〉 1 + |〈a|b〉|² ⎠ ⎝β₁⎠ ⎝β₁⎠.
This gives the equation α₁ + α₁|〈a|b〉|² + 2β₁〈a|b〉 = E₁α₁, and therefore the ratio between α₁ and β₁,
͟β͟₁͟ = ͟E͟₁͟ ͟-͟ ͟1͟ ͟-͟ ͟|͟〈͟a͟|͟b͟〉͟|͟²͟, or
α₁ 2〈a|b〉
β₁ = ͟α͟₁͟(͟E͟₁͟ ͟-͟ ͟1͟ ͟-͟ ͟|͟〈͟a͟|͟b͟〉͟|͟²͟)͟
2〈a|b〉
Using the normalization condition, the values of each constant can be obtained. Plugging the value for α₁ into the equation reveals a quadratic equation.
