The Hamiltonian for a certain two-level system is
H^=ϵ(∣1⟩⟨1∣−∣2⟩⟨2∣+∣1⟩⟨2∣+∣2⟩⟨1∣),
where ∣1⟩, ∣2⟩ is an orthonormal basis and ϵ is a number with the dimensions of energy. Find its eigenvalues and eigenvectors (as linear combinations of ∣1⟩ and ∣2⟩). What is the matrix H representing H^ with respect to this basis?