Let | ψ ψ ⟩ ⟩ = c 1 | ψ ψ 1 ⟩ ⟩ + c 2 | ψ ψ 2 ⟩ ⟩ , where c 1 and c 2 are complex(in general) constants, be any linear combination of | ψ ψ 1 ⟩ ⟩ and | ψ ψ 2 ⟩ ⟩ . Then, H ^ ^ | ψ ψ ⟩ ⟩ = H ^ ^ ( c 1 | ψ ψ 1 ⟩ ⟩ + c 2 | ψ ψ 2 ⟩ ⟩ ) = c 1 H ^ ^ | ψ ψ 1 ⟩ ⟩ + c 2 H ^ ^ | ψ ψ 2 ⟩ ⟩ = E ( c 1 | ψ ψ 1 ⟩ ⟩ + c 2 | ψ ψ 2 ⟩ ⟩ ) = E | ψ ψ ⟩ ⟩ which shows that | ψ ψ ⟩ ⟩ is an eigenstate of H ^ ^ with the same eigenvalue E.