Consider an unperturbed initial system which can be represented by the following Hamiltonian operator: H ^ ^ = − − ℏ ℏ 2 2 m ∇ ∇ 2 + V , where V is the potential energy of the system and m the mass of a particle. Solving Schrödinger equation for a wave-function | Ψ Ψ ⟩ ⟩ for the set of separable solutions, we get the following eigenequation for the time-independent Schrödinger equation: H ^ ^ ψ ψ ( 0 ) | ψ ψ n 0 ⟩ ⟩ = E n 0 | ψ ψ n 0 ⟩ ⟩ . Here, the superscripts represent the perturbation level of correction where 0 represents the unperturbed system and any integers n > 0 represent the levels of correction to the system (e.g. E n 1 represents the first order correction to the eigenenergies due to the perturbation). The subscript on the Hamiltonian operator indicates the basis which the Hamiltonian matrix is represented. In this case, it is represented by the { | ψ ψ n 0 ⟩ ⟩ | n ∈ ∈ N } basis set. This allows us to put the Hamiltonian in a diagonal matrix form where the diagonal elements are the only non-zero matrix elements of this operator.