Suppose we perturb the system by a linear addition of a new interaction or perturbation Hamiltonian H ^ ^ i n t , so our new Hamiltonian will take the following form: H ^ ^ = H ^ ^ ψ ψ ( 0 ) + λ λ H ^ ^ i n t , where λ λ represents the perturbation strength and runs from 0 to 1. This new interaction Hamiltonian may have off-diagonal elements V' which are non-zero and thus making the same calculations as above for the matrix elements gives: ⟨ ⟨ ψ ψ m 0 | H ^ ^ i n t | ψ ψ n 0 ⟩ ⟩ = V ′ . This result states that by adding a perturbation to the unperturbed Hamiltonian, there is a non-zero probability chance of the eigenstates transitioning between one another. An example of this can be with the electron wave-function in the hydrogen atom and the electrons in helium. If the electron wave-fucntion eigenstates can be represented by the unperturbed Hamiltonian, where V -potential energy represents the proton-electron interaction; the helium atom's Hamiltonian will look similar to the hydrogen with the extra term that now represents the new electron-electron interactions since helium has one more electron than hydrogen. In fact, this electron-electron interaction can be succinctly represented by the H ^ ^ i n t = V ′ new interaction term. This additional perturbation of energy to the initial hydrogen system (the addition of a new electron) will cause the eigenstates to transition since the electromagentic interaction will cause coupling between the electrons.