Write the Schrodinger equation for third order approximation of perturbation theory

The Schrödinger equation is a fundamental equation in quantum mechanics that provides a way to calculate the wave function of a physical system and describes how it behaves. However, it seems like you're asking for the third order approximation of perturbation theory.

Perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an additional "perturbing" Hamiltonian representing a weak disturbance to the system.

The third order energy correction in non-degenerate perturbation theory is given by:

E_n^(3) = ∑(m≠n) |<ψ_m^(0)|H'|ψ_n^(0)>|^2 * (1/(E_n^(0) - E_m^(0))) - ∑(l≠n, m≠n) <ψ_n^(0)|H'|ψ_m^(0)><ψ_m^(0)|H'|ψ_l^(0)><ψ_l^(0)|H'|ψ_n^(0)> / ((E_n^(0) - E_m^(0)) * (E_n^(0) - E_l^(0)))

Here, E_n^(0) is the unperturbed energy, H' is the perturbation Hamiltonian, and |ψ_n^(0)> and |ψ_m^(0)> are the unperturbed wavefunctions. The sums are over all states m and l, excluding the state n.

Please note that this is a quite advanced topic in quantum mechanics and requires a good understanding of the principles of quantum mechanics and mathematical skills.