Why do you have factors of "n" all over the place? There should be only an "m" and in each term is has a specific value (like m=2!) so you should get numbers for all the matrix elements (and most of them are zero)Schwarzschild90 said:
Perturbation theory is a mathematical tool used to approximate solutions to problems that cannot be solved exactly. It involves breaking down a complex problem into smaller, more manageable parts and solving them one at a time. In the context of quantum mechanics, perturbation theory is used to find corrections to the energy levels of a system that is slightly perturbed from its known or ideal state.
Second order correction in perturbation theory is the second term in the series expansion of the perturbation solution. It takes into account the effects of the perturbation to the first order correction and calculates a more accurate approximation of the solution. Second order correction is usually more accurate than first order correction, but it may not be enough for highly perturbed systems.
The main difference between first and second order correction is the level of accuracy in the approximation of the solution. First order correction only considers the effects of the perturbation to the first order, while second order correction takes into account the effects of the perturbation to the second order. Therefore, second order correction is more accurate than first order correction.
Second order correction is necessary when the perturbation is large and first order correction is not accurate enough. In some cases, second order correction may also be necessary for small perturbations if the system is highly sensitive to perturbations.
Second order correction is limited in its accuracy, especially for highly perturbed systems. In some cases, higher order corrections may be needed to obtain a more accurate solution. Additionally, perturbation theory is only applicable for small perturbations and may not work for large perturbations or strong interactions.