# Where can I find perturbation theory formulas up to fourth order?

• jeffreydk
In summary, the conversation discusses the search for perturbation theory formulas for energies and states up to fourth order. One person recommends Nicholas Wheeler's Higher-Order Spectral Perturbation for fourth and fifth order calculations, while another suggests using a variational matrix to implicitly include higher order terms. Both solutions are for non-degenerate perturbation theory.
jeffreydk
Does anybody happen to know where to find the perturbation theory formulas for the energies and states up to fourth order? I have to do a calculation up to this order and don't want to have to derive them if I don't have to (I know that Wikipedia has high order energies, but they only have the states up to second order).

Thanks very much for your help.

Well, my QM book recommends for fourth and fifth order: Niholas Wheeler's Higher-Order Spectral Perturbation. I haven't read this myself so I'm not sure what exactly is in there...

As far as I know, this is also just for non-degenerate perturbation theory.

Thank you very much for the suggestion. I am working with a non-degenerate case so I'll take a look.

In my own experience, when it becomes necessary to go beyond 2nd order PT, I generally just set up the variational matrix (in the basis of unperturbed states) and diagonalize it ... that gets all (or at least many) of the higher order PT terms in there implicitly. I don't know if that would be workable for you, but it is probably not any more work than working out the fourth order PT terms ...

## 1. What is perturbation theory?

Perturbation theory is a method used in physics and mathematics to approximate the solutions to complex problems by breaking them down into simpler ones. It involves introducing a small parameter into the equations to make them easier to solve.

## 2. What are perturbation theory formulas used for?

Perturbation theory formulas are used to solve problems that involve small changes or disturbances in a system. These formulas allow us to approximate the solutions and understand the behavior of the system without having to solve the entire problem.

## 3. How does perturbation theory work?

Perturbation theory works by expanding the equations of a system in terms of a small parameter, usually denoted as ε. This allows us to solve the problem in a series of approximations, with each term becoming more accurate as ε becomes smaller.

## 4. What are the limitations of perturbation theory?

Perturbation theory is only valid for small perturbations, and as the perturbation becomes larger, the approximations become less accurate. Additionally, it may not work for systems with large nonlinearities or chaotic behavior.

## 5. How is perturbation theory used in different fields of science?

Perturbation theory is used in a variety of fields, including physics, chemistry, and engineering. In physics, it is used to study the behavior of particles and systems in quantum mechanics and classical mechanics. In chemistry, it is used to understand the electronic structure of molecules. In engineering, it is used to analyze the stability and performance of systems such as airplanes and bridges.

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