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

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Discussion Overview

The discussion centers on the search for perturbation theory formulas for energies and states up to fourth order, specifically in the context of quantum mechanics. Participants explore resources and methods for obtaining these formulas without deriving them from first principles.

Discussion Character

  • Exploratory, Technical explanation, Homework-related

Main Points Raised

  • One participant requests sources for perturbation theory formulas up to fourth order, noting the limitations of existing resources like Wikipedia.
  • Another participant suggests Niholas Wheeler's "Higher-Order Spectral Perturbation" as a potential resource for fourth and fifth order, specifically mentioning its focus on non-degenerate perturbation theory.
  • A third participant expresses gratitude for the suggestion and confirms their work is within the non-degenerate case.
  • One participant shares their personal approach of using a variational matrix to implicitly include higher order terms, suggesting it may be as efficient as calculating the fourth order terms directly.

Areas of Agreement / Disagreement

Participants generally agree on the need for resources related to fourth order perturbation theory, but there are differing approaches to handling higher order terms, with no consensus on the best method.

Contextual Notes

Limitations include the focus on non-degenerate perturbation theory and the potential absence of comprehensive resources for higher order states.

jeffreydk
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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.
 
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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 ...
 

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