The title says it all: I'm confused about the various approximations and their orders. In time-independent perturbation everything is quite explicit and obvious, but, for example, how would it be with the WKB approximation? How do I say what is first and second order?
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I'll just explain the general idea that one has to keep in mind when setting up a perturbation expansion. One generally has to go through the following steps:
Understand what you're expanding around. In quantum field theory, for example, one must be careful to expand around the vacuum of the theory (this is essentially what gives rise to the Higgs mechanism); in quantum mechanics, one can often regard the situation as a perturbation of a simpler system, such as a harmonic oscillator.
Once you've identified what you're expanding around, it should also be clear how the problem at hand differs from the "base case". In what sense can it be regarded as a perturbation? Now, parametrize the deviation from the base case by some parameter. This will be called the expansion parameter.
Now, write out your problem in terms of the base case plus (an infinite series of) terms depending on powers of your expansion parameter. The powers of your expansion parameter define the order of each term.
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$\begingroup$ Could you show a simple example? $\endgroup$ Commented Oct 3, 2015 at 11:02
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$\begingroup$ @LowFieldTheory examples and further elaboration is readily found in pretty much any textbook on quantum mechanics. For example, Griffiths' book treats perturbation theory in detail in chapters 6-11. $\endgroup$– DanuCommented Oct 3, 2015 at 11:05