Method of successive approximation

[Tsunami]

Last semester in my course QM 2 we discussed the Rayleigh-Schrödinger perturbation theory. A very elegant theory, based on a general principle: when terms are depending on a certain factor to the nth order, where the factor is very small if not infinitesimally small, you can collect the terms that have the same nth order dependence, and that state that:

the terms of the 0th order are the most prominent
the terms of the 1th order are second most prominent
etc.

(I'm a bit clumsy at wording this in English, my apologies.)

Now, this semester I'm working on a individual project about thermoacoustics, and I'm using a similar method of perturbation theory - the principle of order division is the same, only the context is different. I've been using this trick without ever questioning where it came from - however, it's time to write a report about it, so I feel I really should know where it came from. However, I can't seem to find a good site about this, and considering I don't even know the name of such a theory, it's not easy to look for it either.

So, my question is: what is the name of the abstract theory that describes this method of succesive approximation? It's always nice if you can provide a lucid link as well, but providing me a name would be a great help already. Thanks.

 

[vanesch]

Last semester in my course QM 2 we discussed the Rayleigh-Schrödinger perturbation theory. A very elegant theory, based on a general principle: when terms are depending on a certain factor to the nth order, where the factor is very small if not infinitesimally small, you can collect the terms that have the same nth order dependence, and that state that:

the terms of the 0th order are the most prominent
the terms of the 1th order are second most prominent
etc.

I can't really help you with the name, sorry. It is really the core of perturbation theory, but I can't really put a name on it. You could look up Fredholm or Volterra integral equations, but I don't know if that's what you're looking for.

However, I'd like to point out something else. The fact that you can equate the different orders has nothing to do with your parameter being "small". The smallness of the parameter only comes into cut off the series.
But the idea behind the perturbation series is in fact different. Instead of solving ONE problem, you solve a WHOLE family of problems, parametrized in your small parameter (say, lambda). If you can now write out equations for that whole family of problems, then things should be ok, because the solution you're looking for is then found by putting in your "correct" value of lambda. But if you leave lambda "free" because you look for the whole family of solutions, and you arrive at an equation:

blabla + lambda . blah1 + lambda^2 . blah2 +... = blublu + lambda. bluh1 +...

then you are in fact solving a POLYNOMIAL equation in lambda, which looks trivial:

(blabla - blublu) + lambda . (blah1 - bluh1) + lambda^2 (blah2 - bluh2) + ... = 0.

Now, for a polynomial to be 0 FOR ALL VALUES OF LAMBDA, this can only happen when all of its coefficients, individually, are 0.

So from this follows that:
blahblah = blublu
blah1 = bluh1
...

which are nothing else but your perturbative equations of different order.

cheers,
Patrick.

 

[thepatientmental]

The method of successive approximation, also known as perturbation theory, is a powerful tool in many areas of physics and mathematics. It allows us to solve complex problems by breaking them down into simpler parts that can be solved iteratively. This method is based on the idea that when dealing with a small or infinitesimal factor, we can collect terms with the same order of dependence and treat them as a whole, with the terms of the lowest order being the most prominent. This approach is commonly used in quantum mechanics, thermoacoustics, and many other fields.

The specific theory you are referring to in your question is called Rayleigh-Schrödinger perturbation theory. This theory is a mathematical framework that allows us to calculate the energy levels and wavefunctions of a quantum system that is slightly perturbed from a known system. It is a generalization of the perturbation theory developed by Lord Rayleigh and Erwin Schrödinger, hence the name.

You can find more information about this theory in many textbooks and online resources. Here are a few links that might be helpful:

- An article on the method of successive approximation by the University of Colorado Boulder: https://www.colorado.edu/physics/phys4810/phys4810_fa18/lecture_notes/lec22.pdf
- A lecture on Rayleigh-Schrödinger perturbation theory by MIT OpenCourseWare: https://ocw.mit.edu/courses/physics...-fall-2013/lecture-notes/MIT8_05F13_Lec22.pdf
- A video explaining the basics of perturbation theory by Khan Academy: https://www.khanacademy.org/science/physics/quantum-physics/quantum-structures/v/perturbation-theory
- A comprehensive book on perturbation theory by G. Barton titled "Elements of Green's Functions and Propagation: Potentials, Diffusion, and Waves": https://books.google.com/books/abou...d_Propag.html?id=Jz6qBgAAQBAJ&source=kp_cover

I hope this helps you in your research and understanding of this important method in physics. Best of luck with your project!