Timeline for Is there a general math term for the idea behind the WKB and similar methods that assume slowly varying sources?
Current License: CC BY-SA 4.0
26 events
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| Mar 2, 2023 at 3:58 | vote | accept | tparker | ||
| Feb 28, 2023 at 0:05 | history | edited | tparker | CC BY-SA 4.0 |
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| Feb 27, 2023 at 18:04 | answer | added | Roger V. | timeline score: 3 | |
| Feb 27, 2023 at 2:30 | comment | added | ZeroTheHero | @tparker still both references are valid and useful I would think. | |
| Feb 27, 2023 at 2:25 | comment | added | tparker | @ZeroTheHero Oh, to clarify, I'm not talking about all the matching of solutions stuff that comes up with WKB near the classical turning points. I'm talking about the simpler procedure described above, where the nature of the constant-source solutions doesn't necessarily change dramatically at any points - more like the slowly-varying-envelope approximation. So maybe we aren't thinking about the same thing after all. | |
| Feb 27, 2023 at 1:54 | comment | added | ZeroTheHero | @tparker In my limited experience with multi scale analysis i found this technique does not use the same tools as WKB. In particular there is none of this matching of solutions business that is pretty typical with WKB. | |
| Feb 27, 2023 at 1:16 | answer | added | ZeroTheHero | timeline score: 2 | |
| Feb 27, 2023 at 0:56 | comment | added | tparker | @ZeroTheHero Thanks, that's helpful; if you want to write an answer then I'll accept. I hadn't heard much about "multi-scale analysis" before, but that also seems to be relevant to this general concept. | |
| Feb 27, 2023 at 0:41 | comment | added | hyportnex | set $s - \hat s=\eta$ and let $|\eta| \le \eta'$ for $x \in \mathbf I[\eta']$, then expand $g$ for small $\eta'$: $g(s,x)=g(\hat s + \eta, x) \approx g(\hat s) + \frac{\partial g}{\partial s}\eta +... $ | |
| Feb 27, 2023 at 0:34 | comment | added | ZeroTheHero | @tparker WKB-like stuff can be found in many math texts on ODEs. Of course it’s famous for its QM history but it does have very wide applications (to acoustics or fluids for instance) and there is an order by order scheme for higher WKB corrections. See for instance the chapter on WKB and Related Methods in Introduction to Perturbation Methods by Mark Holmes, or most other text of perturbative solutions to ODE’s | |
| Feb 27, 2023 at 0:23 | comment | added | tparker | @Mauricio I don't think that "semiclassical methods" is the term that I'm looking for - that's specific to quantum physics, but I'm looking for a more general mathematics PDE framing. This same technique applies in many settings outside of quantum physics. | |
| Feb 27, 2023 at 0:22 | history | edited | tparker | CC BY-SA 4.0 |
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| Feb 27, 2023 at 0:17 | comment | added | tparker | @ZeroTheHero The Wikipedia page on the WKB approximation seems narrower than the general concept I discuss (which is not a series expansion), but perhaps they turn out to be equivalent, or the WKB is the best way of formalizing the idea. But I would guess that such a simple and general concept - which can apply to any PDE and has applications in many areas ofclassical physics - was discovered hundreds of years before the WKB approximation in quantum mechanics. It's true that Steven Strogatz is more a mathematician than a physicist, but his example of WKB is directly taken from quantum physics. | |
| Feb 26, 2023 at 23:38 | comment | added | tparker | @jacob1729 From what I can tell, the method of variation of parameters is unrelated, and its validity is independent of whether the source function varies slowly or fast. | |
| Feb 26, 2023 at 23:31 | comment | added | tparker | @hyportnex I've never seen that formalized; do you have a reference? And how is $|\hat{s} - s(x)|$ a parameter? Isn't it a real-valued function? | |
| Feb 26, 2023 at 22:10 | comment | added | Mauricio | I would call WKB part of the more general semiclassical methods ... | |
| Feb 26, 2023 at 22:05 | history | edited | Qmechanic♦ |
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| Feb 26, 2023 at 21:05 | comment | added | hyportnex | the small parameter is $|\hat s - s(x)|$ where $\hat s$ is the presumed constant for which $Dg = \hat s$ and then your approximation is $f \approx g(s(x),x)$ | |
| Feb 26, 2023 at 20:37 | comment | added | tparker | @JonCuster I think that many specific applications of this technique can also be formulated as applications of perturbation theory, but I think in general it’s conceptual different. It isn’t obvious to me what exactly is the small parameter in general, or what the next term in a general perturbative expansion would be. | |
| Feb 26, 2023 at 20:07 | comment | added | jacob1729 | In some cases this might be related to variation of parameters | |
| Feb 26, 2023 at 20:06 | comment | added | ZeroTheHero | Aren't they all called "WKB", as suggested by Wikipedia. Even more math-type people seem comfortable with this terminology. | |
| Feb 26, 2023 at 19:55 | comment | added | Jon Custer | Perturbation theory? | |
| Feb 26, 2023 at 19:46 | comment | added | tparker | @MariusLadegårdMeyer Yeah, this is technically a math question, but since I'm fuzzily characterizating this technique in terms of examples from physics, I think it's just as much a physics question, and physicists might be more likely than mathematicians to understand what I'm asking. | |
| Feb 26, 2023 at 19:44 | history | edited | tparker | CC BY-SA 4.0 |
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| Feb 26, 2023 at 19:43 | comment | added | Marius Ladegård Meyer | Ask (also) at math.stackexchange perhaps? | |
| Feb 26, 2023 at 19:39 | history | asked | tparker | CC BY-SA 4.0 |