Perturbation Theory

Hmm if I'm understanding this correctly, (and recalling correctly,) there was a chapter at the end of Kleppner & Kolenkow that dealt with this in the n=2 case: Basically assume the solution is a power series and truncate it past x^2. Well, you know how to solve diffeq's like this, so do it and see the solutions. This was offered as a reason why so many things display periodic or seemingly periodic behavior. If the perturbations (x) are small enough to ignore the terms above x^3, then it's just like the spring equations with dampening/forcing.

Carl Bender lecture series is excellent [0]<p>[0] <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=LYNOGk3ZjFM&amp;list=PLzcd6SoIscwjHuWRE38UXWG92uq0Sy4UF">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=LYNOGk3ZjFM&amp;list=PLzcd6SoIsc...</a>

One of the few college lectures I can remember 15 years later is the day our Physical Chemistry teacher taught us this technique to approximate solutions to quantum mechanical problems.<p>You get done solving out a problem not only does math feel like magic, your paper looks like some kind of annotated spell.

Looking at the Wikipedia entry for perturbation theory in quantum mechanics makes me sad. I used to understand these concepts. I still recognize some things like V for potential energy and bra kets, but I&#x27;ve forgotten how they work.<p>My line of work has nothing to do with physics whatsoever. But I sometimes get the urge to purchase an overpriced textbook on Amazon with a solutions manual and revisit QM, for old times sake.

This is used in the programming book Structure and Interpretation of Classical Mechanics [1]. With some of the same authors as the famous Structure and Interpretation of Computer Programs (SICP).<p>[1] <a href="https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Structure_and_Interpretation_of_Classical_Mechanics" rel="nofollow">https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Structure_and_Interpretation...</a>

Back in 2016 me and some collaborators used perturbation theory to calculate the stable modes in a nonlinear PDE that describes lasers.<p>see &quot;appendix C&quot; on p17 of the pdf in <a href="https:&#x2F;&#x2F;arxiv.org&#x2F;abs&#x2F;1611.01676" rel="nofollow">https:&#x2F;&#x2F;arxiv.org&#x2F;abs&#x2F;1611.01676</a><p>The basic idea is that we started with a basis of solutions to the linear equation (d=0), then expanded the nonlinear equation order-by-order in the small parameter d. The math was pretty messy since it was a system of linear equations (involving both E and B fields) but conceptually it was pretty straightforward. It was pretty cool, and definitely very satisfying when the resulting eigenvalues agreed nearly exactly with numerical simulation.

An interesting application of perturbation theory which I used to work with was a software application called 3DVH, which was used in radiation treatments (for cancer for example). Its inputs were the planned geometrical distribution of the radiation absorbed in the patient, and an array of measured dose points around a simulated patient, done before treatment. The output was a fairly accurate estimate of the actual dose distribution in the patient, taking into account the differences between the planned and measured doses. The reason you might want this is because if the difference between planned dose and reality was cutting it close to say, an organ or a major nerve, you&#x27;d really like to know which side of that margin your treatment was going to be on.<p>As long as the difference between the planned and measured doses was small (like, under 3%) then perturbation theory was valid for this use case, but if it got bigger, then the estimate was increasingly inaccurate. During development, software verification and validation took quite a while because of the complexities of the data sets combined with establishing the limits where perturbation theory was no longer valid. There was a lot of debate on where to set warning messages versus plain old disabling the output as invalid.<p>Training new users always seemed to bring up perturbation theory, because of this limit. Like, the first thing almost every physicist did was to create a huge dose difference to see how it would handle it. The software would pop up a message saying the difference was too big for perturbation theory to apply and so the output was disabled (this was a choice made for patient safety; better to display no data than bad data). Then the new user would ask why it wasn&#x27;t working. And I would have to remind them about how perturbation theory worked...<p>Here&#x27;s a few papers about it in case anyone is interested:<p><a href="https:&#x2F;&#x2F;pubmed.ncbi.nlm.nih.gov&#x2F;22225277&#x2F;" rel="nofollow">https:&#x2F;&#x2F;pubmed.ncbi.nlm.nih.gov&#x2F;22225277&#x2F;</a><p><a href="https:&#x2F;&#x2F;pubmed.ncbi.nlm.nih.gov&#x2F;22830756&#x2F;" rel="nofollow">https:&#x2F;&#x2F;pubmed.ncbi.nlm.nih.gov&#x2F;22830756&#x2F;</a>

&gt; The perturbative expansion is created by adding successive corrections to the simplified problem.<p>Wait, so the Theory of Epicycles is morally a Perturbation Theory? (disregarding matching evidence from relativistic effects yadda yadda).<p>In that sense —morally speaking— Epicycles at the time doesn&#x27;t sound that like it was that crazy to do scientifically. Generally speaking, Epicycles would be on the same level as how perturbation theory is used in quantum mechanics.

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