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February 19

De Broglie Wavelengths of particles

Why do the de Broglie wavelengths of different particles converge at high energies? As per the following table:

Energy	Photon Wavelength (m)	Electron Wavelength (m)	Proton Wavelength (m)
1KeV	1.24x10^-9	3.88x10^-11	9.06x10^-13
100KeV	1.24x10^-11	3.70x10^-12	9.06x10^-14
1MeV	1.24x10^-12	8.73x10^-13	2.86x10^-14
100MeV	1.24x10^-14	1.23x10^-14	2.79x10^-15
500MeV	2.48x10^-15	2.48x10^-15	1.14x10^-15
1GeV	1.24x10^-15	1.24x10^-15	7.31x10^-16
100GeV	1.24x10^-17	1.24x10^-17	1.23x10^-17
1TeV	1.24x10^-18	1.24x10^-18	1.24x10^-18

It's clear that as the energy increases, the wavelength of the electron becomes closer to that of a photon until they're essentially the same at around 100MeV. The proton also converges with the other two at around 100GeV. Why? As a first guess I'd say it has to do with the contribution of the rest mass to the total energy being overshadowed by the massive kinetic energy contribution. Am I on the right track? 202.155.85.18 (talk) 02:55, 19 February 2019 (UTC)[reply]

Yes. At high enough kinetic energy, the contribution of the rest mass becomes unimportant. Dragons flight (talk) 10:36, 19 February 2019 (UTC)[reply]

See also Matter wave#Special relativity for the math. --Jayron 32 12:16, 19 February 2019 (UTC)[reply]

This is already well answered, but for completeness note that the rest mass for the photon is 0, for the electron is 0.5109989461(31) MeV/c^2 = 9.10938356(11)×10−31 kg, and for the proton is 938.2720813(58) MeV/c^2 = 1.672621898(21)×10−27 kg. The "MeV/c^2" is meant to remind us that energy IS mass (E=mc^2), but also the units of electron volt can (as described in that article) be riddled through directly to the kilogram equivalent. Anyway, note that when you get to 0.5 MeV the electron wavelength starts to converge and when you get to 1 GeV that's where the proton starts to converge. Wnt (talk) 15:52, 20 February 2019 (UTC)[reply]

Germ theory

Why did it take so long for the germ theory to be accepted by the medical community? Was it because nobody really made the connection between the tiny critters which Leeuwenhoek saw in his microscope, on one hand, and contagious disease on the other? Or was it because of something else? 2601:646:8A00:A0B3:7177:1C34:7814:8A72 (talk) 04:45, 19 February 2019 (UTC)[reply]

Nobody had any evidence that the theory was true until Robert Koch provided the evidence. --Guy Macon (talk) 06:39, 19 February 2019 (UTC)[reply]

Not true. The evidence kept piling up during the 19th century, but (cf. Germ_theory_of_disease#Development) Snow and Pasteur provided fairly solid evidence against spontaneous generation around 1850-1860. Arguably earlier experiments were already some evidence for the germ theory. Tigraan^{Click here to contact me} 12:45, 19 February 2019 (UTC)[reply]

(edit conflict)Actually, it was Louis Pasteur who provided the evidence that first sold the medical establishment on the essential correctness of germ theory; Koch significantly refined the understanding of the nature and biophysical mechanisms of bacteria, however. To address another factor that has some bearing on the OP's question, there had long been some resistance among physicians to the notion that they could be unwitting vectors for infection of patients; read, for example, the very sad tale of Ignaz Semmelweis. Snow ^{let's rap} 12:51, 19 February 2019 (UTC)[reply]

Yeah, I know that one -- he discovered a link between puerperal fever (sp.?) and doctors delivering babies right after doing autopsies and made them wash their hands in between (thereby solving this problem), and the other doctors ran him out of town for this! (Which is what I was thinking when I asked about "something else" -- by "something else" I meant first and foremost vested interests that would be negatively affected by the acceptance of the germ theory.) 2601:646:8A00:A0B3:7177:1C34:7814:8A72 (talk) 03:39, 20 February 2019 (UTC)[reply]

It is likely that the failure to accept the germ theory, which goes back to Athanasius Kircher (at least) is rooted in the failures to elucidate natural selection and genetics until the mid-1800s. This failure, in turn, is rooted in the Abrahamic religions. Abductive (reasoning) 08:16, 19 February 2019 (UTC)[reply]

{{citation needed}} on that last point. shoy (reactions) 19:01, 19 February 2019 (UTC)[reply]

It's because the Romans hadn't managed to invent the microscope. Germ theory was proposed by Marcus Terentius Varro: "Marcus Terentius Varro (116–27 B.C.E) believed that disease occurred due to minute creatures too small for the naked eye to see." But at the time this could not be verified. You could only verify what you can perceive which led to miasma theory. Once you have an established theory, the burden of proof for something else to replace that becomes quite high, it was then no longer good enough to be able to observe microbes for miasma theory to be replaced by germ theory. Had the microscope been invented around 100 B.C.E. then germ theory would likely have been accepted immediately after people could see these "minute creatures too small for the naked eye to see" postulated by Marcus Terentius Varro. Count Iblis (talk) 10:09, 19 February 2019 (UTC)[reply]

Because Correlation does not imply causation. We know today that certain pathogens cause certain diseases because we know the mechanisms by which they do so; that is we can find the actual cause. However, just knowing that a factor is present at the same time as the disease doesn't mean that the factor causes the disease. Do antibodies cause infectious disease? Because they are present all the time? Why did we not draw the conclusion that antibodies were the cause? See, before people could reach the conclusion that germs actually caused the diseases is that they had to find the causal relationship and not just a correlative one. And that took a lot of science. --Jayron 32 13:34, 19 February 2019 (UTC)[reply]

Want to see some really interesting correlations? Click on this link: [ http://www.tylervigen.com/spurious-correlations ]]. --Guy Macon (talk) 19:27, 19 February 2019 (UTC)[reply]

From our Robert Koch article: "His work with anthrax is notable in that he was the first to link a specific microorganism with a specific disease". Yes, we were learning a lot about microorganisms and disease prior to that point, but one can hardly blame anyone for being skeptical before anyone linked a specific microorganism with a specific disease. It's easy to look back with 20-20 hindsight. And even after he showed the link, there was still a question of whether the link is causal; see Jayron32's comment above. Fun fact #1: at one time it was widely believed that fleas caused good health. That's because if you get sick enough the fleas abandon ship. But healthy people had plenty of fleas. So the fleas must cause the good health, right? Fun fact#2: for the last few centuries, the average price of imported rum and the average salary of presbyterian ministers have tracked each other. When one goes up or down, so does the other. Cause and effect? Actually, both tend to track the rate of inflation and the health of the economy. --Guy Macon (talk) 19:27, 19 February 2019 (UTC)[reply]

Want to see some really interesting correlations? Click on this link: [ http://www.tylervigen.com/spurious-correlations ]]. --Guy Macon (talk) 19:27, 19 February 2019 (UTC)[reply]

I stubbed a Marcus von Plenciz article, which links to his published theory from 1762. Catch is that it is guarded by Latin, though odds are a translation can be found with more effort. Wnt (talk) 01:39, 20 February 2019 (UTC)[reply]

Thanks, everyone! So the reason was as I thought, that nobody really made the connection because it was so hard to prove it and to rule out the alternative hypotheses (such as miasmas)? 2601:646:8A00:A0B3:7177:1C34:7814:8A72 (talk) 03:39, 20 February 2019 (UTC)[reply]

Well, just to think out loud, well, suppose you told someone that "it's not miasmas, it's little animals". Now what? Do you know if the little animals can fly like birds from village to village, or how they select a nest, or if they all can swim, or if they are dispersed by flowers with their pollen, or thrive on the vapors of a miasmal swamp? To say "germ theory" is one thing, but it implies some practical idea that germs hang out in mucus, but can survive on surfaces, and be transferred specifically to an eye or a nose more readily than they were to the finger that first touched them, but sometimes they're sprayed out as aerosols in a sneeze ... there's actually a lot of specialist knowledge coded into the ways we think about germs every day. Conversely, there's much specialist knowledge we still don't have on a routine basis, such as whether a disinfectant is really working under various real-life situations like wiping a table or using a hand sanitizer, and this still spreads much disease. Wnt (talk) 15:33, 20 February 2019 (UTC)[reply]

"it was so hard to prove it and to rule out the alternative hypotheses" is true about all science. There's a very common and stubbornly-held misconception that scientific progress happens in great, cataclysmic, paradigm-shifting discoveries made by singular scientists working all alone. It's part of the legacy of the unfortunate Great man theory kind of historiography. In reality, the people we know as the "great discoverers" are really better termed "great science communicators" or "great science PR people". Which is not to say what they did isn't important, but guys like Einstein and Pasteur and Pauling and Newton and the like didn't do it alone, and much of the science "grunt work", the actual performing of experiments, writing reports, publishing papers, etc. etc. was done by thousands of individual scientists, working in anonymity. These guys who become famous either a) hit on an idea that was later verified by all of those thousands of scientists (so mostly luck) or b) synthesized existing data and observations into a novel interpretation, that was also later verified by thousands of scientists. It takes any scientific theory decades and decades, and with thousands and thousands of experiments, and many scientists working on it, before a theory becomes established. Germ theory was no different in that regard. One dude having a "Eureka" moment, and then everyone just believing them is not how science works. It's slow, messy, and gradual. --Jayron 32 15:51, 20 February 2019 (UTC)[reply]

The germ of the theory (to coin a phrase) was set out by Girolamo Fracastoro in 1546. He also gave syphilis its name. 2A00:23C5:318D:5200:5813:75AF:A7A3:BCF (talk) 16:37, 20 February 2019 (UTC)[reply]

Another important point regarding germ theory in specifics: the other competing theory, miasma theory, held on so long because worked, though we know now it worked mostly accidentally, it still met the requirements for a valid theory at the time: when people operated under the idea that the theory was correct, it made predictions that were verifiable. The idea that people got sick from the smells in their world (i.e. stinky smells made you sick) led them to try things to clean up those smells, and that actually cured diseases, though not for the reasons they thought. The greatest health benefit of the industrial revolution was the cleaning up of water sources in urban areas by the installation of sanitary sewers to remove human waste from drinking water. The motivation towards the construction of those sewers was to get rid of the stink (literally, the miasma or bad air) associated with human waste. They thought the stink made you sick. When they cleaned up the sewerage, people stopped getting sick. When they took better personal hygiene, by washing more often (to remove the smells associated with human sweat), people got sick less. The observations verified the the theory, and working within the framework of the theory produced confirming results. We now know today that those remedies worked because they removed harmful bacteria and other pathogens from people, which meant they didn't contract those diseases. In other words, cleaning up all the shit meant people stopped getting sick, but not because the smell of the shit made you sick, but because bacteria that thrive in the shit did. But they didn't know that at the time. They had a theory, they worked in the theory, and their results confirmed the theory. That's another reason why they were resistant to change it. --Jayron 32 17:22, 20 February 2019 (UTC)[reply]

Very true -- the miasma theory was a classic case of "close but no cigar", and it was so close that nobody could really tell that it was wrong in any way (because the conditions which were believed to spread miasmas were also as a rule conductive to the spread of germs and thus were strongly correlated with disease!) 2601:646:8A00:A0B3:7177:1C34:7814:8A72 (talk) 07:01, 21 February 2019 (UTC)[reply]

If they thought stink caused all those infectious diseases (that explains the plague doctor beak filled with perfume) and the Romans they came from loved bathing then why'd they bathe once a year and dump their poop in the street? To avoid suspicion of being non-Christian? And at most 1 Christian holiday could've been attended without B.O., couldn't they have at least agreed to wash like Jesus before the major holidays without being suspected of being cryptojews? Sagittarian Milky Way (talk) 14:35, 21 February 2019 (UTC)[reply]

Dude, what are you on about?!? How on earth are you conflating the Inquisition with 18th century European medical theory with the Roman Empire? You've put 2000 years of history into a blender, hit "frappe" and vomited this out? I don't even know where to begin correcting you... --Jayron 32 15:39, 21 February 2019 (UTC)[reply]

So why did the descendents of Western Roman Empire citizens stop bathing if they believed bad, stinky air caused all the epidemics? If they couldn't sustain the bathing infrastructure why didn't they just soap up and rinse in a stream? Did feudalism just make people used to castle sieges where water had to be conserved? And then it got associated with "infidels" washing so they'd rather meet God in the afterlife than copy them? That's just a guess. Sagittarian Milky Way (talk) 16:50, 21 February 2019 (UTC)[reply]

The span of history between miasma theory and the fall of the Western Roman Empire is over a millennium. No reasonable person would look to find a direct, single, causal connection between such temporally distant and unconnected events. Please get some perspective. --Jayron 32 16:58, 21 February 2019 (UTC)[reply]

Also, per our article on Miasma theory, miasma (μίασμα, ancient Greek: "pollution". AKA "bad air" or "night air" emanated from rotting organic matter and from sewage. Body odor is not the same thing. Plus, those who never bathe are often unaware of the fact that they or other non-bathers have an odor. --Guy Macon (talk) 17:08, 21 February 2019 (UTC)[reply]

That's true but it's also true that people who take one or more showers a day will stink horribly if they have to skip a few days, while people who never take showers only have mild body odors that are easily handled by using deodorants. This is due regular showers with soap eliminating your skin microbiome, which may cause adverse health effects down the line. Count Iblis (talk) 20:41, 21 February 2019 (UTC)[reply]

Note that the way most people go about maintaining their personal hygiene does not improve health at all. It's sanitation that has had a huge impact on health by eliminating diseases like cholera, but we've taken our war against microbes too far. Life on Earth is mostly plants and microbes, it would not be possible for animals to exist if as a rule, microbes were pathogens. The fraction of pathogenic microbes is extremely small, just like the fraction of people in your neighborhood who are criminals. But even a small fraction of pathogens requires us to have an immune system, just like you need to lock your home despite the negligible fraction of burglars among the general population. Just like you do rely on the vast majority of the people in your neighborhood to be normal, helpful people that you can safely interact with, the same is true for the microbes the body encounters. We rely on a symbiotic relationship with many microbes on our skin and in our intestines for our health. Count Iblis (talk) 20:41, 21 February 2019 (UTC)[reply]

Jordan canonical form

If you've got a linear ODE system ${\bf {x}}'={\bf {Ax}}$ (this being the science desk let's assume it comes from a physical system), you can usually diagonalize the matrix A and get solutions $a_{1}e^{\lambda _{1}t}+\cdots$ . Physically such terms look like periodic or inward- or outward-spiralling orbits depending on the sign of the real part of the eigenvalue. But in math, because of the Jordan canonical form, an arbitrary system can also have solutions with terms like like $t^{i}a_{k}e^{\lambda _{k}t}$ . OTOH, the "Jordan box" is very numerically unstable and random matrices are (I think) almost surely diagonalizeable.

My question: do these tⁱ... solution terms actually appear in some relevant way in physics or engineering? Like do you have to put in components to deal with them in electrical circuits? Or are they just a mathematical artifact from a non-theoretical point of view? Thanks. 173.228.123.166 (talk) 08:13, 19 February 2019 (UTC)[reply]

But we have a math desk. Abductive (reasoning) 08:17, 19 February 2019 (UTC)[reply]
- But it's not a math question: I know the math solution. It's a question of whether this type of solution comes up in science and engineering, or whether it's more along the lines of a mathematical counterexample that isn't physically significant. That's why I asked at the science desk. 173.228.123.166 (talk) 09:32, 19 February 2019 (UTC)[reply]
If 2x2 matrices count, damped harmonic oscillators can exhibit critical damping which does involve one such solution. Because it corresponds to the fastest convergence without overshooting, it has some applications (example). Tigraan^{Click here to contact me} 12:52, 19 February 2019 (UTC)[reply]
Such solutions correspond to resonances. Ruslik_Zero 20:27, 19 February 2019 (UTC)[reply]

Thanks! I'll try to work out some examples like an LC filter. 173.228.123.166 (talk) 22:42, 19 February 2019 (UTC)[reply]

I tried searching for "Jordan normal form" "Schroedinger" and got this thing about Jordan's normal form and SUSY. [1] I would be lying if I said I'd even tried to understand what it means, even if it has some phrases like "particle in a box" that sound familiar. Wnt (talk) 01:48, 20 February 2019 (UTC)[reply]