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January 1

Mathematics of quantum mechanics

What mathematical topics (such as vector calculus) should I learn to completely understand the mathematics of quantum mechanics? Britannica User (talk) 07:33, 1 January 2014 (UTC)[reply]

It depends on what you mean by Quantum Mechanics. Linear algebra, fourier transforms, partial diff eqs., advanced calculus, and a smattering of complex analysis will work for most of it. For the advanced stuff, it's all over the place - fibre bundles, differential geometry, functional analysis, lie theory, etc. I'd recommend starting out with linear algebra and advanced calculus, then following that into more advanced topics in analysis; most of what you cover will be useful, or at least give you the right type of thinking. However, a word of caution, I know tons of relevant mathematics, yet have difficulty when learning physics topics; physicists approach math differently and use it differently; personally, I think learning to think in physics is more of a hurdle than the mathematics (,which, honestly, is fairly simple.) --There are many books geared towards covering "The mathematics of physics", leaf through a few and grab one that references quantum stuff; it'll cover most of what you want, and they're usually pretty readable; after that, you'll know enough that you can figure out what you need to brush up on.Phoenixia1177 (talk) 08:32, 1 January 2014 (UTC)[reply]

For example, physics undergraduates at Stanford University take this curriculum. Those five or six required math classes will basically provide an introduction to multivariable calculus and solutions for simple differential equations. Advisors will strongly recommend more mathematics. I highly recommend more classes on analysis and solutions of differential equations. I highly recommend a numerical methods class.

I also highly recommend many many many more classes studying physics. As you will hear repeatedly, you must think like a physicist if you want to understand other physicists. "Lay-persons" create unnecessary conceptual obstacles when describing "strange quantum-mechanical phenomena." Commonly, the "strange" part has nothing to do with the quantization of the system, but is a direct consequence of the ordinary analytical methods of classical physics. So, the more physics you study, the better-equipped you are to overcome those conceptual obstacles. Excellent examples of this abound: wave particle duality is commonly called a quantum-mechanical phenomenon; but if you study the mathematical methods of Fourier analysis, you can just as easily use the same math to study conventional physical systems! In fact, Euler and Laplace and Fourier were equating and transforming macroscopic systems into equivalent wave-representations hundreds of years before Planck! Nobody called it the phonon until someone decided "particles need a suffix -on" - but that never stopped Bessel from fully analyzing them! Popular science writers of the last century seem to have decided to emphasize an arbitrary division between the methods of physics used for atomic systems and those used for larger systems. It's the same mathematics. And the incredibly mind-blowingly obvious thing is, it's the same physics, too, whether you're studying subatomic particles or not. So make sure you learn and understand classical macroscopic physics, before you attempt the more difficult, noisy, and expensive-to-experiment realm of subatomic physics.

Some university students will spend a year abroad to expose themselves to new cultures and ideas and learn to see the world differently. My recommendation is to spend a year in the s-domain. Learn to speak the language, learn to think in a different way, and accomplish all the same tasks you ordinarily do at home, but using somebody else's methods. Try taking a humanities class, but force yourself to stay immersed in the s-domain. Just don't try to talk your way out of any speeding-tickets. When it's time to finally come home, you may find a little culture shock, because even though the experience will be in the past, you will always carry a little piece of the experience forward with you. Nimur (talk) 17:15, 1 January 2014 (UTC)[reply]

You misread the Stanford website. There are only 3 required math classes plus one elective. --Bowlhover (talk) 22:02, 1 January 2014 (UTC)[reply]

The answers above are more intimidating than helpful. To completely understand quantum mechanics at an undergraduate physics level, you only need vector calculus, basic linear algebra, and basic differential equations. Any good quantum mechanics textbook will teach you the rest. --Bowlhover (talk) 21:53, 1 January 2014 (UTC)[reply]

Which quantum mechanics textbook did you read that led you to believe that? I challenge you to name a quantum physics textbook that is either "good" or that teaches you the prerequisite mathematics - let alone "the rest!" I just searched the archive for "good quantum mechanics textbook" and - no joke - MediaWiki reported an HTTP error, "bad request, gateway timeout..." Nimur (talk) 07:56, 2 January 2014 (UTC)[reply]

First I got started with this video and watched other videos on quantum mechanics by DrPhysicsA. Now, I am reading two good Wikibooks (Linear algebra and Calculus). I think that after reading these two books and some other extra topics I will be able completely understand the mathematics of quantum mechanics. Britannica User (talk) 08:49, 2 January 2014 (UTC)[reply]

No, you won't; unless you mean the bare bones basics. If you want to have even a half way decent understanding of the mathematics, you really need to invest some time in working through a couple intermediate level texts on functional analysis and group theory. Both subjects are integral (ha!) to getting a lot of the stuff going on. You should really be able to free ramble a bit about the niceties of convergence in Banach Spaces and say things like, "Blah blah blah root system blah blah lie algebra blah representation theory", if you're going to approach more than a basic treatment. I'm not trying to be a jerk, just being realistic, it's not a simple subject and isn't something you can aspire to after the mathematics equivalent of light reading; though, being a self-studier myself, I can definitely see why it might appear otherwise, I had the same perspective once upon a time. Finally, Nimur really gave the best advice: study way more physics; the hurdle is not mathematics, the hurdle is a deep second nature understanding of physics - even if you learn every relevant theorem of the math, you won't find yourself having an easy time, you'll just find yourself having an easy time with relevant mathematical branches. Good Luck:-)Phoenixia1177 (talk) 05:31, 3 January 2014 (UTC)[reply]

This is really what I needed. Thank you to all for helping me. Britannica User (talk) 04:15, 2 January 2014 (UTC)[reply]

For group theory stuff, these: [1], are free and I remember them as being nice enough. From OCW: [2] and [3] might be worth looking over (see lecture notes); it's not physics oriented based per se, but it gives an idea of the type of mathematics wanted, and is readable enough to get started. If I can get some free time, I'll dig around and see if there aren't some better free online resources at hand - or, perhaps better, some useful inexpensive textbooks to suggest.Phoenixia1177 (talk) 06:40, 3 January 2014 (UTC)[reply]

Astronomy (Is it possible for a planet to revolve around in a twin planet system?)

Is it possible for a planet to revolve around in a twin planet system? — Preceding unsigned comment added by Akshay.C.S (talk • contribs) 15:08, 1 January 2014 (UTC)[reply]

Yes, and the Pluto-Charon system might qualify, in that the center of rotation (the barycenter) is between the two, not within Pluto. (The only reason I said "might" is that Pluto and Charon are now called dwarf planets, rather than standard planets.) StuRat (talk) 15:21, 1 January 2014 (UTC)[reply]

Dwarf planets are planets. Unless they have declared dwarf animals not to be animals? The OP's question might be whether the surfaces of the planets would be tidally locked toward each other, as the moon's is to the earth. μηδείς (talk) 15:37, 1 January 2014 (UTC)[reply]

According to the first line of our dwarf planet article: "A dwarf planet is a planetary-mass object that is neither a planet nor a satellite". StuRat (talk) 15:40, 1 January 2014 (UTC)[reply]

Yes, that was the dumbest nomenclature decision in the history of science. Guaranteed to confuse everyone for ever more in its defiance of whatever natural logic there is in the English language, which normally says that "<adjective><noun> is some type of <noun>". In this case, not so. Dwarf planets are not any kind of planets, apart from, er, the name. -- Jack of Oz ^{[pleasantries]} 17:43, 1 January 2014 (UTC)[reply]

There are many cases where the "<adjective><noun> is some type of <noun>" rule does not apply. An outhouse is not a house, a starfish is not a fish, and a prairie dog is not a dog. StuRat (talk) 11:34, 3 January 2014 (UTC)[reply]

I thought it was a clever shiboleth. Anyone who cares to debate about the definition of "planet" is almost certainly not contributing anything to further the frontiers of human knowledge. It's a quick way for real astronomers to immediately spot people who should be ignored. Nimur (talk) 17:51, 1 January 2014 (UTC)[reply]

What monstrous rubbish, Nimur. Let's all call the Moon a comet from now on, or the rhinoceros a bird, or the eucalypt a vertebrate - because anyone who argues the toss about such petty straw-splitting definitional pedantries is just wasting everyone's time. Yeah, right. -- Jack of Oz ^{[pleasantries]} 23:22, 2 January 2014 (UTC)[reply]

And a planet in Portugês is pt:Planeta; and in Italian, it's it:Pianeta; and in Chinese, zh:行星, and in Japanese, ja:惑星. Et cetera. And with Wikipedia, you can learn all the different ways people call it in all sorts of languages, but you still might know nothing about planets. All you've really learned is something about humans, and what humans call things. It's the difference between knowing the name of something, and knowing something. Nimur (talk) 06:30, 3 January 2014 (UTC)[reply]

You're missing my point, which is about nomenclature, not about the properties of Pluto. The IAU decided to come up with a definition of planet for the first time. The definition they settled on excluded Pluto. Ergo, Pluto and similar celestial bodies should have a label that at the very least does not imply they are a "type" of planet. It would be open to a reasonable person who knew nothing of the back story to assume that a dwarf planet is a planet in every way but has the additional characteristic of being significantly smaller than most. And that would be a wrong assumption. The fault lies with the misleading nomenclature "dwarf planet", because the IAU themselves have said that Pluto is not a planet at all. -- Jack of Oz ^{[pleasantries]} 08:00, 3 January 2014 (UTC)[reply]

The answer to the original question is of course yes, there are many stable configurations for the three body problem. So, it is possible for a "twin planet system" to exist, subject to certain constraints. Nimur (talk) 17:57, 1 January 2014 (UTC)[reply]

Aren't the earth and the moon a "twin planet" system? ←Baseball Bugs ^{What's up, Doc?} carrots→ 18:19, 1 January 2014 (UTC)[reply]

The Earth, Moon, and Sun constitute a nearly perfect three body problem. You have to work very hard to make your math accurate enough to notice effects of any other solar system body. If you measure very carefully, and your numerical methods are absolutely pristine, you may detect a perturbation from the planet Jupiter. With more effort, you can continue to clarify your methods and measurements until you detect the effects of all other bodies in the solar system, and in principle, beyond the solar system too. In practice, long before those effects matter, you also need to account for the fact that planets are nonrigid and nonspherical and have nonuniform density. Finally, if you are even more careful with your methodologies, you may detect libration or orbital precession errors due to the perturbations of general relativity.

What would be awesome is if there were a free online resource that walked a user through these calculations, in order of decreasing significance. I'll see what I can find. Nimur (talk) 19:00, 1 January 2014 (UTC)[reply]

For a starting point, at least, we've got the math that helped find Neptune. Nimur (talk) 19:12, 1 January 2014 (UTC)[reply]

To answer Bugs directly, no, they are not a twin-planet system, because the point about which the Moon and Earth rotate, the barycenter doesn't lie between the surface of the two bodies, but is 1,000 miles below the surface. The Earth wobbles about this point on a monthly basis. The barycenter of the Sun-Jupiter system lies outside the surface of the Sun, however, so one could playfully call ours a failed two-star system. μηδείς (talk) 00:28, 3 January 2014 (UTC)[reply]

However, the center of mass of the whole Solar System varies between being inside and outside the Sun. PrimeHunter (talk) 02:58, 3 January 2014 (UTC)[reply]

Note that the Moon's orbit is expanding and expected to reach 1.6 times its current radius in 15 billion years,^[4] while only about 1.3 times the radius would put the barycenter 1.3 times further from the center of the Earth, i.e. above the surface. So in, I dunno, 7 billion years Earth and Moon will be a double planet by this definition. I guess sometimes twins can be a slow delivery, no? Wnt (talk) 03:12, 3 January 2014 (UTC)[reply]

Interesting, I didn't know about the whole solar system as opposed to the Jove-Sol system. Wnt's right, I meant to make the point about the moon slipping away, but forgot. μηδείς (talk) 03:18, 3 January 2014 (UTC)[reply]

Rutile Inclusions.

Why are the Rutile inclusions in Quartz crystals golden when Rutile by itself is not ? Is this due to some optical phenomena or some chemical interaction or something else ? I've wondered about this for years ! Thank You ! — Preceding unsigned comment added by Vughole (talk • contribs) 20:23, 1 January 2014 (UTC)[reply]

Well, rutile itself is variable in color, depending on various impurities; a standard description characterizes it as "reddish brown, red, pale yellow, pale blue, violet, rarely grass-green; black if high in Nb–Ta; red to brownish red, pale yellow in transmitted light, deep brown to green with high Nb–Ta; in reflected light, gray with bluish tint, with white, yellow, or red internal reflections". The inclusions in this photo don't look particularly golden, but thin inclusions viewed by transmitted light are bound to appear lighter in color than a sizable chunk of the mineral. Deor (talk) 00:17, 2 January 2014 (UTC)[reply]

How thick are these crystals? I'm thinking that you get gold color in thin microtome sections for electron microscopy at about 60-90 micrometers by purely structural factors,^[5] and these fibers also seem really thin, aren't they? Wnt (talk) 03:17, 3 January 2014 (UTC)[reply]

whats that down to why something is mirror polished but not smooth?

hey.I wonder why some mirror polished or high gloss polished or even chromed surfaced have a really high reflexion but arent smooth? for me smooth means that my hand or fingers slide away without any resistance in all pressures so my hand slides even in the highest pressures away. so please anwer this question thankyou! — Preceding unsigned comment added by Saludacymbals (talk • contribs) 21:31, 1 January 2014 (UTC)[reply]

An educated guess: the incident surface is on the metaled rear surface, if that is relatively smooth, then a good quality image is produced. The front surface much less affects reflection, what it does affect, is refraction, but that has very little bearing on the image quality. It has to have a very rough front surface to cause the image to be noisy. Please note that I'm talking in relative terms. Plasmic Physics (talk) 22:52, 1 January 2014 (UTC)[reply]

Paradoxically, smooth surfaces have high friction. Rough surfaces can seem to have high friction because they bite into each other, like teeth catching on teeth. But when this doesn't happen, like a rough surface riubbed against a mirror, the slippage will be more free. There are devices called gauge blocks which are so smooth they will stick to each other due to the friction, see also statics and coefficient of friction. μηδείς (talk) 01:09, 2 January 2014 (UTC)[reply]

International Year of Crystallography

Can you check and see if the syntax of this page is correct, please? --Daniele Pugliesi (talk) 22:27, 1 January 2014 (UTC)[reply]

I've done some basic cleanup work. I've assumed that ONU should be disambiguated to United Nations. Tevildo (talk) 23:49, 1 January 2014 (UTC)[reply]

Thank you very much. Yes, the disambiguation is correct. --Daniele Pugliesi (talk) 02:25, 4 January 2014 (UTC)[reply]

Why does traveling faster than the speed of light make time go backwards?

When you plug a velocity greater than c into the lorentz factor, you get a complex number, not a negative number. 130.56.71.53 (talk) 23:57, 1 January 2014 (UTC)[reply]

See Tachyonic antitelephone. Red Act (talk) 00:20, 2 January 2014 (UTC)[reply]

It doesn't, it just means that you arrive before your image, sort of like a photonic version of a sonic boom. Plasmic Physics (talk) 21:40, 2 January 2014 (UTC)[reply]

The ability to travel faster than the speed of light implies the ability to arrive before you left if you can do it in any frame of reference. If there is just a single frame of reference at any given point that you can travel FTL in relation to, then it is not possible to go backwards in that frame of reference at that point, though you may appear to do so in other frames of reference. For example, suppose you can go FTL at any speed relative to the cosmic microwave background, which is approximately "stationary", in regard to normal matter, at any given point in space. Then at no point do you go backward, and if you had a little clock sitting at nearby points that had been synchronized back near the Big Bang when they were close together, you'd always go from an earlier time reading to a later time reading for different clocks as you moved. However, if you could go "FTL" but not quite backward in time relative to an observer tens of billions of light years away, who thinks your home galaxy is very heavily redshifted, then you could arrive much earlier (according to a nearby clock) than when you left. Remember, to a ten billion year old photon, the moment it was emitted and the moment it is received are the same instant (and the same position!). See relativity of simultaneity, as I don't seem to be writing this very well. Wnt (talk) 23:16, 2 January 2014 (UTC)[reply]