Thinking like a physicist

@mathematicsprof on Twitter recently tweeted a link to a page asking what it's like to understand advanced mathematics. There are a number of very interesting answers there, but one interested me particularly. I can't figure out if there's a way to link to it directly, but I'll quote it here:

From Jack Miller:

 

A two part question to determine if you "think like a mathematician," from Prof. Eugene Luks, Bucknell University, circa 1979.

Part I: You're in a room that is empty except for a functioning stove and a tea kettle with tepid water in it sitting on the floor.  How do you make hot water for tea?
Answer to Part I: Put tea kettle on stove, turn on burner, heat until water boils.

Part II: Next, you're in another room that is empty except for a functioning stove and a tea kettle with tepid water in it sitting on a table.  How do you make hot water for tea?
Non-mathematician's answer to Part II: Put tea kettle on stove, turn on burner, heat until water boils.
Mathematician's answer to Part II: Put the tea kettle on the floor. 

Why?  Because a solution to any new problem is elegantly complete when it can be reduced to a previously demonstrated case.

 This might be why I'm studying physics more than maths. I can see why putting the kettle on the floor solves the problem rather elegantly - I think it's a nicer solution than the "non-mathematician's answer" up there - but it's not how I would solve the problem. Isn't it obvious that the table is negligible in this situation, so that Part II is reduced to Part I?

Mathematicians aren't, I think, supposed to say things are negligible. Assuming that the table is negligible isn't rigourous. It does, however, get to the right solution without (explicitly, at least) going via the floor. It's still elegant (if you can get over the idea of neglecting the table) and it takes less effort.

Perhaps it's related to the idea that physics is not so much about working out how to describe some given bit of the universe as it is about working out which bits of the universe we can describe and doing so. This is usually expressed in terms of finding symmetries, from what I've seen and heard. Here, I think the system is invariant under the introduction of the table, which is a symmetry.

There are probably other ways of solving the problem, too. I think it's a very interesting exercise in how people think!

With Recurring Kittens

One.

I wrote my first exam for this semester yesterday. I think it went reasonably well; it was oddly satisfying to throw out the thirty-plus pages of notes I'd generated over the last few days. (Not the actual notes I made in class - I keep those. But I tossed the duplicates that ensured the class notes did actually (mostly) transfer to my memory.) There is now some danger that I begin to feel that having worked so hard for that course, I needn't worry about the others. Which would be foolish. To combat this assumption, I'm going to try to finish a section of Quantum Mechanics between writing each take here. (I'm stealing the general idea from here.)

Two.

I came across this writing competition yesterday, which asks for writing inspired by Benjamin Franklin's quote "If you would persuade you must appeal to interest rather than intellect." Now, physics is indubitably awesome because it allows us to harness the power of mathematics to understand the nature of the universe. It's pretty cool that way. But part of the fun - especially, perhaps, when it comes to revision - is things like my Q.M. lecturer's comment that there are physics-loving kittens who cry every time students try to explain the uncertainty principle without explicitly stating that it's fundamental. Doesn't sound plausible, you say? Well neither does quantum mechanics.

I don't think those two ideas are properly linked up there. If I figure out how to link them properly, I might have a competition entry.

Three.

I am going to use part three to write about the idea of determinism in classical mechanics so that I can refer to that idea in part four. It's pretty interesting for it's own sake too, though. Determinism starts with the sort of idea that if I know that a train leaves the station at two o'clock and travels at a constant speed of 60 km/h (I have no idea how fast trains actually travel) in a constant direction, I can tell you exactly where it will be at three o'clock. Of course, the train needn't travel at a constant speed in a constant direction, so I could  be quite wrong. When it comes to Newton's laws of mechanics, however, the only thing that can cause a particle to change its state of motion is some kind of external force (that's basically Newton's first law). So if I know all the particles and all the forces they can exert, I can work out everything that will happen. This raises some rather interesting questions about free will, since although practically nobody could know what every single particle in the universe is doing, the idea that it's theoretically possible is rather creepy. Quantum mechanics saves the day here, though: it turns out that even theoretically it's not possible to know everything about even one particle. Of course, that's rather weird in its own right.

Four.

This painting is awesome.

Il castello di Bentheim (Jacob Van Ruisdael)

I don't think you can explain why a painting is awesome by describing particles that fly around colliding and absorbing one another. They can be deterministic classical particles or random quantum particles, but they don't explain things like beauty. Or truth. (You can talk about my perception of truth in terms of particles in my brain, perhaps, but not about truth itself.) And they're not supposed to. That's why physics isn't metaphysics. It's kind of obvious in some ways and awfully hard to hold onto in others. Part of the appeal of physics (apart from the kittens) is trying to understand things. We understand more and more stuff, at a more and more fundamental level, as we go on, but at some point, in some directions, it has to stop working. Which is just as well, on the whole, but can be a little disappointing in the moment.

Five.

 It is a mark of something, I'm sure, that I've quite lost track of my reading list. Probably the amount of work involved in a final year maths/physics course. I don't actually know if I've read (well, finished) anything (that's a book) since Silver on the Tree, although I suspect I haven't. And I don't remember exactly when I read that, so that it's an altogether sorry state of affairs. However, in my efforts to do something meaningful and productive that does not involve calculating the probability amplitude function for a particle on a ring for the umpteenth time, I realised that my average reading rate for the year is still a book a week. And I definitely read more academic papers than I used to, which ought to count for something. Perhaps not altogether a sorry state of affairs, then. But it's rather odd to say that I can't remember the last time I finished reading a book.

Six.

Do you still remember the physics-loving kittens? I suggested that such a thing might be implausible, but through the wonders of Google image search and The Particle Zoo, I have now found such kittens. Behold:

You can click the image for more quarky quirky physics humour with kittens.

If you have not encountered The Particle Zoo before, it's where you go to buy a universe in a box. No, seriously.

Seven.

And that's that. I did not study a section of quantum mechanics between each post, because I couldn't make up my mind about how to define a section. So I did some studying and some writing and it all got mixed up together. As long as I don't start drawing kittens in my exam paper, I think that's okay. I missed the boat for keeping quantum mechanics off the blog a while ago.

Feminine(?) Role Models

Preface: Apparently the people at The Aperiodical are not as great as I think they are. You should make your own judgement, though. (In my undenied naivete, I think it's cool that somebody mentioned me on a podcast. Two, actually.)

 So recently some people at Michigan university did some research finding that feminine maths and science role models are not inspiring. I've been wondering for a little while why the article bothers me. Perhaps it's because I exhibit most of the characteristics that they say put girls off studying STEM (science, technology, engineering, mathematics, if I recall correctly) subjects. Which is, you know, kind of sad. I would not like to think that by wearing pretty skirts and getting good marks I send out a message that other people can't . . . can't what? Do mathematics? Understand physics? That sounds plain ridiculous.


Part of it is that the study was actually fairly limited -- it only looked a middle school girls. More significantly, from what I understand, 'feminine' here means something more like 'glamourous' than, say, 'womanly'. (The introduction to the paper proper refers to "pink-laptop-toting ‘‘Computer Engineer Barbie,’’" for instance.) That's not my immediate interpretation of feminine, but perhaps it's other people's. (If you're reading this, I'd love to know what you think.) I can believe that most girls don't think they have the potential to be the real-life version of Computer Engineer Barbie. I for one don't even want to. Does that make me unfeminine?

I like pretty dresses and I have a formidable collection of hair ribbons, but I can't be bothered to go through a fifteen minute make-up routine every morning. I own pretty high heeled sandals, but I don't wear them often. For one thing they're impractical for running up to the Physics department on the second (I think that's third in the USA, where people count funny) floor of Science block. For another, I'd feel overdressed when I arrived amidst the rest of the jeans-and-sneaker-clad students. If that's unfeminine, then I guess all those conclusions about STEM-successful women being unfeminine are probably true. Only I'd be inclined to think that most women are rather unfeminine then. (In fact this is semi-almost-implied in both the press release above and the paper proper.) Motherhood certainly doesn't sound feminine by this metric, which strikes me as rather odd.

If feminine means putting a fair amount of time and effort into satisfying certain perceptions of beauty, then being feminine means having less time to be good at anything else. It also means that anyone with a different idea of what beauty is will rather dislike being labelled as feminine. It wouldn't surprise me that many women who are successful in any number of other fields qualify as 'unfeminine'. On the other hand if being feminine is simply being typically female, I don't think it's nearly as likely to scare girls away from maths and science. (Of course STEM-successful women may still be perceived as unfeminine, which is not cool, but not something I really want to get into here.)

I do think the study is interesting and even useful, but I don't like the lack of definition of 'feminine'. It seems to mean something different every time I see it, until I find myself wondering how I've ended up in a world where it's referred to as counter-stereotypical for a woman to be feminine.

Perhaps I'm overthinking things, but that does strike me as a little odd.

Redefining Survival Mode

Every once in a while one is forced to go into survival mode, I think. It's simply not possible to do everything that one would like to, or even to do most of it altogether properly. So one ends up doing an okay job of the critical stuff until things improve. For instance, I may or may not have recently been heard to say 'I don't understand that, but it doesn't matter -- it's not in the exam.' It's not that I don't care about understanding things. It's that I have a fairly good argument for understanding the things that are in the exam first.

However, I've noticed that my definition of what exactly constitutes this survival mode seems to be changing. Survival mode used to mean sitting and staring tiredly at my cup of tea instead of getting on with my work. Now I sit and stare tiredly at my cup of tea, contemplating the curl of the tea velocity field and wondering if heat transfer would affect that velocity field. And what would the scalar heat field superimposed (as a colour map, say) on the velocity field look like? And if heat flow was treated as a vector? I may even pull up something like Mathematica or Gnuplot to get an idea of what those things would look like. All this still instead of the work I'm supposed to do, although I won't take it any further than that.

It still feels like survival mode, but it seems qualitatively different from the other kind of survival mode. I'm not quite sure where it comes from. Perhaps it's the increased 'mental fitness' after a couple of years of university maths and science; a gradual change in the way I think about things that's just highlighted by the fact that I'm survival mode-ing. Perhaps half a dozen things. I'd need to create an ensemble of identical systems and observe their evolution over time to be sure.

Whatever the cause, I think it's a kind of interesting phenomenon. I should probably stop staring tiredly at the computer screen and go figure out how to use a Cornu spiral, though.

This describes Fresnel diffraction. I'm still figuring out how.

I like bullet points

• It's somehow less intimidating to write a bunch of not necessarily related points out of whatever vaguely interesting soup is floating around in my head than to pound out a nice set of linky paragraphs with a common theme. Let's not talk about why I might intimidate myself about posting to my own little blog in this corner of the internet. Test season and rationality do not have a high level of overlap.
• Let's not talk about test season either. I'd rather tell you that I started listening to the Math/Maths podcast and it's really awesome. Also, it makes me feel like I should actually remember to write about mathsy physicsy stuff on here more often. I thin we can call that a double win. The podcast assumes you know a little bit about maths (or math, for the Americans), but it certainly doesn't expect you to be at research level in anything. I like listening to something with a bit of meat to it without feeling like I've bitten off more than I can chew!
• I'm not sure whether or not that was a mixed metaphor.
• I've finished the first two of my final year courses! Our computational physics courses are largely based around actually writing code, so there's no final theory exam. The general consensus is that continuous assessment is actually more work than otherwise (I write a three hour theory exam for my 16 credit theoretical courses; I wrote a four hour final practical test for an 8 credit computational course), but it's lovely to be finished already!



Hydrogen molecule ion orbitals.

We get to make pretty pictures in comp. phys. too. Like this one, showing the electron orbitals (where the electron is most likely to occur) of a hydrogen molecule ion. This one was done in Mathematica, which is a wonderful tool for crunching through maths that's technically doable but not very much fun. Also, it draws pretty pictures.

(I haven't taken the care with formatting that I would in a proper report, so if there's anything odd about the image, that's probably why! You can look up molecular orbitals or the linear combination of atomic orbitals model if you're really interested in seeing the science done properly.)

• We spent some time yesterday trying to get Mathematica to draw rank two tensors for us, before realising that it was a rather silly idea. A rank zero tensor is just a scalar, or point on the number line, so it's pretty easy to understand. A rank one tensor is a vector (in R³), which you can visualise as an arrow in three dimensional space. A rank two tensor maps one three dimensional vector to another, which means, as far as I understand it, that you'd need a nine-dimensional blackboard to draw it out. Unfortunately (or perhaps fortunately), Mathematica doesn't have a Plot9D command. I can't imagine why not.
• One of the best parts about getting this far into my degree is that as a class we both know each other well enough and are sufficiently interested in physics that between lectures we (sometimes!) do stuff like trying to draw (potentially impossible) things in Mathematica or work out the details of a proof we glossed over in class. See also: hitting 'random' repeatedly on xkcd, and looking at graphs showing that the exponential growth rate for yoghurt is higher than that of gingerbeer or sourdough by a ridiculous amount.
• Are there physicsy versions of things like Math/Maths and Aperiodical? I can find stuff about science-in-general or maths-in-particular easily enough, perhaps because I already know where some of it is, but physics-in-particular doesn't seem to be very well represented. I don't know if maths gets more attention on it's own because it's sometimes excluded from 'science', if it's just considered more awesome than physics or if I just happen to have stumbled upon the online maths community and have yet to discover the physicsy* analogue.
 

*I have now used 'physicsy' three times. This makes it a real word. To quote the estimable Lewis Carroll (in The Hunting of the Snark) "I have said it thrice: // What I tell you three times is true."