# 1/137

welcome to my world

## University Physics Competition!

So this is the post about the physics competition that I promised. It’s a week late, I know – I had underestimated how much work I had put aside for the competition.

See our paper here, by the way.

The competition is comprised of two problems. You can work in teams of up to three people to solve one of the two problems. By “solve” I mean a) analyze the problem as in-depth as possible and b) write a paper detailing your analysis, all of which must be done within 48 hours. You can think of it as similar to Moody’s Mega Math challenge, but for physics (and 48 hours instead of 14). I think the competition itself is pretty new. It started last year, doesn’t seem to be very well known, and probably doesn’t give very exciting prizes. Oh well. It was pretty awesome anyway.

I decided to work with my (really damn smart) friends Matei Ionita and Yifei Zhao in Columbia’s physics building, Pupin. The whole experience was very intense. We must have slept for something like 8 hours total, but the process of solving a non-trivial physics problem with good friends was a LOT of fun. As for the results of our hard work… well, it exceeded our expectations by a wide margin.

We had decided which problem we were going to work on in the first ten minutes of the competition:

Problem B. Shooting a Basketball for Three Points

In the game of basketball, a player scores three points by successfully making a shot from beyond the three point line, which is 6.2 meters away from the basket in international games, such as the Olympics.  Suppose a player in an Olympic basketball game is at the three point line standing at a point making an angle 45 degrees to the principal axes of the court.   What initial ball velocities and spins will result in a successful shot from this point?

I’m sure every team approached this team slightly differently, but here is what we did (if you want more details, read our rough but pretty complete paper):

1. Set up the equations of motion of the ball while in the air (gravity, buoyancy, drag, and Magnus)
2. Calculate how the velocity and spin of the ball change when it collides with either the rim or the backboard
3. Write a Mathematica simulation to model the actual physical setup given
4. Analyze our data

Step 1 went by relatively quickly, although we weren’t able to take into account the effect of air viscosity on the angular velocity of the basketball. Surprisingly, even our good friend Goo G Le couldn’t help us there. Step 2 (the part of our paper on solving integral equations) was by far the most difficult part of the project. If I remember correctly, we worked on it from Saturday afternoon to Sunday at 6 am. Step 3 just consisted of tedious and painful debugging (but was pretty awesome once we got it working) and step 4 we must have spent all of 30 minutes on (the thirty minutes before the paper was due, that is).

In the end, we essentially had a (relatively bug-free) free throw simulation which we hoped was physically accurate. Here are a few pictures:

We proceeded to run the simulation tens of thousands of times (don’t you just love Mathematica?), with the basketball having a different initial state in each run. We ended up with contour plots that looked something like this:

Broadly, these contours represent the points in the domain of the basketball’s possible initial configuration space (i.e. $v_yv_z\omega_x$ space). The layering we see is most likely due to the distinct trajectories the ball can make; for example, the ball can go directly in, or it can hit the backboard, or perhaps the backboard and rim, etc. In other words, suppose you have a set of initial conditions that make the basketball go directly in the hoop. A small change in the initial conditions might change the trajectory drastically, which is what produces the layered structure (or at least that’s what we think).

Anyway, that’s about all for the competition. Feel free to critique our paper and/or our calculations! … just try to keep in mind that we did this in 48 hours, heh.

For my next post… hmm. I might talk about some cool extra dimensions stuff that Prof. Greene did in our QM class a few weeks ago, or I might write a bitabout some open science stuff I’ve been thinking about recently.

## It’s been a while

So WordPress tells me that I haven’t blogged in months… it’s amazing how time flies in college, what with the interesting classes, people, and of course, New York City.

My courses are surprisingly interesting given my freshie status:

• Intro. to QM: Mostly cool because Brian Greene is our professor. Other than that, I have yet to learn anything really new…
• Honors Math: Essentially calculus with linear algebra, but from an analysis point of view. This class is definitely my hardest class – (proof-based) math is hard. Thankfully we just finished single-variable calculus and moved on to linear algebra, which is pretty intuitive.
• Ordinary Differential Equations: Tedium. Not to mention that I already knew most of the stuff. We’re getting to the interesting stuff like systems, etc. but I’m regretting signing up for this class. I recommend that physics major just pick up ODEs while doing other physics. Taking a class on it is pretty meh.
• Masterpieces of Western Literature: This is a pretty interesting (required core) class where we read and analyze works that have essentially transformed Western society into what it is today. So far – halfway into the semester – we’ve read: the Iliad, the Odyssey, the Hymn to Demeter, the three parts of Aeschylus’ Oresteia, Sophocles’ Oedipus Rex, Euripides’ Medea, and excerpts from Herodotus’ Histories and Thucydides’ History of the Peloponnesian War. I’d say this is my second-hardest class (after Honors Math)
• Frontiers of Science: To sum it up in a sentence… this is a required core class that is designed for non-science majors (but required for ALL students) that promises not to be an exercise in memorization and yet somehow manages to be nothing but that.

Aside from classes, I’ve met a bunch of really cool, really smart people. In fact, I’ll be participating in the University Physics Competition later today with two of the smartest freshman I’ve met… 48 hours of physics, here we come!

As for extracurriculars, I’m not doing much at this point. I wanted to do badminton and martial arts, but I only got my badminton racket from home, and I’ve been too lazy to decide whether to continue karate or to try something new. I am doing something interesting though – I’m working with a sophomore, Ariq Azad (who, I must admit, is doing most of the heavy-lifting), to develop a Columbia mobile app. We’ve just started, but it’s pretty fun so far! We hope to start releasing alpha versions sometime soon (we planned to work/release today but I realized the physics competition starts tonight and not on Saturday).

Anyway, I should go stock up on food and the like for during the competition, so I guess I’ll end my post here. By the way, I’ll probably post the paper we have by the end of the competition, Sunday evening-ish.

But oh, one last thing – I’ve gone back to using Windows recently (because Ubuntu has somehow halved my battery life and because I don’t like Unity) so I cleaned it up a bit, installed Rainmeter, and voila:

My desktop is orders of magnitude nicer than before

## India! And QCD for the layman…

Hey y’all. I’ll be India for about 3.5 weeks (total inter/intranational commute being like half a week, jeez) meeting relatives, etc. It’ll be a lot of fun! There are only really 2 downsides – no guarantee of an internet connection and, of course, the toasty weather (whose idea was it to go to India in June, anyway?!).

My flight is tomorrow, and I have a paper to write about my research last summer by the first of July. Sure I can finish it up on the plane, etc. but I’m not sure if I’ll be able to send it to the professor who wants it so… I’ll be up pretty late today, haha. Honestly, I doubt I’ll be able to finish it (why is my summer so busy?) but oh well.

Here‘s the work-in-progress. I’m trying to make it understandable to anyone who knows even a bit of physics/mathematics. Feedback would be greatly appreciated (especially corrections, hah).

## Wikibooks and arXiv

Yesterday I posted a blurb on open books and after talking to a few people, I’ve realized that there are a few different methods of implementation when it comes to collaborative books: you can be like Wikibooks and, in general, have books completely open to editing, or you could take the arXiv route and have chapters contributed and then ultimately put together (i.e. different people post different chapters on arXiv or something and there is a separate site detailing the layout/path of the material).

Neither of these seem particularly great. The first might be a little too open for serious textbook writing purposes, whereas the latter might lack collaboration, and thus cohesion. Is there some middle ground?

I found the Stacks Project today while waiting for Mathematica to do integrals, and I found it interesting in that the maintainer offers a nice way of contributing – version control. One can clone the book repository, make edits, and then send the patch to the maintainer via email. Using version control is actually, now that I think about it, a pretty obvious solution, though I dunno how it would affect contributions.

Of course, that relies on the contributor’s knowledge of the version control software, which may or may not be at a useful level, although I think that’s a pretty small issue. What I find rather odd, though, is the reliance on email for the patches. Personally, I’d find that rather annoying – couldn’t one have a ‘main’ branch that (only) moderators can merge with the ‘working’ branch (that anyone can edit)? This would require some sort of access control, of course.

Anyway, those were a few of my brainstorms today. Meanwhile, I think I’ll start working on a layman-oriented description of the summer research we did last year (partly for fun, and partly for this.

## Open books

First of all – sorry for the recent lack of posts. Things have been pretty crazy recently, what with graduation and work and stuff. Not to mention my month-long vacation starting next week…

So today my mentor and I were discussing where to buy (physics) textbooks for college – apparently Amazon is often better than university bookstores – and we ended up talking about how odd it is that many entry-level books priced so exorbitantly. It would make sense if, say, a book on an obscure part of quantum field theory were expensive, because chances are it won’t be a New York Times bestseller. But at that point, is the writer making enough money in the first place to justify going through the whole publishing process? Somehow, I don’t see many scientists relying on royalties for a non-trivial portion of their income. But of course, I don’t know much about this stuff – feel free to correct me.

Anyway, the fact remains that many science textbooks are infinitely expensive even though most students have a finite amount of money. What if books were free? There are quite a few free textbooks available online (my dear friend ‘t Hooft, for example, has useful links) but these are mostly solitary efforts (i.e. lecture notes). What if there were a peer-reviewed, completely volunteer-run, online repository in which people could collaborate to create good physics textbooks (think arXiv, kinda) for use in academic settings. My mentor, for example, thinks that there isn’t a good standalone textbook on nuclear physics – what if you invited a number of physicists to consolidate lecture notes, etc. and try to compile a respectable online book? Mr. A in America, could write chapter 1 of a book on topic B, while Mr. C in China, an eloquent writer on topic D, could write chapter 2. Of course, there’d have to be project editors/directors who would work to maintain cohesiveness and whatnot, but that’s just peanuts.

In the age of open source/free software (I’m writing this on Ubuntu, using free blogging software!) it seems inevitable that the status quo cannot be maintained. Think about how unnecessary publishing in a journal is, now that arXiv exists – you still get your work out there, albeit informally.

I think plenty of people would be willing to come together in an organized manner and contribute their knowledge and expertise out of the goodness of their hearts. But maybe that’s just me. What d’you guys think?

P.S. A related discussion here.

Hey guys. I might not be posting interesting stuff as often as I (and hopefully you) would like for the next few days or so.

We are working on finally getting our paper out the door in terms of actual publication this week. To be honest, after the introduction of arXiv, publishing in a journal seems to be more of a conventional formality than anything else.

Anyway, I reread Witten’s “Baryons in the 1/N expansion” for the first time in about a year, and I’m happy to say that I understand almost all of his treatment of non-relativistic (i.e. heavy compared to QCD’s typical energy scale) baryons, their scattering, and their excited states. Of course, I guess this isn’t saying much, because Witten hand-waves quite frequently – for example, he claims that the true Hamiltonian of a single baryon can be approximated as a mean-field Hamiltonian and that the error in this assumption is sufficiently small that it can be treated as a perturbation.  This is true, and subsequent calculations yield accurate energies. However, it turns out that the change in the wavefunction due to the error cannot be treated as a perturbation and thus accurate calculations for the wavefunction cannot be performed at leading order. Or at least, that’s what I understand to be true (completely different from what is true, heh). According to my mentor, this is a rather “disturbing” issue with the mean-field formalism.

Speaking of which, I should probably pick up a book or two on mean-field theory/Hartree-Fock interactions, so I’ll have some idea what I’m talking about…

Nonetheless, I like Witten’s paper because all the arguments that are given are very physical – i.e. understandable to the (relatively speaking) layman. Anyway, due to my newfound kinda-understanding, I’m thinking of perhaps writing a few posts on what large N QCD is (or what QCD is, for that matter), etc. that can hopefully be understood by any intelligent reader with an open (read: gullible) mind. Maybe I can also explain our project!

Well, I guess that’s it for today. Working everyday is rather tiring, probably because my total daily commute is on the order of three hours (bleh), so I think I’ll be sleeping early. Expect another post early next week, I guess.

P.S. I thought this was humorous: goo.gl/NMhrx. Ah, Reddit.

## Integration by integration under the integral sign

In my last post I briefly talked about Feynman’s favorite technique of integration – integration by differentiation. It’s pretty cool, and the thing is, it’s fairly intuitive. The integration technique I’m gonna show you next is nowhere near as intuitive but is nonetheless pretty damn awesome. I couldn’t find very many examples of it, so let’s jump right in.

Imagine you’re walking down a dark street one cold winter’s night when a man (or a woman – I won’t discriminate) in a mask points a gun to your head and says, “Solve the following integral, or else.”

$\int_0^1 (x^b-x^a)/ \ln(x)\,dx$

Now I don’t know about you, but I probably wouldn’t be able to do this integral under such pressure (actually, I’m not sure if I’d be able to do it anyway, seeing as Wolfram|Alpha can’t do it, and parts doesn’t work). But if I were Feynman or Witten (I’ve heard he’s a decently smart guy) or someone, here’s something I might do:

First, let’s look at the much easier integral

$\int_0^1 x^\alpha\,dx=1/(\alpha+1)\, {\rm for\,} \alpha > -1$.

Now let’s (yes, plural) have strokes of genius and multiply both sides of the above equation by $d\alpha$ and integrate from a to b:

$\int_a^b\int_0^1x^\alpha\,dx d\alpha=\int_a^b 1/(\alpha+1)\,d\alpha$

Caveat: As a reader pointed out, it’d be nice to choose values of a and b such that the integral actually converges. On the left, since integration is commutative given that we consort only with sufficiently civilized functions, we can switch the order of integration and perform the inner integral. On the right, we can simply carry out the integration. We now get:

$\int_0^1 (x^b-x^a)/ \ln(x)\,dx = \ln\left|(b+1)/(a+1)\right|.$

… HOLY CRAP.

I completely agree with you if you say this is not the most obvious way of approaching this integral. However, in hindsight, I can see why it works (powers and the logs that pop out when you differentiate)… but still.

Anyway, that’s all I have for today – I’m a bit tired after an especially fun/full-of-learning workday. If you have questions/ideas/statements-of-blown-minds, feel free to comment!

P.S. Anyone have any suggestions in terms of introductory (or slightly higher) books on quantum mechanics? I’m currently going through Griffiths, and for the most part it’s surprisingly clear and concise. It’s sometimes a bit muddy, however. For instance, in the section about the free particle, I felt like he didn’t do a spectacular job explaining that the eigenstates you get are inherently unphysical (and that you have to use an envelope/Fourier transform to get a general but also “localized” solution [or at least, that’s what I gathered from the stuff people said]). Apparently Columbia uses French and Taylor during freshman year.

P.P.S I just can’t get over the title of this post…

## Integration by differentiation of parameter

This a neat little technique that I’m surprised we never learned in high school calculus. Here’s how it goes:

Suppose you want to compute:

$\int_0^\infty dx\,xe^{-\lambda x}.$

This integral can easily be done using typical techniques, but does anyone actually enjoy integration by parts? Instead let’s first look at the easier integral:

$\int_0^\infty dx\,e^{-\lambda x}=\frac{1}{\lambda}$.

Now compare this integral to the original – this one is missing a factor of x in the integrand. Is there any way of producing an $x$? The answer is yes – differentiate both sides of the above equation in respect to $\lambda$:

$\frac{\partial}{\partial\lambda}\int_0^\infty dx\,e^{-\lambda x}=-\int_0^\infty dx\,xe^{-\lambda x}=-\frac{1}{\lambda^2}$

… and we’re done! The original integral is simply equal to $\frac{1}{\lambda^2}.$ Of course, this might have been longer than integrating by parts, but now suppose you want to solve:

$\int_0^\infty dx\,x^2e^{-\lambda x}.$

Well all you really have to do is take another derivative, so you essentially get a whole class of integrals for free! In fact, you can easily derive a  formula for any integral of that form. This technique can be applied to a huge variety of different sets of integrals – try the following, for example:

$\int_0^\infty dx\, x^ne^{-\lambda x^2}$ where $n$ is an integer greater than 0.

Feynman has an interesting story to tell about this technique in his book, Surely You’re Joking, Mr. Feynman!:

One thing I never did learn was contour integration. I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me.

One day he told me to stay after class. “Feynman,” he said, “you talk too much and you make too much noise. I know why. You’re bored. So I’m going to give you a book. You go up there in the back, in the corner, and study this book, and when you know everything that’s in this book, you can talk again.” So every physics class, I paid no attention to what was going on with Pascal’s Law, or whatever they were doing. I was up in the back with this book: Advanced Calculus, by Woods. Bader knew I had studied Calculus for the Practical Man a little bit, so he gave me the real works—it was for a junior or senior course in college. It had Fourier series, Bessel functions, determinants, elliptic functions—all kinds of wonderful stuff that I didn’t know anything about. That book also showed how to differentiate parameters under the integral sign—it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn’t do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me.

Questions/interesting ideas? Let me know!

## First day of work!

Today was my first day of work this summer at University of Maryland’s theory group for quarks, hadrons, and nuclei (TQHN)!

I actually worked there last summer as well, conducting theoretical research on QCD, so it was nice to see the some of the same faces I saw last year  (some of the work we did can be found here). I remember when I first started working last year it took a few days to get acclimatised to the environment, but when you return to a place where you’ve worked before everything feels nice and familiar. Today I walked into the physics building 20 minutes before my mentor did and just chatted with the graduate students who I had worked with/learned from last year. And… when my mentor showed up, he just joined the conversation, haha. I love my workplace – it’s uber-chill (and full of terrible math/physics jokes).

Anyway, from what I remember of last summer, I kinda sorta understood what we was doing when we was solving some quantum many-body problem, but not really. So today, after looking back at all the math and physics we did, I’ve made it a goal to try to understand the subject more thoroughly. This is a pretty insane goal, so I’m going to be tackling it in small steps, reading Griffiths’ introductory QM book during commutes, constantly Wiki-ing, and actually learning some math (I’m currently on Chapter 7).

I also plan to post any particularly interesting math/physics-y things I learn about, mostly because writing things out helps me solidify my ideas, and also because there’s just a lot of cool stuff out there that -people- nerds should know! Maybe my first post will be about integration by differentiation; it’s a cool technique that, unfortunately, isn’t taught in high school but is surprisingly useful.