Mathematical Functions

I recently encountered two extremely useful online resources for "special functions". I mean Bessel functions, Legendre Functions, and the like.

1) NIST online library
2) Abramowitz and Stegun online
This book is a classic, its available online now !

I came across these posts while reading the cosmic variance blog.

MIT Integration Bee

Today, I helped organize the qualifying round of MIT's annual Integration Bee competition. It was fun to see people working through the (reasonably hard) integration problems, 25 integrals to be done in 20 minutes. Immediately after, we had to grade the papers, which did not take too long since there was not much to correct - there were lots of blanks. The highest score was 8/25, and the ten people with 3 or more points get to compete in the final round on Wednesday, Jan 27.

This year's organizer was Prof. Abhinav Kumar (who did a triple major at MIT when he was an undergrad), and Lauren, a sophomore, was the other student helper. The official webpage is here. Earlier versions were organized by Todd Kemp.

2010 problems and answers
2009 problems and answers
2007 problems and answers.

Problems this year were much harder than last year !
Here is a short video from the 2009 final round.

AdS/CFT

I have been working on a project involving AdS/CFT correspondence. Finally, after 8 months of work, I have been able to convince myself that I now have fair understanding of this concept. The main problem was, as usual, finding good pedagogical references. The field is only about 12 years old, so it is not covered in any standard textbooks on either string theory or condensed matter (atleast as far as I know - if you know a book that has a good discussion of it, please let me know).

Just for the record, I am putting here three good references that have helped me learn the subject:

1) Sean Hartnoll. Lectures on holographic methods for condensed matter physics. http://arxiv.org/abs/0903.3246

2) John McGreevy. Holographic duality with a view toward many-body physics. http://arxiv.org/abs/0909.0518

3) Chris Herzog. Lectures on Holographic Superfluidity and Superconductivity. http://arxiv.org/abs/0904.1975

so(4) symmetry of the Hydrogen Atom

Today, I happened to look into the term paper that I wrote for 8.06 (the third semester of Quantum Mechanics at MIT). I read it after almost 8 months, and was pleased to discover that I had written something that was clear and understandable.

The paper discusses how the Schrodinger spectrum of Hydrogen can be derived using the conserved Laplace-Runge-Lenz vector. I summarize the classical case, discuss Lie Algebras, derive the so(4) symmetry in Hydrogen and also calculate

1) the energy eigenvalues,
2) their degeneracy,
3) the permitted angular momentum values for a given energy state, and
4) the ground state wave function.

References are included in which the excited state wave-functions are constructed using ladder operators for so(4), and so(4) symmetry also holds in the relativistic Dirac Hamiltonian.

You can download and read the paper here.

Equations of Fluid Flow

Recently, as I have been searching the libraries for good introductory references on the phenomena of superconductivity and superfluidity, I stumbled upon a book called Superfluid Hydrodynamics by S.J. Putterman.

I really liked the style and tone of the author since everything was written in a very clear, pedagogical way. Much to my delight, the last chapter deals with superconductivity, and the parallels and differences between the two "super" phenomena.

The book begins with a review of reversible fluid flow in normal fluids. The second chapter introduces viscosity. I really liked the way it was presented, so I have written out a summary, listing all the main concepts and equations. You can download the pdf file.

A Very Brief Introduction to the Standard Model of Particle Physics

Credits: I prepared this note jointly with Mohit Jindal.

"Physics" is the study of Nature and natural phenomena. We see matter all around us. It comes in myriad incarnations. One of the most important discoveries of physics has been to establish, beyond doubt, that all matter—all objects around us (and indeed, our own bodies)—are made up of simpler, tiny constituents, replicated in enormous quantities.

The tiniest known matter particles are called “elementary particles”. Of course, new understanding might reveal that some of the particles that we currently think of as “elementary” are composite particles, made of something else. We think of “elementary” particles as being indivisible, incapable of being broken into smaller pieces. In this article, I will summarize the facts about structure of matter, as we know it today.

Bosons and Fermions

All particles in the universe can be divided into two classes, Bosons and Fermions. They are distinguished by a number that is called their “spin”. Spin is a very fundamental property of all matter particles, just as important as their mass or charge. If the spin of the particle is 0 or 1 or 2 or 3, etc., it is a boson. If the spin of the particle is 1/2 or 3/2 or 5/2 or 7/2, etc., it is a fermion. No other spin is possible.

The Matter Particles

There are two classes of elementary particles that make up matter: leptons and quarks. All of them are fermions, having spin 1/2.

There are six leptons, divided into three families

electron (e)	muon (μ)	tau (τ)
electron neutrino	muon neutrino	tau neutrino

The particles in the top row all have electric charge -1. All the neutrinos are uncharged.

There are six quarks, again divided into three families

up (u)	charm (c)	top (t)
down (d)	strange (s)	bottom (b)

The particles in the top row all have electric charge 2/3, the ones in the bottom row have charge -1/3.

All leptons and quarks have a non-zero mass, although the mass of neutrinos is very, very small.

The Force Carriers

There are four basic types of forces that we know of. Each of them is transmitted by certain particles. The Standard Model just deals with three of them. These forces, along with their force carriers are listed in the following table. The fourth force, the force of gravity, is not included.

Force	Electromagnetic	Weak	Strong
Carrier Particle	photon	W+, W-, Z	gluons

All the force carrying particles listed in the above table have spin 1, and are thus bosons. The photon and the gluons (there are 8 of them) are exactly massless. The W's and Z particles have mass. It is precisely because of the fact the W's and Z particles have a mass, that we need a Higgs particle in the theory. Higgs particle has spin 0, and allows the W's and Z to have a mass.

So all in all, there are six leptons, six quarks, photon, gluons, W's, Z and their anti-particles. These are the elementary particles. Everything else in this world is non-elementary, made up of a combination of these particles! Experiments done in various labs across the world have found all the particles listed above, but not the Higgs particle. That is yet to be found, and finding the Higgs is one of the main objectives of the LHC.

The electromagnetic force is experienced by all the particles that carry electric charge. All the leptons and quarks experience the weak force. The strong force, however, is felt only by the quarks. Note that the three neutrinos have a zero charge and thus can experience only the weak force.

Study Tips for IIT-JEE

Finally, I have even found an answer to the second question that people often ask me: How to study for IIT-JEE ? Upfront: I am not good at recommending people how to study because I have not observed enough students who have done well in the exam. So, I have often tried to evade that question. But I was delighted to find that Arvind Chauhan, who was my physics teacher for IIT-JEE has put up a fascinating answer here.

There are no second thoughts as to the quality of teaching of Chauhan Sir, he was simply remarkable. The first testimonial on this page is from me (!). His vast experience with students preparing for IIT-JEE and his remarkable teaching qualities mean that his recommendations have been well thought of and are really effective.

I have tried to answer the first question, which books to use for IIT-JEE on a previous blogpost. You can read it here.