Monthly Archive for August, 2007

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Dispersion and Band Structure – Notes

Published on August 25, 2007 in Physics. 0 Comments Tags: , , , , .

The purpose of this essay is the revise, clarify and elucidate concepts in Dispersion and Band Structure.

A dispersion relation is a relation between Energy and Momentum. For a free particle E = \frac{p^2}{2m}. The dispersion relation is generally between E and k. (Note that p=\hbar k. This always holds. It comes from De Broglie’s equation \lambda=\frac{h}{p} \Rightarrow \frac{2 \pi}{\lambda} = k = \frac{p}{\hbar}. For a crystal, using the nearly free electron model, the relation is a (modified) parabola (just the free particle) with some “breaks” in it. For diagram see here (look at the Extended Zone Scheme part of the diagram). The breaks occur at k=\frac{\pi}{a},\frac{2\pi}{a} etc. Why? For k=\frac{\pi}{a} \Rightarrow a = \frac{\lambda}{2}. This is a standing wave situation where the group velocity of the electron wave becomes zero (left traveling reflected wave superimposes on right traveling incoming wave. The atoms represent the “walls” for the reflection). So its not possible to have k=\frac{\pi}{a} and a break occurs at that point. Now the a = \lambda is a similar situation, so the k=\frac{\pi}{a} point also has a break in the parabola and so on. Note that the reduced zone scheme is just a compact way of representing the band structures. Depending on what curve you are, you can read off the energy but to get the right k you need to add right number of 2\pi/a terms.

Angular Momentum Coupling – Basics

Published on August 24, 2007 in Physics. 0 Comments Tags: , , , , , , .

Let us say we have a Helium atom with two electrons. Total angular momentum of the system is a constant of motion. Assume that the electrons and protons do not have spin and there is only interaction between the protons and electrons. Assume two cases:
(1) The electrons do not interact. In this scenario, the angular momentum of each electron is a constant of motion.
(2) The electrons interact. In this scenario while the total angular momentum is a constant of motion as usual, the individual angular momentums are not. This is because if an electron increases its radius (thus increasing its angular momentum) the other electron has to reduce its radius so that total angular momentum remains constant.

In scenario (2) we say that the two angular momentums have coupled with each other. The interaction between the electrons introduces a “coupled” term in the Hamiltonian. So now the Hamiltonian is not commutative with the individual angular momentums any more.

Another good way to understand angular momentum coupling is to think of spin-orbit coupling in the Hydrogen Atom. This is a coupling between the intrinsic angular momentum (spin) of the electron and its orbital angular momentum. The mechanism of the coupling arises in the following fashion:

An electron in its rest frame sees the proton rotate around it. This rotating proton is a current that exerts a magnetic field on the electron. The electron has a dipole moment (by virtue of having spin) so it interacts with the magnetic field induced by the proton rotation. In the Hamiltonian we get a term involving the orbital and spin terms of the angular momentum. Now \mathbf{L} and \mathbf{S} are not longer constants of motion because of this “cross term”. Instead, we have \mathbf{J} = \mathbf{L} + \mathbf{S} as a constant of motion. (Why? \[H, \mathbf{J}\]=0 \Rightarrow eigenstate of H is an eigenstate of \mathbf{J}. Lets say a system that does not exchange any energy is in some eigenstate. This is eigenstate corresponds to some specific \mathbf{J}. Thus \mathbf{J} is constant of motion).

Now because of the reasoning above, \mathbf{L} and \mathbf{S} are no longer constants of motion while \mathbf{J} is. Note that interestingly \mathbf{S}^2 and \mathbf{L}^2 are still constants of motion. (Note: S^2 = \hbar^2 \frac{1}{2}(\frac{1}{2} + 1)). This makes sense because \mathbf{S}^2 is a scalar. The magnitude of the intrinsic angular momentum of an electron never changes.

In conclusion, coupling is when two angular momentums via some interaction do not remain constants of motion anymore. Their values “couple” as they lose constancy but their sum remains constant as in above example.

Useful References:
(1) Griffiths, Introduction to Quantum Mechanics, 2nd Edition, pg 283
(2) http://en.wikipedia.org/wiki/Spin-orbit_coupling
(3) http://en.wikipedia.org/wiki/Angular_momentum_coupling

Retail chain attacked by politicians. Then asked to close. Only in India

Published on August 23, 2007 in Opinion, Commentary & Reviews and Politics & Law. 0 Comments Tags: , , , , , .

ril_logo.jpgReliance Fresh is a vegetable chain owned by Mukesh Ambani’s Reliance Industries. Its a modern food chain that offers low prices, fresh vegetables and a superior shopping experience. Its a breath of fresh air in India’s highly fragmented and obsolete retail industry. It was attacked by a Member of Parliament. Now its been asked to close in UP.

Well, first some background. The government is UP is new. The Samajwadi Party (SP) recently lost to the Bahujan Samaj Party (BSP) in a landslide. The BJP has been demolished too. So we can understand the frustration of the opposition parties. They would like to create a controversy and destabilize Mayawati’s government.

Its easy to whip up fear among the millions of poor in UP. A big corporate giant is out to exploit farmers and kill small traders. They will send all the small traders out of business with their low prices. After they have done that they will raise prices for the consumer and drop procurement prices for the farmer. All very predictable. But the sad part is the its all very untrue.

As of 2007 retail and food chains account for a vanishing fraction of total sales in India (unlike advanced economies). They do not represent a threat to the millions of small traders as a whole now and for the forseeable future. In the localities they are present, they can negatively affect the small traders. But they provide so many benefits as a whole to the economy that its certainly worth it in the long run.

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India now Nokia’s second market

Published on August 23, 2007 in Economics/Business. 0 Comments Tags: , , , , .

“Mobile phone maker Nokia says India has overtaken the US to become its second largest market in terms of sales.”

Wow! The Indian Elephant is making its presence felt across the world…slowly but surely. China is not the only new kid on the block!

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‘Overwhelmingly likely’ that life began in space

Published on August 21, 2007 in Science & Technology. 0 Comments Tags: , , .

Recent probes inside comets show it is overwhelmingly likely that life began in space, according to a new paper by U.K. scientists.

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Laser breakthrough could lead 50,000 times faster hard drives!

Published on August 14, 2007 in Opinion, Commentary & Reviews and Physics. 0 Comments Tags: , , , .

Physicists in Netherlands and Japan are the first to flip the value of a magnetic memory bit by firing a very short pulse of circularly-polarized laser light at it. The result could lead to the development of low-cost and ultrafast all-optical magnetic hard disk drives

Normal hard drives use magnetic induction to write and magneto-resistive effects to read. Magneto-optical (MO) drives are different from normal hard drives as they use different principles. This development relates to them. In a MO drive, reading is achieved through the Kerr effect which is similar to the Faraday effect (light reflected off a magnetized surface has changed polarization). In a normal MO drive, writing is achieved by shining a laser onto a bit. The laser heats the bit to the Curie point (thus destroying the magnetization of the bit). An external magnetic field is now applied to set the bit to desired value (1 or 0). The magnetization of the bit needs to be destroyed because of problems with hysteresis. For instance, if the bit was 0 and you wanted to make it 1 you would need a different magnetic field than if the bit was 1 and then you wanted to make it 0. Since keeping track of which bit was 1 and which bit was 0 would defeat the whole purpose ,we first just “erase” the bit by heating it with a laser to its Curie point.

Notice that an external magnetic field is required in the above explanation. In the research conducted above, researchers use a very short laser pulse to simultaneously heat the bit to near the Curie point and use the EM field of the laser beam (circularly polarized light) to flip the value of the bit.

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The Safest Seat on an Airplane (Hint: It Isn’t First Class)

Published on August 14, 2007 in Science & Technology. 0 Comments

If want to survive a plane crash…sit at the BACK of the aircraft. Check out why…

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