Month: July 2009

Dirac equation is one of the big achievements of 20th century physics, taking quantum mechanics into the real of special relativity, serving as a foundation of quantum field theory.

Paul Dirac himself describes his motivation for writing his Relativistic Electron Equation, as he calls it.

There was thus a real difficulty in making the quantum mechanics agree with relativity. That difficulty bothered me very much at the time, but it did not seem to bother other physicists, for some reason which I am not very clear about.

By asking a question that nobody cared about, he was able to write an equation that not only connected special relativity and quantum mechanics, but gave much more than that. In his own words,

Now, I found out that this equation gives the electron a spin of a half a quantum and also gives it a magnetic moment, and this spin and magnetic moment are in agreement with obsevartion. […] The new theory still allows negative energies […].

He was very troubled by the negative energy solutions. After all, energy is bounded from below, everybody knew that there must be a lowest state of energy that could be called “zero” with no energies under it. This energy state was associated with the vacuum state. Dirac being himself, in order to fix the negative energy issue, decided to redefine vacuum.

Previously, people thought of the vacuum as a region of space that is completely empty, a region of space that does not contain anything at all. Now we must adopt a new picture. […] we must set up a new picture of the vacuum in which all the negative energy states are occupied and all the positive energy states are unoccupied.

Redefining nothingness takes cojones.

We can get a departure from the vacuum state in two ways: one way is to bring attention on the holes. Well, one can look into the question of how a “hole” will move if there is an electromagnetic field present. And, it moves in roughly the same way as the elctron that fills up that “hole” would move. […] these “holes” move as though they had positive energies and positive charges instead of the usual negative charge of the electron; the “holes” appear as a new kind of particle having a positive charge.

In his new theory, the negative-energy particles could be treated as new particles with positive energies that behave like electron’s evil twins with positive charge.

Let’s recap. Dirac asked a question nobody cared about, then to answer it he made up an equation that gave negative energies that didn’t make sense. Instead of discarding his solution or his question, he reinvents the concept of vacuum and to make it work in a consistent manner, he required to invent a new unobserved physical particle. Dirac’s equation requires the existence of the anti-matter particle, the positron. That is a lot of “if”s. He was unable to fully articulate the prediction of the particle, his explanation follows.

[…] I did not dare to put forward that idea, because it seemed to me that if this new kind of particle (having the same mass as the electron and an opposite charge) existed, it would certainly have been discovered by experimenters. […] That, of course, was really quite wrong of me; it was just lack of boldness.

Still Paul was able to stick to his guns because his equation was, according to him, beautiful. And of course, no only positrons were experimentally discovered not long after, the concept of “holes” as mathematical solutions to the “lack” of electrons is a central idea in solid state physics. Without Dirac Holes there wouldn’t be transistors, for example.

Now, in a recent paper, an experimental group announced they can manipulate the quantum properties of holes just like electrons in materials.

A Coherent Single-Hole Spin in a Semiconductor

Semiconductors have uniquely attractive properties for electronics and photonics. However, it has been difficult to find a highly coherent quantum state in a semiconductor for applications in quantum sensing and quantum information processing. We report coherent population trapping, an optical quantum interference effect, on a single hole. The results demonstrate that a hole spin in a quantum dot is highly coherent.

The abstraction of “Dirac’s Hole”, only introduced for pure mathematical aesthetical reasons is now an accepted, essential, physical particle that is used and manipulated at will. Only a theoretical giant like Dirac could have pulled that off.

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The only thing in this world that gives orders… is balls.
-Tony Montana

We take the constant rate of increase in computing power for granted. One of the founders of Intel, Moore, estimated once the rate of the number of transistors $$n$$ in an integrated chip to be exponential, doubling every year. The smaller size of these transistors allows manufactures to pack more and more in each chip every year, making processors more powerful, memory capacities larger, and overall, computers faster.

Moore’s law can be stated as: $$n(t)=2^{frac{t-t_0}{2}}$$ where $$n(t)$$ is the number of transistors on a chip at some time in the future $$t$$, $$n(t_0)$$ is the known number of transistors for a time $$t_0$$.

This exponential trend cannot be sustained forever, at some point the transistor will be so small that quantum effects will dominate and the laws of physics that make the transistor work will change dramatically, effectively making them stop working. J. Powel looked at this in The Quantum Limit to Moore’s Law (subscrition required). Here, I reproduce a similar calculation with compatible results.

The question I am interested in is: given all things equal, what is a good estimate for the year that we will reach the quantum regime where transistors won’t be able to get any smaller?

First, we need one data point of the number of transistors in an integrated circuit at a certain year. For this, I used Intel’s own data that says that in 2007 they could manufacture transistor of lenght $$45nm=45times10^{-9}m$$. By assuming that the number of transistors is inversely proportional to the size of the transistor, we can estimate $$n(t_0)$$.

Now, we need to estimate the number of transistors in an integrated circuit just before they transistor reaches the quantum limit. For this, I chose the transistor to be of the order of the Compton wavelength of the electron, or 10^{-12} m . The Compton wavelength $$lambda=frac{h}{m_e c}$$ is the dimension of an electron from Heisenberg’s uncertainty principle. If you are in this regime, the electron behaves mostly as a wave, not as a particle and the electronic properties of the transistor will change, making it unusable. It is a fundamental physical limitation of the size of a transistor.

With this numbers at hand, I solved for $$t$$, and found out the year where Moore’s law will break:

Year $$2038$$ is when Moore’s law will not hold anymore.

The purpose of this excersize is NOT to predict the end of the computer growth as we know it, as this is bound to fail, but to stress the need for understanding electron transport in the quantum regime. The quantum regime is something we will reach in my lifetime, and a new theory of electronics in the fully quantum regime is needed before we get there.

But, also, this number misteriously agrees with XKCD’s end-of-the-world prediction in a manner different from what XKCD intended.

Creepy.

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mobilis in mobili