Archive for the ‘Math’ Category

My ten cents on Quantum Mechanics

Wednesday, August 25th, 2010

You can read all sort of new age weirdness and academic pomposity associated with quantum mechanics. Controversy rages on up to this day about the fundamentals of this cornerstone of modern physics.

So what is so special about QM. My interest in it comes from the mathematical framework it was built on. I think it’s an interesting way to learn new mathematics, and indeed it has proven to be just that. My experience has been that once you finally get the hang of it, all the mystery you’ve read about it seems to diffuse in to thin air.

It is easy to deflate QM, if for some strange reason that is your goal. The basic premise the quantum is built on is diffusion. Yes, the Schrödinger wave equation is nothing more than a mathematical description of diffusion in imaginary time. All the nonintuitive aspects of QM are closely related to the complex plane. Schrödinger equation was an important step in the development of the theory, but is actually just a small part of the framework and it is incomplete without the concept of what a measurement is.

Of all the books written about QM I’ve studied, the best introduction by far is in the good old communist era Landau and Lifshitz book Course of Theoretical Physics Volume 3: Quantum Mechanics (Non-relativistic Theory). It gets right to the point; it is impossible to measure properties of fundamental particles with absolute precision. The reason is simple and intuitive. Measurements of fundamental particles have to be made with some sort of force mediating way. In the case of measuring the position of an electron you can use light. The more precision you want, the shorter wavelength you’ll end up using. This in turn means that the photon carries more energy. When the photons interact and scatter with the electron you want to measure, the more the electron recoils from this energy. Repeated measurements will end up jittering the particle more and more, adding to the uncertainty of the measured path. Consecutive lower energy, bigger wavelength photons end up jittering the particle’s path less, but the wavelength dictates the uncertainty in the measurement.

This ends up being reproduced by the basic QM vector space framework. The Heisenberg uncertainty principle in the mathematics stems from the commutation relations of the observables, which in the simplest case here are the position and momentum of the particle.

The above video illustrates the unitary time evolution and diffusion of a simple potential well quantum system in both the position and momentum spaces. The initial gaussian probability distribution moves and spreads out in space with time. The distribution cancels out with itself in this bound system, in free space the wave packet would diffuse indefinetly. You don’t actually exactly know where the particle is until you measure it, and collapse the wave function to some specific state. For example measuring the position would pick a definite position from the distribution at random with some probability and the time evolution would diffuse from that point on, until a new measument is made.

Is it then such a mystery that QM manages to be backed up by experimental measurements, when the whole theory is rigged up to do exactly that?

Quantum fetishism

Tuesday, August 3rd, 2010

Quantum sells in the supermarket of ideas.

Ode to the founder of cybernetics

Thursday, July 15th, 2010

Average of sin^100(x).

Thursday, June 10th, 2010

I found an interesting math problem on The Reference Frame blog, concerning solving the average of the 100th power of sin(x). Mr. Motl, the blog author seemed to think that there is no fast 5 minute way to solve this problem, only to discover several exact attacks on the comments to complement his approximation.

All it takes is some fundamental concepts in complex analysis. To be fair, it took me more than 5 minutes to work it thru and correct some errors along the way. I’d say something like 10-15 minutes. Have to say to my defense though, that I did the work under being influenced by sleep deprivation (unfortunately I suffer from a sleep disorder).

Here’s a fast way to work out an exact result. First recognizing that sin(x)=(exp(i*x)-exp(-i*x))/(2*i) gets you (1/2^100)*sum(k=0..100)(100!/(k!*(100-k)!)*exp(i*x*(100-2*k)) after applying the 100th power and some binomial magic. Expanding the sum you can group the terms to form cosines in the form of (exp(i*x)+exp(-i*x))/2, whose average is ofcourse zero. The only term that contributes to the average is the exp(0)/2 term (since the binomial expansion is symmetric in this way) and you end up with 100!/(2^100*50!*50!) as the answer.

Unpopular science

Tuesday, May 4th, 2010

I’m a big fan of pop science tv-shows; currently my playlist is loaded with Stephen Hawking’s new popular tv-series. I think it’s healthy to keep things in balance though, so why not add some counter weight with something that the physics community seems to consider very unpopular indeed.

David Bohm’s views about the foundations of quantum mechanics gave him a lot of flack from the physics community. Here he is in a long 5 part interview:

Lorenz model

Tuesday, April 27th, 2010

To complement the FPU post below, here’s a similar approach to the well known Lorenz model used in chaos studies. There’s an increasing Rayleigh number used in the computation of this clip.

It’s obvious that there are very interesting regions of nonchaotic behaviour in the model.

Fermi-Pasta-Ulam

Tuesday, April 27th, 2010

Current studies have led me to meddle with the famous FPU (Fermi-Pasta-Ulam) problem. Unsuprising event, since it is one of the cornerstones of the study of computational physics.

It was one of the first problems that was tackled not using analytic math tools, but using high speed digital computing. The reason behind this sort of approach was the difficulty of dealing with nonlinear equations; something that is near impossible to deal with exact analytical attacks. Digital computers and numerical analysis however is the ideal tool to conduct these sort of chaotic computational experiments with.

A lot has been written about the FPU problem (try the wikipedia article for a decent summary), but an immediate way to grasp the problem is by hearing how it sounds. The system described in the problem consists of masses coupled together, the usual scalar wave equation with nonlinear coupling terms added. Here the initial gaussian pulse oscillates in the system without damping and with increasing nonlinearity.

Another example is done with a custom VST plugin. The system is driven with two pulse oscillators.

More automata music.

Sunday, October 11th, 2009

I’ve been kept busy by Schrödinger, Fourier and Dirac lately, but I had some free time finally and put together a more refined way to derive musical (or so) structures out of one dimensional cellular automata system. Partly inspired by the excelent lectures on early finnish experimental electronic music scene at the local media art museum (see http://mansedanse.com/events_fi.html).

The algorithm quantizes the chromatic scale down to any arbitrary scale and picks up two notes to be played. This produces a more music-like result than the total chaos of applying the whole automata state straight to the chromatic scale.. though I’m not saying that it can’t produce interesting results.

http://www.punainen.org/~biotek/cell0011.mp3

http://www.punainen.org/~biotek/cell0012.mp3

Here is the MATLAB code responsible for these, keep in mind though that the quantizer code was written in the middle of the night (I think you can tell) so there are probably some glitches to it.

http://www.punainen.org/~biotek/automata1.m

Old Connection Machine promotional videos.

Monday, August 31st, 2009

Connection machines were a line of parallel supercomputers built by Thinking Machines Corporation. Notably Stephen Wolfram and Richard Feynman were involved in the early years of the corporation.

Here are some delightfully academic and stiff Thinking Machines Corp. promo videos from Youtube. The first one features some great footage from a lattice gas automata fluid dynamics model (I can’t believe it took a person year for the LGA model, I put one together in a week or so on MATLAB. My ego is pleased.)

Ironically Thinking Machines Corp. went bankcrupt in 1994, when parallel computing is a hot commodity today in 2009.

Automata music

Tuesday, August 25th, 2009

What would be better application for one dimensional cellular automata than autogenerative electronic music. I know Wolfram Tones offers something along these lines but I set out to experiment on my own first with just the bare rules applied to the chromatic scale. Couple of important rules rendered on the chromatic scale:

http://www.punainen.org/~biotek/rule30.mp3

http://www.punainen.org/~biotek/rule110.mp3

These does seem to hold a eerie quality to them, not that anyone would recognize them as music. Next I cherrypicked a good set of rules which seemed to work nice enough when applied in a random order. I think I used the c major scale for this one.

http://www.students.tut.fi/~heikkara/autobach4002.mp3

That sounds a lot more like music, almost emotional at times. This is a Processing program playing midi notes to a Roland Jx-3p.