DATA-log

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 sells in the supermarket of ideas.

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:

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.

Lorenz0001 by janne808

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

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.

Fpu1 by janne808

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

Fpu2 by janne808

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.

Stephen Wolfram and George Johnson on ‘A New Kind Of Science’ (ie. on automata and computation.)

http://bloggingheads.tv/diavlogs/8986

Anthony Aguirre and Clifford Johnson chit chat about string theory vs. field theories, coupled with some cosmology.

http://www.bloggingheads.tv/diavlogs/22011

Sean Carroll and Mark Trodden from Cosmic Variance blog on the topic of cosmology, dark energy and other interesting issues.

http://bloggingheads.tv/diavlogs/21709

Peter Woit and Sabine Hossenfelder on various sociological issues concerning young theoretical physicists.

http://bloggingheads.tv/diavlogs/12950

Since googling up NASA’s online VLF receiver mp3 stream (seems to be defunct now, fortunately there are other streams available.. see http://abelian.org/vlf/) I wanted to try to receive these signals myself. So I came up with some schematics on the internet for a basic vlf receiver and built and tested a couple of these.

E-field receiver clip 1

E-field receiver clip 2

The sferics are crisp, loud and clear, the circuits work very well if the conditions are right. Small enough to fit a pocket, it’s interesting to walk aroud the area I live and listen in to the various weird signals eminating from all sorts of electronics and machinery. I’ve yet to record auroral activity, but that is next on the agenda.

I want to write a little bit more about my studies on waves and automata models. I wrote a vastly improved TLM code on MATLAB which now includes for example first order absorbing boundaries. It is important to distinct this approach from a mathematical model, this is a analogous physical system to wave propagation. You could think of it as using computer memory element grid as an discrete analogy to the vacuum.

This sort of physical modelling and computation was first used by a Hungary born mathematician and electrical engineer Dr. Gabriel Kron in 1943 while working for General Electric (see paper called ‘Equivalent Circuits to Represent the Electromagnetic Field Equations’ on Physical Review Vol.64 Numbers 3-4 1943.) The approach involved analog computing in the form of a RLC network. The approach was then picked up by P.B. Johns and R.L. Beurle (see paper ‘Numerical solution of 2-dimensional scattering problems using a transmission-line matrix’ on Proc. IEE, Vol. 118, No. 9, 1971, pp. 1203-1208) applied to ‘computors’ as they were then called.

The Johns and Beurle numerical method involves applying a simple scattering automata rule to a discrete node grid. This doesn’t exactly involve integration in the sense that a discretisized mathematical model would; only arithmetic needed is addition (floating or fixed-point) for summing up the node incidence and reflection time-step impulses involved in the scattering rule (which are directly derivable from normalized unitary impedance electrical node network.)

The simple TLM here is configured for the classic Young double slit experiment. Although this particular setup could be thought to just propagate the electric field, the corresponding magnetic field can be derived aswell together with different permittivity and permeability coefficients to model material properties.