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?
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The radio system currently supports some interactivity in the form of synthesized speech shoutout and request functionality. More could be done with it however. Current future plans include webcrawling information and music file extraction and also some form of soundcloud integration where the system analyzes the music tags and processes the music into a radio stream.
I think the internet can offer new insights into this old aging medium, in the form of commune driven interactivity and automatic computational analysis.
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.
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: