Saturday, December 12, 2015

"Quantum Dot" (revisited)

I don't know. I write about politics, history, genetics or intelligence and the readers line up. If I write about Christmas cakes or central heating, crowds read my work. But let it be physics and you can count the page-views on your fingers and toes.

Back in 1973 I was a teacher-training student in London (secondary maths since you ask, and just booted out of Warwick University - another story). I was also a member of the International Marxist Group .. and broke. I suggested to the appropriate senior comrade that I might be excused endless and onerous political duties during the summer to get a job on a building site. The ostensible reason was to proselytize amongst the urban proletariat but even I knew that was idiotic; my real reason was to make some serious money. Amazingly, the IMG bought it.

In 1973 the economy was booming and a construction site down from the Elephant and Castle was prepared to take me on - I had never done building work in my life. To say I was a weedy, incompetent fool amongst all those tough, muscled and very working-class builders would be to massively understate it. I didn't have the first clue, couldn't be trusted to follow the simplest instruction and the only thing I got really good at was drinking five pints at lunchtime (it was very hot and you habituate fast).

I lasted out my time, made my money and never discussed politics. Years later I wrote a story which combined the self-pity of the effete intellectual with a bigging up of my own intellectual stature. Yes, Reader, that story was published on this very blog (read it after you've finished with my intro here).


Just a note on the physics of "Quantum Dot". In 2007 I hadn't yet taken the OU quantum theory course (SM358) (which I only got around to in 2009) so in retrospect I'm surprised at how much I got right - I've only had to make a few tweaks to update it.

The central physics idea is the time evolution of a well localised particle (an electron) placed at the centre of a quantum dot. In hand-wavy terms (as in the story) the evolution is pretty obvious, but actually, what does the wave function do?

At time of writing I wouldn't have had a clue, but the usual trick is to model such an initial wave function as a narrow Gaussian, approximating a Dirac delta function - and then apply the Schrödinger equation for the time evolution. The maths is difficult but I did find this surprisingly accessible paper which investigates the relative contributions of the energy eigenfunctions. For an initial narrow Gaussian there are a lot, with amplitudes falling away for the higher energy eigenstates, as you would expect. David Etlinger's analysis is for the infinite square well, but the interior results won't be qualitatively different for a quantum dot.

In any event, since we now know the amplitude for each energy eigenstate, we just add them all together. Turns out my description:
"its wave function is just smeared around: small-scale choppy, large-scale pretty much flat - in configuration space, that is."
is a fair description of the emerging state (p. 17).

OK, now I've put you off, go read the story! You may then spot the connection with the first paragraph above.

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