This pre-Thanksgiving break lecture is purely for your own edification--this material won't show up on the test.

The fact that light acts like a wave is well-known: interference phenomena are common, from the colors shining off a CD to oil drops on concrete to the double-slit experiment a peacock's tail to Newton's Rings. You can play with wave and interference phenomena online using Paul Falstad's Ripple Tank Applet. and Olympus Microscope's double-slit experiment. The weirdest part about wave phenomena is destructive interference: two out-of-phase waves can cancel each other out to give nothing (like in noise-cancelling headphones). Destructive interference is what cancels out most colors in an oil sheen, and what creates the dark spots in the double-slit experiment.

In 1924 Louis de Broglie (in a 9-page PhD thesis in physics) proposed that even matter acts as waves, and gives an equation for the wavelength. This bizarre idea is now universally accepted--interference of electrons was demonstrated experimentally three years later by Davisson, Germer, and Thomson. 1926 Erwin Schrödinger invented the "Schrödinger Equation" to describe how wavefunctions for light and matter change as time progresses. This is still the defining equation in quantum mechanics.

This was a very curious time, because Millikan's oil-drop experiments (in 1909-1913) had shown very clearly that electrons were particles, not waves--he'd even measured the mass and charge of an electron. Einstein had shown in 1905 that the photoelectric effect was best explained by considering light as a series of particles (called photons), not waves. So we've got the strange situation where experiments are telling us two totally contradictary things.

In a nutshell, the wave/particle contradiction boils down to this:

- Wavelike: Electrons have to be waves because they can interfere
with one another. Particles shouldn't ever cancel each other out
(destructive interference), but electrons and photons do cancel each
other out (e.g., in the double-slit experiment).

- Particlelike: Electrons have to be particles because they're
"quantized": the mass and charge of a clump of electrons is always
measured to be an integer multiple of the mass and charge of a single
electron. Waves should be able to dissipate smoothly, so you
should be able to capture half an electron wave, but in real
experiments, this just never happens!

Wave-like properties seem insane when applied to matter. You can't somehow roll together two ball bearings (at the right angles perhaps?) so that when they collide you're left with zero ball bearings (destructive interference).

The current explanation of these two separate facts is, essentially, two separate theories:

- Wavelike: Electrons and photons really are described by a wavefunction that evolves through time according to Schrödinger's
equation. A single electron's wavefunction can cancel itself out
in places, as can the (coupled) wavefunctions of two or more
"entangled" electrons. Thus electrons and photons really act like
waves. The official name for this process is "unitary evolution".

- Particlelike: Electrons and photons are always measured as discrete particles, with probability according to the square of the wavefunction.
That is, when acting in a wavelike way, the particle extends across
space and sits in a superposition of different locations, but when you
look at it, it makes up its mind and is actually observed in a single
spot. This means the very act of observation ("measuring")
perturbs the system, and makes it act a different way (the Heisenberg
Uncertainty Principle). This amazing fact can itself be
experimentally verified--adding an electron detector to the double-slit
experiment destroys the interference pattern! Nobody's clear on
exactly what constitutes a "measurement", but it's clear anything that
we can see or directly measure causes stuff to act like particles; it
only acts like waves when "nobody's watching". The official name
for this process is "state-vector reduction" or "wavefunction collapse".

If we connect wires between two adjacent wells, the electrons that (maybe) are in the wells interact as waves. If we connect circuitry between the wells, the electrons will seep out into the circuitry--so we can hence perform calculations on these quantum bits! If we're very careful (and this is the hard part), the calculations will stay wavelike.

Why do you care if your quantum computation stays wavelike? Well, imagine you've got a 1000-qubit register where all the bits are in the superposition of 0 and 1. If you feed that register into a quantum circuit, the circuit's output is the superposition of all the outputs of all 1000-bit inputs. If you now carefully collapse the circuit's output to a particular value, the inputs will also collapse due to entanglement. But this means you can now trivially find the input for any output, or invert any function!

For example, consider multiplication (Peter Shor described a theoretical quantum multiplication circuit in 1994). On a normal machine, it's easy (polynomial time in the number of digits) to multiply two long numbers, but it's really hard (exponential time in the number of digits) to find the factors of the product of two numbers. But on a quantum machine, it's polynomial time for either one!

The real practical problem with quantum computers is that of "decoherence"--premature wavefunction collapse. If the electrons involved interact with too much of the outside world, they cease to behave in a wavelike fashion too early, and then you're back to normal circuitry. The largest quantum computers actually built so far have only two or three qbits, which means they're not at all useful--even grade schoolers can already factor 3-bit numbers. With larger quantum computers manufacturing defects, thermal noise, and leakage current destroy wavefunctions before they can do useful work. But a huge variety of electron (and other particle) containment devices are being tried and built, so useful quantum computers may someday be built. Or perhaps there is some physical limitation on the scale of wavelike effects, and hence quantum computers will always be limited to a too-small number of bits or too-simple circuitry.

- Typical computers lock up about once a day (including flight computers, laptops, digital cameras, etc.) and must be rebooted.

- Human beings have never been observed to lock up, and have never
needed to be rebooted. This is a good thing, because a spaceship
with an insane or catatonic crew might have trouble returning to Earth.

- Many independent decision-makers. Examples: A cell is a
complicated collection of separate but interacting proteins. A
market is a collection of interacting buyers and sellers. A
gravel pile is a collection of interacting pebbles. Results: no
central decision-maker means no single point of failure, so the loss of
any few small pieces doesn't affect the overall result.

- Dynamic equilibrium--lots of small-scale stuff is changing all
the time, but the overall averages remain quite constant.
Examples: the metabolic rate of a cell is the result of all its pieces,
but it's quite predictable overall. A market's overall prices
tend to remain fairly stable. A gravel pile's average slope can
be predicted to within a few degrees.

The standard machine design responses are top-down social control measures like signs, fines, fences, periodic cleanup drives, or guards; these are extremely expensive per bottle collected. One could also imagine complicated mechanical devices like a bottle-collecting pump, dredge or skimmer, which has the problem that it'll immediately clog up with non-bottle stuff (algae, plants, sticks, rocks).

Ecosystem design offers many alternatives. One is a to design a small bottle-eating critter, like an insect; but living things have their own drawbacks, like escape, extinction, disease, etc. Consider the option of periodically dumping a box of "sticky BBs" into the pond. These would be tiny (1mm) spheres made of a special plastic that sticks only to polyethelene (which the bottles are made of). How would this help eliminate bottles?

- Wind blows the stuff in the pond around.

- Whenever a bottle runs into a BB, the BB sticks to the bottle.

- Hence after a while most of the bottles in the pond accumulate a few BBs.
- When two (BB-covered) bottles run into each other, the BBs stick to
each other. Since the BBs are stuck to the bottles, that means the
bottles stick together.

- After a while, all the bottles in the pond will be glued together into one big glob. (See the bizarre video game Katamari Damacy.)

This is exactly the strategy your body uses to eliminate known invaders like viruses and bacteria (bottles). Your immune system engineers a sequence that literally physically sticks to these invaders, and attaches two copies of the sequence to a Y-shaped protein called an antibody (BB). Each side of the antibody sticks to one invader, which (a) coats invaders in antibodies, which may keep them from operating properly and (b) sticks invaders together into bigger clumps of invaders and antibodies, which can't cause as much trouble as separate invaders and are easier to eliminate completely.

People are also trying to combine quantum and biological computing--using electrons on biological molecules to represent quantum states. There is even speculation (from Rodger Penrose, see "The Emperor's New Mind" and "Shadows of the Mind") that mammal brains use some sort of quantum computation to do their amazing work.