This one is really pretty embarrassing. LEDs, being very simple devices made of semiconductors, are about as in my wheelhouse as anything you can buy at Wal-Mart these days. Even so, until I got around to looking it up a few minutes ago I had no idea how white (polychromatic) LEDs made light in a whole bunch of wavelengths at once.
First off, a quick primer on how LEDs (Light Emitting Diodes) work. I've simplified a lot here for lucky folks who haven't been subjected to solid-state physics, but you'll get the gist. Use the handy diagram below to follow along.
|Simplified energy band diagram of a semiconductor. If you've never seen one of these before I sincerely envy you.|
If you're willing to ignore some quantum mechanics (being an engineer, I am always willing to ignore some quantum mechanics) your basic single-color LED is a pretty simple thing. All semiconductors have something called a bandgap, which is kind of a condemned energy region where electrons aren't allowed to be (see previously-referenced handy diagram). Most of the time, electrons hang out in an energy regime below the bandgap called the valence band (1), because like everything else in nature they are lazy (you can work out a staggering amount of physics from the fact that basically everything in the universe, from electrons to stars to your author, just wants to get to the lowest possible energy state and stay there). Giving the electrons a metaphorical kick in the ass (in the form of some injected energy, like electrical current) will knock a few of them up across the bandgap and into the region above it (2), which for various reasons is called the conduction band.
Since it took energy to get there, our slacker electrons aren't going to want to stay in the conduction band any longer than they have to. A lot of them will eventually drop back down across the bandgap into the valence band (3). Doing this causes them to lose energy, in this case an amount of energy exactly equal to the bandgap (this entire process is not unfamiliar to anyone who's ever tried to drag me out of bed in the morning, minus the part where I emit light). If you're using the right kind of semiconductor, that energy will be released as a photon (light), with a wavelength equal to whatever the bandgap energy was (4). So now you've got (and I'm vastly oversimplifying the explanation here, caveat emptor) a semiconducting device that emits light equal to the semiconductor's bandgap energy when you run enough current through it. Neat, right?
So what color would you like your LED to be? You can set that by your choice of semiconductor material: if you want a device that emits low-frequency (red, infrared) light, pick one with a low bandgap energy; for a blue or UV LED, go with a high-bandgap semiconductor. Through the magic of alloyed compound semiconductors, you can design materials with pretty much any bandgap you want (within reason) these days, so have fun.
You may have already caught the problem with white LEDs at this point. If not, the clue is in the phrase "bandgap energy," singular. As in one specific energy. You can make the bandgap be one of many different energies, and therefore get lots of different possible colors, but at the end of the day you have to pick one energy, which means picking one color. As a result, LEDs tend to be very monochromatic, which is usually a pretty useful property. Unfortunately white light, by definition, is a combination of pretty much every energy/wavelength/color in the visible spectrum, so to make a white LED you'd need to somehow design a semiconductor with many bandgap energies at once. For various complicated reasons, this is impossible. So how do they work?
The answer, naturally, turns out to be "by cheating DUH." White LEDs are a bit of a misnomer, as the diode itself is not emitting white light. Instead, you've got a device consisting of a diode tuned to emit blue or ultraviolet (high-energy) light, coated in a material called a phosphor. You might remember phosphors from the old-timey pre-flatscreen TV you finally just got around to throwing out; that thing worked by zapping a phosphor-coated piece of glass with an electron beam to make the phosphors light up. The canny reader will have guessed by now that phosphors will take energy thrown at them and re-radiate it as light. Again, like LEDs, you can tune what that light looks like by the composition of the phosphor, but the big difference here is that you're not limited to a single color; phosphorescence can be broadband (i.e. white) if you want it to be. You'll also get more energy/light out if you put more energy/light in, which is why for best results you'll want to use a LED that emits high-energy (blue or UV) photons for this.
So put it all together and you've got a blue or UV LED coated in a phosphor material tuned to emit light at a whole shit-ton of visible wavelengths. Blue/UV light injects energy into phosphor, phosphor repsonds by re-radiating the energy as white light, QE-fucking-D. For added fun, you can tune the composition of the phosphor to vary the spectrum, in case you want more of a natural yellow than a true white, for example.
So to summarize: white LEDs are actually phosphor-coated blue LEDs, a handy workaround for the fact that you can't generally get LEDs to emit more than one wavelength of light. I actually don't feel as bad for not getting this now, since it turned out to be basically a hack rather than a gross misunderstanding of physics on my part.
As always thanks to Wikipedia for educating my overeducated ass.