## Saturday, January 9, 2010

### Uncertainty Principle: Intrinsic Property or Measurement Effect?

At last night's Saint Louis Skeptics in the Pub, the conversation came---as it always must---to physics. Saint Gasoline and unBeguiled wanted to know whether the position-momentum uncertainty principle is a real property of a quantum particle, or simply a limitation of our ability to measure both the position and momentum of quantum particles. I guess, even more strongly than "they wanted to know," they were arguing for the latter, more intuitive answer.

As an example of the kind of thinking they argued for, take this quote from the Wikipedia article on the uncertainty principle:
Heisenberg's microscope
One way in which Heisenberg originally argued for the uncertainty principle is by using an imaginary microscope as a measuring device. He imagines an experimenter trying to measure the position and momentum of an electron by shooting a photon at it.
If the photon has a short wavelength, and therefore a large momentum, the position can be measured accurately. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. If the photon has a long wavelength and low momentum, the collision doesn't disturb the electron's momentum very much, but the scattering will reveal its position only vaguely.
If a large aperture is used for the microscope, the electron's location can be well resolved (see Rayleigh criterion); but by the principle of conservation of momentum, the transverse momentum of the incoming photon and hence the new momentum of the electron resolves poorly. If a small aperture is used, the accuracy of the two resolutions is the other way around.
The trade-offs imply that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower bound, which is up to a small numerical factor equal to Planck's constant.

Now, I don't think this thinking is exactly wrong. I mean, if one constructed this experiment I don't doubt that it would validate the results predicted by the uncertainty principle. However, I think the model behind it is flawed. What we imagine in this gedankenexperiment is a little ball that is the electron being hit by a little ball that is the photon and the two bouncing off each other.

What this sort of explanation ignores is the fact that matter behaves as a wave.

I may need to do some argumentation to convince you (specifically Dustin) that matter waves aren't just a mathematical convenience for calculations but are, in fact, the actual nature of matter. If I need to do that, I'll do it later. Today is not the day to explain matter waves. Here, though, is a very quick argument: we know ordinary waves (water, sound, etc.) do certain things like reflection and refraction and interference and whatever. Some of those things are done only by waves and nothing else. We see quantum particles doing those same things. Here's a handy chart.

So let's simply accept for now that electrons are, in almost all circumstances, wavelike. I think we all have heard many times that quantum particles are both particulate and wavelike at times, and might take it for granted. We sort of skip over the fact that electrons are waves without really understanding what that implies, so let's investigate. This will involve a little math, but stick with me. It shouldn't be that bad.

Let us assume the electron has as its wavefunction pretty much the simplest wave you can get, a sine wave. This is a periodic function, which means that if you are at any point and you move some special distance away, everything about the function will look the same. This distance is called the wavelength, denoted by lambda. There is a relationship between wavelength and momentum called the de Broglie relation, which says momentum is equal to Planck's constant divided by wavelength.

p = \frac{h}{\lambda}


So for an electron with a sine wavefunction, we know it has a definite momentum.

What about position? To find the position of the electron, we need to square the wavefunction and integrate it over all space. (Don't worry, I won't make you sit through that.) When we do that with a sine, the result we get is meaningless. It says there is a smeared out probability to find the electron everywhere, and no place is more probable than any other. Thus there is no helpful position information we can get out of this wavefunction.

This is all summarized nicely in this picture I stole.

To make some kind of meaningful statement about position, then, the electron can't be in a state with a wavefunction that is just a sine wave. What we can do is make a "wave packet" by adding together a few different sine waves. If we pick the wavelengths and amplitudes properly, we should be able to get a decently localized position. However, by adding together different sine waves we have introduced more than one wavelength. With more than one wavelength we don't know exactly what the momentum is. Again, I stole a picture.

Now, looking over this example I gave, is this a measurement effect? Most emphatically NO. If you know the momentum exactly, it is not the case that you just can't measure the position, but the idea of position doesn't even make sense. Look back at that youtube video I posted. That is a wave with a well-defined wavelength, and therefore a well-defined momentum. I ask you: where is that wave? Is that wave in a particular place? At a particular position? No it is not. The wave is spread out over all space. If the wave had a less-precise momentum, as in perhaps there are several waves of different wavelength added together, we would see waves only in some constrained area and there would be no waves outside that area. In that case, it is meaningful to say that the wave is in a particular place, because it is localized to an area.

If we accept that matter actually has wave properties (and this is well-established both theoretically and experimentally, but perhaps that's a topic for another day), the uncertainly between position and momentum comes as an immediate consequence. By virtue of how position and momentum are defined, they are not completely compatible. This incompatibility has nothing to do with what you can measure. It has to do, as I outlined here, with what the concepts of position and momentum mean when applied to waves.