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As I understand it, when physicists talk about something behaving both like a particle and a wave, what they mean is that it has momentum like a particle, but its position is determined probabilistically by a wave function.

What I, as someone ignorant of quantum mechanics, gather from experiments such as the one with the double slit is that quanta will continue to have an indeterminate position until they are disturbed ("observed"); I believe the phrase used is "collapsing the wave function".

My question is the following: what does it mean to disturb something whose position is not yet well defined? What exactly is it that the "observer" interacts with?

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As I understand it, when physicists talk about something behaving both like a particle and a wave, what they mean is that it has momentum like a particle, but its position is determined probabilistically by a wave function.

That's not quite accurate. It would be better to say that it interacts like a particle but propagates like a wave. In particular, while a quantum object is not interacting with anything, we represent it by a wave (a.k.a. wavefunction). The way in which the wavefunction changes over time is well described by the Schroedinger equation. Given a wavefunction, you can probabilistically determine a position and also probabilistically determine a momentum.

However, when the object interacts with a classical measuring device (pretending for the moment that such a thing exists), that interaction occurs at a single point. The position at which it occurs is distributed according to the probability distribution you can extract from the wavefunction that the particle had prior to the interaction. This makes interaction a rather odd event, because immediately before the interaction, the object had a spread-out wavefunction, but afterwards, it has a wavefunction completely localized to a single point. The Schroedinger equation cannot account for that type of change in the wavefunction, so in this model, something else has to be going on. Whatever it is, we call it collapse of the wavefunction.

Admittedly, the presence of some mysterious process that isn't described by a known evolution equation is disconcerting. What it is generally taken to mean in reality is that the model which predicts this wavefunction collapse is insufficient. Many scientists expect that taking into account that the measuring device is a quantum object itself will go a long way toward resolving this problem. If you're interested, much of the work being done in this area falls under quantum decoherence.

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Wouldn't you say that the two electrons in a helium atom interact? But they show no trace of particle-likebehavior, except when another particle such as an energetic photon, passes by? - Thus it would be better to say that quantum objects behave like a wave all the time, and that sometimes, their behavior can also be interpreted in terms of particles. This is just like in classical optics, where waves always work, while particle rays only sometimes. –  Arnold Neumaier Jul 24 '12 at 13:44
@ArnoldNeumaier That would be another example of the quantum detector. –  David Z Jul 24 '12 at 17:22
I meant to question your assertion that ''it interacts like a particle but propagates like a wave''. It interacts only_rarely like a particle, namely only when there is a discrete ionization event, such as when a photon ionizes a helium atom, or when a buckyball ionizes an atom in the detector. –  Arnold Neumaier Jul 24 '12 at 17:30
Ah, I see what you're saying. In my usage "interaction" refers to exactly those kinds of discrete events. –  David Z Jul 24 '12 at 19:07
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Sayng that the position of a particle is not well-defined only means that a mathematically exact point where a particle is does not exist. Similarly, the momentum of a particle is not well-defined, but this only means that there is no mathematically exact direction into and speed with which a particle is moving.

However, the mean position and mean momentum of a particle is perfectly well-defined at all times; this is exploited in the Ehrenfest equations to relate quantum dynamics and classical dynamics. So are the standard deviations from the mean. Thus one can know approximately where a particle is. (Otherwise it would be impossible to do experiments.)

In some experiments such as the double slit, the standard deviation is much bigger than the mean, which means that the prediction where the particle will appear on the screen becomes very inaccurate, and indeed, randomness prevails, though with some structure left, visible in the interference patttern.

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It's important to accept that this question is meta-physical in nature. Otherwise was there would be some method of empirically confirming that the position was well defined prior to interaction. That would not only require going back in time to measure something before you measure it, you'd also have to interact with the particle without disturbing it.

It is the observer's knowledge of the particle's position which is defined prior to measurement. This aspect of quantum mechanics is no different from the classical setting since measurements always come with error bars attached.

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