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Let me give you an example:
A particle unobserved has a variety of possible positions and momentums at any given instant.
The layman's view is that we simply don't know what the particle is doing, but it does have a specific position and a specific momentum.
To the quantum physicist, the particle is actually in what is called a "superposition" of states corresponding to all of those positions and momentums. It may have a larger amount (amplitude) of one state than another, but the idea still holds; the particle cannot be neatly described as having a specific position or momentum.
Now suppose you want to measure the position of the particle.
To the layman, you simply look for the particle and, having found it, write down its position; the momentum is unaffected, because you haven't touched the particle in any way.
To the quantum physicist, measuring the particle's position collapses the superposition into a single position state. This changes the particle from a quantum perspective. Note that the observer does not have control of how the superposition collapses, although if a state has a higher amplitude in the superposition, it is more likely that the superposition will collapse into that state.
The most famous change is to the momentum. The momentum states are not the same as the position states; when the particle is collapsed into a specific position state, it misses all of the momentum states, so you still have no idea what the momentum is. In fact, the more you know about the position, the less you can know about the momentum.

The confusion here is about what the superposition means, and what collapsing the superposition does.
Quantum mechanics doesn't actually tell us if the superposition has a physical existence or if it's just a figment of the mathematics. Nor does it give any clue as to what actually occurs when the superposition collapses. It just assumes the collapsing as an axiom and nobody's been able to figure out a really good interpretation.

that is... almost what she's saying. you're closer though. To make a measurement we must alter the system, but this isn't a limitation of our technology, it is a property of nature. We are constrained by the uncertainty principle length where:

(delta)x * (delta)p >= h/2pi

I hope that reads well, (delta)x is the uncertainty in the position measurement and (delta)p is the uncertainty (or error) in the momentum measurement, and the multiplication of both has to be greater than or at least equal to Planck's length.

As Layra-chan said, any system that we are interested in measuring exists in a superimposed state, that is the reality of its existence while unobserved, it is a ghost. To probe the system is to disturb it and have the system collapse to one of its possibilities. Say a particle that had 30% 4m/s velocity 20% 5m/s velocity and 50% 1m/s velocity, once seen has only 100% 1m/s velocity.

This collapse is independent of the fineness of the probe, the mere presence of a probe (for without one we cannot observe) will collapse these superimposed states onto one of them, called an eigenstate.

sou da ne domokun

Yokata ne! nihonjin janai demo nihongo o benkyoushiteimasu... eeto ano anata no saigo kanji wa wakaranainda. Boku kanji no jouzu ga naindakara kanji no benkyou hon o kaimashita. Sochira gakkou naka de benkyoushimasu no kawari ni futsu gakkou no benkyoumono hehehe. Kono konpyuuta de wa nihongo kakimono o dekinai no gomen ne!
EDIT:
Ano kanji ga sagashimashita Ano bun wa watashi no nihongo wa ii desu ka
Hai iida yo mrgreen

whee

I bet you've been waiting a while to pull out that pun.

Tá sé mí-muinte a beidh ag labhairt i teanga ná tuigeann daoine eile.

And that was so completely helpful.

When you calculate a wavefunction, you're not measuring, you're deducing based on previous measurements. A measurement ought to be mechanical and immediate, and hence would fall under the quantum mechanical definition of measurement/observation.

I still wouldn't call plugging data into other things a measurement; you already have the data, so the physical system itself is now secondary. The data could have been gathered from anywhere; it could have been made up. I consider this to be a calculation, since the system no longer needs to be present.
A measurement as I tend to view it relies completely on the existence of the system; no previously collected data regarding the system is necessary, but the system itself is the focus and the object. I never point out that position and momentum need to be known, because that isn't true.
I'd contrast the situations as the difference between interpreting an image and gathering images on ones retina. The first requires thought but no source, in that the world that the image came from is irrelevant as long as a brain is present; the second requires a source but no thought, in that a mindless, disembodied eye could do it, but it requires a world to be viewed.

I would perhaps go as far as to define measurement as requiring interaction with the particles and waves scattering off of the system; this certainly doesn't contradict the (scientific) classical paradigm of measurement. The only difference between the classical and quantum paradigms regarding measurement is in the effects of the measurement, not how it is carried out.