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So I learned that after a measurement of, lets say the position of the wavefunction of a particle is made, if another measurement of the position of the particle is made right away, you should get the same result because the wavefunction has collapsed.

How can you tell the difference between the observer measuring the wavefunction which causes the collapse vs. something else that measured the wavefunction and collapsed it already and you are measuring it the second time around?

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There's a more detailed model of the experiment in which you aren't making the same measurement twice. You say it yourself, the two measurements are made at different times. It's better, IMO, to get out of thinking in terms of the first approximation that is quantum mechanics, instead moving to thinking in terms of the second approximation that is quantum field theory.

In a quantum field theory, there is a difficult question of just what you might be measuring. If we think in terms of QFT being a field theory, it's not at all clear that you're measuring the position of a single particle, because in field terms you're just measuring a statistic of the field in two different places in space-time (constructing an ensemble of data points or some other way of relating probabilities to statistics needs some care in this conceptual background, of course). Even if we think in terms of QFT being a particle theory, which IMO, and in the opinion of an increasing number of papers in the literature, goes somewhat against the current of the formalism, the two measurements may be measurements of the positions of different particles, not of two positions of the same particle.

I would further reduce Marek's comment that you don't need to tell when there is a collapse. If you're an instrument-level experimentalist, aka a lab technician, all you care about is that there is an analogue signal. You might notice that the signal has more-or-less discrete transitions between different levels or different types of behavior, and because of that you might start to record when those discrete events occur. Only if you care about theory, and want to give an explanation for the discrete events, do you mention collapse (although "collapse" is more an encoded description of the observed discrete structure than an explanation). I think it's only if you care about theory that you have particular wishes as to what features of the analogue signal you will convert to logbook or digital representation or put into a [published paper, which we can suppose will include particular interest in any identifiable discrete features there might be in the analogue signal.

Bjorn's Answer appeared while I wrote this. It's a good way to start discussing more detailed models of what measurements do (it's Useful!) in quantum mechanical terms. I can get carried away by QFT ways of thinking. In the QM sort of vein, you might try the book "Operational Quantum Physics" by Paul Busch, Marian Grabowski, and Pekka J. Lahti (don't be misled by the "Operational", it's far more theory than experimental, and also not elementary).

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What Marek wrote is accurate with for example a photon polarization measurement (the canonical example is that an additional polarizer after a first polarizer won't change the intensity if they are oriented the same way) but in the OPs example, actually if you measure the position accurately the first time, you will collapse the wavefunction into an uncertainty of the momentum so the next position measurement will not be entirely predictable and hence you will not measure the same thing. This by itself does not allow you to realize that the wavefunction already collapsed though.

However, if you allow your position measurements to be done in different places, you can exploit the fact that your collapsed wavefunction has lost some of its components which can be crucial for subsequent interference and hence detect the difference you are asking about, at a certain efficiency at least.

Consider the typical dual-slit experiment and the fact that a position measurement of which slit the particle goes through causes a loss of interference at the detection screen. You can put a detector at a fringe which should be completely dark and if you do detect a particle there, you can infer that the wavefunction collapsed before the both slits could interfere (and therefore conclude that a measurement was done at certain places in the experiment).

This would be considered a form of "interaction-free measurement" of which the bomb-detection experiment is the most popular I think - again in that, you can detect the presence of a position-measurement (a photon-sensitive bomb going off) in a certain place in the experiment by the loss of interference effects in front of later detectors.

Both of these examples are examples of counterfactual measurements, where you draw conclusions about something you don't explicitely measure, and hence provide a loophole to your question.

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As you correctly say in first paragraph, measuring twice is the same thing as measuring just once. Therefore you can't distinguish these two alternatives. But there is no need to tell. If you are an experimentalist, all you care about is that the wavefunction has collapsed and there is some number shown on your measuring apparatus. You don't care how precisely was the collapse realized.

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+1 Everything in a few words ! –  Frédéric Grosshans Apr 10 '11 at 20:04
    
@Frédéric: thanks, but seeing other answers I realized I've taken the question too literary with measurement being essentially a constant projection operator on the Hilbert space. In reality, subsequent measurements can differ in number of ways, mainly because of imperfect apparatuses and interaction with environment between measurements. So I prefer other answers more. –  Marek Apr 10 '11 at 20:39
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