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I'm about to go into my 4th and final year of my undergraduate physics degree and after all the quantum mechanics we've done so far, I still get this nagging feeling that I'm answering homework and exam problems blindly. Apologies for the length of this question by the way.

For example, take the infinite square well problem. The particle is confined in a 1-dimensional region of space in an infinite potential. It's easy enough to mess around with the Schrodinger equation and out pops a nice, simple little wavefunction, say, [tex] \psi (x) = Asin(n\pi x/a) [/tex] for a particle confined between x=0 and x=a. Now apparently this can be used to determine how likely you are to find the particle in a specific position x, if you simply square the wavefunction.

I have never understood where this came from. No text book I have read has explained why this is done or why it works. Also, in the infinite square well example that I have used, it seems to me to give a bit of a dubious result.

When you square this wavefunction, you find that at x=0 and x=a, you get [tex]\psi^2 = 0[/tex]. Assuming that squaring the wavefunction really does give you the probability of the particle's position, then this result of zero probability at the walls of the well makes sense - since how could the particle possibly escape an infinite potential and cross over to this impossible-to-reach region? This much I am happy with.

However, take the n=2 (first excited) eigenstate. At x=a/2 we are going to get [tex]sin^2(\pi) = 0[/tex] which means that we have a zero probability of finding the particle right in the middle of the well. How does that make sense? This does not seem like a sensible result to me. What is wrong with being in the middle of the well when you are at this energy level? Does this not suggest that the probability interpretation of the square of the wavefunction is less than perfect?

Oh, another thing I have been wondering about is to do with Heisenburg's uncertainty principle. I have watched a few quantum documentaries (eg. with that Michio Kaku guy) on the internet and they always go on about quantum particles existing in 2 places at once, but they seem to be aimed at a less-educated audience so they just state their results with no explanation.

The understanding I have about this and the uncertainty principle is that, if you have particle sitting still in some position, you have to bounce a photon off it to detect it. When the photon leaves the particle, the particle recoils under momentum conservation, so by measuring the position of the particle you have altered its momentum - hence the position and momentum cannot both be known with absolute certainty at the same time.

My question about all of this is, are these documentaries being inaccurate/misleading when they say that the particle is in two positions at the same time?

Just because our imperfect method of measuring introduces this uncertainty in a particle's position, surely that doesn't mean the particle literally exists in 2 positions at once, right?

Shouldn't these documentaries really be trying to say that there is a probability of finding the particle exclusively in each position?

Is there really an intrinsic uncertainty in a particle's position, or is it just that human methods of collecting position data introduce this uncertainty that in reality is not there?

I'm not sure if I've explained what I mean here (hence the long, rambling question ^_^) but I appreciate any attempt at an answer.

Thanks.

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# The probability interpretation of the wavefunction?

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