What most physicists refer to as "particle" is very different from what lay people understand by that term. If you ask physicists at the LHC about particles, they will explain what I've already mentioned, because what I'm saying is far from being revolutionary.
You can count individual quanta of any kind (photons, electrons, etc.), and you can measure their quantum collapse. But that does not mean they are localized "particles" the way Dirac liked to think about them.
Of course you can do that, if we couldn't detect state collapses then they wouldn't be a staple of quantum mechanics. If you measure a rotating wave function over and over then it wont rotate since it will collapse into the same state again, while if you let it be it can rotate into another state giving you another measurement result.
This works since rotations aren't linear, small rotations are quadratic and hence will almost always result in the original state. You can also use this technique to rotate a state by making many measurements slowly changing the axis, so each measurement results in a small rotation.
Edit: But you are right that we can't see the history of state collapses, but they are definitely required for our current theories to work as you get the wrong experimental results without them in the theory.
>> Of course you can do that, if we couldn't detect state collapses then they wouldn't be a staple of quantum mechanics.
No. If you shoot pairs of entangled particles in opposite directions, someone receiving one stream of particles can take or not take measurements thereby collapsing or not collapsing the wave function of the particles going in the other direction. If you could tell the difference between a particle with a collapsed wave function and one without, this could be used for FLT communication. Bottom line is we can't tell if a wave function is "collapsed" or not. It's not a real event.
> if you let it be it can rotate into another state giving you another measurement result.
But then following your prior reasoning, that's just another collapse. So if the only way to measure is to collapse then pmkahler is right: there is no way to discern a collapsed wave function from a non collapsed one.
> But then following your prior reasoning, that's just another collapse.
But it isn't random, if we know how fast it rotates then the second time we measure it we can get a close to exact result.
The most famous experiment for this is the double slit experiment. Normally when you fire particles through you will get an interference pattern on the other side since the particles passes through like a wave. Measuring it in one will collapse the wavefunction and therefore destroying the inference pattern, so now the particles mostly just travels straight and creates a distribution of hits as if it passed through just a single slit.
Edit: Can look at this picture from wikipedia showing the difference between single and double slit, just measuring at one of the slits will even cause particles passing through the other slit to go much straighter.
You can count individual quanta of any kind (photons, electrons, etc.), and you can measure their quantum collapse. But that does not mean they are localized "particles" the way Dirac liked to think about them.