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Total Randomness or Not?

  1. Feb 4, 2009 #1
    Hi, I have a quick question regarding the working of quantum physics. As I understand it, it's not possible to be 100% sure of a particle's location until we look at it. The question is though: how unsure are we of this? In other words, how many possible positions could the particle be in until we look at it - is it a question of millions, trillions, or, an infinite number of positions? Is there some constraint to the number of positions the particle could be in or are there no laws in the quantum world - is it all totally random?
     
  2. jcsd
  3. Feb 4, 2009 #2
    Hi. Actually as I understood your question is about the interpretation of wave function in quantum physics.

    Let's consider a single particle which can move only in one direction for example. The motional state of the particle can be described giving a wave function Psi(x,t) at a certain time t. This is a function of time and space. You interprete this saying that if you measure the position X of the particle at time t, the probability that you get result x0 is proportional to Psi(x0,t).

    Anyway there can be imagined situations in which the wave function is peaked around one particular point x1. Thus the measure of the position practically gives always the value x1. This happens we say when the particle is in an eigenstate of the position operator of eigenvalue x1. In general however the particle is not in an eigenstate of the position (this means that the particle has not the property: its position is this). But it doesn't mean neither that the particles is in several positions at the same time. This is a completely wrong statement. You simply do not know where it is until you measure it. People say sometimes in that state the particle does not possess the element of reality called position.

    After you measure you know the position with some uncertainty given by your measure apparatus of course.
     
  4. Feb 4, 2009 #3
    Before measurement, the particle can be nearly anywhere. The distribution however is not totally random, instead it is given by the particle's wavefunction. The wavefunction can be zero on certain locations (such as the edges of an infinitely deep well) but it is generally spread out to infinity. However, the chance of finding the particle (given by the absolute square of the wavefunction) is will generally be much larger at 'more probable' positions.

    If you look at a particle in a (not infinitely large) box for example, the chance of finding it in the center is much larger than finding it at the edges (assuming it is in the ground state). However, it can be found at the edges (the chance is not zero for a real, non infinite box), and it can even be found outside the box even though it has not got enough energy to 'jump out of' the box (this is called tunneling).

    The wavefunction is basically just a measure of the probability of finding the particle when you measure it. It doesn't tell you anything about the actual position before or after you measured it. However, when you do measure the particle, one interpretation of QM is that the wavefunction 'collapses' into a sharp spike (at the location where you found the particle). It soon spreads out again however, but if you measure the particle a second (third, fourth...) time quickly enough, you are most probable of finding it roughly at the same location as before, because the wavefunction is nearly zero everywhere except where you found it before!


    So in other words, there are contraints on the position of the particle, but (generally, not always) it is true that there are an infinite number of possible positions (it can be anywhere). It is just that it is very very unlikely to find it at places where the wavefunction is very small.
     
  5. Feb 4, 2009 #4
    I think the question is about the precision of the particle location rather than a general question. Will the position be to an infinite number of decimal places? The mathematics makes usual convergence assumptions.
     
  6. Feb 4, 2009 #5
    According to Heisenbergs uncertainty principle, a particle never has an exactly defined location. Even if you could measure the particle with an infinitely precise apparatus, you would not know it's exact location. Obviously, that single measurement would yield one definite answer, but if you had multiple particles, all in exaclty the same state, and you would measure the position of each particle with your infinitely accurate apparatus, you would still get different answers.
     
  7. Feb 4, 2009 #6
    I do not understand what do you mean that if you measure the position with arbitrary precision you can never say the particle IS in the position you measured.
     
  8. Feb 4, 2009 #7
    What the first quote should say is that even if you had an infinitely precise apparatus....you would still not be able to measure it's exact location. In other words, the inability to get a precise at an exact location is a function of the ambiguity of the particle itself, not the exactness of the apparatus.

    A way to think about this is to imagine an electron, say, in ever tighter confinement....as the confinement collapses, the motion of the particle becomes increasingly energetic..it just won't cooperate and sit still for a nice measure.
     
  9. Feb 4, 2009 #8

    ZapperZ

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    That is not correct. For example, I can measure as accurately as I want the position of any electrons, and the "uncertainty" here is only limited by the technology of the instrument, not by the HUP. Why? Because I can continue to wait for technology to advance and my uncertainty of that position will continue to improve, no matter where that electron came from or what it was doing before I measure it.

    What most people misunderstand about the HUP is that it is not the accuracy of what one measure in a single measurement. That isn't the HUP, and one look at it one can see that it involves not only a statistical set of measurements (not just one), but also our ability to predict the next one. It involves what happened before the measurement in terms of the non-commuting observable, i.e. if you've made an accurate determination of a position, then your ability to predict its momentum becomes fuzzier. It tells you nothing about the accuracy of a single measurement, be it of the position, or the momentum. This has nothing to do with the HUP but rather the accuracy of the instrument.

    Zz.
     
  10. Feb 4, 2009 #9
    So it's true that you can measure the position of a particle to infinite precision (assuming you had the technology). But if you try to make the same measurement over and over (on many particles in the same state, not one particle repeadetely), you don't get that same position each time?
    (That's basically what I was trying to say, did I get that correct?)
     
  11. Feb 4, 2009 #10
    Thanks for the answers. I have thought about the implications of this and have come up with another couple of questions:

    1. Given that one has measured the particle's position at time t = 0, then at time t = 1 the particle's position would be very likely to be at some point close to the position it was at time t = 0 but it could however (despite its very remote probability) be at another point very far removed. Now, can this work for instance in a particle in one's body? In other words, could a particle in one's body be measured at time t = 0 to be in one's body and time t = 1 outside of one's body? Similarly, could particles outside of one's body suddenly be floating somewhere in the organs?

    2. In quantum mechanics, can particles travel faster than the speed of light?

    Any insights would be very much appreciated.
     
  12. Feb 4, 2009 #11

    ZapperZ

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    Not getting the same position each time is not identical to saying each single measurement can't be as accurate as possible. A spread in "x" does not imply a spread in "Delta(x)".

    Zz.
     
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