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Quantum Mechanics of wave function and probability rationals

  1. Oct 8, 2009 #1
    1. The problem statement, all variables and given/known data
    The wave function in state n=2 is given: W2(x)=(2/L)^(1/2)sin(2pix/L) with boundaries x=0 and x=L
    at x=L/2, W2(L/2)=0, which means that the probability of finding the particle in a small region about x=L/2 is zero. Nevertheless, there is equal probability to find the particle in the left half of the box as in the right half.
    How is this possible if the particle has no way of passing through the point x=L/2? is this an unresolvable paradox in quantum mechanics?

    2. Relevant equations
    P(x)=integral of (w(x))^2 dx


    3. The attempt at a solution
    I approached this question more in the mathematic path.
    here is what I wrote
    Probability is defined such that dx cannot equal to 0. therefore, even though W2(L/2)=0, it is only the amplitude at x=L/2 which implies x=0. However, dx cannot be 0 if we are looking for the probability, and therefore a non-zero valued dx would have P(x) yield a value that is close to but never 0. therefore, there is a probability of particle passing through point x=L/2 even though it is likely to be very small.

    Can anyone check my answer?
    Thanks alot
     
  2. jcsd
  3. Oct 8, 2009 #2

    jambaugh

    User Avatar
    Science Advisor
    Gold Member

    If I put a ping-pong ball in a box (length L). Shake it around and then put a divider through a slot (at L/2) to split the box into two halves...
    Granted my example actually prevents the ball from crossing the midpoint whereas in the quantum version you quote this is not impossible. Nonetheless the logic of the question is the same. Whether it can cross a point has nothing to do with the probability of it being found on either side.

    Furthermore, in the quantum case the statement "the particle has no way of passing through the point at x=L/2" is not in fact correct. A quantum can cross a point and yet have zero probability of being found at that point (see for example tunneling).
     
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