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Harmonic Probability

  1. Oct 21, 2010 #1
    1. The problem statement, all variables and given/known data

    A particle oscillates with simple harmonic motion along the x-axis with a displacement amplitude, a, and spends a time, dt, in moving from x to x+dx. Show that the probability of finding it between x and x+dx is given by P(x) = dx/(pi(a^2 - x^2)^1/2)

    2. Relevant equations

    x = asin(wt)
    v = wacos(wt)

    3. The attempt at a solution

    I'm really kind of lost on how to begin this one. I know I have to set up some sort of integral involving the time it takes to go from x to x+dx which would be dt. I know that to get a probability function you've got to normalize the integral. If someone could point me in the right direction, it would be much appreciated.

    P.S. I know there is another thread on this, but I started a new one because I don't really understand the other thread. Thanks.
  2. jcsd
  3. Oct 21, 2010 #2
    You've got the right idea. You need an expression for dx in terms of dt, because dt is the measure of probability: the more time spent in that interval x and x+dx, the greater the probability of finding the particle at x.

    You listed v = aw cos wt, but v = dx/dt. So this looks like a good starting point. Next thing to think about: if the particle spends time dt between x and dx, what probability does this correspond to? We know that within half a period, the particle has been through every possible location between -a and a only once, and then the motion repeats. The total probability of the particle being between -a and a within this half a period is 1, so what is the probability corresponding to the time interval dt?

    So to summarise, begin with v = dx/dt, and use the fact that dt has a relation to probability, and perform a substitution along the way to get the right expression.
  4. Oct 21, 2010 #3
    okay, so dx = awcos(wt)dt
    also, the probability for dt will be 2 i think.

    i believe i will need 1 = 2dt/T because there is a probability of two for the entire period since it goes through each position twice. then i make some substitutions. dt = dx/awcos(wt). i can pretty much see how i have to do this now. it makes a lot more sense now. thanks!
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