Harmonic Probability Homework Solution

[poobar]

Homework Statement

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)

Homework Equations

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

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.

 

[quenderin]

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.

 

[poobar]

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!