Expectation values

  • Thread starter Pacopag
  • Start date
1. The problem statement, all variables and given/known data
A particle moves in a sequence of steps of length L. The polar angle [tex]\theta[/tex] for each step is taken from the (normalized) probability density [tex]p(\theta)[/tex]. The azimuthal angle is uniformly distributed. Suppose the particle makes N steps.
My question is how do I find the expectation value (say [tex]<z^2>[/tex] for example).

2. Relevant equations
Usually for a probability density p(x) we have
[tex]<x^m>=\int x^m p(x) dx[/tex].

3. The attempt at a solution
I think that I can get the values for one step. eg.
[tex]<z^2>=\int_0^\pi (Lcos(\theta))^2p(\theta)d\theta={L^{2}\over 2}[/tex]
Note: the density [tex]p(\theta)[/tex] is normalized.
I just don't know how to treat N steps. Do I just multiply the one-step result by N?
What is [tex]p(\theta)[/tex]? Is it given?
Oh ya. Sorry. It is
[tex]p(\theta) ={2 \over \pi}cos^2({\theta \over 2})[/tex]

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