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Probability of some state for oscillator

  1. Dec 26, 2008 #1

    LMZ

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    1. The problem statement, all variables and given/known data
    having a math oscillator, with
    [tex]\varphi(t) = \varphi_0 cos(\frac{2 \pi}{T})t[/tex]
    [tex]T=2 \pi \sqrt{\frac{l}{g}}[/tex]

    find the probability to find [tex]\varphi[/tex] in interval [tex][\varphi, \varphi + d\varphi][/tex]

    2. Relevant equations
    http://en.wikipedia.org/wiki/Liouville's_theorem_(Hamiltonian)


    3. The attempt at a solution
    [tex]dw=\int \rho dp d \varphi[/tex]

    trying using liuville theorem:
    [tex]\frac{\partial}{\partial \varphi}(\rho \dot{\varphi}}) + \frac{\partial}{\partial p}(\rho \dot{p}) = 0[/tex]
    [tex]\frac{\partial \rho}{\partial \varphi} \dot{\varphi}+\rho \frac{\partial \dot{\varphi}}{\partial \varphi} + \frac{\partial \rho}{\partial p} \dot{p}+\rho \frac{\partial \dot{p}}{\partial p} = 0[/tex]
    [tex]\frac{\partial \dot{\varphi}}{\partial \varphi} = 0; \frac{\partial \dot{p}}{\partial p} = m r(sin(\varphi)-cos(\varphi))[/tex]
    [tex]\rho = \frac{2 \frac{\partial \rho}{\partial t}}{m r(sin(\varphi)-cos(\varphi))}[/tex]

    don't know if I'm right till now. Next I think to introduce probability density (rho) in integral for probability (dw), and don't know how to integrate.

    thanks for your time, I appreciate this!
     
  2. jcsd
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