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Quantum pendulum

  1. Oct 3, 2011 #1
    Hy

    I am trying to solve problem of quantum pendulum in region of unstable equilibrium.
    I am doing it in Heiseberg interpretation of QM. The equation of motion that I am getting is
    [tex] \dot{\dot{\theta}} = \omega^2\theta [/tex],
    and the solution is in form of :

    [tex] x (t) = A\cosh(\omega t) + B\sinh (\omega t) [/tex].

    With some starting connditions I can get A i B, that is simple. But problem arose when I am computing standard deviations od for example
    [tex] (\delta x )^2 = <(x - <x>)^2 >[/tex]
    I am getting imaginary numbers, and time dependence. Time dependence is OK, because it is Heisenberg picture, but whay imagenery part in this standard deviations. State is :
    [tex] 1/{\sqrt{\sigma{sqrt{2\pi}e^{ip_0 x}e^{-\frac{(x-x_0)^2}{4\sigma^2}[/tex].
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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