# Quantum pendulum

1. Oct 3, 2011

### jackdamiels

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
$$\dot{\dot{\theta}} = \omega^2\theta$$,
and the solution is in form of :

$$x (t) = A\cosh(\omega t) + B\sinh (\omega t)$$.

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
$$(\delta x )^2 = <(x - <x>)^2 >$$
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 :
$$1/{\sqrt{\sigma{sqrt{2\pi}e^{ip_0 x}e^{-\frac{(x-x_0)^2}{4\sigma^2}$$.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution