Wavefunction Graph

This graph shows the (unnormalized) $\alpha = \text{constant}$ slice of the mixmaster wavefunction $\Omega \left( \alpha, \beta_{\pm} \right) = e^{-S \left( \alpha, \beta_{\pm} \right)}$, over the $\beta_{+} \beta_{-}$-plane. The animation runs over values of $\alpha$ from $-2$ to $2$.

Calculating the normalization constant $\mathcal{N} = \mathcal{N} \left( \alpha \right)$ for each slice such that

\begin{align} \int_{-\infty} ^{\infty} \int_{-\infty} ^{\infty} \mathcal{N} ^{2} e^{-2S \left( \alpha , \beta_{\pm} \right)} d \beta_{+} d \beta_{-} = 1 \end{align}

by a Monte Carlo routine yields the following values of $\mathcal{N}$, plotted as a function of $\alpha$:

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