  We present a numerical study of the quantum action previously introduced as a
parametrisation of Q.M. transition amplitudes. We address the questions: Is the
quantum action possibly an exact parametrisation in the whole range of
transition times ($0 < T < \infty$)? Is the presence of potential terms beyond
those occuring in the classical potential required? What is the error of the
parametrisation estimated from the numerical fit? How about convergence and
stability of the fitting method (dependence on grid points, resolution, initial
conditions, internal precision etc.)? Further we compare two methods of
numerical determination of the quantum action: (i) global fit of the Q.M.
transition amplitudes and (ii) flow equation. As model we consider the inverse
square potential, for which the Q.M. transition amplitudes are analytically
known. We find that the relative error of the parametrisation starts from zero
at T=0 increases to about $10^{-3}$ at $T=1/E_{gr}$ and then decreases to zero
when $T \to \infty$. Second, we observe stability of the quantum action under
variation of the control parameters. Finally, the flow equation method works
well in the regime of large $T$ giving stable results under variation of
initial data and consistent with the global fit method.
