  After a short review of the quantum mechanics canonically associated with a
classical real valued random variable with all moments, we begin to study the
quantum mechanics canonically associated to the \textbf{standard semi--circle
random variable} $X$, characterized by the fact that its probability
distribution is the semi--circle law $\mu$ on $[-2,2]$. We prove that, in the
identification of $L^2([-2,2],\mu)$ with the $1$--mode interacting Fock space
$\Gamma_{\mu}$, defined by the orthogonal polynomial gradation of $\mu$, $X$ is
mapped into position operator and its canonically associated momentum operator
$P$ into $i$ times the $\mu$--Hilbert transform $H_{\mu}$ on $L^2([-2,2],\mu)$.
In the first part of the present paper, after briefly describing the simpler
case of the $\mu$--harmonic oscillator, we find an explicit expression for the
action, on the $\mu$--orthogonal polynomials, of the semi--circle analogue of
the translation group $e^{itP}$ and of the semi--circle analogue of the free
evolution $e^{itP^2/2}$ respectively in terms of Bessel functions of the first
kind and of confluent hyper--geometric series. These results require the
solution of the \textit{inverse normal order problem} on the quantum algebra
canonically associated to the classical semi--circle random variable and are
derived in the second part of the present paper. Since the problem to
determine, with purely analytic techniques, the explicit form of the action of
$e^{-tH_{\mu}}$ and $e^{-itH_{\mu}^2/2}$ on the $\mu$--orthogonal polynomials
is difficult, % aaa ask T if it is solved the above mentioned results show the
power of the combination of these techniques with those developed within the
algebraic approach to the theory of orthogonal polynomials.
