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 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 ${\mathrm{\Gamma }}_{\mu }$, 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 $e^{it{P}^2/2}$, respectively, in terms of Bessel functions of the first kind and of confluent hyper-geometric series. These results require the solution of the 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 $e^{-t{H}_{\mu }}$ and $e^{-it{H}_{\mu }^2/2}$ on the $\mu$-orthogonal polynomials is difficult, 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.

在简要回顾了与所有矩的经典实值随机变量规范相关的量子力学之后，我们开始研究与标准半圆随机变量规范相关的量子力学$X$, 特点是其概率分布为半圆定律$\mu$在$[-\mathrm{2个},\mathrm{2个}]$. 我们证明，在识别${\mathrm{大号}}^{\mathrm{2个}}([-\mathrm{2个},\mathrm{2个}],\mu )$与$\mathrm{1个}$-mode 交互 Fock 空间${\mathrm{\Gamma }}_{\mu }$, 由正交多项式等级定义$\mu$,$X$映射到位置算子及其规范关联的动量算子$P$进入$我$倍$\mu$-希尔伯特变换$H_{\mu }$在${\mathrm{大号}}^{\mathrm{2个}}([-\mathrm{2个},\mathrm{2个}],\mu )$. 在本文的第一部分，在简要描述了更简单的情况之后$\mu$-谐波振荡器，我们找到一个明确的动作表达式，在$\mu$-平移群的半圆类似物的正交多项式${\mathrm{电子}}^{我吨P}$和自由进化的半圆模拟${\mathrm{电子}}^{我吨P^{\mathrm{2个}}/\mathrm{2个}}$，分别根据第一类贝塞尔函数和合流超几何级数。这些结果需要解决与经典半圆随机变量正则相关的量子代数的逆正规阶问题，并在本文的第二部分中推导出来。由于问题是用纯粹的分析技术来确定动作的显式形式${\mathrm{电子}}^{-吨H_{\mu }}$和${\mathrm{电子}}^{-我吨H_{\mu }^{\mathrm{2个}}/\mathrm{2个}}$在$\mu$- 正交多项式是困难的，上述结果显示了这些技术与在正交多项式理论的代数方法中开发的技术相结合的力量。