  In the 1980s, Viennot developed a combinatorial approach to studying mixed
moments of orthogonal polynomials using Motzkin paths. Recently, an alternative
combinatorial model for these mixed moments based on lecture hall paths was
introduced in arXiv:2311.12761. For sequences of orthogonal polynomials, we
establish here a bijection between the Motzin paths and the lecture hall paths
via a novel symmetric lecture hall graph. We use this connection to calculate
the moments of the position operator in various separable quantum systems, such
as the quantum harmonic oscillator and the hydrogen atom, showing that they may
be expressed as generating functions of Motzkin paths and symmetric lecture
hall paths, thereby providing a quantum interpretation for these paths. Our
approach can be extended to other quantum systems where the wavefunctions are
expressed in terms of orthogonal polynomials.
