Quantum interpretation of lattice paths

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.