Quantum interpretation of lattice paths

TL;DR

This paper establishes a quantum interpretation of lattice paths by connecting Motzkin and lecture hall paths through a new graph, enabling the calculation of quantum system moments via combinatorial models.

Contribution

It introduces a bijection between Motzkin and lecture hall paths using a symmetric graph, linking combinatorics with quantum physics for the first time.

Findings

Moments of quantum systems are expressed as generating functions of lattice paths.

The bijection simplifies calculations of quantum moments for harmonic oscillator and hydrogen atom.

The approach extends to other quantum systems with orthogonal polynomial wavefunctions.

Abstract

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.