Quantum vs Classical Birth and Death Processes; Exactly Solvable Examples

TL;DR

This work introduces a coinless quantisation of birth and death processes by mapping the classical generator $L$ to a quantum Hamiltonian $\\mathcal{H}=-\\Phi^{-1}L\\Phi$ using the stationary distribution. It proves that the classical and quantum BD systems share the full spectrum and, when $B(x)$ and $D(x)$ align with Askey-scheme orthogonal polynomials, the eigenvectors are these polynomials and the eigenvalues are analytic, enabling exact, solvable dynamics. The paper catalogs 16 continuous-time and 11 discrete-time explicitly solvable BD examples, many with integer spectra and periodic dynamics, and provides explicit amplitude formulas in terms of orthogonal polynomials, highlighting a deep link between reversible Markov chains, spectral theory, and one-dimensional quantum systems. This framework yields a versatile, parameter-rich toolkit for exact analysis of reversible BD processes and quantum walks, with potential applications in simulation and quantum-classical correspondence.

Abstract

A coinless quantisation procedure of continuous and discrete time Birth and Death (BD) processes is presented. The quantum Hamiltonian H is derived by similarity transforming the matrix L describing the BD equation in terms of the square root of the stationary (reversible) distribution. The quantum and classical systems share the entire eigenvalues and the eigenvectors are related one to one. When the birth rate B(x) and the death rate D(x) are chosen to be the coefficients of the difference equation governing the orthogonal polynomials of Askey scheme, the quantum system is exactly solvable. The eigenvectors are the orthogonal polynomials themselves and the eigenvalues are given analytically. Many examples are periodic since their eigenvalues are all integers, or all integers for integer parameters. The situation is very similar to the exactly solvable one dimensional quantum mechanical systems. These exactly solvable Markov chains contain many adjustable free parameters which could be helpful for various simulation purposes.