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A coinless quantisation procedure of continuous and discrete time Birth andDeath (BD) processes is presented. The quantum Hamiltonian H is derived bysimilarity transforming the matrix L describing the BD equation in terms of thesquare root of the stationary (reversible) distribution. The quantum andclassical systems share the entire eigenvalues and the eigenvectors are relatedone to one. When the birth rate B(x) and the death rate D(x) are chosen to bethe coefficients of the difference equation governing the orthogonalpolynomials of Askey scheme, the quantum system is exactly solvable. Theeigenvectors are the orthogonal polynomials themselves and the eigenvalues aregiven analytically. Many examples are periodic since their eigenvalues are allintegers, or all integers for integer parameters. The situation is very similarto the exactly solvable one dimensional quantum mechanical systems. Theseexactly solvable Markov chains contain many adjustable free parameters whichcould be helpful for various simulation purposes.

Quantum vs Classical Birth and Death Processes; Exactly Solvable Examples