Asymptotic low-temperature behavior of the classical path probability density for a Morse oscillator

A closed-form asymptotic low-temperature expression for the thermally averaged probability density of an ensemble of Morse oscillators is derived within the framework of a classical path approximation due to Miller. The resulting equation is of the correct quantum form [ρβ(x,x) =exp(−βε0)‖ψ0‖2], where ψ0 differs from the exact ground-state wave function by a simple multiplicative factor. Hence, the normalized probability distribution function converges to the exact result as T→0. The incomplete treatment of quantum effects afforded by the classical path approximation is manifested mostly in the Boltzmann factor, which converges to the pure harmonic result (ε0=ℏω/2).