Abstract
The Lie--Trotter product formula is a foundational approximation for the quantum partition function, yet obtaining rigorous error bounds for the unbounded Hamiltonians common in physics remains a significant challenge. This paper provides a quantitative error analysis for this approximation across two key systems. For a particle in a smooth, periodic potential, we establish an optimal convergence rate of O(1/N²) for both the partition function and thermal averages, where N is the number of imaginary time steps. We then extend this analysis to the more challenging case of a confining potential on R, proving a nearly optimal rate of O((log N+1)^3/2/N²). The derived error bounds provide a firm mathematical foundation for the high-order accuracy of path integral simulations in quantum statistical mechanics.