Lecture
This lecture discusses path integral methods, focusing on the convergence of observables in quantum systems. The instructor begins by addressing the practical question of how many replicas are needed for convergence in path integrals, using a simple harmonic oscillator as a model system. The Hamiltonian for the harmonic oscillator is derived, emphasizing its purely harmonic nature. The lecture explains how to calculate the partition function and the importance of the number of replicas in achieving low error rates in energy calculations. The instructor highlights the need for a sufficient number of replicas to ensure convergence, particularly for properties like potential energy and heat capacity. The discussion then shifts to computational techniques, including ring polymer contraction and high-order path integral approaches, which aim to reduce the computational cost associated with simulating large numbers of replicas. The lecture concludes with practical implications for simulating systems like liquid water, emphasizing the balance between accuracy and computational efficiency.