Monte Carlo Methods and its applications in quantum mechanics

Abstract

Accurately determining the ground state properties of quantum systems, especially in the absence of analytical solutions, remains a central challenge in computational physics. In this work, we explore Monte Carlo methods, based on stochasting sampling, to study analytically solvable models to validate the method. Then we apply it to more complex systems, investigating how Monte Carlo method handles physical proprieties, singularities, and simulation parameters tuning. We begin with the Variational Monte Carlo (VMC) method, which provides an upper bound to the ground state energy through stochastic evaluation of expectation values. Sampling is performed using both the Metropolis and Metropolis–Hastings algorithms. To optimize the trial wave function, we apply gradient-based minimization and particle swarm optimization (PSO), targeting either the energy or the variance, depending on the energy landscape. To reduce the dependence on the quality of the trial wave function, we then introduce the Diffusion Monte Carlo (DMC) method, which projects out the ground state by simulating the imaginary-time evolution governed by the Green’s function of the Hamiltonian. We apply DMC to various systems; the harmonic oscillator, the Morse potential, and the hydrogen atom, which exhibits a divergence at the origin. We analyze the convergence of DMC with respect to key parameters such as the time step ∆τ and the number of walkers, highlighting the method’s accuracy, stability, and sensitivity to these choices. Finally we expand by introducing importance sampling in the framework of DMC, which can be done by incorporating a guiding function implemented using the Metropolis algorithm, this procedure is often referred to by hybrid Monte Carlo. Then, on we give an insight on the fixed node approximation, a necessary procedure to address the fermionic sign problem in DMC. The results confirm the effectiveness of quantum Monte Carlo techniques in capturing the essential physics of both simple and interacting quantum systems.