Quantum Monte Carlo

Intro

Apply monte carlo techniques to solve quantum mechanical calculations such as the finding the ground state of a quantum system. Monte carlo methods are used to solve the high-dimensional integrals of quantum problems.

• Variational monte carlo: Use a parametrized trial function and optimize it
• Diffusion monte carlo: Exploit the similarity between the Schrödinger equation and a diffusion equation, using monte carlo methods to simulate a classical diffusion
• Path-integral monte carlo: Path integrals map a quantum problem onto a classical system with an increased number of dimensions. The classical many-particle system can be analyzed using monte carlo integration.

Variational monte carlo methods

Use a variational method to find the ground state of a Hamiltonian. Use a trial function with a parameter or multiple parameters and optimize them such that the total energy is at its minimum. To calculate the expectation value of the energy, a high dimensional integral has to be solved for many-body systems. Use monte carlo integration for this calculation.

Variational method

1. Construct a trial wave function $\psi_\alpha(R)$, which depends on the variational parameters $/alpha = (\alpha_1, ..., \alpha_N)$ and the space coordinates $R$ of all particles of the system.
2. Evaluate the expectation value of the energy for a specific $\alpha$ by employing $$ \langle E \rvert = \frac{\langle\psi_\alpha \rvert H \lvert \psi_\alpha\rangle}{\langle \psi_\alpha\rvert\psi_\alpha \rangle} $$
3. Vary the parameters $\alpha$ and find the value that minimize the expectation value, corresponding to the ground state.