12 Quantum Monte Carlo methods
12.1 Introduction
In Chapters 1 to 4 we studied methods for solving the Schrödinger equation for
many-electron systems. Many of the techniques described there carry over to other
quantum many-particle systems, such as liquid helium, and the protons and neut-
rons in a nucleus. The techniques which we discussed there were, however, all of
a mean-field type and therefore correlation effects could not be taken into account
without introducing approximations. In this chapter, we consider more accurate
techniques, which are similar to those studied in Chapter 10 and are based on using
(pseudo-)random numbers – hence the name ‘Monte Carlo’ for these methods.
In Chapter 10 we applied Monte Carlo techniques to classical many-particle sys-
tems; here we use these techniques for studying quantum problems involving many
particles. In the next section we shall see how we can apply Monte Carlo tech-
niques to the problem of calculating the quantum mechanical expectation value of
the ground state energy. This is used in order to optimise this expectation value by
adjusting a trial wave function in a variational type of approach, hence the name
variational Monte Carlo(VMC).
In the following section we use the similarity between the Schrödinger equation
and the diffusion equation in order to calculate the properties of a collection of
interacting quantum mechanical particles by simulating a classical particle diffusion
process. The resulting method is calleddiffusion Monte Carlo(DMC).
Then we describe the path-integral formalism of quantum mechanics, which is
a formulation elaborated by Feynman, based on ideas put forward by Dirac[1],
in which a quantum mechanical problem is mapped onto a classical mechanical
system (at the expense of increasing the number of degrees of freedom). This
classical many-particle system can then be analysed using methods similar to those
used inChapter 10. This is called thepath-integral Monte Carlomethod (PIMC).
The last section of this chapter is dedicated to a stochastic technique, based on
diffusion Monte Carlo, for diagonalising the transfer matrix of a lattice spin model
372