The Monte-Carlo technique for the calculations of functional integral in two one-dimensional quantum-mechanical problems had been applied. The energies of the bound states in some potential wells were obtained using this method. Also some peculiarities in the calculation of the kinetic energy in the ground state had been studied.
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 neutrons 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 systems; 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 techniques 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 called diffusion Monte Carlo (DMC).
The feasibility of using Monte Carlo methods in finite fermion systems has been investigated. The basic tool is a (recently suggested) functional integral representation of $Z=\mathrm{tr}[\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}II)]$ using real coherent states of Slater determinants. Conditions for using the discretized (N-point lattice) version as a starting point for Monte Carlo integrations are studied. The limit of large $\ensuremath{\beta}$ is achieved only with large $\ensuremath{\surd}N$. For a two-fermion system numerical calculations of the ground-state energy agree farily well with the exact value.
We present novel Monte Carlo methods for treating the interacting shell model that allow exact calculations much larger than those heretofore possible. The two-body interaction is linearized by an auxiliary field; Monte Carlo evaluation of the resulting functional integral gives ground-state or thermal expectation values of few-body operators. The ``sign problem'' generic to quantum Monte Carlo calculations is absent in a number of cases. We discuss the favorable scaling of these methods with nucleon number and basis size and their suitability to parallel computation.
We have recently developed a formalism for carrying out Monte Carlo calculations on systems with coupled boson-fermion degrees of freedom. Here we present the results obtained using these techniques on some specific systems. First the quantum-mechanics problem of a single boson coordinate linearly coupled to a fermion is studied. The fermion Green's function and the boson coordinate displacement are calculated and compared with exact results. Extending this analysis to a one-dimensional array, a field theory in one space and one imaginary time dimension is investigated. The fermion and boson Green's functions are calculated and compared with analytic results obtained in the limit where the bare boson frequency is much larger or much smaller than the characteristic fermion energies.
We present results for a variety of Monte Carlo annealing approaches, both classical and quantum, benchmarked against one another for the textbook optimization exercise of a simple one-dimensional double well. In classical (thermal) annealing, the dependence upon the move chosen in a Metropolis scheme is studied and correlated with the spectrum of the associated Markov transition matrix. In quantum annealing, the path integral Monte Carlo approach is found to yield nontrivial sampling difficulties associated with the tunneling between the two wells. The choice of fictitious quantum kinetic energy is also addressed. We find that a ``relativistic'' kinetic energy form, leading to a higher probability of long real-space jumps, can be considerably more effective than the standard nonrelativistic one.