文档介绍：Lecture Notes, Como 1995 September 6, 1998
Quantum Theory
HANS DE RAEDT
Institute for Theoretical Physics and Materials Science Centre
University of Groningen, h 4
NL-9747 AG Groningen, herlands
E-mail: ******@
ABSTRACT
The purpose of this set of lectures is to introduce the general concepts that are
at the basis of puter simulation algorithms that are used to study the be-
havior of condensed matter quantum systems. The emphasis is on the underlying
concepts rather than on specific applications. Topics treated include exact diago-
nalization techniques, variational methods, Trotter-Suzuki formulae, and Feynman
path integrals. Some of the concepts covered are illustrated in an application to
electron-phonon models.
1. Introduction
There is a broad consensus that the Schr¨odinger equation for the time-dependent
wave function |Φ(t)i
∂
i¯h |Φ(t)i = H|Φ(t)i , (1)
∂t
and the expression for the thermal expectation value of an observable X
Tr e−βHX
hXi = , (2)
Tr e−βH
describe the dynamic and thermodynamic properties of a system, modelled by a
Hamiltonian H, that is in thermal equilibrium at an inverse temperature β=
1/kBT . Therefore the rules for formulating the problem are fixed. What remains
is to find ways of solving (1) puting (2).
For classical many-body systems, we can use Molecular Dynamics or Monte
Carlo methods to study the model behavior. There are almost no restrictions on
the kind of classical models that can be treated in this manner. Unfortunately,
for quantum systems the situation is not as good as in the classical case. From
numerical point of view, the only “complication” is that, one way or another, quan-
—1—
tum mechanics is a theory that is formulated in terms of objects (operators, or,
for our purposes, matrices) that do not mute. plication
has tremendous consequences for it makes the construction and justification of al-
gorithms for simulating quantum systems a difficult task. It is