Chapter 14
Quantum Monte Carlo Methods
If, in some cataclysm, all scientific knowledge were to be destroyed, and only one sentence passed on
to the next generation of creatures, what statement would contain the most information in the fewest
words? I believe it is the atomic hypothesis (or atomic fact,or whatever you wish to call it) that all
things are made of atoms, little particles that move around in perpetual motion, attracting each other
when they are a little distance apart, but repelling upon being squeezed into one another. In that one
sentence you will see an enormous amount of information about the world, if just a little imagination
and thinking are applied.Richard Feynman, The Laws of Thermodynamics.AbstractThe aim of this chapter is to present examples of applications of Monte Carlo meth-
ods in studies of simple quantum mechanical systems. We study systems such as the harmonic
oscillator, the hydrogen atom, the hydrogen molecule and the helium atom. Systems with
many interacting fermions and bosons such as liquid^4 He and Bose Einstein condensation of
atoms are discussed in chapters 16 and 17.
14.1 Introduction
Most quantum mechanical problems of interest in for exampleatomic, molecular, nuclear and
solid state physics consist of a large number of interactingelectrons and ions or nucleons.
The total number of particlesNis usually sufficiently large that an exact solution cannot be
found. In quantum mechanics we can express the expectation value of a given operatorÔfor
a system ofNparticles as
〈Ô〉=
∫dR 1 dR (^2) ∫...dRNΨ∗(R 1 ,R 2 ,...,RN)Ô(R 1 ,R 2 ,...,RN)Ψ(R 1 ,R 2 ,...,RN)
dR 1 dR 2 ...dRNΨ∗(R 1 ,R 2 ,...,RN)Ψ(R 1 ,R 2 ,...,RN)
, (14.1)
whereΨ(R 1 ,R 2 ,...,RN)is the wave function describing a many-body system. Although we
have omitted the time dependence in this equation, it is an ingeneral intractable problem.
As an example from the nuclear many-body problem, we can write Schrödinger’s equation as
a differential equation with the energy operatorĤ(the so-called Hamiltonian) acting on the
wave function as
ĤΨ(r 1 ,..,rA,α 1 ,..,αA) =EΨ(r 1 ,..,rA,α 1 ,..,αA)
where
r 1 ,..,rA,
are the coordinates and
α 1 ,..,αA,
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