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A quantum mechanical system in a pure state can be described by single
wave function
, which can be expressed in terms
of eigenvalues and eigenfunctions
of the
Hamiltonian . The corresponding density matrix operator is given by,

(10) 
The density matrix provides a convenient way to extend the study to
finite temperature. Following the principles of statistical mechanics,
one puts the system in contact with a heat bath and assigns classical
probabilities to the quantum mechanical states
, which leads to a thermal density matrix,

(11) 
In the canonical ensemble at temperature , the probabilities are
proportional to the Boltzmann factor
, where
. The density matrix now reads,

(12) 
The expectation of any operator is given by,

(13) 
and the canonical partition function is
. For sake of numerical simulations, it is convenient to change to a
positionspace representation. Introducing the set of coordinates for
a system of particles in dimensions
, the density matrix becomes,

(14) 
For any hermitian Hamiltonian the density matrix is symmetric in its two arguments,

(15) 
The expectation value is given by,
For a free particle in a periodically repeated box of size and volume
, the
density matrix can derived from the exact eigenfunctions of the
Hamiltonian given by plane waves,

(18) 
with kvector
, where is a dimensional integer vector. Hence,
where
for a particle of mass
. Alternatively, this solution can be derived from the Bloch
equation,

(22) 
which is a diffusion equation in imaginary time . The initial condition
is provided by the known high temperature limit,

(23) 
For free particles, the Bloch equation simply reads,

(24) 
The operator can be applied either to the first or to the
second argument in
. In any case, this equation
describes the diffusion of paths in imaginary time. The exact solution
is given by Eq. 2.11. The diffusion constant
is determined by the mass of the particle. It is large for
light particles leading to a fast diffusion in imaginary time, and
small for heavy and therefore classical particles. The width of the
density matrix is given by
, which is related to the frequently used
thermal de Broglie wave length defined as,

(25) 
If this length reaches the order of the interparticle spacing, many
body quantum effects become important. This relation is usually
discussed in terms of the degeneracy parameter,

(26) 
which relates the volume per particle
to the
volume occupied by an individual path . This parameter
defines a temperature scale for the emergence of quantum statistical
effects such as Bose condensation in Bosonic systems and the formation
of a Fermi surface in Fermion systems. The latter process will be
discussed in detail in sections
2.6 and
5.3.
Next: Imaginary Time Path Integrals
Up: Path Integral Monte Carlo
Previous: Path Integral Monte Carlo
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Burkhard Militzer
20030115