Next: Imaginary Time Path Integrals Up: Path Integral Monte Carlo Previous: Path Integral Monte Carlo   Contents

# The Thermal Density Matrix

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 position-space 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,
 (16) (17)

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 k-vector , where is a -dimensional integer vector. Hence,
 (19) (20) (21)

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)

 (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 inter-particle 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   Contents
Burkhard Militzer 2003-01-15