Density matrix
of a state defined on the algebra
of bounded linear operators acting on a Hilbert space
The positive nuclear operator such that
![]() | (1) |
where . Conversely, any state
, i.e. any linear positive
normalized
functional on
, can be represented in the form (1), i.e. it has a density matrix
, which is moreover unique.
The concept of a density matrix arose in statistical physics in defining a Gibbs quantum state. Let a quantum system occupying a finite volume in
be described by the vectors of a certain Hilbert space
, by the Hamiltonian
and, possibly, by some set of mutually commuting "first integrals"
,
. A Gibbs state for such a system is a state on
defined by the density matrix
![]() | (2) |
where is a normalizing factor and
,
are real parameters.
In addition to the density matrix (2), the state of a system in quantum statistical physics may be defined by means of the so-called reduced density matrix. In the simplest case of a system of identical particles (bosons or fermions) described by the vectors of a Fock [Fok] space , the reduced density matrix
of a state
is the set of (in general, generalized) functions
![]() | (3) |
![]() |
where
![]() |
and where ,
, are the creation operators and annihilation operators, respectively, acting in
. If the creation and annihilation operators in
are replaced by some other system of generators
(
is a certain set of indices), then the reduced density matrix
for a state
is defined by analogy with (3) as the set of values of
on all possible monomials of the form
![]() |
The reduced density matrix is particularly convenient for determining the limiting Gibbs state in a -algebra
of so-called quasi-local observables:
(the bar denotes closure in the uniform topology).
References
[1] | L.D. Landau, E.M. Lifshitz, "Statistical physics" , Pergamon (1980) (Translated from Russian) |
[2] | D. Ruelle, "Statistical mechanics: rigorous results" , Benjamin (1974) |
Comments
The statement that any state has a representation (1) has been proved for finite-dimensional
only.
The functions defined by (3) are the quantum analogues of distribution functions.
Density matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Density_matrix&oldid=46627