# Density Matrix

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## Density matrix

A matrix which is constructed as the most general statistical description of the states of a many-particle quantum-mechanical system. The state of a quantum system is described by a normalized wave function ψ(*x, t*) [where *x* stands for all coordinates of the system, and *t* for the time], which satisfies the Schrödinger equation (1), where *H* is

*x, t*) may be expanded in terms of a complete orthonormal set {ϕ(

*x*)}, as in Eq. (2). Then, the

*See*Laser, Quantum mechanics

In quantum statistics, one deals with an ensemble of *N* systems which have the same hamiltonian. If the αth member of the ensemble is in the state ψ^{α} in Eq. (4), the density matrix is defined as the ensemble average, Eq. (5).

*See*Statistical mechanics

## Density Matrix

an operator by means of which it is possible to calculate the average value of any physical quantity in quantum statistical mechanics and, in particular, in quantum mechanics. A density matrix describes a system’s state based on an incomplete set (incomplete in terms of quantum mechanics) of data on the system (*see*MIXED STATE).

## density matrix

[′den·səd·ē ′mā·triks]*describing an ensemble of quantum-mechanical systems in a representation based on an orthonormal set of functions φ*

_{mn}*for any operator*

_{n};*G*with representation

*G*, the ensemble average of the expectation value of

_{mn}*G*is the trace of ρ

*G*.