Density Matrix

density matrix [′den·səd·ē ′mā·triks]

(quantum mechanics)

A matrix ρ_mndescribing an ensemble of quantum-mechanical systems in a representation based on an orthonormal set of functions φ_n;for any operatorGwith representationG_mn, the ensemble average of the expectation value ofGis the trace of ρG.

---

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

(1) 

the hamiltonian of the system, and ħ is Planck's constant divided by 2π. Furthermore, ψ(x, t) may be expanded in terms of a complete orthonormal set {ϕ(x)}, as in Eq. (2). Then, the

(2) 

(3) 

density matrix is defined by Eq. (3), and this density matrix describes a pure state. Examples of pure states are a beam of polarized electrons and the photons in a coherent beam emitted from a laser. 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).

(4) 

(5) 

In general, this density matrix describes a mixed state, for example, a beam of unpolarized electrons or the photons emitted from an incoherent source such as an incandescent lamp. The pure state is a special case of the mixed state when all members of the ensemble are in the same state. 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 (seeMIXED STATE).