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

(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 ψα 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).

density matrix

[′den·səd·ē ′mā·triks]
(quantum mechanics)
A matrix ρ mn describing an ensemble of quantum-mechanical systems in a representation based on an orthonormal set of functions φ n; for any operator G with representation Gmn, the ensemble average of the expectation value of G is the trace of ρ G.
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1] means tracing over the first density matrix while [Tr.
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The density matrix of the setup is a pure type one, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.
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