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.
References in periodicals archive ?
As a first example we consider the matrix function known as Fermi-Dirac density matrix (FDM), which is fundamental to computational quantum chemistry:
1 shows the rapid decay in the Fermi-Dirac density matrix corresponding to a "one-dimensional Anderson model"; see [1].
1 we show results for the approximation of the Fermi-Dirac density matrix for different values of n, [mu], and [beta].
Only couples of Georgiev's states can be collectively zeroed, but which members will enter in the zeroed couples depends on the density matrix of the setup.
Indeed the analysis of the proposed here Georgiev's four-slit experiment, as well as the analysis of Unruh's and Afshar's setups, show that which way claims defined as provable bijections are just another mathematical expression of the underlying density matrix of the setup, and as discussed earlier diagonalized mixed density matrices in standard Quantum Mechanics are possible only if one considers quantum entanglements in the context of Zeh's decoherence theory [9].
The known Hamiltonian defines the corresponding evolution operator U = exp(-iHt), which will transform the density matrix.
The density matrix of neutrons propagating through the described system is calculated from the following expression:
To define the direction of spin before target, one needs to know the density matrix.
The (reduced) density matrix is mixed type one, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.
2](x) should be separately normalized to 1, and as elements in the main diagonal of the density matrix must be taken the statistical probabilities defining the mixture (Zeh [14]).
The RSMDB is a knowledge base established by NovaScreen by (1) drawing on the accumulated biological information locked up in our armamentarium of drugs and failed drugs and (2) creating a high density matrix of the molecular interactions between these drugs/drug-like chemicals and molecular disease-related targets such as receptors and enzymes.
For larger applications, the 2010 can be seamlessly integrated with a Keithley 7000 Series High Density Matrix Switching System.