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 ?
The experimental observations of the present paper encourage us to interpret the existence of events with the Gaussian pseudorapidity distributions of produced particles in central relativistic proton-nucleus interactions as a result of a proton-tube collisions and subsequent formation in the course of an interaction of a droplet of hadronic matter, the quark-gluon plasma, i.e., the primordial high density state, whose expansion and cooling lead to its decay with production of final state particles.
Four different traffic attributes such as distance, traffic density state, travel time and number of intersections were considered for this problem of study.
Each density state was simulated 10 times and the fundamental diagram for CD model and RF model with different values of parameter [gamma] could be obtained, as shown in Figure 2.
The role of little clay within the mix is unlikely significant if the sand particles are in full contact or interlocked at the density state described.
In the following paper the impact is determined by a randomly occurring "false negative" detection error on the quality of the density state estimation of counting systems during large-scale pedestrian events.
In order to further analyze the influence of the stress on ferromagnetic characteristics, the total electron density states are calculated under different stresses.
In this work, a variety of soil density states were also studied.
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Therefore, CSSM is a powerful tool able to explain the behaviour of soil at various density states. It is a globally recognised framework that the critical states for sand and clay are both well established.
This means that the diatomic orientational relaxation takes place outside the Markovian limit, being hindered by an anisotropic potential which presents a certain temporal coherence, producing effects like the apparition in the central part of the spectral density between the P- and R-branches, the so called Q-branch [6], which in the high density states of this study is associated with the long-lived diatomic-solvent spatial correlations and in the low density limit, it leads to the traditional Van der Waals complexes [16].
The bill provides a $14.1 billion package of tax incentives for energy production and conservation and includes a provision that requires high-traffic density states, like New York and California, to use ethanol as a gas additive.

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