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 classic literature ?
And leaning over the air apparatus, he saw that the tap was allowing the colorless gas to escape freely, life-giving, but in its pure state producing the gravest disorders in the system.
When the wine came, too, I thought it flat; and it certainly had more English crumbs in it, than were to be expected in a foreign wine in anything like a pure state, but I was bashful enough to drink it, and say nothing.
The mathematical definition of entanglement varies depending on whether we consider only pure states or a general set of mixed states (Giannetto) (1).
In the course of the development of quantum mechanics it has become clear that the concept of pure states is not sufficient when taking environmental influences causing decoherence effects into account.
A growing body of evidence even suggests that both everyday, familiar water and supercooled water are actually blends of two distinct liquid water forms that are never seen in their pure states.
The book lays the blame for much of the violence on an ideology of ethnic or integral nationalism that has convinced many state builders that only ethnically pure states can modernize and grow strong.
While we may legitimately employ it to explain the nationalist struggles of an oppressed minority like the Kurds, we would be rightly reluctant to see "dignity" at stake in the struggles of Serbs and Croats for their own ethnically pure states.
Thus far, we have been representing pure states by Hilbert space vectors.
Whilst the components of an entangled system cannot individually be regarded as being in pure states, they can invariably be regarded as being in mixed states.
In general, if we think of pure states as limiting cases of mixed states, the mixed state of a physical system embodies everything about the system which affects the probabilities associated with the outcomes of measurements on the system, considered in isolation.
Such mixtures are standardly thought to admit of an 'ignorance interpretation' according to which each member of an ensemble of systems in state W can be said to actually be in one of the pure states represented by a (not necessarily orthogonal) set {[P.
Given this very weak requirement (which I have intentionally formulated so as not to contain any a priori preference for one particular decomposition of W into pure states {[P.