Operator density

A differential equation for the time evolution of the density operator may be derived by taking the time derivative of equation (Al.6.49) and using the TDSE to replace the time derivative of the wavefiinction with the Hamiltonian operating on the wavefiinction. The result is called the Liouville equation, that is. 

In the above discussion of relaxation to equilibrium, the density matrix was implicitly cast in the energy representation. However, the density operator can be cast in a variety of representations other than the energy representation. Two of the most connnonly used are the coordinate representation and the Wigner phase space representation. In addition, there is the diagonal representation of the density operator in this representation, the most general fomi of p takes the fomi... 

The total of tliree interactions of the material with the field can be distributed in several different ways between the ket and the bra (or more generally, between left and right interactions of the field with the density operator). For example, the first temi in equation (Al.6.101) corresponds to all tliree interactions being with the ket, while the second temi corresponds to two interactions with the ket and one with the bra. The second temi can be fiirther subdivided into tliree possibilities that die single interaction with the bra is before, between or after the two interactions with the ket (or correspondingly, left/right interactions of the field with... 

A stationary ensemble density distribution is constrained to be a functional of the constants of motion (globally conserved quantities). In particular, a simple choice is pip, q) = p (W (p, q)), where p (W) is some fiinctional (fiinction of a fiinction) of W. Any such fiinctional has a vanishing Poisson bracket (or a connnutator) with Wand is thus a stationary distribution. Its dependence on (p, q) through Hip, q) = E is expected to be reasonably smooth. Quanttun mechanically, p (W) is die density operator which has some fiinctional dependence on the Hamiltonian Wdepending on the ensemble. It is also nonnalized Trp = 1. The density matrix is the matrix representation of the density operator in some chosen representation of a complete orthononnal set of states. If the complete orthononnal set of eigenstates of die Hamiltonian is known ... 

A diagrannnatic approach that can unify the theory underlymg these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the fonn of the time evolution of the light/matter density operator. (It is recoimnended that anyone interested in advanced study of this topic should familiarize themselves with density operator fonnalism [8, 9, 10, H and f2]. Most books on nonlinear optics [13,14, f5,16 and 17] and nonlinear optical spectroscopy [18,19] treat this in much detail.) Once the density operator is known at any time and position within a material, its matrix in the eigenstate basis set of the constituents (usually molecules) can be detennined. The ensemble averaged electrical polarization, P, is then obtained—tlie centrepiece of all spectroscopies based on the electric component of the EM field. 

Consider an ensemble composed of constituents (such as molecules) per unit volume. The (complex) density operator for this system is developed perturbatively in orders of the applied field, and at. sth order is given by The (complex). sth order contribution to the ensemble averaged polarization is given by the trace over the eigenstate basis of the constituents of the product of the dipole operator, N and = Tr A pp... 

A good introductory treatment of the density operator formalism and two-dimensional NMR spectroscopy, nice presentation of Redfield relaxation theory. 

The experiment starts at equilibrium. In the high-temperature approximation, the equilibrium density operator is proportional to the sum of the operators, which will be called F. If there are multiple exchanging sites with unequal populations, p-, the sum is a weighted one, as in equation (B2.4.31). 

A number of procedures have been proposed to map a wave function onto a function that has the form of a phase-space distribution. Of these, the oldest and best known is the Wigner function [137,138]. (See [139] for an exposition using Louiville space.) For a review of this, and other distributions, see [140]. The quantum mechanical density matrix is a matrix representation of the density operator... 

Molecular weight of gas Gas density Operating temperature Operating pressure Inlet dew point Desired outlet dew point... 

Since we shall also be interested in analyzing the confined fluid s microscopic structure it is worthwhile to introduce some useful structural correlation functions at this point. The simplest of these is related to the instantaneous number density operator... 

The translational microscopic structure of the confined fluid is partially revealed by correlations in the number density operator, given by... 

Thermal conductivity is not a static property of a material. It can vary according to the density, operating temperature and type of gas entrapped within the voids. 

Some Theorems Concerning Particle Density Operators. Applying the theorem of Eq. (8-103) to the last factor of that same equation, we have ... 

In practical applications one uses the definitions, Eq. (8-118), of the population density operators to write Eq. (8-159) in the form ... 

Consider the following operator, defined in any ensemble, and called the density operator ... 

It is worthwhile to consider the same theorem in terms of coordinate space instead of occupation number space. Thus, we may envision an ensemble of systems whose states are X>, and whose distribution probabilities among these states are w(X). We define the density operator... 

The appropriate expression for the operator H in the above equations is that appearing in Eq. (8-160). Eor a first example, consider an ideal gas without interactions. Assuming that the one-particle wave functions used in the population density operators are the energy eigenfunctions, then the matrix H0llA is diagonal, and we can write... 

As a somewhat exceptional example of this we shall find the rate of change of the ensemble density operator, Eq. (8-186), which we now rewrite more explicitly to bring out its time-dependence ... 

The expectation value of the density operator, and, indeed, all the components of the density matrix, are stationary in time for an ensemble set up in terms of energy eigenstates. IT we use occupation number representation to set up the density matrix, it is at once seen from Eq. (8-187) that it also is independent of time ... 

Number density operator, total, 452 Number density of particles, 3 Numbers, representation in digital computation, 50 Numerical analysis, 50 field of, 50... 

Total electric charge operator, 542 Total energy operator, 506,542 Total momentum operator, 506,542 Total number density operator, 452... 

Quantum statistical mechanics with the concepts of mixed states, density operators and the Liouville equation. 

For a pure state density operator, the Fourier transform of this double-time Green s function yields the spectral representation of the propagator (21)... 

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