Density operator, definition

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

With these definitions, the nonequilibrium density operator for a subsystem of a thermal reservoir of inverse temperature p is [5]... 

For conservative systems with time-independent Hamiltonian the density operator may be defined as a function of one or more quantum-mechanical operators A, i.e. g= tp( A). This definition implies that for statistical equilibrium of an ensemble of conservative systems, the density operator depends only on constants of the motion. The most important case is g= [Pg.463]

The definition of a grand canonical density operator requires that the system Hamiltonian be expressed in terms of creation and annihilation operators. 

Abstract The theoretical basis for the quantum time evolution of path integral centroid variables is described, as weU as the motivation for using these variables to study condensed phase quantum dynamics. The equihbrium centroid distribution is shown to be a well-defined distribution function in the canonical ensemble. A quantum mechanical quasi-density operator (QDO) can then be associated with each value of the distribution so that, upon the application of rigorous quantum mechanics, it can be used to provide an exact definition of both static and dynamical centroid variables. Various properties of the dynamical centroid variables can thus be defined and explored. Importantly, this perspective shows that the centroid constraint on the imaginary time paths introduces a non-stationarity in the equihbrium ensemble. This, in turn, can be proven to yield information on the correlations of spontaneous dynamical fluctuations. This exact formalism also leads to a derivation of Centroid Molecular Dynamics, as well as the basis for systematic improvements of that theory. 

The theorem holds if the exchange-correlation potential VXc equals the functional derivative of the exchange-correlation energy /iXc with respect to the electron density p - an operational definition, which is intrinsic to DFT. 

After having described the expression for the rate constant within the framework of classical mechanics, we turn now to the quantum mechanical version. We consider first the definition of a flux operator in quantum mechanics.2 To that end, the flux density operator (for a single particle of mass to) is defined by... 

We shall do that with the same kind of the operational definition as in Section IV.B.5. Specifically, we assume that each signal-to-noise value emerges as a result of a three-step procedure (1) one decides on the frequency il at which the test would be performed then (2), at double this frequency, the noise spectral power density in the state with no driving field is evaluated, thus yielding... 

The HSAB principle can be considered as a condensed statement of a very large amount of experimental information, but cannot be labelled a law, since a quantitative definition of the intuitive concepts of chemical hardness (T ) and softness (S) was lacking. This problem was solved when the hardness found an exact, and also an operational, definition in the framework of the Density Functional Theory (DFT) by Parr and co-workers [2], In this context, the hardness is defined as the second order derivative of energy with respect to the number of electrons and has the meaning of resistance to change in the number of electrons. The softness is the inverse of the hardness [3]. Moreover, these quantities are defined in their local version [4, 5] as response functions [6] and have found a wide application in the chemical reactivity theory [7],... 

Figure 1. Operational definition of equilibrium contact time (ECT) in the FFT Process fabric (65/36 polyester/cotton sheeting, 3.7 oz/yd2) speed of fabric (100 ft/min) nozzle gap width (0.5 in.) machine contact time (MCT) (0.025 sec) foam (0.15 g/cc density, 46% solids content).

Here we manipulated the left side of the inequality by using the corresponding definition and the Lagrange parameter associated with the Hamiltonian operator H. For the sake of clarity, we have here exhibited the parameter dependence of the appropriate ensemble density operators along with those associated with the respective minimum free energies and entropies corresponding to the two Hamiltonians. 

At this point we have expressed the Hamiltonian, the density operator and the evolution operator in Fourier space. We have introduced an effective Hamiltonian, defined in the Hilbert space of the same dimension 2 as the total time-dependent Hamiltonian itself, and we have shown how to transform operators between the two representations. The definition of the effective Hamiltonian enables us to predict the overall evolution of the spin system, despite the fact that we can not find time-points for synchronous detection, f, where Uint f) = exp —iWe//t In actual experiments the time dependent signals are monitored and after Fourier transformation they result in frequency sideband... 

The definition of the matrix elements of the BMFT Hamiltonian is straightforward, as well as the definition of the initial density operator ... 

An estimated 75 million people are affected by osteoporosis to some degree in the United States, Europe, and Japan. Osteoporosis is a systematic skeletal disease characterized by bone mass and microarchitectural deterioration with a consequent increase in bone fragility and susceptibility to fracture. Operationally, osteoporosis can be defined as a certain level of bone mineral density. The definition of osteoporosis is somewhat arbitrary and is based on epidemiological data relating fracture incidence to bone mass. Uncertainty also is introduced due to variability in bone densitometry measurements. Other clinical measures to assess the skeleton include collagen cross-links (measure of bone resorption) and levels of bone-specific alkaline phosphatase and osteocalcin (bone formation). A list of biochemical markers of bone remodeling is provided in Table 37-3. Measurement of total serum alkaline phosphatase level and urinary hydroxyproline or calcium levels is of limited value. 

The time evolution ofthe density operator can be found from the time evolution of F(Z) and the definition (10.3) ... 

Two other properties of the density operators follow from its definition (10.3). First, it is Hermitian, that is, p (Z) = p(Z). Second it is idempotent, that is, satisfies the property... 

We have already encountered the projection operator formalism in Appendix 9A, where an apphcation to the simplest system-bath problem—a single level interacting with a continuum, was demonstrated. This formalism is general can be applied in different ways and flavors. In general, a projection operator (or projector) P is defined with respect to a certain sub-space whose choice is dictated by the physical problem. By definition it should satisfy the relationship = P (operators that satisfy this relationship are called idempotent), but other than that can be chosen to suit our physical intuition or mathematical approach. For problems involving a system interacting with its equilibrium thermal environment a particularly convenient choice is the thermal projector. An operator that projects the total system-bath density operator on a product of the system s reduced density operator and the... 

Equation (10.111) has the same form as Eq. (10.107), however, the definition (10.105) is replaced by a more general definition involving the time-dependent density operator of the bath. 

From X-ray diffraction the distance between ions (that is, + / ) can be measured with great precision. However, knowing where one ion ends and where the other begins is a more difficult matter. When careful X-ray diffraction measurements have been used to map out the electron density between ions and the point at which the electron density is a minimum is taken as the operational definition of the limits of the ions involved, and the results are... 

First it is necessary to introduce appropriate notation and definitions and to recall some properties of the density operator introduced by von Neumann [10] and Dirac [11]. A general many-electron wavefunction, for a system in stationary state K, will be written where q stands for all the required particle variables (space and... 

We will review here some definitions and properties of reduced density matrices for fermionic systems. The focus will be on the general density operator p and the corresponding Liouville equation... 

Another virtue of Eq. (10) was that it gave not only an exact definition of hordness, but also an operational definition, so that numbers could be obtained. Further, the exact definition was firmly grounded in density functional theory, so that the fundamental meaning of hardness could be explored. 

Show that Eq. (6.26) is correct with the definition of the current density operator as given by Eq. (6.27). 

The operational definition of a statistical-limit molecule is negative. We define it as a system not exhibiting any measurable deviation from the behavior expected for a single s> level coupled to the 1/2 manifold(s), which forms a dense, uniform quasi-continuum, correctly described by average values of coupling constants, <1 5,>, and level densities, . This coupling provides the [s> level with a supplementary width yf given by Eq. (202). Its overall decay rate and fluorescence yield are described as... 

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