Representations of operators

Fock Space Representation of Operators.—Let F be some operator that neither creates nor destroys particles, and is a known function in configuration space for N particles. In symbols such an operator must by definition have the following matrix elements in Fock space ... 

Flow, control of, 265 Flow function on network, 258 Flow, optimal, method for, 261 Fock amplitude for one-particle system, 511 Fock space, 454 amplitudes, 570 description of photons, 569 representation of operators in, 455 Schrodinger equation in, 459 vectors in, 454 Focus, 326 weak, 328... 

Energy-Separable Faber Polynomial Representation of Operator Functions Theory and Application in Quantum Scattering. 

Operators corresponding to physical quantities can also be expanded in terms of irreducible tensors in the quasispin space of each individual shell. To this end, it is sufficient to go over to tensors (17.43) and next to provide their direct product in the quasispin space of individual shells. This procedure can conveniently be carried out for a representation of operators such that the orbital and spin ranks of all the one-shell tensors are coupled directly. Here we shall provide the final result for the two-particle operator of general form (14.57)... 

Representation of Operational Alternatives Using State Task Network... 

State Task Network (STN) representation of operating sequence for binary and multicomponent batch distillation... 

Matrix representation of operators. - If an operator T is defined on the space A = (x), then the quantity TXi is also an element of this space and may be expanded in the form... 

Huang Y., Kouri, D.J. and Hoffman, D.K. (1994) General, energy-separable Faber polynomial representation of operator-functions - theory and application in quantum scattering J. Chem. Phys. 101, 10493-10506. 

Figure 1. Schematic representation of operation of heterogeneous liquid crystal light shutters A) glass beads in a liquid crystal film, B) liquid crystal imbibed in a microporous film, C) encapsulated liquid crystal, D) polymer dispersed liquid crystals.

Equilibrium time correlation function expressions for transport properties can be derived using linear response theory [3]. Linear response theory can be carried out directly on the Wigner transformed equations of motion to obtain the transport properties as correlation functions involving Wigner transformed quantities. Alternatively, we may carry out the linear response analysis in terms of abstract operators and insert the Wigner representation of operators in the final form for the correlation function. We use the latter route here. 

Dimensionless Representation of Operating and Design Parameters 318 7.1.3 Scaling Up and Down 322... 

G. Beylkin, On the Representation of Operators in Bases of Compactly Supported Wavelets, SIAM Journal of Numerical Analysi. s. 29 (1992), 1716-1740. 

Therefore, the two leading terms of the Taylor expansions for the centroid and Kubo correlation functions are the same, the difference between them beginning with the third term (i.e., at order t ). The latter term can be taken as an example of how to evaluate the leading correction term to the centroid correlation function (and thereby demonstrate that the centroid correlation function is a well-defined approximation to the Kubo correlation function). The Gaussian representation of operators in phase space [Eq. (2.61)] proves to be useful but not essential in this analysis. 

The specification super-operator is common in quantum chemical emd physical literature for linear mappings of Fock-space operators. It is very helpful to transfer this concept to the extended states A, B) and define the application of super-operators by the action on the operators A and B. We will see later how this definition helps for a compeict notation of iterated equations of motion and perturbation expansions. In certain cases, however, the action of a super-operator is fully equivalent to the action of an operator in the Hilbert space Y. The alternative concept of Y-space operators allows to introduce approximations by finite basis set representations of operators in a well-defined and lucid way. 

Figure 1.2 Schematic representation of operation, gradients, and fluxes in mixed pro-ton-eiectron-conducting membranes used for dehydrogenation of reformed methane (syngas). Gradients represent quaiitativeiy chemi-...

The representation of operators in incomplete basis sets will cause severe problems in maintaining gauge invariance in the calculation of magnetic susceptibilities. It is generally required to use field-dependent basis functions (see the... 

This chapter deals with the discussion and interpretation of approximate molecular electronic structure methods in terms of propagator concepts. Only situations with fixed nuclear frameworks are considered, and the discussion is limited to the description of states that are close in energy to the normal state of the system. We adopt the view that the main features of the electronic structure of such states can be developed in terms of atomic orbital representations of operators and that only valence shell orbitals need be considered. 

Operator algebra shares the characteristic with matrix algebra indeed, matrices can be considered as representations of operators in a given set of functions (coordinate system). 

The representation of operators by spinors is a rather more delicate matter. By 1935 only one detailed application of Kramers s symbolic method to atomic spectroscopy appears to have been carried out that of Wolfe (1932) to the atomic configuration k. Some of the difficulties of the method are discussed below in section 5.1. For the moment we need only say that the abstract nature of the method did not prove appealing to most theoretical spectroscopists. No one in the 1930 s or since, as far as the writer is aware, has used the functions... 

If the Hartree-Fock equations associated with the valence pseudo-Hamiltonian (167) are solved with extended basis sets, then all the above F are almost basis-set-independent. At the present time, and for practical reasons, most of the ab initio valence-only molecular calculations use coreless pseudo-orbitals. The reliability of this approach is still a matter of discussion. Obviously the nodal structure is important for computing observable quantities such as the diamagnetic susceptibility which implies an operator proportional to 1/r. From the computational point of view, it is always easy to recover the nodal structure of coreless valence pseudo-orbitals by orthogonalizing the valence molecular orbitals to the core orbitals. This procedure has led to very accurate results for several internal observables in comparison with all-electron results. The problem of the shape of the pseudoorbitals in the core region is also important in relativity. For heavy atoms, the valence electrons possess high instantaneous velocities near the nuclei. Schwarz has recently investigated the compatibility between the internal structure of valence orbitals and the representations of operators such as the spin-orbit which vary as 1/r near the nucleus. ... 

The representation of operators on these states can now be found. One can start from one-particle operators, such as the external potential V(r). 

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