Field operators scalar

Equations (ll-377)-(ll-381) for the field operators entail that under time inversion the interaction hamiltonian transforms like a scalar... 

It is possible to perform a systematic decoupling of this moment expansion using the superoperator formalism (34,35). An infinite dimensional operator vector space defined by a basis of field operators Xj which supports the scalar product (or metric)... 

The vectorial operator gradient (with symbol nabla V) allows the passage from scalar to vectorial fields. For scalar the vector V4> (gradient of (fi) is given by... 

For a time-independent scalar potential, the electron-positron field operator, (a ), is expanded in a complete basis of four-component solutions of the time-dependent Dirac equation [19],... 

In order to discuss interacting electrons, it is often advantageous to introduce electron field operators or second quantization, which means that the expansion coefficients as t) of Eq. (3.2) become operators rather than scalars. These operators are non-Hermitian so that also their ad joints aj(t) are needed. These operators are postulated to satisfy the anticommutation relations (at equal times) ... 

A key quantity is the vector potential A f,t) = A f), which satisfies a wave equation analogously to the wave equation for the electron field operators ip ). It is chosen divergence free V A r) = 0. Invoking zero scalar potential and this choice is usually referred to as the Coulomb gauge. 

Such a treatment can, with advantage, be expressed in terms of the superoperators introduced in Eq. (4.19) and in terms of a basis of field operators. The basis of fermion-like operators Xj = a, aj[aja, ,a aja, a ap, - is chosen, such that the electron field operators correspond to the SCF spin orbitals. The field operator space supports a scalar product (XjlXj) = ([A , X,]+) = Tr /9[Xl,Xj]+, where p is the density operator defined in Eq. (4.33). The superoperator identity and the superoperator hamiltonian operate on this space of fermion-like field operators and, in particular, Xi HXj) = [x/, [H,Xj - J. ) = Tt p[xI[H,X ] U. ... 

If we specifically consider the mixing of two single-mode, amplitude-stabilized, first-order coherent waves, both of which are well collimated, parallel, plane polarized along a common unit vector, and normally incident onto a photosensitive material, we may write the positive portion of the electric field operator as the superposition of two scalar fields... 

We consider first the expansion of a free field operator in terms of creation and annihilation operators. For a real scalar field describing quanta of mass IJL we write... 

A few comments on the layout of the book. Definitions or common phrases are marked in italic, these can be found in the index. Underline is used for emphasizing important points. Operators, vectors and matrices are denoted in bold, scalars in normal text. Although I have tried to keep the notation as consistent as possible, different branches in computational chemistry often use different symbols for the same quantity. In order to comply with common usage, I have elected sometimes to switch notation between chapters. The second derivative of the energy, for example, is called the force constant k in force field theory, the corresponding matrix is denoted F when discussing vibrations, and called the Hessian H for optimization purposes. 

Thus, we have expressed the vector field g in terms of a scalar function, U p), by a relatively simple operation. 

Here, /3 and / are constants known as the Bohr magneton and nuclear magneton, respectively g and gn are the electron and nuclear g factors a is the hyperfine coupling constant H is the external magnetic field while I and S are the nuclear and electron spin operators. The electronic g factor and the hyperfine constant are actually tensors, but for the hydrogen atom they may be treated, to a good approximation, as scalar quantities. 

A stress that is describable by a single scalar can be identified with a hydrostatic pressure, and this can perhaps be envisioned as the isotropic effect of the (frozen) medium on the globular-like contour of an entrapped protein. Of course, transduction of the strain at the protein surface via the complex network of chemical bonds of the protein 3-D structure will result in a local strain at the metal site that is not isotropic at all. In terms of the spin Hamiltonian the local strain is just another field (or operator) to be added to our small collection of main players, B, S, and I (section 5.1). We assign it the symbol T, and we note that in three-dimensional space, contrast to B, S, and I, which are each three-component vectors. T is a symmetrical tensor with six independent elements ... 

The divergence operator is the three-dimensional analogue of the differential du of the scalar function u x) of one variable. The analogue of the derivative is the net outflow integral that describes the flux of a vector field across a surface S... 

This may well appear not to produce anything new until the electron is examined in an external electromagnetic field, represented by a scalar potential V, and a vector potential A. The appropriate operators then become... 

A variety of properties can be defined and calculated I will restrict attention to the operators involved in the calculation of dipole polarizabilities and NMR parameters, corresponding to the introduction of a uniform electric field E represented by the scalar potential... 

The differential operator divergence allows the passage from vectorial to scalar fields. For a vector [Pg.810]

Operating on the scalar field rj yields a straightforward result ... 

Discuss the pro s and con s of writing and using the spatial components of the substantial-derivative operator as either one of the two equivalent notations for either a scalar of vector field ... 

Equation (5.38) can be interpreted as the scalar product of a forward-moving density and a backward-moving time-dependent operator. The optimal field at time t is determined by a time-dependent objective function propagated from the target time T backward to time t. A first-order perturbation approach to obtain a similar equation for optimal chemical control in Liouville space has been derived in a different method by Yan et al. [28]. 

This can be extended straightforwardly to angular momentum operators and infinitesimal magnetic field generators. Therefore, a commutator such as Eq. (818) is equivalent to a vector cross-product. If we write Bm> as the scalar magnitude of magnetic flux density, the commutator (818) becomes the vector cross-product... 

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