Expectation value of operator

The density amplitudes can usually be calculated more efficiently than the density operator because they depend on only one set of variables in a given representation although there are cases, such as shown below for the time-dependent Hartree-Fock density operator, where the advantages disappear and it is convenient to calculate the density operator. Expectation values of operators A t) follow from the trace over the density operator, as... 

Barysz, M. and Sadlej, A.J. (1997) Expectation values of operators in approximate two-component relativistic theories. Theoretica Chimica Acta, 97, 260-270. 

Using TFD we are able to find the thermal expectation values of operators. In general, through the Bogoliubov transformation from ai (t), aa(t) to al(P,t),aa(P,t), we find the formula... 

Diagonalization Method to Calculate Expectation Values of Operators Non-Commutative to the Hamiltonian. Vibrational Assignment of HOC1. 

The primary questions that remain to be answered are the following to evaluate the energy and amplitude equations (7) and (21), we need to (i) construct H and (ii) have some means of evaluating expectation values of operators O with the reference o-... 

The occupation number vectors are basis vectors in an abstract linear vector space and specify thus only the occupation of the spin orbitals. The occupation number vectors contain no reference to the basis set. The reference to the basis set is built into the operators in the second quantization formalism. Observables are described by expectation values of operators and must be independent of the representation given to the operators and states. The matrix elements of a first quantization operator between two Slater determinants must therefore equal its counterpart of the second quantization formulation. For a given basis set the operators in the Fock space can thus be determined by requiring that the matrix elements between two occupation number vectors of the second quantization operator, must equal the matrix elements between the corresponding two Slater determinants of the corresponding first quantization operators. Operators that are considered in first quantization like the kinetic energy and the coulomb repulsion conserve the number of electrons. In the Fock space these operators must be represented as linear combinations of multipla of the ajaj... 

N0 can be any integer to which the system of electrons is bound. In Eq. (28a), is the ground-state wave function of the N0-particle system and 4/n0+i that of the (N0 + 1 (-particle system. Thus D, changes discontinuously as Jf crosses an integer. It is a continuous, piecewise-linear operator-valued function of Jf. Consequently, all expectation values of operators are also piecewise-linear functions of Jf. Their first derivatives with respect to Jf are all discontinuous, piecewise-constant functions of Jf. 

Initially, at f < 0, the isolated system is described by the Hamiltonian H0 that does not depend on time. Thus, any expectation value averaged with the density matrix p0 is time-independent. At t > 0, the system is disturbed by an external time-dependent field. Then, the evolution of the expectation value of operator O can be presented as follows ... 

We shall find it very useful to express this information in one matrix. For example, to express the expectation value of operator A succincdy, we combine Eqs. 11.4 and 11.5 and rearrange terms to obtain ... 

By writing the expectation value of operator F for the normalized wave function tp as , Eq. (311) takes the form... 

At a microscopic level, two kinds of functions can be derived from quantum mechanics. On the one hand are those corresponding to global properties such as the energy, expectation values of operators, chemical potential, or global hardness as defined within the conceptual DFT context [35] and on the other hand are local properties such as electron density distributions, the electron localization function ELF [24], local hardness [36,37], and the Fukui functions [38,39],... 

The calculation of expectation values of operators over the wavefunction, expanded in terms of these determinants, involves the expansion of each determinant in terms of the N expansion terms followed by the spatial coordinate and spin integrations. This procedure is simplified when the spatial orbitals are chosen to be orthonormal. This results in the set of Slater Condon rules for the evaluation of one- and two-electron operators. A particularly compact representation of the algebra associated with the manipulation of determinantal expansions is the method of second quantization or the occupation number representation . This is discussed in detail in several textbooks and review articles - - , to which the reader is referred for more detail. An especially entertaining presentation of second quantization is given by Mattuck . The usefulness of this approach is that it allows quite general algebraic manipulations to be performed on operator expressions. These formal manipulations are more cumbersome to perform in the wavefunction approach. It should be stressed, however, that these approaches are equivalent in content, if not in style, and lead to identical results and computational procedures. 

Measurable quantities are expressed by expectation values of operators. The expectation value of the operator O is given by... 

The transformation of the Dirac Hamiltonian to two-component form is accompanied by a corresponding reduction of the wavefunction. As discussed in detail in section 2, the four-component Dirac spinor will have only two nonvanishing components, as soon as the complete decoupling of the electronic and positronic degrees of freedom is achieved, and can thus be used as a two-component spinor. This feature can be exploited to calculate expectation values of operators in an efficient manner. However, this procedure requires that some precautions need to be taken care of with respect to the representation of the operators, i.e., their transition from the original (4 x 4)-matrix representation (often referred to as the Dirac picture) to a suitable two-component Pauli repre-... 

If the expectation value of operator Fa is denoted by (Fa), then equation (34) can be written in the concise form... 

Fock spaces) and corresponding quantum mechanical operators. The connection between different formulations must be that they give the same expectation values of operators. 

As seen from eq. (10.21), the first-order property is given as an expectation value of operators linear in the perturbation. The second-order property contains two contributions, an expectation value over quadratic (or bilinear) operators and a sum over products of matrix elements involving linear operators connecting the ground and excited states. 

Comparison with Eq. (21) indicates that in overlapping basis the elements of the spin-dependent first- and second-order density matrix can be obtained as expectation values of operator strings constructed from biorthogonal creation and annihilation operators ... 

Before we can start with the discussion of time-dependent perturbation theory in the form of response theory, we need to introduce an alternative formulation of quantum mechanics, called the interaction or Dirac representation. In general, several representations of the wavefunctions or state vectors and of the operators of quantum mechanics are equivalent, i.e. valid, as long as the expectation values of operators ( 0 I d I o) or inner products of the wavefunctions ( o n) are always the same. Measurable quantities and thus the physics are contained in the expectation values or inner products, whereas operators and wavefunctions are mathematical constructs used in a particular formulation of the theory. One example of this was already discussed in Section 2.9 on gauge transformations of the vector and scalar potentials. In the present section we want to look at a transformation that is related to the time dependence of the wavefunctions and operators. 

An important application of these rules is the calculation of expectation values of operators in the basis of states with definite J and J. Since these states are obtained by combinations of states with spin S and orbital angular momentum L, we denote them as JLSJ ), which form a complete set of (2/ + 1) states for each value of J, and a complete set for all possible states resulting from the addition of L and S when all allowed values of J are included ... 

The generalized density functions therefore allow us to relate expectation values of operators directly to the electron distribution. It is also worth noting that successive density matrices are related in particular. 

In our discussion, we have so far examined the electron density in the sjrin-orbital and orbital spaces. Let us now consider the electron density in ordinary space. Of particular interest are the expectation values of operators that probe the presence of electrons at particular points in space. Thus, the one-electron first-quantization operator in the form of a linear combination of Dirac delta functions... 

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