Creation operators equation

We define the configuration space Heisenberg creation operator by the equation... 

The functions fk and are the counterparts of the so-called destruction (annihilation) and creation operators in the Heisenberg-Dirac picture. It is noted in anticipation that these operators occur as the solutions a,k(t) = lulkt of the Hamiltonian equation... 

On the other hand, Stecki and Taylor29 study the equation which gives the correlations in the Prigogine theory. The property (44) for the creation operator allows us to develop p 0 (t — t ) about t = 0 in Eq. (48), which gives... 

The creation operators aj are the hermitian adjoint of the operators a . The properties of a can be inferred from the above equations. From Eq. (1.12) the hermitian conjugated operators are seen to satisfy the anticommutation relation... 

Since these two equations hold for any one-determinant wave function, and the functions on the right side of these equations only differ in sign, we arrive at the following anticommutation relation for the creation operators ... 

We now expand the submatrix element of the irreducible tensorial product of creation operators on the right side of this equation by using (5.16), and then go over to CFP according to (15.21)... 

Equation (29) is particularly useful for the quantum theory because we know how to represent the Coulomb gauge vector potential operator in terms of photon annihilation and creation operators since the Green s function g(x,x ) remains a c-numbcr. (29) gives an operator representation of the vector potential in an arbitrary gauge. 

Equation (8) can be rewritten equivalently, in terms of the bath oscillators annihilation and creation operators, as... 

In this expression, hp = pW q) represeiits a matrix element of the one-electron component of the Hamiltonian, h, while (pqWrs) s ( lcontains general annihilation and creation operators (e.g., or ) that may act on orbitals in either occupied or virtual subspaces. The cluster operators, T , on the other hand, contain operators that are restricted to act in only one of these spaces (e.g., al, which may act only on the virtual orbitals). As pointed out earlier, the cluster operators therefore commute with one another, but not with the Hamiltonian, f . For example, consider the commutator of the pair of general second-quantized operators from the one-electron component of the Hamiltonian in Eq. [53] with the single-excitation pair found in the cluster operator, Tj ... 

The contraction rules we examined earlier (cf. Eqs. [92] and [93]) state that since the creation operator is on the left, the contraction is zero unless af, and a both act in the hole space and give 8. This simplifies the one-electron part of the equation to... 

Relative to direct application of the anticommutation relations for annihilation and creation operators, Wick s theorem helps to dramatically reduce the tedium involved in deriving the rather complicated amplitude equations above. However, as illustrated by Eq. [151], Wick s theorem still does not go far enough. Even if the cluster operator is truncated to include only double excitations, the resulting algebra provides many opportunities for error. Wlien even... 

These results seem to be fully satisfactory, both the equations of the first and second moments agree with the classical results, and the eventual equilibrium agrees with thermal considerations. However, the form (26) is not acceptable, as we can see from the following considerations [Hakim 1985 Ambegaokar 1991 Munro 1996], For a summary of the situation see Ref. [Stenholm 1994], In the case of the harmonic oscillator, we introduce the customary annihilation and creation operators by setting... 

Equations (34) and (35) tell us how many-electron states, constructed by letting electron creation operators act on the vacuum state, transform under the elements of the symmetry group of the core Hamiltonian. The vacuum state is assumed to be invariant under the action of these symmetry operations, i.e., R10) = 10). 

Notice that the summation of equation (36) has been simplified. Since each state in the set y appears twice in the sum, coupled with its time-reversed partner, equation (32) can be used to cut the number of terms in half. Strictly speaking, each of the pair creation operators shown in equation (43) ought to have an additional index specifying the particular set of molecular orbitals to which it corresponds. 

Here, just as in equation (43) we have cut the number of terms in the sum in half, since each orbital appears twice in the sum shown in equation (36). In equation (44) we have included the shell index n explicitly in the label of the symmetry-preserving pair creation operator. In the individual one-electron creation operators of equation (44), this index is implied, as is the subshell index /. 

If one constructs the projection operators from the same independent particle basis as is used in the expansion of then the effect is simply to limit the expansion of to include only two-electron configurations involving positive energy spinors. Defining bj, to be a creation operator for the single-particle positive-energy state, the equation... 

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