Hole operator

It should be observed, however, that one cannot in general expect that an arbitrarily chosen basis B in the operator space should satisfy these three conditions, and this applies particularly to the particle-hole operator bases commonly used in the special propagator methods or the EOM approach. In such a case, the new operator basis B(1) defined by the linearly independent elements in the four matrices (2.31) may serve as a new starting basis. Due to the construction, the basis B(,) is automatically closed under adjunction (t). It is also clear that the basis B(l) is closed under multiplication due to the fact that one has the multiplication rule... 

If

truncated basis of order n in the carrier space, then the corresponding basis P = Pk in the operator space is of order m = n2. It may further be shown4 that, if n —> °° and the basis tp = < becomes complete in the carrier space, then the basis P = Pki becomes complete in the HS operator space. It should perhaps be observed that this completeness theorem is somewhat different in nature from the completeness theorems for products of particle-hole operators which are proven to be... 

It should be observed that it does not matter whether one describes the operator space in terms of particle-hole operators in the language of second quantization or in terms of ket-bra operators. Choosing the ket-bra basis P = Pu constructed from the orthonormal set

carrier space as our basis B, one obtains for r = (k, /) and s = (m, n) ... 

The series truncates (exactly) after four commutators regardless of the level at which (T ) T is truncated (if at all). This exact truncation is a result of the fact that // contains at most two-electron operators, which involve four general (particle or hole) operators i j lk. Therefore [//, T] contains at most three general operators, [[f/, T], T] contains two, and [[[[ J... 

A wide variety of shapes is possible. A simple circular die gives a circular product use of a die plate with many holes operating in parallel enables production rates to be enhanced. A circular die with a cylindrical torpedo mounted along the axis of the pipe will give a tube. The shape of the duct determines the external shape of the object being formed. Interchangeable... 

HT 1 Hole Operation. Special procedure is necessary for operation with the HT 1 hole. The coffin is removed from the dolly by the crane and rolled over 180 on a special fixture. It is then lowered on the dolly with the trunnion riser blocks removed. When the coffin and dolly extensions are bolted on, the coffin is ready for use. 

Burnt-out bore-hole Operating and partially bumt-out bore-hole... 

If a Hamiltonian has particle-hole (or charge-conjugation) S3unmetry then it is invariant under the transformation of a particle into a hole under the action of the particle-hole operator, J ... 

The particle-hole excitations, defined in Section 3.5, are eigenstates of the noninteracting Hamiltonian, but they are not eigenstates of the particle-hole operator, J, introduced in Section 2.9.2. To see this, consider the operation of J on the singlet excitation, A ) ... 

The electric dipole operator is antisymmetric with respect to the particle-hole operator and thus it connects states of opposite particle-hole symmetry. The proof is identical to that for the inversion operator. 

In terms of the particle-hole operators a and d (defined as a for the creation and destruction... 

Substitution of the particle-hole operators also in equations (8.7) and (8.8) yields, after elementary calculation, ... 

It should be mentioned that, in deriving equation (8.14), we neglected all terms in H that contain two particle or two hole operators (the wave function contains also only the operators of one electron-hole pair). If one wants to consider also double excitations (two electron-hole pairs) a two-particle Green function must be used and a diagrammatic technique. Further even at the one electron-hole case one has to introduce so-called vertex corrections. This has been done for the most important diagrams, but a description of this rather complicated formalism lies outside the framework of this book. 

In words, for virtual orbitals the holes operators bi", bj act exactly in the same manner as the particle operators aj do, while their role is reversed for occupied orbitals. Operator creates an electron in the virtual space, while it annihilates an electron in the Fermi see. This is equivalent to saying that it creates a hole in HF>. Similarly, operator bj creates an electron in 1HF>, while it annihilates one in the virtual subspace. This particle-hole formalism is in analogy with that of quantum field theory where, for instance, the holes correspond to positrons while the particles are electrons. 

It is now a trivial task to show that the following equations hold for the hole operators ... 

The operators bj (bj) are the creation (annihilation) operators for holes. Shortly, they are called hole operators. The commutation properties of the hole operators bj, bj are obviously identical to those of the particle operators a, and aj. 

Note that the hole operators introduces in this section are different from those in Sect. 5, since the definition of the latters depends on the occupancy of the orbital which they are acting on. No such trick is utilized here. 

Remember now that operators cjy, c y refer to holes rather than electrons. The particle number operator N of Eq. (3.1) can be expressed in terms of hole operators as ... 

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