Annihilation operator definition

The state w, f>s is an eigenstate of N with eigenvalue N, and N is called the total population operator. Because the vector , Os is a function of the time, it is necessary to specify the time at which the creation or annihilation operators are applied, and in some discussions it may be advisable to indicate the time explicitly in the symbol for the operator. For our present discussion it will be sufficient to keep this time dependence in mind. In an expression such as Eq. (8-109), all the creation operators are applied at the same time, and since they all commute, this presents no logical problem. The order of the operators in the definition Eq. (8-107) is important however the opposite order produces a different operator ... 

In formulating the second-quantized description of a system of noninteracting fermions, we shall, therefore, have to introduce distinct creation and annihilation operators for particle and antiparticle. Furthermore, since all the fermions that have been discovered thus far obey the Pauli Exclusion principle we shall have to make sure that the formalism describes a many particle system in terms of properly antisymmetrized amplitudes so that the particles obey Fermi-Dirac statistics. For definiteness, we shall in the present section consider only the negaton-positon system, and call the negaton the particle and the positon the antiparticle. 

This confirms our interpretation of the operators 6,6 and d,d as creation and annihilation operators for particles of definite momentum and energy. Similar consideration can be made for the angular momentum operator. The total electric charge operator is defined as... 

Definition of Normal Product.—Given a product of free field creation and annihilation operators U,X,- -, FF, we define the operator N as... 

I wish to stress that the meaning of the word Hole here is different and far more general than in Many Body Perturbation Theory. Indeed, no specific reference state is required in this definition and the difference between the RO s and the HRO"s follows exclusively from the different order of the creator operators with respect to the annihilator operators in E and in E respectively. 

The definition of a grand canonical density operator requires that the system Hamiltonian be expressed in terms of creation and annihilation operators. 

Assuming that all the 1-electron levels are arranged in some definite order, then, since nk is 1 or 0, we can define the annihilation operator ck by... 

With these definitions the creation operators a, rcj) transform as spherical tensors under rotation. The annihilation operators do not. However, it is easy to construct operators that do transform as spherical tensors [Eq. (1.23)]. These will be denoted by a tilde and written as... 

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

It should be stressed that in the literature one can come across a wide variety of notations for creation and annihilation operators. In this book we follow the authors [14, 95] who attach the sign of Hermitian conjugation to the electron annihilation operator, but not to the electron creation operator. Although the opposite notation is currently in wide use, it is inconvenient in the theory of the atom, since it is at variance with the common definitions of irreducible tensorial quantities. 

The mathematical apparatus of the angular momentum theory can be applied to describe the tensorial properties of electron creation and annihilation operators in the space of occupation numbers of a certain definite one-particle state a). It follows from (13.29) and (13.30) that the operators... 

To proceed further, we have to know how to handle the products of creation and annihilation operators. It is Wick s theorem which tells us how to deal with the products of these operators. Before presenting Wick s theorem we have to introduce some necessary definitions and relations. The creation and annihilation operators satisfy the anticommutation relation... 

We have utilized the fermion creation and annihilation operators denoted a a, and apa, respectively. These operators act on the electron in the pth orbital with the projected spin a. The set 0p(r)) represents the molecular orbitals and the last term, hauc, in Eq- (13-3) is the nuclear repulsion energy. We use the following definitions of the one-electron excitation operator... 

This new definition of normal ordering changes our analysis of the Wick s theorem contractions only slightly. Whereas before, the only nonzero pairwise contraction required the annihilation operator to be to the left of the creation operator (cf. Eq. [84]), now the only nonzero contractions place the q -particle operator to the left of the -particle creation operator. There are only two ways this can occur, namely. 

Using Eq. (30), the definition of the antisymmetrized two-electron integral, and the anticommutation relation of fermion annihilation operators, Eq. (33) may be written... 

A useful test on the consistency of the definition of the transformation operator for creation operators and for annihilation operators is provided... 

Having established that creation and annihilation operators are rank 1 covariant and contravariant tensors, respectively, with respect to the operator ( )L,S, we can define an rath-rank boson operator as consisting of a like number of fermion creation and annihilation operators. Then the normal product of an rath-rank boson operator is a natural definition for the irreducible tensor. 

The term M p,is the eph coupling constant, and ba is the annihilation operator of the mode a, whose frequency and normal mode coordinate are represented by Q,a and Qp, respectively. The sites for electrons i( T) coupled with phonons are restricted to the C region or a subpart of C. The focused modes should be sufficiently localized on the molecule in term of their definition. Practically, these internal modes can be calculated by means of a frozen-phonon approximation, where displaced atoms are atoms in the c region (or its subpart) denoted as a vibrational box though a check for convergence to the size of the vibrational box is necessary [90]. 

From these definitions of electron creation and annihilation operators, the following anticommutator relations may be derived ... 

Let us stress that the operational definition of the quantum phase of radiation [47] is also based on the use of bilinear forms in the photon operators. In the simplest form, the idea of the operational approach to the phase difference can be illustrated with the aid of the two-port interferometer shown in Fig. 11 (see Refs. 14 and 47 for more detailed discussion). The two incident monochromatic (or quasimonochromatic) light beams are combined by a symmetric beamsplitter oriented at 45° to each beam. The resultant intensities emerging from each output port are measured by the two photodetectors connected with a comparator (computer) as in the Hanbury-Brown-Twiss interferometer [85] (also see Refs. 14, 15, and 86). Following Noh et al. [47], we denote by a and 2 the photon annihilation operators, describing the field at the two input ports, and by a and 04 the corresponding operators at the two output ports. Then... 

This decomposition suggests that at least in principle the //o-problem can be solved. The additional interaction Hamiltonian H (t) is assumed to be localized in time, i.e. //i ( r t — oo) = 0. The explicitly time-independent Hamiltonian Ho can be diagonalized through a proper definition of creation/annihilation operators a-s)f/a-s) corresponding to noninteracting particles of the species (s) and characterized by a set of quantum numbers denoted by a label i ... 

A proper definition of (quasi-)particle-creation and (quasi-)particle-annihilation operators an and a is provided by diagonalization of the (time-independent) unperturbed part Ho = //ext+Z/e-e of the total Hamiltonian. After the iteration is performed (e.g. on the Dirac-Fock level) the latter may be cast into the form... 

Obviously, any excitation of a core orbitals or excitation into a virtual orbital always leads out of the model space owing to our definitions above. The other terms of the wave operator, in contrast, which include only creation and annihilation operators of the (core-)valence orbitals, may result in either internal or external excitations, in dependence also of the particular basis function [Pg.195]

If applied to the reference state normal order enables us immediately to recognize those terms which survive in the computation of the vacuum amplitudes. The same applies for any model function and, hence, for real multidimensional model spaces, if a proper normal-order sequence is defined for all the particle-hole creation and annihilation operators from the four classes of orbitals (i)-(iv) in Subsection 3.4. In addition to the specification of a proper set of indices for the physical operators, such as the effective Hamiltonian or any other one- or two-particle operator, however, the definition and classification of the model-space functions now plays a crucial role. In order to deal properly with the model-spaces of open-shell systems, an unique set of indices is required, in particular, for identifying the operator strings of the model-space functions (a)< and d )p, respectively. Apart from the particle and hole states (with regard to the many-electron vacuum), we therefore need a clear and simple distinction between different classes of creation and annihilation operators. For this reason, it is convenient for the derivation of open-shell expansions to specify a (so-called) extended normal-order sequence. Six different types of orbitals have to be distinguished hereby in order to reflect not only the classification of the core, core-valence,... orbitals, following our discussion in Subsection 3.4, but also the range of summation which is associated with these orbitals. While some of the indices refer a class of orbitals as a whole, others are just used to indicate a particular core-valence or valence orbital, respectively. 

With this definition of the orbital indices, we now say that any (given) string of creation and annihilation operators obeys an extended normal-order sequence,... 

The treatment of orbital overlap in conjunction with the use of nonorthogonal basis sets deserves particular attention in treatments in terms of electron field operators. The definition of creation and annihilation operators and their anticommutation rules are basic for this development. Let s( ) be a set of atomic spin orbitals used to define the creation operators... 

It follows from the last equation that in order to calculate E. we have to know the result of the operation of A on 0) (i.e., on the linear combination of determinants), which comes down to the operation of the creation and annihilation operators on the determinants, which is simple. It can also be seen that we need to apply the operator S to 0), but its definition shows that this... 

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