Creation operator, definition

The appropriate quantum mechanical operator fomi of the phase has been the subject of numerous efforts. At present, one can only speak of the best approximate operator, and this also is the subject of debate. A personal historical account by Nieto of various operator definitions for the phase (and of its probability distribution) is in [27] and in companion articles, for example, [130-132] and others, that have appeared in Volume 48 of Physica Scripta T (1993), which is devoted to this subject. (For an introduction to the unitarity requirements placed on a phase operator, one can refer to [133]). In 1927, Dirac proposed a quantum mechanical operator tf), defined in terms of the creation and destruction operators [134], but London [135] showed that this is not Hermitean. (A further source is [136].) Another candidate, e is not unitary. 

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 ... 

The proof is by induction. It is clearly true for two factors since then it reduces to the definition of the contraction symbol. Furthermore, it is sufficient to prove the theorem under the assumption that Z is a creation operator and that all the operators UV XY are destruction operators. If UV- - -XY are all destruction operators and Z is a creation operator, we may then add any number of creation operators to the left of all factors on both sides of Eq. (10-196) within the N product, without impairing the validity of our theorem, since the contraction between two creation operators gives zero. If on the other hand Z is a destruction operator and UV - - - are creation operators, then Eq. (10-196) reduces to a trivial identity... 

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... 

Definition 15 A -body operator is a Hermitian operator that can be represented as a polynomial of degree 2 A in the annihilation and creation operators, and is of even degree in these operators. In addition, a A -body operator must be orthogonal to all k — l)-body operators, all k — 2)-body operators,. .., and all scalar operators, with respect to the trace scalar product. 

To bring the vector space to live, we introduce some elementary operators, creation operators, through the definitions... 

The properties of the creation operators can be deduced from the above definitions. Operating twice with at on an occupation vector gives... 

Most formulations of MCSCF theory are based on the second quantization formalism. We therefore review briefly in this section the basic definitions of the annihilation and creation operators, and the expansion of quantum mechanical operators in products of them. 

Let us look at operator at, which is the Hermitian conjugate of the creation operator. Using the definition of the Hermitian conjugated operator... 

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 number of occupied orbitals with ju < )5 for the (iV—l)-electron configuration on the right-hand side of (3.135) is one fewer than for the original configuration p). We apply the definition (3.128) of the creation operator 4 to (3.135) to obtain... 

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. 

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

Here, we discuss only a few basic properties of generalized squeezed vacuum, given by (78). By definition, it is properly normalized for arbitrary dimension of the Hilbert space. There are several ways to prove that the generalized squeezed vacuum goes over into the conventional squeezed vacuum ( C)) in the limit of s —> oo. By definition (78), one can conclude that the property lim< x C)(5 = 0)(oo) = 10) holds, since the FD annihilation and creation operators go over into the conventional ones lim< x as = a and lim< x a a. One can also show, at least numerically, that the superposition coefficients (81) approach the coefficients bn for the conventional squeezed vacuum Iims x bn = bn for n = 0,..., s. We apply another method based on the calculation of the scalar product (ClOo) - We show the analytical results for C < 1 only. We have found the scalar product between conventional and generalized squeezed vacuums in the form (for even, v)... 

From the definition of i Fi it can be readily seen that the inter-dimensional states are degenerate under the transformation given by (n, l, D) —> (n, l 1, D 2). Very recently [106], similar eigenspectral properties of the 2D isotropic harmonic oscillator, centrally enclosed in an axially symmetric box with impenetrable walls, have been derived using the annihilation and creation operators and the infinitesimal operators of the SU(2) group. Extension to the three dimensional case using the SU(3) group has been also completed [107],... 

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... 

The same relation holds when applied to a general state with any particle number. A state can now be defined from which the single-particle states can be created thereof, by applying creation operators, one is able to create states with any number of particles. This state will be denoted by 10) and will be called the vacuum state. Thus, the following definition is used ... 

The increasing use of heavy water (D2O) in studies of reaction mechanism, in nuclear technology and in medical research has created a demand for accurate measurements of acidity in this medium. As the glass electrode responds satisfactorily to deuterium ion in D2 O, this need is provided by modern pH meters. The operational definition of pH also allows the creation of a scale, the pD scale, where the solvent is heavy water. Reference values have been assigned to three selected buffers in heavy water (Paabo Bates, 1969). Standard pD values for these buffers for various temperatures are given in Table 6.8. When pH meters are standardized against these buffers in heavy water solution, the meter readings are in pD units. 

From this definition, it is evident that application of creation operator Y, to the Fermi vacuum is equivalent to annihilation of a particle (or creation of a hole) in l o). The effect of Y, on the Fermi vacuum state is the creation of a particle (or annihilation of a hole) in o) The effect of Y on the Fermi vacuum is the creation of a particle in the virmal spin-orbitals and finally, the effect of Y on o) is the annihilation of a particle in virtual spin-orbitals. Thus, for example, a singly excited Slater determinant ) can be described as... 

It is easily seen from this definition that such an operator is interpreted as coupled products of annihilation and creation operators of one-electron states. Therefore, due to the orthogonality of one electron states, the reduced matrix element is equal to 1 when to the right (and to this left) the same one electron state is created and annihilated. 

The creation operators aj, for nonorthogonal spin orbitals are defined in the same way as for orthonormal spin orbitals (1.2.5). As for orthonormal spin orbitals, the anticommutation relations of the creation operators and the properties of their Hermitian adjoints (the annihilation operators) may be deduced from the definition of the creation operators and from the inner product (1.9.2). However, it is easier to proceed in the following manner. We introduce an auxiliary set of symmetrically orthonormalized spin orbitals... 

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... 

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

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