Annihilation operators, quantum

At the end of Section 8.16 we mentioned that the Fock representation avoids the use of multiple integrations of coordinate space when dealing with the many-body problem. We can see here, however, that the new method runs into complications of its own To handle the immense bookkeeping problems involved in the multiple -integrals and the ordered products of creation and annihilation operators, special diagram techniques have been developed. These are discussed in Chapter 11, Quantum Electrodynamics. The reader who wishes to study further the many applications of these techniques to problems of quantum statistics will find an ample list of references in a review article by D. ter Haar, Reports on Progress in Physics, 24,1961, Inst, of Phys. and Phys. Soc. (London). 

As a simple model, we confine our attention just to a single mode Ha(t) of the Hamiltonian (23). Note that neither any instantaneous eigenstate of Ha(t) is an exact quantum state nor e-/3ii W is a density operator. To calculate the thermal expectation value of an operator A, one needs either the Heisenberg operator Ah or the density operator pa(t) = UapaUa Now we use the time-dependent creation and annihilation operators (24), invariant operators, to construct the Fock space. 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

In Chapter 1 we have discussed the familiar realization of quantum mechanics in terms of differential operators acting on the space of functions (the Schrodinger wave function formulation, also called wave mechanics ). A different realization can be obtained by means of creation and annihilation operators, leading to an algebraic formulation of quantum mechanics, sometimes called matrix mechanics. For problems with no spin, the formulation is done in terms of boson creation, b (, and annihilation, ba, operators, satisfying the commutation relations... 

In the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrodinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate, r2,... and momentum p, p2,..., boson creation and annihilation operators, b ia, bia. The index i runs over the number of relevant degrees of freedom, while the index a runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i j,... 

The Majorana operators My annihilate one quantum of vibration in bond / and create one in bond j, or vice versa. 

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]. 

Here a and are the usual oscillator creation and annihilation operators with bosonic commutation relations (73), and 0i,..., 1 ,..., 0Af) denotes a harmonic-oscillator eigenstate with a single quantum excitation in the mode n. According to Eq. (80a), the bosonic representation of the Hamiltonian (79) is given by... 

In the usual formalism of quantum mechanics, the first quantization formalism, observables are represented by operators and the wave functions are normal functions. In the method of second quantization, the wave functions are also expressed in terms of operators. The formalism starts with the introduction of an abstract vector space, the Fock space. The basis vectors of the Fock space are occupation number vectors, with each vector defined by a set of occupation numbers (0 or 1 for fermions). An occupation number vector represents a Slater determinant with each occupation number giving the occupation of given spin orbital. Creation and annihilation operators that respectively adds and removes electrons are then introduced. Representations of usual operators are expressed in terms of the very same operators. 

The sums of products of CFP obey some additional relations. In fact, the operators of particle number N, of orbital L and spin S momenta are expressed in terms of tensorial products of electron creation and annihilation operators - relationships (14.17), (14.15) and (14.16), respectively. We can expand the submatrix elements of such tensorial products using (5.16) and then go over, using (15.21) and (15.15), to the CFP. On the other hand, these submatrix elements are given by the quantum numbers of the states of the lN configuration. Then we obtain... 

The concept of quasispin quantum number was discussed in the Introduction and Chapter 9 (see formulas (9.22) and (9.23)). Now let us consider it in the framework of the second-quantization technique. We can introduce the following bilinear combinations of creation and annihilation operators obeying commutation relations (14.2) - the quasispin operator ... 

Formula (17.16) is the most general form of the two-electron matrix element in which all four one-electron wave functions have different quantum numbers. We shall put it into general formula (13.23), whereupon the creation and annihilation operators will be rearranged to place side by side those second-quantization operators whose rank projections enter into the same Clebsch-Gordan coefficient. Summing over the projections then gives... 

E = V x C. This implies an interesting interpretation of the Hopf index n, since that helicity is equal to the classical expression of the difference between the numbers of right-handed and left-handed photons contained in the field Nr — Nr (defined by substituting Fourier transform functions for creation and annihilation operators in the quantum expression). In other words, n = Nr — Nr- This establishes a relation between the wave and the particle understanding of the idea of helicity, that is, between the curling of the force lines to one another and the difference between right- and left-handed photons contained in the field. 

One of the most important concepts of quantum chemistry is the Slater determinant. Most quantum chemical treatments are made just over Slater determinants. Nevertheless, in many problems the formulation over Slater determinants is not very convenient and the derivation of final expressions is very complicated. The advantage of second quantization lies in the fact that this technique permits us to arrive at the same expressions in a considerably simpler way. In second quantization a Slater determinant is represented by a product of creation and annihilation operators. As will be shown below, the Hamiltonian can also be expressed by creation and annihilation operators and thus the eigenvalue problem is reduced to the manipulation of creation and annihilation operators. This manipulation can be done diagrammatically (according to certain rules which will be specified later) and from the diagrams formed one can write down the final mathematical expression. In the traditional way a Slater determinant I ) is specified by one-electron functions as follows ... 

The quantity (—l)s gives us the correct sign of the determinant. For purposes of quantum chemistry it is more convenient to specify the state vectors in such a way that only the occupied spin-orbitals are listed in the vector. Therefore instead of Inln2. ..) we write A iA2 . ) In this case, we define the annihilation operator XA, as... 

This representation among others removes one more inconsistency in quantum chemistry one generally deals with the systems of constant composition i.e. of the fixed number of electrons. The expression eq. (1.178) allows one to express the matrix elements of an electronic Hamiltonian without the necessity to go in a subspace with number of electrons different from the considered number N which is implied by the second quantization formalism of the Fermi creation and annihilation operators and on the other hand allows to keep the general form independent explicitly neither on the above number of electrons nor on the total spin which are both condensed in the matrix form of the generators E specific for the Young pattern T for which they are calculated. 

We give here a brief account of the quantum dynamics in L-space explicitly for the system described above. The relation between H- and L-space is summarized in Table 1. The creation operator in the //-space becomes a pair of creation superoperators, and similarly for the annihilation operator, in L-space. They are denoted by... 

To develop the quantum theory of electromagnetic radiation, it is useful to reformulate the harmonic-oscillator problem in terms of creation and annihilation operators, following a derivation due to Dirac. The Schrodinger equation (5.9) can be written... 

For obvious reasons, and a are known as step-up and step-down operators, respectively. They are also called ladder operators since they take us up and down the ladder of harmonic-oscillator eigenvalues. In the context of radiation theory, and a are called creation and annihilation operators, respectively, since their action is to create or annihilate a quantum of energy. 

Here B (Bq) is the Bose creation (annihilation) operator of the collective intensive vibrational quantum with an energy of hi q and quasi-momentum of hq. 

In order to rigorously describe the nonlinear interaction between the weak pulsed fields, we now turn to the fully quantum treatment of the system. The traveling-wave electric fields can be expressed through single mode operators as j(z, t) = dj(t) Cqz (j = 1, 2), where uj is the annihilation operator for the field mode with the wavevector kp + q. The singlemode operators a and aq possess the standard bosonic commutation relations... 

The importance of these creation and annihilation operators results from the fact that quantum-mechanical operators may be written as an expansion in products of these operators. These products are then said to form an operator basis. For example, an arbitrary one-electron operator has the form... 

The simplest way to show the principal difference between the representations of plane and multipole photons is to compare the number of independent quantum operators (degrees of freedom), describing the monochromatic radiation field. In the case of plane waves of photons with given wavevector k (energy and linear momentum), there are only two independent creation or annihilation operators of photons with different polarization [2,14,15]. It is well known that QED (quantum electrodynamics) interprets the polarization as given spin state of photons [4]. The spin of photon is known to be 1, so that there are three possible spin states. In the case of plane waves, projection of spin on the... 

In turn, the monochromatic multipole photons are described by the scalar wavenumber k (energy), parity (type of radiation either electric or magnetic), angular momentum j 1,2,..., and projection m = —j,..., / [2,26,27]. This means that even in the simplest case of monochromatic dipole (j = 1) photons of either type, there are three independent creation or annihilation operators labeled by the index m = 0, 1. Thus, the representation of multipole photons has much physical properties in comparison with the plane waves of photons. For example, the third spin state is allowed in this case and therefore the quantum multipole radiation is specified by three different polarizations, two transversal and one longitudinal (with respect to the radial direction from the source) [27,28], In contrast to the plane waves of photons, the projection of spin is not a quantum number in the case of multipole photons. Therefore, the polarization is not a global characteristic of the multipole radiation but changes with distance from the source [22],... 

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... 

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