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. [Pg.452]

Let at(q) and a(q) be the creation and annihilation operators of cavity photons, and B and B, be the creation and annihilation operators of an excitation on the ith molecule . We do not account for the dipole-dipole interactions between different molecules , as we consider the coupling by light to be the principal coupling mechanism. Thus the Hamiltonians of noninteracting cavity photons and molecular excitations are [Pg.288]

Q < I — y, here the a,- form a (finite-dimensional) basis of annihilation operators, y is the 1-RDM, and I is the identity matrix of the appropriate size. If the density matrices are known to be real symmetric then the may be assumed real, otherwise they should be assumed complex. For fixed particle number [Pg.94]

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]. [Pg.96]

Here, ancj are the creation and annihilation operators, respectively, for phonons in mode q = (q, r), where q denotes the vector of the phonon and r is the branch label. The energy of these phonon modes is given by u>q. Furthermore, the single-molecule Hamiltonian as well as the intermolecular transfer interaction are still considered to be operators in phonon space. [Pg.413]

Consider first the case where P and Q are simple creation and annihilation operators, e.g. P = af and Q — a . The residue <0 P nj>

In order to define excitation operators, one need not start from the creation and annihilation operators one can instead simply require that action of, for example, aP on a Slater determinant >1) with occupied and (for p q) ij/p unoccupied replaces by ij/p. Otherwise it annihilates >1). [Pg.295]

By computing the commutators of the components of the quasispin operator with electron creation and annihilation operators, we can directly see that the latter behave as the components of a tensor of rank q = 1/2 in quasispin space and obey the relationship of the type (14.2) [Pg.145]

When one takes the expectation value of the conmutator/anticonmuta-tor of a string of p annihilating operators and a string of p creator operators one obtains a series of terms involving products of Kronecker deltas and of q — 1, q — 2,.,.,1-RDM s elements. [Pg.38]

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. [Pg.60]

Accordingly, the diagonal elements of the P, Q, and G Hamiltonians should give results consistent with the (2, 2) conditions. This occurs. For example, using the commutation rules for creation and annihilation operators, the diagonal elements of the P Hamiltonian become [Pg.467]

We shall denote the creation and annihilation operators for a negaton of momentum p energy Ep = Vp2 + m2 and polarizations by 6 (p,s) and 6(p,s) respectively. In the following, by the polarization we shall always mean the eigenvalue of the operator O-n, where O is the Stech polarization operator and n some fixed unit vector. We denote the creation and annihilation operators for a positon (the antiparticle) of momentum q energy = Vq2 + m2, polarization t, by d (q,t) and [Pg.540]

To discern the algebraic structure of the new normal ordered product, we write the ordinary product as unitary transforms of Q-products in ordinary normal order. Since Q s are sums of products of odd number of cre-ation/annihilation operators (as follows from eq.(27)), we shall use different [Pg.179]

In eq.(28), we have first rewritten qi s in terms of Q,-, and then reordered the product QiQjQkQi as a sum of normal products, using as the vacuum. The traditional Wick s theorem applied to the products of Q, s will lead to pair contractions in the traditional sense between groups of creation-annihilation operators in one Q, with one or more Qy s. If these contractions completely exhaust all the operators present in the composites Q,-, Qj- etc. involved in the contraction, we denote them by bars and centred or filled circles. Joining by some operators will lead to terms with carets and open circles. Thus, the second and the third term in the braces involve incomplete contractions. The second term has connections between operators of Q and Qj and between Qk and Qi. The third term involves connections between Q,- and Q and between Qj and Q(. The fourth term involves complete connections between all the operators of Qi and Qj. The fifth term involves contraction of all the operators of Q,- with those of Qj and of some between Qk and Q . The sixth term involves complete contractions between operators of Q,- and Qj and of Qk and Qi. The seventh term QiQjQkQi indicates that all the operators of Qii Qji Qk and Qi are contracted among themselves which cannot be factored out to pairs such as QiQj QkQi etc. [Pg.180]

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 [Pg.85]

Expanding the quantity q in (3.90) with respect to deviations from equilibrium up to quadratic terms and introducing normal coordinates the Hamiltonian Hl can be written as a sum of Hamiltonians which correspond to harmonic oscillators in their normal coordinates. Then we use the phonon creation and annihilation operators, i.e. the operators 6 r and 5qr (q is the phonon wavevector and r indicates the corresponding frequency branch) and obtain the Hamiltonian Hl in the form [Pg.69]

The rewriting of commutators and anticommutators is guided by the simple rule that the particle rank of the operator should be reduced. The particle rank of an operator consisting of a string of p creation and q annihilation operators is 2 (p+q). A reduction in the particle rank by one can [Pg.56]

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