Operator phonon creation

Or is the frequency of the harmonic oscilator and b) are boson (phonon) creation (annihilation) operators. In order to use the perturbation theory we have to split the Hamiltonian (16) onto the unperturbed part Hq and the perturbation H ... 

Here a is a dimensionless constant, 5p(R) is the density fluctuation of the medium at the position R (the center of symmetry of the benzoic acid dimer), 0)D is the Debye frequency, and N is the number of acoustic modes, cot = 7 sound k, (bk) is the Bose operator of creation (annihilation of a acoustic phonon with the wave vector k). In the localized representation we have... 

Here Efn(0) refers to the /th excited state of a free molecule in the crystal a n + (aQ is the Bose operator of creation (annihilation) of an intramolecular vibrational excitation in the nth molecule M2(k) refers to the energy of an optical phonon with the wave vector k connected with proton oscillations in the O H O bridge (bk) is the Bose operator of phonon creation (annihilation) and is the coupling energy between the molecular excitation and phonons. 

The interaction in non-metals (e.g. ionic crystals or covalent semiconductors) will be now expressed in terms of phonon creation ay and destruction a j operators related to the pnonon amplitudes in (2.2) by... 

Considering Equation 6.38 again, we need to transform the Hamiltonian expression. Thus, if cos(k) and ss(k) are the frequency and the polarization vector for the classic modes with polarization s and wave vector k, respectively, we can define the phonon creation (aks+) and annihilation ( /,s ) operators as... 

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... 

The first two terms are diagonalized by introducing the phonon creation and annihilation operators (see (Cohen-Tannoudji et al. 1977)) ... 

We shall always be interested in longitudinal vibrations of quasi-one-dimensional systems, so the vector notation can mostly be omitted in subsequent discussions if one refers to the component of a vector parallel to the chain axis. The phonon amplitudes m, can be expressed in terms of phonon creation and annihilation operators as ... 

We now quantize the vibrational modes r and write the normal mode coordinates in terms of the phonon creation/annihilation operators... 

Here ak a ) is the annihilation (creation) operator of an exciton with the momentum k and energy Ek, operator an(a ) annihilates (creates) an exciton at the n-th site, 6,(6lt,) is the annihilation (creation) operator of a phonon with the momentum q and energy u) q), x q) is the exciton-phonon coupling function, N is the total number of crystal molecules. The exciton energy is Ek = fo + tfcj where eo is the change of the energy of a crystal molecule with excitation, and tk is the Fourier transform of the energy transfer matrix elements. 

In terms of the creation-annihilation electron and phonon operators the Hamiltonian can be cast as follows ... 

Here d ,dl and airaj are annihilation and creation operators for the QD electrons and phonons, respectively. As in case (1), Mq is a semiconductor electron-phonon constant and a>fD is a phonon frequency. A-D is the energy of noninteracting electrons and 3 is a Coulomb integral. 

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