Quasiparticle Transformations

As we have shown in [21,14] this quasiparticle transformation leads from crude adiabatic to adiabatic Hamiltonian. The Hamiltonian (39) is adiabatic Hamiltonian. Note that the force constant for harmonic oscillators is given as second derivative of Escf at point R . We shall call the corresponding phonons the adiabatic phonons. 

Contents Introduction. - Concept of Creation and Annihilation Operators. -Particle Number Operators. - Second Quantized Representation of Quantum Mechanical Operators. - Evaluation of Matrix Elements. - Advantages of Second Quantization. - Illustrative Examples. - Density Matrices. -Connection to Bra and Ket Formalism. - Using Spatial Orbitals. - Some Model Hamiltonians in Second Quantized Form. - The Brillouin Theorem. -Many-Body Perturbation Theory. -Second Quantization for Nonorthogonal Orbitals. - Second Quantization and Hellmann-Feynman Theorem. - Inter-molecular Interactions. - Quasiparticle Transformations. Miscellaneous Topics Related to Second Quantization -Problem Solutions. - References -Index. 

In a broad sense, a quasiparticle transformation consists of an arbitrary transformation of the creation/annihilation operators and it is usually introduced for mathematical convenience. In most applications, the transformations involve one-particle operators, but sometimes two-particle combinations are useful, too. Such transformations will be briefly reviewed in Sects. 16.1 and 16.2, respectively, while in Sect. 16.3 an application to the latter will be presented. 

These are the conditions for the quasiparticle transformation to be a canonical transformation. Some special cases of such transformations we have seen previously. If, for example, Bij, = 0, one has ... 

The transformation of Eq. (16.4) is not really a quasiparticle transformation, since it just reflects a transformation of the underlying orbital space If creates an electron on orbital Xi than creates one on = X/t ikXk- The canonical condition of Eq. (16.5) means that the transformation matrix A is unitary. In Sect. 13 we have also seen non-unitary basis set transformations of the form of Eq. (16.4), for which Eq. (16.5) does not hold, and which do not leave the anticommutation rules invariant. 

Another special case of the general quasiparticle transformation is when the coefficients Aj = 0. This corresponds to interchanging the creation and annihilation operators. The canonical conditions of Eq. (16.3) then require matrix B to be unitary. With B being the unit matrix, this is the particle-hole transformation we have considered in Sect. 10.2. 

Mixing of the above two limiting cases can sometimes be useful, too. One of the most important quasiparticle transformations of this type is the so-called Boguliubov transformation applied in the standard theory of superfluidity and superconductivity. These theories are out of the scope of this book but we shall make an exception below to explain the merit of the Boguliubov transformation (Landau et al. 1980). 

The analysis of the physical meaning of these results leads to an explanation of low-temperature superconductivity, but we stop the treatment here having reached the aim to show how the Boguliubov quasiparticle transformation of Eq. (16.13) resulted in a useful energy formula. 

An interesting feature of Eq. (16.17) is worth mentioning. It is seen that two fermion operators are collected in this equation to form a pair of zero resulting spin. Such pairs are called Cooper pairs in the theory of superconductivity. Equation (16.17) can also be considered as a quasiparticle transformation which is, however, very different from the single-particle transformations we have seen previously. Two-electron quasiparticle transformations will be introduced in the next section in a somewhat different context. 

Nonlinear quasiparticle transformations involving a product of two creation operators can be called two-particle transformations. The form of two-particle transformations we shall consider here is given as ... 

Let us study now the formal aspects of the quasiparticle transformation of Eq. (16.28). The quasiparticle annihilation operators are defined by ... 

As it is well known proper many body methods including Feynman diagrammatic techniques, developed in elementary particle physics, were transferred to solid-state physics many years ago. The introduction to quantum chemistry followed later, but only on the electronic level. So the question then appears Is it possible to formulate the full quantum chemical electron-vibrational Hamiltonian in a second quantization formalism The answer is negative. In fact the author did spend many years attempting to construct ideal representations by means of appropriate quasiparticle transformations (cf. equivalent FrOhlich type unitary transformations), but all variants, being either adiabatic- or nonadiabatic representations, did indeed fail. The reason lies actually on a deeper level than one would initially imagine. 

The fundamentals of the approach to be presented here were already obtained in the authors PhD thesis in 1986 [12] and in an unpublished work from 1988 [13]. In full analogy with the solid-state electron-phonon interaction development, a similar apparatus for quantum chemistry was built utilizing the second quantization quasiparticle concept of the electron-vibrational Hamiltonian. This was a more complex operation than first anticipated. The formulation proceeded stepwise, i.e. first the crude representation, then the adiabatic, and finally the nonadiabatic one. As later recognized, quasiparticle transformations that leads to individual representations were in fact nothing but the full quantum chemistry equivalent of Frohlich transformation used in solid-state physics. 

Our aim is now to find the most general group of quasiparticle transformations for the electron fermion and the hypervibration boson operators, binding individual representations of the total Hamiltonian. The author in his thesis on this topic [12] proposed two transformations - the first of the adiabatic type, dependent on... 

The advantage of the quasiparticle transformations lies in the fact that they are more transparent than the global transformation of the whole Hamiltonian. The first transformation in (28.32) with generator Si is equivalent to the adiabatic quasiparticle transformation from the crude into the adiabatic representation, defined through new quasiparticles in adiabatic representation with double bar... 

DiagonaUzation of the terms which contain boson operators in the first order gives us equations for the first order coefficients of the unknown operators c and c of quasiparticle transformations (28.33) and (28.37)... 

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