Boguliubov transformation

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). 

This is called the Boguliubov transformation. To maintain the commutation rules of Eq. (16.6) for operators bp bp the coefficients should obey the condition ... 

We can see that the Boguliubov transformation diagonalized the model Hamiltonian of Eq. (16.7) exactly. 

The problem of this interacting Fermi gas is more complicated than the weakly interacting Bose system treated previously, and the Boguliubov transformation of Eqs. (16.13-15) does not diagonalize the Hamiltonian of Eq. (16.12) exactly. Let us still transform this Hamiltonian by expressing it in terms of operators b, b". The result is ... 

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. 

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