Projektpraktikum - TU Graz - Institut für Theoretische Physik ...

The goal, to diagonalize this Hamiltonian, will be achieved by using a
Bogoliubov transformation:
ˆbk = ukâk + vkâ †
−k
ˆ† b k = u∗kâ †
k + v∗ kâ−k
ˆb−k = u−kâ−k + v−kâ †
k
ˆ† b −k = u∗−kâ †
−k + v∗ −kâk
(2.15)
The parameters uk and vk have to be determined in a way that, assuming
one set of operators already obeys the commutation relation 2.3 (this operators
will be âk and â †
) also obeys it. The last
k ), the other set (ˆbk and b †
k
requirement for the coefficients will be that the Hamiltonian gets diagonal.
Nevertheless this turns out to be possible only for indices k = 0, because for
k = 0:
(A − 2t) ˆb †
0 ˆ
b0 + B ˆb0 ˆb0 + ˆb †
0 ˆb †
0
(µ + 2t) ˆ b †
0 ˆ b0 +
µ + 2t
2
=
ˆb0 ˆb0 + ˆb †
0 ˆb †
0 .
A Bogoliubov transformation is for such an arrangement of coefficients not
possible. This can also be seen by looking at equation 2.21 where k = 0
would lead to a singularity, because of dividing by zero.
The state k = 0 corresponds to the ground state where in 2.4 the creation
and annihilation operators have been replaced by the condensate amplitude
φ(R), where
R |φ(R)|2 = N0.
Since we are interested in BEC, where a big fraction of particles is in the
ground state, there is no big difference between the Fock-states:
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