Editing Bogoliubov transformation (section)

== Multimode example ==
The [[Hilbert space]] under consideration is equipped with these operators, and henceforth describes a higher-dimensional [[quantum harmonic oscillator]] (usually an infinite-dimensional one).

The [[ground state]] of the corresponding [[Hamiltonian (quantum mechanics)|Hamiltonian]] is annihilated by all the annihilation operators:

:<math>\forall i \qquad a_i |0\rangle = 0.</math>

All excited states are obtained as [[linear combination]]s of the ground state excited by some [[creation operators]]:

:<math>\prod_{k=1}^n a_{i_k}^\dagger |0\rangle.</math>

One may redefine the creation and the annihilation operators by a linear redefinition:

:<math>a'_i = \sum_j (u_{ij} a_j + v_{ij} a^\dagger_j),</math>

where the coefficients <math>u_{ij},v_{ij}</math> must satisfy certain rules to guarantee that the annihilation operators and the creation operators <math>a^{\prime\dagger}_i</math>, defined by the [[Hermitian conjugate]] equation, have the same [[commutator]]s 
for bosons and anticommutators for fermions.

The equation above defines the Bogoliubov transformation of the operators.

The ground state annihilated by all <math>a'_i</math> is different from the original ground state <math>|0\rangle</math>, and they can be viewed as the Bogoliubov transformations of one another using the operator–state correspondence. They can also be defined as [[squeezed coherent state]]s. BCS wave function is an example of squeezed coherent state of fermions.<ref>{{cite journal | last=Svozil | first=K. |author-link=Karl Svozil| title=Squeezed fermion states | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=65 | issue=26 | date=1990-12-24 | issn=0031-9007 | doi=10.1103/physrevlett.65.3341 | pages=3341–3343| pmid=10042844 | bibcode=1990PhRvL..65.3341S }}</ref>