- Bogoliubov transformation
In
theoretical physics , the Bogoliubov transformation, named afterNikolay Bogolyubov , is aunitary transformation from aunitary representation of somecanonical commutation relation algebra orcanonical anticommutation relation algebra into another unitary representation, induced by anisomorphism of the commutation relation algebra. The Bogoliubov transformation is often used to diagonalize Hamiltonians, which yields the steady-state solutions of the correspondingSchr%C3%B6dinger equation . The solutions ofBCS theory in a homogeneous system, for example, are found using a Bogoliubov transformation.Single bosonic mode example
Consider the canonical commutation relation for
bosonic creation and annihilation operators in the harmonic basis:left [ hat{a}, hat{a}^dagger ight ] = 1Define a new pair of operators:hat{b} = u hat{a} + v hat{a}^dagger:hat{b}^dagger = u^* hat{a}^dagger + v^* hat{a}
where the latter is the
hermitian conjugate of the first. The Bogoliubov transformation is a canonical transformation of these operators. To find the conditions on the constants "u" and "v" such that the transformation remains canonical, the commutator is expanded, viz.:left [ hat{b}, hat{b}^dagger ight ] = left [ u hat{a} + v hat{a}^dagger , u^* hat{a}^dagger + v^* hat{a} ight ] = cdots = left ( |u|^2 - |v|^2 ight ) left [ hat{a}, hat{a}^dagger ight ] .
It can be seen that u|^2 - |v|^2 = 1 is the condition for which the transformation is canonical. Since the form of this condition is reminiscent of the hyperbolic identity, the constants "u" and "v" can be parameterized as
:u = e^{i heta_1} cosh r ,!:v = e^{i heta_2} sinh r ,! .
Multimode example
The
Hilbert space under consideration is equipped with these operators, and henceforth describes a higher-dimensionalquantum harmonic oscillator (usually an infinite-dimensional one).The
ground state of the corresponding Hamiltonian is annihilated by all the annihilation operators::forall i qquad a_i |0 angle = 0
All excited states are obtained as
linear combination s of the ground state excited by some creation operators::prod_{k=1}^n a_{i_k}^dagger |0 angle
One may redefine the creation and the annihilation operators by a linear redefinition:
:a'_i = sum_j (u_{ij} a_j + v_{ij} a^dagger_j)
where the coefficients u_{ij},v_{ij} must satisfy certain rules to guarantee that the annihilation operators and the creation operators a^{primedagger}_i, defined by the
Hermitian conjugate equation, have the samecommutator s.The equation above defines the Bogoliubov transformation of the operators.
The ground state annihilated by all a'_{i} is different from the original ground state 0 angle and they can be viewed as the Bogoliubov transformations of one another using the
operator-state correspondence . They can also be defined assqueezed coherent state s.In
physics , the Bogoliubov transformation is important for understanding of theUnruh effect ,Hawking radiation andBCS theory , among many other things.ee also
References
* J.-P. Blaizot and G. Ripka: Quantum Theory of Finite Systems, MIT Press (1985)
* A. Fetter and J. Walecka: Quantum Theory of Many-Particle Systems, Dover (2003)External links
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