Many Body bogoliubov transformation

In summary, the occupation of each single-particle state with wave vector k =/= 0 in the ground state is given by the expression nk = <0|bk†bk|0>, where b and b† are bogoliubov transformations defined as bk = cosh(θ)ak - sinh(θ) a†-k and bk† = cosh(θ)a†k - sinh(θ)a-k. This differs from the occupation in the vacuum state, as the state |0> is the vacuum of the annihilation operators ak, but not of the bogoliubov operators bk.
  • #1
Hl3
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Homework Statement


The occupation of each single-particle state with wave vector k =/= 0 in the ground state is given by nk = <0|bkbk|0>
where b and b† are bogoliubov transformaition. Find an expression for nk.

bk = cosh(θ)ak - sinh(θ) a-k
bk = cosh(θ)ak - sinh(θ)a-k

Homework Equations

The Attempt at a Solution


I don't fully understand the notaition with zeros. I believe nk would equal to 0, however question asks for the expression for nk. thnaks in advance
 
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  • #2
No, it is not. The state ##|0>## is the vacuum of the annihilation operators ##a_k##. That is,
$$a_k|0>=0$$ for all ##k##, but
$$b_k|0>\neq 0$$ (unless ##\theta =0##).
 

Related to Many Body bogoliubov transformation

1. What is a Many Body Bogoliubov Transformation?

A Many Body Bogoliubov Transformation is a mathematical technique used in the study of quantum mechanics and many-body systems. It involves transforming a set of operators, known as Bogoliubov operators, into a new basis in order to simplify the equations describing the system.

2. What is the purpose of a Many Body Bogoliubov Transformation?

The purpose of a Many Body Bogoliubov Transformation is to simplify the mathematical description of a many-body system, making it easier to solve and analyze. It is often used in the study of superfluidity, Bose-Einstein condensates, and other quantum systems.

3. How does a Many Body Bogoliubov Transformation work?

A Many Body Bogoliubov Transformation involves finding a set of transformations that diagonalize the Hamiltonian (energy operator) of the system. This results in a simplified form of the Hamiltonian, making it easier to solve and analyze the system.

4. What are some applications of Many Body Bogoliubov Transformation?

Many Body Bogoliubov Transformation has various applications in the study of quantum systems, including superfluidity, Bose-Einstein condensates, and quantum phase transitions. It is also used in the study of collective excitations and fluctuations in many-body systems.

5. Are there any limitations to Many Body Bogoliubov Transformation?

While Many Body Bogoliubov Transformation is a powerful mathematical tool, it does have some limitations. It is most effective for systems with a large number of particles, and may not accurately describe systems with strong interactions or in extreme conditions such as high temperatures or high densities.

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