Many Body bogoliubov transformation

[Hl3]

Homework Statement

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

b_k = cosh(θ)a_k - sinh(θ) a^†_-k
b_k^† = cosh(θ)a^†_k - sinh(θ)a_-k

Homework Equations

The Attempt at a Solution

I don't fully understand the notaition with zeros. I believe n_k would equal to 0, however question asks for the expression for n_k. thnaks in advance

 

[eys_physics]

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 $\theta =0$).