Bogoliubov transformation / Interpretation of diagonalized Hamiltonian

[Abigale]

Hey,

I consider a diagonalized Hamiltonian:

$H=\sum\limits_{k} \underbrace{ (\epsilon_{k} u_{k}^2 -\epsilon_{k} v_{k}^2 -2\Delta u_{k} v_{k} )}_{E_{k}}(d_{k \uparrow}^{\dagger}d_{k \uparrow} + d_{k \downarrow}^{\dagger}d_{k \downarrow}) +const$
with fermionic creation and annihilation operators.

From solution I know that: $E_{k} =\sqrt{\Delta^2 +\epsilon_{k}^2}$ but how can I get this result?





Things I even know is that: $u_k^2 + v_k^2 =1$ and:
$\sum\limits_k <br /> <br /> \underbrace{(<br /> -2\epsilon_k u_k v_k +\Delta v_k^2 -\Delta u_k ^2<br /> )}_{\stackrel{!}{=}0}<br /> <br /> (d_{k \uparrow}^{\dagger}d_{k \downarrow}^{\dagger} + d_{k \downarrow}d_{k \uparrow})$.

Thank you guys!

 

[DrDu]

Things I even know is that: $u_k^2 + v_k^2 =1$ and:
$(<br /> -2\epsilon_k u_k v_k +\Delta v_k^2 -\Delta u_k ^2<br /> )=0$.

This are two equations for the two unknowns u and v. Solve for them and put into the defining equation for E_k!